From owner-reliable_computing [at] interval [dot] louisiana.edu Fri Mar 1 11:38:01 2002 Received: (from daemon@localhost) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) id g21Hc1k11639 for reliable_computing-outgoing; Fri, 1 Mar 2002 11:38:01 -0600 (CST) Received: from bologna.vision.caltech.edu (bologna.vision.caltech.edu [131.215.134.19]) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) with ESMTP id g21Hbt411635 for ; Fri, 1 Mar 2002 11:37:56 -0600 (CST) Received: from peking.vision.caltech.edu (peking [131.215.134.18]) by bologna.vision.caltech.edu (8.9.3/8.8.7) with ESMTP id JAA09877; Fri, 1 Mar 2002 09:37:49 -0800 Received: (from arrigo@localhost) by peking.vision.caltech.edu (8.9.3+Sun/8.9.1) id JAA12508; Fri, 1 Mar 2002 09:37:42 -0800 (PST) X-Authentication-Warning: peking.vision.caltech.edu: arrigo set sender to arrigo [at] vision [dot] caltech.edu using -f From: Arrigo Benedetti MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Message-ID: <15487.48230.153413.854985 [at] gargle [dot] gargle.HOWL> Date: Fri, 1 Mar 2002 09:37:42 -0800 To: reliable_computing [at] interval [dot] louisiana.edu CC: berz [at] msu [dot] edu Subject: Taylor models (repost) X-Mailer: VM 6.92 under Emacs 20.4.1 Reply-To: arrigo [at] bologna [dot] vision.caltech.edu Sender: owner-reliable_computing [at] interval [dot] louisiana.edu Precedence: bulk Dear all, during the last few days I have studing the work by Martin Berz and his students about Taylor models since I believe that my optimization code could benefit from them. The cost function that I am trying to optimize using a modified version of the CToolbox GlobalOptimize module is given by the ratio of two 8th order polinomials. Due to the overestimation produced during the computation of the interval extensions of the function, jacobian and hessian by the naive application of interval arithmetic or even the mean value forms, the number of boxes that should be generated before the monotonicity and concavity tests can be effective at deleting boxes is around 10^40 (!!!). On the other end, I know that the cost function is actually quite smooth, and because of the way it is formulated I also know that there are several terms that cancel each other out, giving rise to overestimation. Taylor models seem to be a good cure for this problem, so I have started to implement them in a Matlab prototype that I have written using INTLAB. I have a question about the data structure used to represent a Taylor model though. In the works of Berz the coefficients of the polynomial part are always represented by floating point points numers. Shouldn't these be intervals? Since the coefficients are manipulated every time the Taylor models are combined in arithmetic operations or used as arguments in elemetary functions, how can I get verified result if intervals are not used for the coefficients? Thanks in advance, -Arrigo -- Dr. Arrigo Benedetti e-mail: arrigo [at] vision [dot] caltech.edu Caltech, MS 136-93 phone: (626) 644-3757 Pasadena, CA 91125 fax: (626) 795-8649 From owner-reliable_computing [at] interval [dot] louisiana.edu Fri Mar 1 13:40:40 2002 Received: (from daemon@localhost) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) id g21Jedh11901 for reliable_computing-outgoing; Fri, 1 Mar 2002 13:40:39 -0600 (CST) Received: from mail7.wi.rr.com (mkc-162-160.kc.rr.com [24.94.162.160]) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) with ESMTP id g21JeY411897 for ; Fri, 1 Mar 2002 13:40:35 -0600 (CST) Received: from Marquette.edu ([65.29.171.35]) by mail7.wi.rr.com with Microsoft SMTPSVC(5.5.1877.537.53); Fri, 1 Mar 2002 13:38:21 -0600 Message-ID: <3C7FD90C.4070106 [at] Marquette [dot] edu> Date: Fri, 01 Mar 2002 13:39:56 -0600 From: "Dr. George Corliss" Reply-To: George.Corliss [at] Marquette [dot] edu Organization: Marquette University, EECE User-Agent: Mozilla/5.0 (Windows; U; Windows NT 5.1; en-US; rv:0.9.4) Gecko/20011128 Netscape6/6.2.1 X-Accept-Language: en-us MIME-Version: 1.0 To: arrigo [at] bologna [dot] vision.caltech.edu CC: reliable_computing [at] interval [dot] louisiana.edu, berz [at] msu [dot] edu Subject: Re: Taylor models (repost) References: <15487.48230.153413.854985 [at] gargle [dot] gargle.HOWL> Content-Type: text/plain; charset=us-ascii; format=flowed Content-Transfer-Encoding: 7bit Sender: owner-reliable_computing [at] interval [dot] louisiana.edu Precedence: bulk Arrigo, I hope Martin will also respond, but here is my understanding. > during the last few days I have studing the work by Martin Berz and > his students about Taylor models since I believe that my optimization > code could benefit from them. > The cost function that I am trying to optimize using a modified > version of the CToolbox GlobalOptimize module is given by the ratio of > two 8th order polinomials. > Due to the overestimation produced during the computation of the > interval extensions of the function, jacobian and hessian by the naive > application of interval arithmetic or even the mean value forms, > the number of boxes that should be generated before the monotonicity > and concavity tests can be effective at deleting boxes > is around 10^40 (!!!). Yes, Taylor models should help. > > On the other end, I know that the cost function is actually quite > smooth, and because of the way it is formulated I also know that there > are several terms that cancel each other out, giving rise to > overestimation. > Taylor models seem to be a good cure for this problem, so I have > started to implement them in a Matlab prototype that I have written > using INTLAB. Agreed. > > I have a question about the data structure used to represent a Taylor > model though. In the works of Berz the coefficients of the polynomial > part are always represented by floating point points numers. > Shouldn't these be intervals? No. > Since the coefficients are manipulated > every time the Taylor models are combined in arithmetic operations or > used as arguments in elemetary functions, how can I get verified > result if intervals are not used for the coefficients? Martin represents a quantity as a floating point coefficient polynomial plus a constant bound interval remainder. When you add (for example) two Taylor models, all round off errors that could occur in the coefficients are lumped into the remainder interval. Often, the remainder interval is a couple orders of magnitude larger than round off in the leading coefficients, so throwing in a little extra does not hurt. You should also have going for you that you are working well within the radius of convergence of the Taylor series, so the magnitudes of the floating point coefficients (and hence their round off errors) drop off rapidly. If you have not done so, you might read the papers by Martin's team and by Baker Kearfott on Taylor models in Automatic Differentiation: From Simulation to Optimization, appearing recently from Springer. Dr. George F. Corliss Electrical and Computer Engineering Haggerty Engineering 296 Marquette University P.O. Box 1881 Milwaukee, WI 53201-1881 USA George.Corliss [at] Marquette [dot] edu Office: 414-288-6599; Dept: 288-6820; Fax: 288-5579 From owner-reliable_computing [at] interval [dot] louisiana.edu Mon Mar 4 13:12:53 2002 Received: (from daemon@localhost) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) id g24JCrs08046 for reliable_computing-outgoing; Mon, 4 Mar 2002 13:12:53 -0600 (CST) Received: from tlo ([206.158.98.80]) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) with ESMTP id g24JCeF08042 for ; Mon, 4 Mar 2002 13:12:40 -0600 (CST) Received: from 206.158.98.80 ([206.158.98.80]) by tlo with Microsoft SMTPSVC(5.0.2195.4453); Sat, 2 Mar 2002 03:50:08 -0800 Content-type: text/html Date: Sat, 02 Mar 2002 03:48:57 -0800 From: advloans [at] yahoo [dot] com Subject: ADVLoans.com Your Rates Are Too High! To: reliable_computing [at] interval [dot] louisiana.edu Message-ID: X-OriginalArrivalTime: 02 Mar 2002 11:50:08.0468 (UTC) FILETIME=[66CDC140:01C1C1E0] Sender: owner-reliable_computing [at] interval [dot] louisiana.edu Precedence: bulk ADVLoans.com

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From owner-reliable_computing [at] interval [dot] louisiana.edu Mon Mar 4 13:23:56 2002 Received: (from daemon@localhost) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) id g24JNtc08143 for reliable_computing-outgoing; Mon, 4 Mar 2002 13:23:55 -0600 (CST) Received: from bologna.vision.caltech.edu (bologna.vision.caltech.edu [131.215.134.19]) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) with ESMTP id g24JNoF08139 for ; Mon, 4 Mar 2002 13:23:50 -0600 (CST) Received: from peking.vision.caltech.edu (peking [131.215.134.18]) by bologna.vision.caltech.edu (8.9.3/8.8.7) with ESMTP id LAA13998; Mon, 4 Mar 2002 11:23:27 -0800 Received: (from arrigo@localhost) by peking.vision.caltech.edu (8.9.3+Sun/8.9.1) id LAA13556; Mon, 4 Mar 2002 11:23:15 -0800 (PST) X-Authentication-Warning: peking.vision.caltech.edu: arrigo set sender to arrigo [at] vision [dot] caltech.edu using -f To: reliable_computing [at] interval [dot] louisiana.edu CC: berz [at] msu [dot] edu Subject: Re: Taylor models (repost) References: <15487.48230.153413.854985 [at] gargle [dot] gargle.HOWL> <3C7FD90C.4070106 [at] Marquette [dot] edu> From: Arrigo Benedetti Date: 04 Mar 2002 11:23:15 -0800 In-Reply-To: <3C7FD90C.4070106 [at] Marquette [dot] edu> Message-ID: Lines: 57 User-Agent: Gnus/5.0808 (Gnus v5.8.8) Emacs/20.4 MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Sender: owner-reliable_computing [at] interval [dot] louisiana.edu Precedence: bulk "Dr. George Corliss" writes: > Arrigo, > > I hope Martin will also respond, but here is my understanding. > [...] > > > > I have a question about the data structure used to represent a Taylor > > model though. In the works of Berz the coefficients of the polynomial > > part are always represented by floating point points numers. > > Shouldn't these be intervals? > > No. > > Martin represents a quantity as a floating point > coefficient polynomial plus a constant bound interval > remainder. When you add (for example) two Taylor models, > all round off errors that could occur in the coefficients > are lumped into the remainder interval. Often, the > remainder interval is a couple orders of magnitude larger > than round off in the leading coefficients, so throwing > in a little extra does not hurt. You should also have > going for you that you are working well within the > radius of convergence of the Taylor series, so the > magnitudes of the floating point coefficients (and hence > their round off errors) drop off rapidly. > > If you have not done so, you might read the papers by > Martin's team and by Baker Kearfott on Taylor models > in Automatic Differentiation: From Simulation to > Optimization, appearing recently from Springer. > > I have read the papers by Berz and his students posted on their web sites which I believe include the papers that you are referring to. I found the most comprehensive account on the implementation of Taylor models in Chap. 5 of the PhD thesis of Makino, available at http://www.beamtheory.nscl.msu.edu/makino/phd.html Maybe I missed something, however I still do not see how, for a Taylor model of order n, one can get rigorous bounds for the parts of order k<=n of the polynomial (called order bounds intervals by Makino) if the coefficients of the polynomial are stored and traeted as non-interval quantities. I am looking forward to hear from Berz too. Best, -Arrigo Benedetti -- Dr. Arrigo Benedetti e-mail: arrigo [at] vision [dot] caltech.edu Caltech, MS 136-93 phone: (626) 644-3757 Pasadena, CA 91125 fax: (626) 795-8649 From owner-reliable_computing [at] interval [dot] louisiana.edu Mon Mar 4 14:43:18 2002 Received: (from daemon@localhost) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) id g24KhHC08355 for reliable_computing-outgoing; Mon, 4 Mar 2002 14:43:17 -0600 (CST) Received: from carbon.cudenver.edu (wlodwick [at] carbon [dot] cudenver.edu [132.194.10.4]) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) with ESMTP id g24KhCF08351 for ; Mon, 4 Mar 2002 14:43:12 -0600 (CST) Received: from localhost (wlodwick@localhost) by carbon.cudenver.edu (8.8.8/8.8.8) with ESMTP id NAA23397; Mon, 4 Mar 2002 13:41:11 -0700 (MST) Date: Mon, 4 Mar 2002 13:41:11 -0700 (MST) From: To: cc: Subject: Re: ADVLoans.com Your Rates Are Too High! In-Reply-To: Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: owner-reliable_computing [at] interval [dot] louisiana.edu Precedence: bulk On Sat, 2 Mar 2002 advloans [at] yahoo [dot] com wrote: > > > > ADVLoans.com > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > >
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> > > --------------------------------------------------------------------- Weldon A. Lodwick University of Colorado at Denver (303) 556-8462 (office) Department of Mathematics - Campus Box 170 (303) 556-8442 (secretary) P.O. Box 173364 (303) 556-8550 (fax) Denver, Colorado USA 80217-3364 email: weldon.lodwick [at] cudenver [dot] edu {Web site - home page} URL:http://www-math.cudenver.edu/~wlodwick --------------------------------------------------------------------- From owner-reliable_computing [at] interval [dot] louisiana.edu Tue Mar 5 00:40:06 2002 Received: (from daemon@localhost) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) id g256e5v09423 for reliable_computing-outgoing; Tue, 5 Mar 2002 00:40:05 -0600 (CST) Received: from mail.epost.de ([64.39.38.76]) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) with ESMTP id g256dwF09419 for ; Tue, 5 Mar 2002 00:39:59 -0600 (CST) Received: from TP570Berz (12.245.210.11) by mail.epost.de (5.5.052) (authenticated as martin.berz [at] epost [dot] de) id 3C6B20EC003A9DAB for reliable_computing [at] interval [dot] louisiana.edu; Tue, 5 Mar 2002 07:39:49 +0100 Reply-To: From: "Martin Berz" To: "Reliable_Computing@Interval. Louisiana. Edu" Subject: Re: Taylor models (repost) Date: Tue, 5 Mar 2002 01:39:50 -0500 Message-ID: MIME-Version: 1.0 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit X-Priority: 3 (Normal) X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook IMO, Build 9.0.2416 (9.0.2911.0) Importance: Normal X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2600.0000 Sender: owner-reliable_computing [at] interval [dot] louisiana.edu Precedence: bulk Arrigo, > > "Dr. George Corliss" writes: > > > Arrigo, > > > > I hope Martin will also respond, but here is my understanding. > > > [...] > > > > > I have a question about the data structure used to represent a Taylor > > > model though. In the works of Berz the coefficients of the polynomial > > > part are always represented by floating point points numers. > > > Shouldn't these be intervals? > > > > No. > > > > Martin represents a quantity as a floating point > > coefficient polynomial plus a constant bound interval > > remainder. When you add (for example) two Taylor models, > > all round off errors that could occur in the coefficients > > are lumped into the remainder interval. Often, the > > remainder interval is a couple orders of magnitude larger > > than round off in the leading coefficients, so throwing > > in a little extra does not hurt. You should also have > > going for you that you are working well within the > > radius of convergence of the Taylor series, so the > > magnitudes of the floating point coefficients (and hence > > their round off errors) drop off rapidly. What George writes is quite to the core. In typical examples, the width of the remainder bound is say 10^-10, but the error in each coefficient is in the range of 10^-16 in double precision. The sharpest way to estimate the effects of the floating point coefficient errors would indeed be to view them in each arithmetic step as intervals with initially zero width, then perform the polynomial coefficient arithmetic in interval arithmetic, and then sum up all errors that are made by representing the still very narrow resulting intervals by their floating point midpoints again. However, this is great overkill because of the size difference of 10^-16 and 10^-10; so you'd be working with high precision on something with an overall contribution that is relatively unimportant. Because of this, some rather crude and fast estimate based on maximum magnitude of coefficients and their number is completely sufficient. > > > > If you have not done so, you might read the papers by > > Martin's team and by Baker Kearfott on Taylor models > > in Automatic Differentiation: From Simulation to > > Optimization, appearing recently from Springer. > > > > > > I have read the papers by Berz and his students posted on > their web sites which I believe include the papers that you > are referring to. I found the most comprehensive account on > the implementation of Taylor models in Chap. 5 of the PhD > thesis of Makino, available at > http://www.beamtheory.nscl.msu.edu/makino/phd.html > Maybe I missed something, however I still do not see how, > for a Taylor model of order n, one can get rigorous bounds > for the parts of order k<=n of the polynomial (called order > bounds intervals by Makino) if the coefficients of the > polynomial are stored and traeted as non-interval quantities. > I am looking forward to hear from Berz too. > > Best, > -Arrigo Benedetti Well, of course the coefficients do not represent the EXACT Taylor coefficients of the function in question, but only floating point approximations to those. However, in the entire argument for making the arithmetic rigorous, including the rather involved business of the order bounds in Makino's dissertation, it is not at all necessary that they be EXACT coefficients! All that matters is that in each step, the function is indeed rigorously contained in the tube formed by whatever the coefficients are and the remainder bound. For purists, and at the expense of nomenclature overkill, it may be perhaps most accurate to refer to the method as "near-Taylor Model" or "Weierstrass Model". Actually if you think about it really carefully, conceptually this turns out to be very similar to the range bounding problem with plain intervals, as opposed to the point evaluation with intervals. In the case of range bounding, you compute f ( [a,b] ), where a and b are significantly far apart, by interval arithmetic. However, a and b are floating point numbers, and so we need to take care of the error in their arithmetic; well, conventionally this error is lumped into the overall width of the interval in each step (by simple forced rounding). Also in this case you would be able to perform your arithmetic by consistently viewing the endpoints themselves to be treated in point interval arithmetic. And only in the end you would compute a grand total interval spanned by the INTERVAL [a] and the INTERVAL [b]. But, if a and b are far apart as they are in the range bounding problem, this is just similarly unnecessary overkill, and nobody does it that way. Incidentally, in the case of the Taylor models, this idea that the coefficients don't have to be exact Taylor coefficients can perhaps be illustrated and actually exploited in practice in some cases where the underlying function is NOT differentiable, say for the absolute value or a step function. One can still do beneficial Taylor model work if one only makes sure that whatever the intrinsic is, the inclusion in whatever polynomial and remainder bound you choose is rigorous. As a particular example, consider the case of the abs function, and let's say your domain is say from -.01 to 1.01, centered around .5. Obviously the abs is not differentiable and hence not Taylor expandable over the domain; yet you can still rigorously describe it with the "Taylor" model x + [-.02,.02], or with a little thought even with a smarter polynomial to make the remainder tighter yet. One can then continue computation with this enclosure, and if there's dependency, obtain a result that is likely sharper than if you had just taken a plain interval to describe the function. We hope this helps, and greetings to all, Kyoko and Martin From owner-reliable_computing [at] interval [dot] louisiana.edu Tue Mar 5 08:03:26 2002 Received: (from daemon@localhost) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) id g25E3P110509 for reliable_computing-outgoing; Tue, 5 Mar 2002 08:03:25 -0600 (CST) Received: from mail7.wi.rr.com (mkc-162-160.kc.rr.com [24.94.162.160]) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) with ESMTP id g25E3JF10505 for ; Tue, 5 Mar 2002 08:03:20 -0600 (CST) Received: from Marquette.edu ([65.29.171.35]) by mail7.wi.rr.com with Microsoft SMTPSVC(5.5.1877.537.53); Tue, 5 Mar 2002 08:01:26 -0600 Message-ID: <3C84D021.4060102 [at] Marquette [dot] edu> Date: Tue, 05 Mar 2002 08:03:13 -0600 From: "Dr. George Corliss" Reply-To: George.Corliss [at] Marquette [dot] edu Organization: Marquette University, EECE User-Agent: Mozilla/5.0 (Windows; U; Windows NT 5.1; en-US; rv:0.9.4) Gecko/20011128 Netscape6/6.2.1 X-Accept-Language: en-us MIME-Version: 1.0 To: Arrigo Benedetti CC: reliable_computing [at] interval [dot] louisiana.edu, berz [at] msu [dot] edu Subject: Re: Taylor models (repost) References: <15487.48230.153413.854985 [at] gargle [dot] gargle.HOWL> <3C7FD90C.4070106 [at] Marquette [dot] edu> Content-Type: text/plain; charset=us-ascii; format=flowed Content-Transfer-Encoding: 7bit Sender: owner-reliable_computing [at] interval [dot] louisiana.edu Precedence: bulk Arrigo, >>>I have a question about the data structure used to represent a Taylor >>>model though. In the works of Berz the coefficients of the polynomial >>>part are always represented by floating point points numers. >>>Shouldn't these be intervals? > Maybe I missed something, however I still do not see how, > for a Taylor model of order n, one can get rigorous bounds > for the parts of order k<=n of the polynomial (called order > bounds intervals by Makino) if the coefficients of the > polynomial are stored and traeted as non-interval quantities. Perhaps a very simple example can help Let a(t) be contained in the Taylor Model A = 1 + 1 t + [1, 2] x 10^-16 b(t) in B = -1 + 0.5 t + [2, 5] x 10^-16 for t in [-10, 10], say. Taylor models are usually higher degree and multivariable, but a simple example may help answer the floating point vs. interval question. In exact arithmetic a+b is in A + B = 0 + 1.5 t + [3, 7] x 10^-16 In practice, we can build into the + operation the knowledge that 0 = A[0] + B[0] could be in error by 1 ULP (or 1/2) 1.5 = A[1] + B[1] could be in error by 1 ULP Hence, the error in the Taylor polynomial 0 + 1.5 t for t in [-10, 10] must be in [-1, 1] (1 + 1x10) ULP For simplicity, if 1 ULP = 10^-16, we lump this error with the Taylor remainder: a+b is in A + B = 0 + 1.5 t + [-7, 18] x 10^-16 We have guaranteed bounds using floating point coefficients. We also have canceled two nearly equal quantities without a gross overestimation. Dr. George F. Corliss Electrical and Computer Engineering Haggerty Engineering 296 Marquette University P.O. Box 1881 Milwaukee, WI 53201-1881 USA George.Corliss [at] Marquette [dot] edu Office: 414-288-6599; Dept: 288-6820; Fax: 288-5579 From owner-reliable_computing [at] interval [dot] louisiana.edu Tue Mar 5 09:28:38 2002 Received: (from daemon@localhost) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) id g25FSbT10742 for reliable_computing-outgoing; Tue, 5 Mar 2002 09:28:37 -0600 (CST) Received: from mailgate.rz.uni-karlsruhe.de (exim [at] mailgate [dot] rz.uni-karlsruhe.de [129.13.64.97]) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) with ESMTP id g25FSVF10738 for ; Tue, 5 Mar 2002 09:28:33 -0600 (CST) Received: from math.uni-karlsruhe.de (iamlapc9.mathematik.uni-karlsruhe.de [129.13.114.109]) by mailgate.rz.uni-karlsruhe.de with esmtp (Exim 3.33 #1) id 16iGrb-00021C-00; Tue, 05 Mar 2002 16:28:27 +0100 Message-ID: <3C84E4D8.45D04CED [at] math [dot] uni-karlsruhe.de> Date: Tue, 05 Mar 2002 16:31:36 +0100 From: Markus Neher X-Mailer: Mozilla 4.78 [de]C-CCK-MCD DT (Windows NT 5.0; U) X-Accept-Language: de MIME-Version: 1.0 To: George.Corliss [at] Marquette [dot] edu CC: reliable_computing [at] interval [dot] louisiana.edu Subject: Re: Taylor models (repost) References: <15487.48230.153413.854985 [at] gargle [dot] gargle.HOWL> <3C7FD90C.4070106 [at] Marquette [dot] edu> <3C84D021.4060102 [at] Marquette [dot] edu> Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Sender: owner-reliable_computing [at] interval [dot] louisiana.edu Precedence: bulk George, personally, I think that Taylor models are the best tools in the market for the validated solution of IVPs or for other problems where the wrapping effect plays an important role. For the computation of a single range bound, however, the wrapping effect should not be critical. Considering range bounds, can Taylor models improve centered forms? Centered forms converge quadratically. They also take care of the dependency problem so that quadratic convergence is maintained for the evaluation of terms such as f(x) + (-f(x)), where direct interval evaluation yields a large overestimation of the true range. To make this more specific, let me repeat the first part of Arrigo's original question: >>> I still do not see how, >>> for a Taylor model of order n, one can get rigorous bounds >>> for the parts of order k<=n of the polynomial Assume that k=n=6 and that you have three variables. Is there a practical algorithm to evaluate the polynomial part of the Taylor model accurate enough to obtain range bounds that converge to the true range with order 6? Best regards, Markus ----- Dr. Markus Neher Institut fuer Angewandte Mathematik, Universitaet Karlsruhe, Germany From owner-reliable_computing [at] interval [dot] louisiana.edu Tue Mar 5 10:19:43 2002 Received: (from daemon@localhost) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) id g25GJgv10892 for reliable_computing-outgoing; Tue, 5 Mar 2002 10:19:42 -0600 (CST) Received: from mercury.Sun.COM (mercury.Sun.COM [192.9.25.1]) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) with ESMTP id g25GJaF10888 for ; Tue, 5 Mar 2002 10:19:37 -0600 (CST) Received: from engmail1.Eng.Sun.COM ([129.146.1.13]) by mercury.Sun.COM (8.9.3+Sun/8.9.3) with ESMTP id IAA08374; Tue, 5 Mar 2002 08:19:19 -0800 (PST) Received: from phys-mpkmaila (phys-mpkmaila.Eng.Sun.COM [129.146.18.131]) by engmail1.Eng.Sun.COM (8.9.3+Sun/8.9.3/ENSMAIL,v2.1p1) with ESMTP id IAA29668; Tue, 5 Mar 2002 08:19:18 -0800 (PST) Received: from gww (gww.Eng.Sun.COM [129.146.78.116]) by mpkmail.eng.sun.com (iPlanet Messaging Server 5.2 (built Jan 13 2002)) with SMTP id <0GSI002Q9DCRY4 [at] mpkmail [dot] eng.sun.com>; Tue, 05 Mar 2002 08:19:39 -0800 (PST) Date: Tue, 05 Mar 2002 08:19:13 -0800 (PST) From: William Walster Subject: Re: Taylor models (repost) To: George.Corliss [at] Marquette [dot] edu, markus.neher [at] math [dot] uni-karlsruhe.de Cc: reliable_computing [at] interval [dot] louisiana.edu Reply-to: William Walster Message-id: <0GSI002QADCRY4 [at] mpkmail [dot] eng.sun.com> MIME-version: 1.0 X-Mailer: dtmail 1.3.0 @(#)CDE Version 1.4.2 SunOS 5.8 sun4u sparc Content-type: TEXT/plain; charset=us-ascii Content-transfer-encoding: 7BIT Content-MD5: QbZDQAWw6Xy22AptlNdIFw== Sender: owner-reliable_computing [at] interval [dot] louisiana.edu Precedence: bulk >Date: Tue, 05 Mar 2002 16:31:36 +0100 >From: Markus Neher >Subject: Re: Taylor models (repost) >To: George.Corliss [at] Marquette [dot] edu >Cc: reliable_computing [at] interval [dot] louisiana.edu >MIME-version: 1.0 >Content-transfer-encoding: 7bit >X-Accept-Language: de > >George, > >personally, I think that Taylor models are the best tools in the market >for the validated solution of IVPs or for other problems where the >wrapping effect plays an important role. For the computation of a single >range bound, however, the wrapping effect should not be critical. > >Considering range bounds, can Taylor models improve centered forms? >Centered forms converge quadratically. They also take care of the >dependency problem so that quadratic convergence is maintained for the >evaluation of terms such as f(x) + (-f(x)), where direct interval >evaluation yields a large overestimation of the true range. I believe this is *not* correct. Neither centered forms nor Taylor models handle the effects of dependence over large argument intervals (boxes). They are both asymptotic results that work well "in the small" or "the medium", but can be worse than simple interval evaluation "in the large". We still need methods to use to get even "reasonable" range bounds of expressions over large intervals (boxes). > >To make this more specific, let me repeat the first part of Arrigo's >original question: > >>>> I still do not see how, >>>> for a Taylor model of order n, one can get rigorous bounds >>>> for the parts of order k<=n of the polynomial > >Assume that k=n=6 and that you have three variables. Is there a >practical algorithm to evaluate the polynomial part of the Taylor model >accurate enough to obtain range bounds that converge to the true range >with order 6? > >Best regards, > >Markus > >----- > >Dr. Markus Neher >Institut fuer Angewandte Mathematik, Universitaet Karlsruhe, Germany From owner-reliable_computing [at] interval [dot] louisiana.edu Tue Mar 5 10:45:58 2002 Received: (from daemon@localhost) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) id g25GjvC11022 for reliable_computing-outgoing; Tue, 5 Mar 2002 10:45:57 -0600 (CST) Received: from imf20bis.bellsouth.net (mail120.mail.bellsouth.net [205.152.58.80]) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) with ESMTP id g25GjqF11018 for ; Tue, 5 Mar 2002 10:45:52 -0600 (CST) Received: from u8174 ([65.81.243.152]) by imf20bis.bellsouth.net (InterMail vM.5.01.04.05 201-253-122-122-105-20011231) with SMTP id <20020305164658.DGZH9003.imf20bis.bellsouth.net@u8174>; Tue, 5 Mar 2002 11:46:58 -0500 Message-Id: <2.2.32.20020305164506.009931c0 [at] pop [dot] louisiana.edu> X-Sender: rbk5287 [at] pop [dot] louisiana.edu X-Mailer: Windows Eudora Pro Version 2.2 (32) Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" Date: Tue, 05 Mar 2002 10:45:06 -0600 To: Markus Neher , George.Corliss [at] Marquette [dot] edu From: "R. Baker Kearfott" Subject: Re: Taylor models (repost) Cc: reliable_computing [at] interval [dot] louisiana.edu Sender: owner-reliable_computing [at] interval [dot] louisiana.edu Precedence: bulk Markus, Taylor models do a better job of taking care of the dependency problem than centered forms, in some cases, over somewhat larger boxes. For fixed Taylor model and fixed quadratic form, both the Taylor model and quadratic form exhibit "quadratic convergence" as the box size is made smaller. Note that a quadratic form is a degree-1 Taylor polynomial. What I'm saying is that higher degree Taylor polynomials may sometimes do a better job over larger boxes. The issue is not the convergence order, since the polynomial part is also evaluated with interval arithmetic (and hence involves some, but often less overestimation). (In ODE, the Taylor models are used somewhat differently.) Also, there is no theorem (yet) about when the Taylor models do a better job, and, indeed, they sometimes do not do a significantly better job. I've got two papers with comparisons, one with A. Arazyan that appeared in the proceedings of AD 2000, and one with Bill Walster that is presently under review. Of course, I'm open to hearing of new results in the area :-) Best regards, Baker At 04:31 PM 3/5/02 +0100, Markus Neher wrote: >George, > >personally, I think that Taylor models are the best tools in the market >for the validated solution of IVPs or for other problems where the >wrapping effect plays an important role. For the computation of a single >range bound, however, the wrapping effect should not be critical. > >Considering range bounds, can Taylor models improve centered forms? >Centered forms converge quadratically. They also take care of the >dependency problem so that quadratic convergence is maintained for the >evaluation of terms such as f(x) + (-f(x)), where direct interval >evaluation yields a large overestimation of the true range. > >To make this more specific, let me repeat the first part of Arrigo's >original question: > >>>> I still do not see how, >>>> for a Taylor model of order n, one can get rigorous bounds >>>> for the parts of order k<=n of the polynomial > >Assume that k=n=6 and that you have three variables. Is there a >practical algorithm to evaluate the polynomial part of the Taylor model >accurate enough to obtain range bounds that converge to the true range >with order 6? > >Best regards, > >Markus > >----- > >Dr. Markus Neher >Institut fuer Angewandte Mathematik, Universitaet Karlsruhe, Germany > > --------------------------------------------------------------- R. Baker Kearfott, rbk [at] louisiana [dot] edu (337) 482-5346 (fax) (337) 482-5270 (work) (337) 981-9744 (home) URL: http://interval.louisiana.edu/kearfott.html Department of Mathematics, University of Louisiana at Lafayette Box 4-1010, Lafayette, LA 70504-1010, USA --------------------------------------------------------------- From owner-reliable_computing [at] interval [dot] louisiana.edu Tue Mar 5 11:45:52 2002 Received: (from daemon@localhost) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) id g25HjpL11203 for reliable_computing-outgoing; Tue, 5 Mar 2002 11:45:51 -0600 (CST) Received: from bologna.vision.caltech.edu (bologna.vision.caltech.edu [131.215.134.19]) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) with ESMTP id g25HjkF11199 for ; Tue, 5 Mar 2002 11:45:46 -0600 (CST) Received: from peking.vision.caltech.edu (peking [131.215.134.18]) by bologna.vision.caltech.edu (8.9.3/8.8.7) with ESMTP id JAA25482 for ; Tue, 5 Mar 2002 09:45:45 -0800 Received: (from arrigo@localhost) by peking.vision.caltech.edu (8.9.3+Sun/8.9.1) id JAA13979; Tue, 5 Mar 2002 09:45:31 -0800 (PST) X-Authentication-Warning: peking.vision.caltech.edu: arrigo set sender to arrigo [at] vision [dot] caltech.edu using -f To: reliable_computing [at] interval [dot] louisiana.edu Subject: Re: Taylor models (repost) References: <15487.48230.153413.854985 [at] gargle [dot] gargle.HOWL> <3C7FD90C.4070106 [at] Marquette [dot] edu> <3C84D021.4060102 [at] Marquette [dot] edu> From: Arrigo Benedetti Date: 05 Mar 2002 09:45:31 -0800 In-Reply-To: <3C84D021.4060102 [at] Marquette [dot] edu> Message-ID: Lines: 53 User-Agent: Gnus/5.0808 (Gnus v5.8.8) Emacs/20.4 MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Sender: owner-reliable_computing [at] interval [dot] louisiana.edu Precedence: bulk "Dr. George Corliss" writes: > Arrigo, > > >>>I have a question about the data structure used to represent a Taylor > >>>model though. In the works of Berz the coefficients of the polynomial > >>>part are always represented by floating point points numers. > >>>Shouldn't these be intervals? > > Maybe I missed something, however I still do not see how, > > for a Taylor model of order n, one can get rigorous bounds > > for the parts of order k<=n of the polynomial (called order > > bounds intervals by Makino) if the coefficients of the > > polynomial are stored and traeted as non-interval quantities. > Perhaps a very simple example can help > > > Let a(t) be contained in the Taylor Model A = 1 + 1 t + [1, 2] x 10^-16 > b(t) in B = -1 + 0.5 t + [2, 5] x 10^-16 > for t in [-10, 10], say. > > Taylor models are usually higher degree and multivariable, > but a simple example may help answer the floating point > vs. interval question. > > In exact arithmetic > a+b is in A + B = 0 + 1.5 t + [3, 7] x 10^-16 > > In practice, we can build into the + operation the > knowledge that > 0 = A[0] + B[0] could be in error by 1 ULP (or 1/2) > 1.5 = A[1] + B[1] could be in error by 1 ULP > Hence, the error in the Taylor polynomial > 0 + 1.5 t for t in [-10, 10] must be in [-1, 1] (1 + 1x10) ULP > > For simplicity, if 1 ULP = 10^-16, we lump this error > with the Taylor remainder: > a+b is in A + B = 0 + 1.5 t + [-7, 18] x 10^-16 > > We have guaranteed bounds using floating point > coefficients. We also have canceled two nearly > equal quantities without a gross overestimation. > Everything is clear now. I also assume that the sum of the ULP is automatically taken care of in the implementation of Taylor models by Berz et al. in COSY. Thanks much, -Arrigo -- Dr. Arrigo Benedetti e-mail: arrigo [at] vision [dot] caltech.edu Caltech, MS 136-93 phone: (626) 644-3757 Pasadena, CA 91125 fax: (626) 795-8649 From owner-reliable_computing [at] interval [dot] louisiana.edu Tue Mar 5 18:17:45 2002 Received: (from daemon@localhost) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) id g260HiS11874 for reliable_computing-outgoing; Tue, 5 Mar 2002 18:17:44 -0600 (CST) Received: from postmarq.mu.edu (maia.mu.edu [134.48.1.6]) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) with ESMTP id g260HZF11870 for ; Tue, 5 Mar 2002 18:17:37 -0600 (CST) Received: from Marquette.edu ([134.48.92.93]) by postmarq.mu.edu (Netscape Messaging Server 4.15 marquette Dec 7 2001 06:47:59) with ESMTP id GSIZH400.C9X; Tue, 5 Mar 2002 18:17:28 -0600 Message-ID: <3C856040.2000500 [at] Marquette [dot] edu> Date: Tue, 05 Mar 2002 18:18:08 -0600 From: "Dr. George F. Corliss" Reply-To: George.Corliss [at] Marquette [dot] edu Organization: Marquette University User-Agent: Mozilla/5.0 (Windows; U; Windows NT 5.0; en-US; rv:0.9.4) Gecko/20011128 Netscape6/6.2.1 X-Accept-Language: en-us MIME-Version: 1.0 To: Markus Neher CC: reliable_computing [at] interval [dot] louisiana.edu Subject: Re: Taylor models (repost) References: <15487.48230.153413.854985 [at] gargle [dot] gargle.HOWL> <3C7FD90C.4070106 [at] Marquette [dot] edu> <3C84D021.4060102 [at] Marquette [dot] edu> <3C84E4D8.45D04CED [at] math [dot] uni-karlsruhe.de> Content-Type: text/plain; charset=us-ascii; format=flowed Content-Transfer-Encoding: 7bit Sender: owner-reliable_computing [at] interval [dot] louisiana.edu Precedence: bulk Markus, Others have addressed other aspects. Let me attempt > Assume that k=n=6 and that you have three variables. Is there a > practical algorithm to evaluate the polynomial part of the Taylor model > accurate enough to obtain range bounds that converge to the true range > with order 6? Many of us have asked repeatedly about bounding the polynomial part, and I think I have finally figured out the essence of Martin's answers: Wrong question. I think I understand why. Taylor models might have asymptotic properties, but their strength is in addressing the dependency problem outside the asymptotic range. Bill is still looking for breakthroughs for the "big box" problem. The general problem of bounding a say 6th degree polynomial in say 6 variables is very hard. If we take that as the problem, there probably are no good answers. Martin thrives on "Good enough." A degree one, 6 variable function is easy to bound. By looking at signs of first order terms, you can tell which corners to evaluate. So we get bounds from the first order terms, then add an enclosure of contributions we could get from terms of order 2 and higher. Key: If the Taylor models are going to do you any good anyway, the series is probably converging, so second and higher order terms are probably rather small. Hence, we put crude bounds on possible contributions from orders 2 and higher, we use those to inflate the order 1 bounds. Sometimes, this will perform poorly, but often it seems to perform very well. Dr. George F. Corliss Electrical and Computer Engineering Haggerty Engineering 296 Marquette University P.O. Box 1881 Milwaukee, WI 53201-1881 USA George.Corliss [at] Marquette [dot] edu Office: 414-288-6599; Dept: 288-6820; Fax: 288-5579 From owner-reliable_computing [at] interval [dot] louisiana.edu Wed Mar 6 06:36:29 2002 Received: (from daemon@localhost) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) id g26CaS213434 for reliable_computing-outgoing; Wed, 6 Mar 2002 06:36:28 -0600 (CST) Received: from mailgate.rz.uni-karlsruhe.de (exim [at] mailgate [dot] rz.uni-karlsruhe.de [129.13.64.97]) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) with ESMTP id g26CaNF13430 for ; Wed, 6 Mar 2002 06:36:23 -0600 (CST) Received: from math.uni-karlsruhe.de (iamlapc9.mathematik.uni-karlsruhe.de [129.13.114.109]) by mailgate.rz.uni-karlsruhe.de with esmtp (Exim 3.33 #1) id 16iaeZ-0001s1-00; Wed, 06 Mar 2002 13:36:19 +0100 Message-ID: <3C860E03.84282521 [at] math [dot] uni-karlsruhe.de> Date: Wed, 06 Mar 2002 13:39:32 +0100 From: Markus Neher X-Mailer: Mozilla 4.78 [de]C-CCK-MCD DT (Windows NT 5.0; U) X-Accept-Language: de MIME-Version: 1.0 To: George.Corliss [at] Marquette [dot] edu CC: reliable_computing [at] interval [dot] louisiana.edu Subject: Re: Taylor models (repost) References: <15487.48230.153413.854985 [at] gargle [dot] gargle.HOWL> <3C7FD90C.4070106 [at] Marquette [dot] edu> <3C84D021.4060102 [at] Marquette [dot] edu> <3C84E4D8.45D04CED [at] math [dot] uni-karlsruhe.de> <3C856040.2000500 [at] Marquette [dot] edu> Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Sender: owner-reliable_computing [at] interval [dot] louisiana.edu Precedence: bulk George, this is meant as an answer to your response and those of others. I agree to what Baker Kearfott and Bill Walster wrote in their postings: The big box problem is the real issue. So far, neither centered forms nor Taylor models have provided a solution for this. > Others have addressed other aspects. Let me attempt > > > Assume that k=n=6 and that you have three variables. Is there a > > practical algorithm to evaluate the polynomial part of the Taylor model > > accurate enough to obtain range bounds that converge to the true range > > with order 6? > > Many of us have asked repeatedly about bounding the > polynomial part, and I think I have finally figured > out the essence of Martin's answers: Wrong question. Is it? How do we measure the quality of Taylor model based range enclosures compared to other forms? If order of convergence is measured by the amount of range overestimation as the box size tends to zero, then a Taylor model consisting of a polynomial part of order n and a small remainder interval doesn't provide an n-th order range enclosure, unless there is a way to evaluate the polynomial part exactly. When the polynomial is evaluated by some kind of Horner's scheme, the range overestimation can be much larger than the width of the interval term, so that above definition of order of convergence does not correspond with the order of the polynomial. As Baker pointed out, the evaluation of the polynomial *may sometimes* do a better job than direct interval evaluation or than centered forms. Hopefully, future research will turn this observation into a theorem. Best regards to all, Markus From owner-reliable_computing [at] interval [dot] louisiana.edu Wed Mar 6 07:04:07 2002 Received: (from daemon@localhost) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) id g26D47J13567 for reliable_computing-outgoing; Wed, 6 Mar 2002 07:04:07 -0600 (CST) Received: from mail.epost.de ([64.39.38.76]) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) with ESMTP id g26D42F13563 for ; Wed, 6 Mar 2002 07:04:02 -0600 (CST) Received: from TP570Berz (12.245.210.11) by mail.epost.de (5.5.052) (authenticated as martin.berz [at] epost [dot] de) id 3C6B20EC003F6DFF for reliable_computing [at] interval [dot] louisiana.edu; Wed, 6 Mar 2002 14:03:57 +0100 Reply-To: From: "Martin Berz" To: Subject: RE: Taylor models (repost) Date: Wed, 6 Mar 2002 08:03:52 -0500 Message-ID: MIME-Version: 1.0 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit X-Priority: 3 (Normal) X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook IMO, Build 9.0.2416 (9.0.2911.0) Importance: Normal In-Reply-To: <3C856040.2000500 [at] Marquette [dot] edu> X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2600.0000 Sender: owner-reliable_computing [at] interval [dot] louisiana.edu Precedence: bulk Markus, Bill, Baker, George, Arrigo, here are some more comments on various fragments. > Markus writes: > > personally, I think that Taylor models are the best tools in the market > for the validated solution of IVPs or for other problems where the > wrapping effect plays an important role. For the computation of a single > range bound, however, the wrapping effect should not be critical. Whether the dependency problem or wrapping effect matters for range bounding depends greatly on the example function you are considering. There are various cases where this matters greatly and prevents any of the conventional "box exclusion" methods to begin excluding boxes. One that we have been working on is the normal form defect problem from dynamical systems theory. Another one is the famous Gritton's function that Baker posted on this forum more than a year ago. Stimulated by his posting, it was then studied in detail by Baker and ourselves; Baker's work is I believe summarized in the AD2000 book, and ours can be found in a recent paper in Nonlinear Analysis, http://bt.nscl.msu.edu/cgi-bin/display.pl?name=HOINL00 > > Considering range bounds, can Taylor models improve centered forms? > Centered forms converge quadratically. They also take care of the > dependency problem so that quadratic convergence is maintained for the > evaluation of terms such as f(x) + (-f(x)), where direct interval > evaluation yields a large overestimation of the true range. > Just as a small comment, a method can maintain quadratic convergence even if the method can not avoid the dependency problem; these two things are not necessarily tied. Regarding the performance regarding the dependency problem of centered versus Taylor, we haven't actually studied this ourselves, but it sounds like Baker has ... see his comments below. > To make this more specific, let me repeat the first part of Arrigo's > original question: > >>> I still do not see how, >>> for a Taylor model of order n, one can get rigorous bounds >>> for the parts of order k<=n of the polynomial > > Assume that k=n=6 and that you have three variables. Is there a > practical algorithm to evaluate the polynomial part of the Taylor model > accurate enough to obtain range bounds that converge to the true range > with order 6? > > Best regards, > > Markus The answer to Markus' specific question about the local asymptotics is a clear yes. And this yes is even independent of the particular method of bounding of the polynomials. In particular, the simplest way of performing all the "bounding" that is going on in the polynomial is simple evaluation with conventional intervals. Even under this simplest approach, the convergence is of true order 6, and in general of order (n+1). Actually more so, one can even show rigorously that as the domain width h decreases, the new error will even always go down FASTER than h**6, if the implementation of all intrinsics and elementary operations only satisfies inclusion monotonicity. >>> I still do not see how, >>> for a Taylor model of order n, one can get rigorous bounds >>> for the parts of order k<=n of the polynomial I believe while Markus was wondering about the true asymptotic behavior which I just mentioned, the original question of Arrigo was in a slightly different direction. I interpreted his concern to be something like "if the Taylor expansion coefficients are slightly off because they are floating point, then how can the methodology of the order bounds in Makino's dissertation yield true statements about the order bounds". (The order bounds are actually a technical issue concerning how to best implement the arithmetic and are not primarily related to the range bounding problem Markus mentions). To answer Arrigo, unless I am missing some of the intricacies of what Kyoko does in her COSY implementation of the Taylor model approach, Makino's "order bounds" are NOT true order bounds of the original Taylor polynomial. But again this doesn't matter as long as the function is truly contained in the tube given by the Taylor model. Perhaps Kyoko can say more about this since she knows the details best, but after her and my last joint message she went on a trip (to explain COSY things to Beam Physics people at a conference at UCLA) and maybe offline for a week or so. (Actually, wait: Arrigo is at Caltech, perhaps a meeting could be arranged...) > > > Assume that k=n=6 and that you have three variables. Is there a > > practical algorithm to evaluate the polynomial part of the Taylor model > > accurate enough to obtain range bounds that converge to the true range > > with order 6? > George answers: > Many of us have asked repeatedly about bounding the > polynomial part, and I think I have finally figured > out the essence of Martin's answers: Wrong question. > > I think I understand why. Taylor models might have > asymptotic properties, but their strength is in > addressing the dependency problem outside the asymptotic > range. Bill is still looking for breakthroughs for the > "big box" problem. > > The general problem of bounding a say 6th degree polynomial > in say 6 variables is very hard. If we take that as the > problem, there probably are no good answers. Martin > thrives on "Good enough." A degree one, 6 variable > function is easy to bound. By looking at signs of > first order terms, you can tell which corners to evaluate. > So we get bounds from the first order terms, then add > an enclosure of contributions we could get from terms of > order 2 and higher. > > Key: If the Taylor models are going to do you any good anyway, > the series is probably converging, so second and higher > order terms are probably rather small. Hence, we put > crude bounds on possible contributions from orders 2 and > higher, we use those to inflate the order 1 bounds. > > Sometimes, this will perform poorly, but often it seems > to perform very well. > What George says there is quite true, and, as always, George succeeds in expressing it very clearly and probably much better than Kyoko or I could ... As I mentioned above, for Markus' questions about asymptotics the choice of the bounding scheme doesn't matter. As George mentions now, even for the general range bounding question, it doesn't matter all that much. The bottom line is that the resulting Taylor model remainder bound will be usably small IF AND ONLY IF the approximation of the function by its Taylor model converges rapidly with order, and hence the higher orders fall of quickly. But this means precisely that the higher orders are also easier and easier to bound. And to say it again, the Taylor models do NOT solve the Np-hard question of bounding a polynomial over an arbitrary range. > Bill comments to the same topic: > > > > >Considering range bounds, can Taylor models improve centered forms? > >Centered forms converge quadratically. They also take care of the > >dependency problem so that quadratic convergence is maintained for the > >evaluation of terms such as f(x) + (-f(x)), where direct interval > >evaluation yields a large overestimation of the true range. > I believe this is *not* correct. Neither centered forms nor Taylor > models handle the effects of dependence over large argument intervals > (boxes). They are both asymptotic results that work well "in the small" > or "the medium", but can be worse than simple interval evaluation > "in the large". > > We still need methods to use to get even "reasonable" range bounds > of expressions over large intervals (boxes). I believe Bill is quite right with his assessment as well. The "in the large" problem is wide open, and in fact it's at the core of the entire question of global optimization. And it's Np hard ... Besides the dependency problem which we have already discussed, another angle from which I like to look at the conventional intervals, the centered forms, and the Taylor models, is that these are techniques of various orders to attack the same problem, MUCH LIKE OTHER HIGH-ORDER SCHEMES for quadrature, ODE solution, PDE solution etc etc etc. All these high order methods capitalize on perhaps the one and only exploitable intrinsic property that solutions to our numerics problems have: they are infinitely often differentiable, and thus over a reasonable neighborhood, can be represented by their Taylor expansion (or other slightly modified suitable polynomial). This is the only idea the Taylor model approach rests on, and the dramatic precision it can achieve over reasonable domains is just in complete analogy to the precision that a high-order quadrature scheme yields compared to a low order one. Also, in complete analogy, if you make your box or step too large, the high-order schemes may fail even more miserably than low-order schemes. So in this light, there is no big magic in the Taylor models, they only bring into the "verification" business an old trick that has been used forever in non-verified settings, but that so far has been very hard to exploit in a general setting that is possible with the Taylor models. > And Baker writes: > Taylor models do a better job of taking care of the dependency problem > than centered forms, in some cases, over somewhat larger boxes. > For fixed Taylor model and fixed quadratic form, both the Taylor > model and quadratic form exhibit "quadratic convergence" as the box > size is made smaller. > Note that a quadratic form is a degree-1 Taylor polynomial. What > I'm saying is that higher degree Taylor polynomials may sometimes > do a better job over larger boxes. The issue is not the convergence > order, since the polynomial part is also evaluated with interval > arithmetic (and hence involves some, but often less overestimation). > (In ODE, the Taylor models are used somewhat differently.) Also, there > is no theorem (yet) about when the Taylor models do a better job, > and, indeed, they sometimes do not do a significantly better job. > I've got two papers with comparisons, one with A. Arazyan that > appeared in the proceedings of AD 2000, and one with Bill Walster > that is presently under review. So far we have looked at the "small size" problem and concluded that asymptotically Taylor models indeed have high-order properties and avoid dependency; we have looked at the "large size" problem and concluded that asymptotically Taylor models fail like anything else (but perhaps also with higher order and hence more miserably ;-) ), and Baker is now discussing in some detail the "intermediate size" problem. From the analogy to other numerical methods like Runge Kuttas of various orders, we would expect that the higher order method would still have a general tendency to outperform the lower order ones, but nothing is guaranteed here. But to push the analogy with other numerical methods further: what makes these industrial strength is usually the dynamic adjustment of step size (or domain size) and order. In a similar vein this is possible for the Taylor models; in particular it's actually conceptually easy to adjust the order of the method by keeping track of how large the contributions to the high orders are. In the extreme case, they would fall back to zeroth order, i.e. plain intervals. And thus, in this self-adapting way, you'd have something like a "Theorem" that Taylor models are "always" better than intervals (otherwise such a theorem is probably not possible to prove). I hope that Kyoko gets around eventually to implement this fully in her Taylor model package; some rudimentary parts of it exist. Finally, one aspect that hasn't been mentioned is the performance for different numbers of variables, the dimensional curse. Here the good old Taylor expansion, and with it the Taylor models, provides a relief because the number of Taylor coefficients increases only moderately with dimensionality, and if your choice is to raster coarsely with Taylor models or even only slightly more finely with intervals, the Taylor models win. Again, no surprise: we are again just exploiting a symmetry of the underlying functions, namely their high order differentiability, that the intervals cannot exploit. To conclude, after all these discussion, I'd be curious to see Arrigo's original functions that stimulated all of this - so if he could send us some more details, either to all of you or to us at MSU to take a look ourselves, it would be very useful. Greetings, Martin From owner-reliable_computing [at] interval [dot] louisiana.edu Wed Mar 6 09:27:15 2002 Received: (from daemon@localhost) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) id g26FRFr13896 for reliable_computing-outgoing; Wed, 6 Mar 2002 09:27:15 -0600 (CST) Received: from clmboh1-smtp3.columbus.rr.com (clmboh1-smtp3.columbus.rr.com [65.24.0.112]) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) with ESMTP id g26FR9F13892 for ; Wed, 6 Mar 2002 09:27:10 -0600 (CST) Received: from oemcomputer (dhcp065-024-174-102.columbus.rr.com [65.24.174.102]) by clmboh1-smtp3.columbus.rr.com (8.11.2/8.11.2) with SMTP id g26FR3R24447; Wed, 6 Mar 2002 10:27:03 -0500 (EST) Message-ID: <008d01c1c522$fb073cc0$66ae1841 [at] columbus [dot] rr.com> From: "Ramon Moore" To: "Markus Neher" , Cc: References: <15487.48230.153413.854985 [at] gargle [dot] gargle.HOWL> <3C7FD90C.4070106 [at] Marquette [dot] edu> <3C84D021.4060102 [at] Marquette [dot] edu> <3C84E4D8.45D04CED [at] math [dot] uni-karlsruhe.de> <3C856040.2000500 [at] Marquette [dot] edu> <3C860E03.84282521 [at] math [dot] uni-karlsruhe.de> Subject: Re: Taylor models (repost) Date: Wed, 6 Mar 2002 10:24:16 -0500 MIME-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: 7bit X-Priority: 3 X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook Express 5.50.4133.2400 X-MimeOLE: Produced By Microsoft MimeOLE V5.50.4133.2400 Sender: owner-reliable_computing [at] interval [dot] louisiana.edu Precedence: bulk Dear People, It is not at all obvious that the range of values of the polynomial of degree 299 given by P(x)=SUM{[(-1)^k]*[x^(2k+1)]/(2k+1)!}, summed over k from 0 to 149, has a range of values contained in the interval [-1 - 1.5*10^(-13), 1 + 1.5*10^(-13)] for all x in the interval [0, 100] unless you happen to know that P(x) is the first part of the Taylor series of the sine function, up to the x^(299) term. Of course evaluating the polynomial P(x) for large x would be a terrible idea, even though P(x) agrees with the sine function very accurately for all x up to x = 100. Most terms of the polynomial grow exceedingly large for large x, and catastrophic cancellations abound. What we DO instead is make use of our extensive knowledge of the sine function, its periodicity of 2*pi, and the fact that its range of values, for real x, is always contained in [-1, 1], and that is an odd function, sin(-x) = -sin(x), and that it is monotone increasing in [-pi/2, pi/2], etc. We are, of course, not so lucky with extensive analytic knowledge of every polynomial in one or more variables that comes along. The good news is that in any specific area of applied mathematics, there are certain expressions, perhaps not very many, that are always around, always used. Much is usually known about their behavior that can be and is used in numerical computations. I mention this as a reminder that it is not necessary to tackle the completely general problem of efficiently computing rigorous bounds on the range of values of functions defined by arbitrary expressions. I suggest it is a far wiser use of our time to tackle the problem for functions defined by special types of expressions that occur in important application areas. best wishes to all, Ramon Moore ----- Original Message ----- From: "Markus Neher" To: Cc: Sent: Wednesday, March 06, 2002 7:39 AM Subject: Re: Taylor models (repost) > George, > > this is meant as an answer to your response and those of others. > > I agree to what Baker Kearfott and Bill Walster wrote in their postings: > The big box problem is the real issue. So far, neither centered forms > nor Taylor models have provided a solution for this. > > > Others have addressed other aspects. Let me attempt > > > > > Assume that k=n=6 and that you have three variables. Is there a > > > practical algorithm to evaluate the polynomial part of the Taylor model > > > accurate enough to obtain range bounds that converge to the true range > > > with order 6? > > > > Many of us have asked repeatedly about bounding the > > polynomial part, and I think I have finally figured > > out the essence of Martin's answers: Wrong question. > > Is it? > > How do we measure the quality of Taylor model based range enclosures > compared to other forms? If order of convergence is measured by the > amount of range overestimation as the box size tends to zero, then a > Taylor model consisting of a polynomial part of order n and a small > remainder interval doesn't provide an n-th order range enclosure, unless > there is a way to evaluate the polynomial part exactly. > > When the polynomial is evaluated by some kind of Horner's scheme, the > range overestimation can be much larger than the width of the interval > term, so that above definition of order of convergence does not > correspond with the order of the polynomial. > > As Baker pointed out, the evaluation of the polynomial *may sometimes* > do a better job than direct interval evaluation or than centered forms. > Hopefully, future research will turn this observation into a theorem. > > Best regards to all, > > Markus > From owner-reliable_computing [at] interval [dot] louisiana.edu Wed Mar 6 09:32:26 2002 Received: (from daemon@localhost) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) id g26FWPW13984 for reliable_computing-outgoing; Wed, 6 Mar 2002 09:32:25 -0600 (CST) Received: from zeus.polsl.gliwice.pl (root [at] zeus [dot] polsl.gliwice.pl [157.158.1.3]) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) with ESMTP id g26FWGF13979 for ; Wed, 6 Mar 2002 09:32:17 -0600 (CST) Received: from andrzej1 (a18.asystent.polsl.gliwice.pl [157.158.187.18]) by zeus.polsl.gliwice.pl (8.9.3 (PHNE_25183)/8.9.3) with SMTP id QAA13370; Wed, 6 Mar 2002 16:32:13 +0100 (MET) Message-ID: <005001c1c524$14500e90$0200a8c0@andrzej1> From: "Andrzej Pownuk" To: Cc: "Andrzej Pownuk" Subject: Taylor models Date: Wed, 6 Mar 2002 16:32:08 +0100 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_004D_01C1C52C.75D12C00" X-Priority: 3 X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook Express 6.00.2600.0000 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2600.0000 Sender: owner-reliable_computing [at] interval [dot] louisiana.edu Precedence: bulk This is a multi-part message in MIME format. ------=_NextPart_000_004D_01C1C52C.75D12C00 Content-Type: text/plain; charset="iso-8859-2" Content-Transfer-Encoding: quoted-printable Sometimes this method may be useful. Let us consider the following polynomial=20 f(x)=3D 240000 * x - 25000 * x^2 + 3500 * x^3/3 - 25*x^4 + x^5/5 and the interval [x]=3D[-1000, 1] First derivative of the function f has the following form d(1)f(x)=3D 240000-50000*x+3500* x*x-100*x*x*x+x*x*x*x; Interval extension of this function is equal to: d(1)f([-1000, 1]) =3D [-1103310000,1103550240000] We don't know if this function is monotone.=20 Because of that we can calculate second derivative d(2)f(x)=3D -50000 + 7000* x - 300* x*x + 4* x*x*x Interval extension of this function is equal to: d(2)f([-1000, 1]) =3D [-4307050000,4257000] We don't know if this function is monotone.=20 Because of that we can calculate third derivative d(3)f(x)=3D 7000 - 600* x + 12*x*x Interval extension of this function is equal to: d(3)f([-1000,1])=3D[-5600,12607000] We don't know if this function is monotone.=20 Because of that we can calculate forth derivative d(4)f(x)=3D -600+24*x Interval extension of this function is equal to: d(4)f([-1000,1]) =3D [-24600,-576] We can see that the sign of the forth derivative is constant. Because of that third derivative is monotone and=20 extreme value of this function can be calculated using endpoints of given interval. d(3)f(-1000) =3D 12607000 d(3)f(1) =3D 6412 We can see that the sign of the third derivative is constant. Because of that second derivative is monotone and=20 extreme value of this function can be calculated using endpoints of given interval. d(2)f(-1000)=3D -4.3071e+009 d(2)f(1)=3D -43296 We can see that the sign of the second derivative is constant. Because of that first derivative is monotone and=20 extreme value of this function can be calculated using endpoints of given interval. d(1)f(-1000) =3D 1.1036e+012 d(1)f(-1) =3D 193401 We can see that the sign of the first derivative is constant. Because of that function f is monotone and=20 extreme value of this function can be calculated using endpoints of given interval. f(-1000)=3D -2.2619e+014 f(1)=3D 2.1614e+005 Finally f([-1000, 1])=3D[ -2.2619e+014, 2.1614e+005] This is the exact range of the function f. If this procedure fails, we can divide the interval into two parts and repeat procedure again. If the boxes are sufficiently small=20 then we can apply Taylor model. This procedure can be also applied in multidimensional case. Pownuk A., New inclusion functions in interval global optimization of = engineering structures. EUROPEAN CONFERENCE ON COMPUTATIONAL MECHANICS, Cracow, 26 - 29 June 2001, pp.460-461=20 Matlab code: lowerX=3D-1000; upperX=3D 1; x=3Dinterval(lowerX,upperX); 'First order derivative' 240000-50000*x+3500* x*x-100*x*x*x+x*x*x*x 'Second order derivative' -50000 + 7000* x - 300* x*x + 4* x*x*x 'Third order derivative' 7000 - 600* x + 12*x*x 'Fourth order derivative' -600+24*x x=3DlowerX; y4_lower=3D-600+24*x x=3DupperX; y4_upper=3D-600+24*x x=3DlowerX; y3_lower=3D7000 - 600* x + 12*x*x x=3DupperX; y3_upper=3D7000 - 600* x + 12*x*x x=3DlowerX; y2_lower=3D-50000 + 7000* x - 300* x*x + 4* x*x*x x=3DupperX; y2_upper=3D-50000 + 7000* x - 300* x*x + 4* x*x*x x=3DlowerX; y1_lower=3D240000-50000 *x+3500* x*x-100* x*x*x+x*x*x*x x=3DupperX; y1_upper=3D240000-50000 *x+3500* x*x-100* x*x*x+x*x*x*x x=3DlowerX; y0_lower=3D240000 * x - 25000 * x^2 + 3500 * x^3/3 - 25*x^4 + x^5/5 x=3DupperX; y0_upper=3D240000 * x - 25000 * x^2 + 3500 * x^3/3 - 25*x^4 + x^5/5 Regards, Andrzej Pownuk Ph.D., research associate (adjunct) at: Chair of Theoretical Mechanics Faculty of Civil Engineering Silesian University of Technology E-mail: pownuk [at] zeus [dot] polsl.gliwice.pl URL: http://zeus.polsl.gliwice.pl/~pownuk ------=_NextPart_000_004D_01C1C52C.75D12C00 Content-Type: text/html; charset="iso-8859-2" Content-Transfer-Encoding: quoted-printable
Sometimes this method may be = useful.
 
 
 

Let us consider the following polynomial
 
f(x)=3D 240000 * x - 25000 * x^2 + 3500 * x^3/3 - 25*x^4 + = x^5/5
 
and the interval [x]=3D[-1000, 1]
 
First derivative of the function f has the following form
 
d(1)f(x)=3D 240000-50000*x+3500* x*x-100*x*x*x+x*x*x*x;
 
Interval extension of this function is equal to:
 
d(1)f([-1000, 1]) =3D [-1103310000,1103550240000]
 
We don’t know if this function is monotone.
Because of = that we can=20 calculate second derivative
 
d(2)f(x)=3D -50000 + 7000* x - 300* x*x + 4* x*x*x
 
Interval extension of this function is equal to:
 
d(2)f([-1000, 1]) =3D [-4307050000,4257000]
 
We don’t know if this function is monotone.
Because of = that we can=20 calculate third derivative
 
d(3)f(x)=3D 7000 - 600* x + 12*x*x
 
Interval extension of this function is equal to:
 
d(3)f([-1000,1])=3D[-5600,12607000]
 
We don’t know if this function is monotone.
Because of = that we can=20 calculate forth derivative
 
d(4)f(x)=3D -600+24*x
 
Interval extension of this function is equal to:
 
d(4)f([-1000,1]) =3D [-24600,-576]
 
We can see that the sign of the forth derivative is = constant.
Because of=20 that third derivative is monotone and
extreme value of this function = can be=20 calculated
using endpoints of given interval.
 
d(3)f(-1000) =3D 12607000
d(3)f(1) =3D  6412
 
We can see that the sign of the third derivative is = constant.
Because of=20 that second derivative is monotone and
extreme value of this = function can be=20 calculated
using endpoints of given interval.
 
d(2)f(-1000)=3D -4.3071e+009
d(2)f(1)=3D  -43296
 
We can see that the sign of the second derivative is = constant.
Because=20 of that first derivative is monotone and
extreme value of this = function can=20 be calculated
using endpoints of given interval.
 
d(1)f(-1000) =3D  1.1036e+012
d(1)f(-1) =3D  = 193401
 
We can see that the sign of the first derivative is = constant.
Because of=20 that function f is monotone and
extreme value of this function can = be=20 calculated
using endpoints of given interval.
 
f(-1000)=3D -2.2619e+014
f(1)=3D   2.1614e+005
 
Finally
 
f([-1000, 1])=3D[ -2.2619e+014, 2.1614e+005]
 
This is the exact range of the function f.
 
If this procedure fails,
we can divide the interval into = two parts=20 and repeat procedure again.
If the boxes are sufficiently small =
then we=20 can apply Taylor model.
 
This procedure can be also applied in multidimensional case.
 
 
 
Pownuk A., New inclusion functions in interval global optimization = of=20 engineering structures.
EUROPEAN CONFERENCE ON COMPUTATIONAL=20 MECHANICS,
Cracow, 26 - 29 June 2001, pp.460-461
 
 
 
 
 

Matlab code:
 
 
 
 
 
 
 
lowerX=3D-1000;
upperX=3D = 1;
x=3Dinterval(lowerX,upperX);
 
'First order derivative'
240000-50000*x+3500*=20 x*x-100*x*x*x+x*x*x*x
'Second order derivative'
-50000 + 7000* x - = 300*=20 x*x + 4* x*x*x
'Third order derivative'
7000 - 600* x + = 12*x*x
'Fourth=20 order derivative'
-600+24*x
 
x=3DlowerX;
y4_lower=3D-600+24*x
x=3DupperX;
y4_upper=3D-60= 0+24*x
 
x=3DlowerX;
y3_lower=3D7000 - 600* x + = 12*x*x
x=3DupperX;
y3_upper=3D7000=20 - 600* x + 12*x*x
 
x=3DlowerX;
y2_lower=3D-50000 + 7000* x - 300* x*x + 4*=20 x*x*x
x=3DupperX;
y2_upper=3D-50000 + 7000* x - 300* x*x + 4* = x*x*x
 
x=3DlowerX;
y1_lower=3D240000-50000 *x+3500* x*x-100*=20 x*x*x+x*x*x*x
x=3DupperX;
y1_upper=3D240000-50000 *x+3500* = x*x-100*=20 x*x*x+x*x*x*x
 
x=3DlowerX;
y0_lower=3D240000 * x - 25000 * x^2 + 3500 * x^3/3 - = 25*x^4 +=20 x^5/5
x=3DupperX;
y0_upper=3D240000 * x - 25000 * x^2 + 3500 * = x^3/3 - 25*x^4=20 + x^5/5
 

Regards,
 
     Andrzej Pownuk
 

Ph.D., research associate (adjunct) at:
Chair of Theoretical = Mechanics
Faculty of Civil Engineering
Silesian University of=20 Technology
E-mail: pownuk [at] zeus [dot] polsl.gliwice.pl=
URL:=20 http://zeus.polsl.gliwice.p= l/~pownuk
------=_NextPart_000_004D_01C1C52C.75D12C00-- From owner-reliable_computing [at] interval [dot] louisiana.edu Wed Mar 6 12:48:25 2002 Received: (from daemon@localhost) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) id g26ImOp14395 for reliable_computing-outgoing; Wed, 6 Mar 2002 12:48:24 -0600 (CST) Received: from mailbox.univie.ac.at (mailbox.univie.ac.at [131.130.1.27]) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) with ESMTP id g26ImGF14391 for ; Wed, 6 Mar 2002 12:48:17 -0600 (CST) Received: from univie.ac.at (hektor.mat.univie.ac.at [131.130.16.21]) by mailbox.univie.ac.at (8.12.2/8.12.2) with ESMTP id g26Im9qM058116; Wed, 6 Mar 2002 19:48:10 +0100 Message-ID: <3C866469.6CF1FEB7 [at] univie [dot] ac.at> Date: Wed, 06 Mar 2002 19:48:09 +0100 From: Arnold Neumaier Organization: University of Vienna X-Mailer: Mozilla 4.78 [en] (X11; U; Linux 2.4.9-21 i686) X-Accept-Language: en, de MIME-Version: 1.0 To: berz [at] msu [dot] edu CC: reliable_computing [at] interval [dot] louisiana.edu Subject: Re: Taylor models (repost) References: Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Sender: owner-reliable_computing [at] interval [dot] louisiana.edu Precedence: bulk Martin Berz wrote: > One that we > have been working on is the normal form defect problem from dynamical > systems theory. [...] can be found in a recent paper in > Nonlinear Analysis, http://bt.nscl.msu.edu/cgi-bin/display.pl?name=HOINL00 Could you please post a precise description of this and related problems from applications, so that one can use them as test examples? Arnold Neumaier From owner-reliable_computing [at] interval [dot] louisiana.edu Wed Mar 6 13:27:40 2002 Received: (from daemon@localhost) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) id g26JRdM14530 for reliable_computing-outgoing; Wed, 6 Mar 2002 13:27:39 -0600 (CST) Received: from mailbox.univie.ac.at (mailbox.univie.ac.at [131.130.1.27]) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) with ESMTP id g26JRYF14526 for ; Wed, 6 Mar 2002 13:27:35 -0600 (CST) Received: from univie.ac.at (hektor.mat.univie.ac.at [131.130.16.21]) by mailbox.univie.ac.at (8.12.2/8.12.2) with ESMTP id g26JRSqM121538; Wed, 6 Mar 2002 20:27:29 +0100 Message-ID: <3C866DA0.FAA1E6E3 [at] univie [dot] ac.at> Date: Wed, 06 Mar 2002 20:27:28 +0100 From: Arnold Neumaier Organization: University of Vienna X-Mailer: Mozilla 4.78 [en] (X11; U; Linux 2.4.9-21 i686) X-Accept-Language: en, de MIME-Version: 1.0 To: berz [at] msu [dot] edu, reliable_computing [at] interval [dot] louisiana.edu Subject: Taylor models - how to get valid comparisons References: <3C866469.6CF1FEB7 [at] univie [dot] ac.at> Content-Type: text/plain; charset=UTF-7 Content-Transfer-Encoding: 7bit Sender: owner-reliable_computing [at] interval [dot] louisiana.edu Precedence: bulk Martin, > Nonlinear Analysis, http://bt.nscl.msu.edu/cgi-bin/display.pl?name=HOINL00 I looked at the paper. I find it quite uninformative as regards the 6D example. It is a bit strange to compare your method against simple interval evaluation, which is clear not to perform well. Any method can be made to look very well if contrasted with the simplest of all methods. But one would not compare a good optimization method against steepest descent, or a good integrator against computing Riemann sums (which is the analogue of interval evaluation) - so why do you do your comparisons in such a useless way? To see how your method compares with other state of the art techniques, you'd have to compare it at least to techniques that use centered forms with slopes which are available for easy use in Rump's INTLAB (and perhaps to Bernstein polynomial techniques, and to affine arithmetic), and then it is likely that the huge improvement factors claimed boil down to quite a small improvement. (Or a large one - then it would be a fair victory!) Unfortunately, your other publications on Taylor methods also suffer from similar presentation problems. As to the reachable accuracy, I don't believe your claims that the enclosure found is of high order; your results only show that the remainder interval can be enclosed to high order. But you still have to enclose the range of the Taylor polynomial (with real coefficients) by an interval, and if you do it by using the Horner or the power form, you only get an accuracy of O(h^2). Since you usually use such a simple evaluation scheme (at least to present your examples), your claims are just hot air, it seems. I hope this will motivate to present a really convincing demonstration of the quality of your methods... Best wishes, Arnold From owner-reliable_computing [at] interval [dot] louisiana.edu Wed Mar 6 13:39:51 2002 Received: (from daemon@localhost) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) id g26JdpW14630 for reliable_computing-outgoing; Wed, 6 Mar 2002 13:39:51 -0600 (CST) Received: from mailbox.univie.ac.at (mailbox.univie.ac.at [131.130.1.27]) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) with ESMTP id g26JdiF14625 for ; Wed, 6 Mar 2002 13:39:45 -0600 (CST) Received: from univie.ac.at (hektor.mat.univie.ac.at [131.130.16.21]) by mailbox.univie.ac.at (8.12.2/8.12.2) with ESMTP id g26JddqM157668; Wed, 6 Mar 2002 20:39:40 +0100 Message-ID: <3C86707B.55475A90 [at] univie [dot] ac.at> Date: Wed, 06 Mar 2002 20:39:39 +0100 From: Arnold Neumaier Organization: University of Vienna X-Mailer: Mozilla 4.78 [en] (X11; U; Linux 2.4.9-21 i686) X-Accept-Language: en, de MIME-Version: 1.0 To: berz [at] msu [dot] edu, reliable_computing [at] interval [dot] louisiana.edu Subject: Re: Taylor models - how to get valid comparisons References: <3C866469.6CF1FEB7 [at] univie [dot] ac.at> <3C866DA0.FAA1E6E3 [at] univie [dot] ac.at> Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Sender: owner-reliable_computing [at] interval [dot] louisiana.edu Precedence: bulk Arnold Neumaier wrote: > you'd have to compare it at least to techniques that use centered forms > with slopes which are available for easy use in Rump's INTLAB > (and perhaps to Bernstein polynomial techniques, and to affine > arithmetic), Another relevant comparison would be to a modified Moore-Skelboe algorithm, applied twice to get the lower and upper bound. This is a little inefficient since part of the wirk is duplicated, but it has the advantage that it can be done in a simple way since the NEOS server allows one to solve global optimization problems on a box online via the WWW. See http://www-neos.mcs.anl.gov/neos/solvers/GO:GLOBMIN/ From owner-reliable_computing [at] interval [dot] louisiana.edu Fri Mar 8 05:37:14 2002 Received: (from daemon@localhost) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) id g28BbDB12292 for reliable_computing-outgoing; Fri, 8 Mar 2002 05:37:13 -0600 (CST) Received: from mailhost.iitb.ac.in (garbo.iitb.ac.in [203.197.74.149] (may be forged)) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) with SMTP id g28BaP312026 for ; Fri, 8 Mar 2002 05:37:04 -0600 (CST) Received: (qmail 13129 invoked from network); 8 Mar 2002 11:12:59 -0000 Received: from mailscan2.iitb.ernet.in (HELO thisdomain) (144.16.108.202) by mailhost.iitb.ac.in with SMTP; 8 Mar 2002 11:12:59 -0000 Received: from nataraj by iitb.ac.in ; Fri, 08 Mar 2002 17:06:08 +0530 Date: Fri, 08 Mar 2002 17:06:08 +0530 X-Originating-IP: 144.16.100.71 X-Auth-User: nataraj [at] ee [dot] iitb.ernet.in From: "P. S. V. Nataraj" To: Subject: RE: Taylor models - how to get valid comparisons Date: Fri, 8 Mar 2002 17:11:03 +0530 Message-ID: MIME-Version: 1.0 Content-Type: text/plain; charset="Windows-1252" Content-Transfer-Encoding: 7bit X-Priority: 3 (Normal) X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook IMO, Build 9.0.2416 (9.0.2910.0) Importance: Normal In-Reply-To: <3C86707B.55475A90 [at] univie [dot] ac.at> X-MimeOLE: Produced By Microsoft MimeOLE V5.50.4807.1700 Sender: owner-reliable_computing [at] interval [dot] louisiana.edu Precedence: bulk Dear All, For a new inclusion function form having higher order convergence, along with a convergence study of this form vs. that of Berz's Taylor model, pl. see our paper (under review): www.ee.iitb.ac.in/~nataraj/Super_TB_ps.zip In the new form, Bernstein polynomial techniques were used to bound the range of the Taylor polynomial part. We studied in this paper six problems, of dimensions varying from 1 to 6. The new form (along with Taylor model) has also been used in a modified Moore-Skelboe Global Optimization algorithm in our paper (to appear in Jl. Global Optimization) www.ee.iitb.ac.in/~nataraj/GOTB_Final_PS.zip In all our studies, we badly needed a good collection of challenge problems to study the convergence behavior of the various forms. Can somebody suggest a good collection of such problems - preferably with a structure more general than polynomial, and dimensions varying from 1 to (at least) 6 ? Regards, Nataraj .............................................. Prof. P. S. V. Nataraj Systems and Control Engineering Group Department of Electrical Engineering Indian Institute of Technology Bombay 400 076 India Ph: +91-022-5723757 Fax: +91-022-5726263 Email: nataraj [at] ee [dot] iitb.ernet.in -----Original Message----- From: owner-reliable_computing [at] interval [dot] louisiana.edu [mailto:owner-reliable_computing [at] interval [dot] louisiana.edu]On Behalf Of Arnold Neumaier Sent: Thursday, March 07, 2002 1:12 AM To: berz [at] msu [dot] edu; reliable_computing [at] interval [dot] louisiana.edu Subject: Re: Taylor models - how to get valid comparisons Arnold Neumaier wrote: > you'd have to compare it at least to techniques that use centered forms > with slopes which are available for easy use in Rump's INTLAB > (and perhaps to Bernstein polynomial techniques, and to affine > arithmetic), Another relevant comparison would be to a modified Moore-Skelboe algorithm, applied twice to get the lower and upper bound. This is a little inefficient since part of the wirk is duplicated, but it has the advantage that it can be done in a simple way since the NEOS server allows one to solve global optimization problems on a box online via the WWW. See http://www-neos.mcs.anl.gov/neos/solvers/GO:GLOBMIN/ --- Incoming mail is certified Virus Free. Checked by AVG anti-virus system (http://www.grisoft.com). Version: 6.0.325 / Virus Database: 182 - Release Date: 2/19/02 --- Outgoing mail is certified Virus Free. Checked by AVG anti-virus system (http://www.grisoft.com). Version: 6.0.325 / Virus Database: 182 - Release Date: 2/19/02 From owner-reliable_computing [at] interval [dot] louisiana.edu Fri Mar 8 09:46:18 2002 Received: (from daemon@localhost) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) id g28FkIZ18782 for reliable_computing-outgoing; Fri, 8 Mar 2002 09:46:18 -0600 (CST) Received: from mailbox.univie.ac.at (mailbox.univie.ac.at [131.130.1.27]) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) with ESMTP id g28FkD318778 for ; Fri, 8 Mar 2002 09:46:13 -0600 (CST) Received: from univie.ac.at (hektor.mat.univie.ac.at [131.130.16.21]) by mailbox.univie.ac.at (8.12.2/8.12.2) with ESMTP id g28FjouS116070; Fri, 8 Mar 2002 16:45:52 +0100 Message-ID: <3C88DCAE.9BD1F39D [at] univie [dot] ac.at> Date: Fri, 08 Mar 2002 16:45:50 +0100 From: Arnold Neumaier Organization: University of Vienna X-Mailer: Mozilla 4.78 [en] (X11; U; Linux 2.4.9-21 i686) X-Accept-Language: en, de MIME-Version: 1.0 To: "P. S. V. Nataraj" CC: reliable_computing [at] interval [dot] louisiana.edu Subject: Re: Taylor models - how to get valid comparisons References: Content-Type: text/plain; charset=UTF-7 Content-Transfer-Encoding: 7bit Sender: owner-reliable_computing [at] interval [dot] louisiana.edu Precedence: bulk "P. S. V. Nataraj" wrote: > For a new inclusion function form having higher order convergence, along > with a convergence study of this form vs. that of Berz's Taylor model, > www.ee.iitb.ac.in/+AH4-nataraj/Super_TB_ps.zip > www.ee.iitb.ac.in/+AH4-nataraj/GOTB_Final_PS.zip Thanks for the papers. Very nice results! It confirms my suspicion that the Taylor model only has the quadratic approximation property, and it shows a way how to modify it to get truly high order with good efficiency in low dimensions. One correction: You state that the Bernstein form gives exact results for sufficiently small boxes. But this is true only if the extremal value is attained in a box; it is not the case, e.g., for f(x)=x^3-6x in [0,2] since the minimum is irrational. Thus you need slight modifications in your GO algorithm: you must use the computed upper bound for min p(x) instead of min p(x), and probably you must also adjust your criterion for stopping Bernstein subdivision. I'd also recommend that you mention the leading order of the cost of the transformation to Bernstein form, since this may explain the limitations of your method in higher dimensions. > In all our studies, we badly needed a good collection of challenge problems > to study the convergence behavior of the various forms. > > Can somebody suggest a good collection of such problems - preferably with a > structure more general than polynomial, and dimensions varying from 1 to (at > least) 6 ? http://www.mat.univie.ac.at/+AH4-neum/glopt/test.html contains collections of test problems for global optimization. Both the Dixon-Szego test set and the More test set with bounds by Gay are low-dimiensional (up to dimension 12, but some have varying dimensions). Instead of global optimization, you can also find the ranges, and shrink the domains to see how the size of the box affects the overestimation of the range. You can generate arbitrarily challenging test problems with the rational test problem generator described in http://www.mat.univie.ac.at/+AH4-neum/papers.html#ratglob Best wishes, Arnold PS. Do you have a WWW page woth your publications? From owner-reliable_computing [at] interval [dot] louisiana.edu Fri Mar 8 10:41:45 2002 Received: (from daemon@localhost) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) id g28GfjF19024 for reliable_computing-outgoing; Fri, 8 Mar 2002 10:41:45 -0600 (CST) Received: from mail.epost.de ([64.39.38.76]) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) with ESMTP id g28GfZ319020 for ; Fri, 8 Mar 2002 10:41:35 -0600 (CST) Received: from TP570Berz (12.245.210.11) by mail.epost.de (5.5.052) (authenticated as martin.berz [at] epost [dot] de) id 3C6B20EC00478282; Fri, 8 Mar 2002 17:41:05 +0100 Reply-To: From: "Martin Berz" To: "Arnold Neumaier" , Subject: RE: Taylor models - how to get valid comparisons Date: Fri, 8 Mar 2002 11:41:02 -0500 Message-ID: MIME-Version: 1.0 Content-Type: text/plain; charset="utf-7" X-Priority: 3 (Normal) X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook IMO, Build 9.0.2416 (9.0.2911.0) Importance: Normal In-Reply-To: <3C866DA0.FAA1E6E3 [at] univie [dot] ac.at> X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2600.0000 Content-Transfer-Encoding: 8bit X-MIME-Autoconverted: from quoted-printable to 8bit by interval.louisiana.edu id g28Gff319021 Sender: owner-reliable_computing [at] interval [dot] louisiana.edu Precedence: bulk Dear Arnold et al., +AD4- +AD4- +AD4- Nonlinear Analysis, +AD4- http://bt.nscl.msu.edu/cgi-bin/display.pl?name+AD0-HOINL00 +AD4- +AD4- I looked at the paper. I find it quite uninformative as regards the 6D +AD4- example. Well, the 6D example cannot be described as nicely and compactly as the Gritton problem also studied in the paper, which is merely a one variable high-order polynomial that can be written down analytically. As is described, the 6D example is +ACI-industrial strength+ACI-, a composition of three polynomials of order 10 in six variables. That makes it impossible to list in detail, since each of these polynomials has many thousands of coefficients. However, we are happy to provide you or anyone else who asks the set of coefficients to try whatever method they want on it, which would certainly be interesting. From the past we also have tools in FORTRAN, COSY language, C or C+-+- that we can dig out that evaluates the polynomials. The theory behind how these polynomials are obtained and why they are important can for example be found in a recent book of mine, Modern Map Methods in Particle Beam Physics, Academic Press, 1999, ISBN 0-12-014750-5, posted at http://bt.nscl.msu.edu/cgi-bin/display.pl?name+AD0-AIEP108book . The relevant part is the theory of +ACI-normal forms+ACI- in Chapter 7. However, as with perhaps many other heavy duty problems from other fields, you have to be prepared to invest a fair amount of time to get to the bottom of the theory. +AD4- It is a bit strange to compare your method against simple +AD4- interval +AD4- evaluation, which is clear not to perform well. Any method can be made +AD4- to look very well if contrasted with the simplest of all methods. But +AD4- one would not compare a good optimization method against steepest +AD4- descent, or a good integrator against computing Riemann sums (which is +AD4- the analogue of interval evaluation) - so why do you do your comparisons +AD4- in such a useless way? +AD4- +AD4- To see how your method compares with other state of the art techniques, +AD4- you'd have to compare it at least to techniques that use centered forms +AD4- with slopes which are available for easy use in Rump's INTLAB +AD4- (and perhaps to Bernstein polynomial techniques, and to affine +AD4- arithmetic), +AD4- and then it is likely that the huge improvement factors +AD4- claimed boil down to quite a small improvement. (Or a large one - then +AD4- it would be a fair victory+ACE-) Unfortunately, your other publications on +AD4- Taylor methods also suffer from similar presentation problems. Well, first of all the paper you are looking at has a strict page limit, and we had to throw away nearly half of what we wanted in there already+ADs- and then it is in a field outside of traditional interval methods, where using many different methods you introduce may make you loose more of the audience. We did, however, make several comparisons in the direction you mention. First, we evaluated the system with linear Taylor models which have quadratic convergence and linear cancellation of dependencies. Hence except for the cancellation, they provide similar behavior to centered forms and to affine arithmetic (see also Baker's comments from a few days ago - by the way, Baker, did you study your system with higher order Taylor models as well?). The resulting second order method certainly helps somewhat compared to the intervals, but is still far from sufficient to crack this problem. I suppose when Kyoko is back from her meeting, she may possibly be able to dig out some detailed numbers and provide them. Regarding the question of optimization, we tried several of the packages including Rump's and some earlier(?) version of Globsol, but they were all limited by the size of the buffer because no intervals could be excluded due to overestimation. I guess these tools have some of the state of the art methods you mention built in to the extent they have proven useful. If you know how to do it with NEOS, in particular how to make the interface to evaluate the function in this complicated case, I think this would be very worthwhile to pursue. +AD4- +AD4- As to the reachable accuracy, I don't believe your claims that the +AD4- enclosure found is of high order+ADs- your results only show that the +AD4- remainder interval +AD4- can be enclosed to high order. But you still have to enclose the range +AD4- of the Taylor polynomial (with real coefficients) by an interval, and if +AD4- you +AD4- do it by using the Horner or the power form, you only get an accuracy of +AD4- O(h+AF4-2). Since you usually use such a simple evaluation scheme (at least +AD4- to +AD4- present your examples), your claims are just hot air, it seems. Well, may I propose to try to keep the air cool by not hyperventilating? First, so far we usually don't even get to the point of explaining sophisticated polynomial bounding because just getting across the basic Taylor model idea takes all the time. And again, in most cases the enclosure of the polynomial turns out to offer not so much additional gain for the overall accuracy, once you add the minimum level of sophistication of using an intrinsic square operation for even orders. Some representative examples of this are given, for example, in Makino's dissertation http://bt.nscl.msu.edu/cgi-bin/display.pl?name+AD0-makinophd p. 119-123. Keep in mind too that you never have to range bound any intermediate results, you leave them in the Taylor model form. That's one of the tricks that helps so much in the verified integrators. However, there are various extensions of this range bounding question+ADs- for example we know how to range bound general quadratic forms and in case of univariate Taylor models, since we know how to get zeros from polynomials of up to fourth order up to arithmetic order, we can rigorously bound a polynomial up to fifth order. In any case, what's left to bound is some kind of higher order part+ADs- and even if you use just plain intervals for that, the overestimation of the true range is usually in the range of a few percent only. And since you do this only once at the very end but never in intermediate steps, this is your total overestimation. However, if you REALLY want to, and for the sake of the theoretical argument, you CAN ALSO BOUND THE POLYNOMIAL of the Taylor model with an overestimation that truly scales with order (n+-1) like the remainder. Of course this comes with the standard caveat that you have to stay sufficiently away from the floor due to machine precision, and it ONLY works for Taylor polynomials where the contributions of the higher orders fall off quickly, which however is what we have in Taylor models. There are at least two methods that we know that allow you to do that. The first method is outlined in Makino's dissertation p. 128-130, it is iterative and in practical cases can reduce the polynomial overestimation below the remainder bound. It is based on the fact that the linear part locally dominates, which when combined with a crude estimate of all nonlinear contributions, provides a way to narrow down the regions where maxima can occur. +AD4- +AD4- I hope this will motivate to present a really convincing demonstration +AD4- of the quality of your methods... +AD4- +AD4- Best wishes, +AD4- +AD4- Arnold +AD4- Sure, we are always motivated +ADs--) To this end, we'd be more than happy to work with anybody who wants to try his/her favorite tool on the normal form defect problem and see what happens. Best wishes, Martin P.S.: While I was typing this, I saw the posting of Nataraj+ADs- however, I couldn't open the .ps files, Ghostview 4 stopped somewhere early on in both papers and I couldn't print them either - can others open them? Can Nataraj perhaps post a pdf? From owner-reliable_computing [at] interval [dot] louisiana.edu Fri Mar 8 11:09:39 2002 Received: (from daemon@localhost) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) id g28H9c519148 for reliable_computing-outgoing; Fri, 8 Mar 2002 11:09:38 -0600 (CST) Received: from clmboh1-smtp3.columbus.rr.com (clmboh1-smtp3.columbus.rr.com [65.24.0.112]) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) with ESMTP id g28H9I319144 for ; Fri, 8 Mar 2002 11:09:19 -0600 (CST) Received: from oemcomputer (dhcp065-024-174-102.columbus.rr.com [65.24.174.102]) by clmboh1-smtp3.columbus.rr.com (8.11.2/8.11.2) with SMTP id g28H9AE16519; Fri, 8 Mar 2002 12:09:10 -0500 (EST) Message-ID: <000e01c1c6c3$90d9cdc0$66ae1841 [at] columbus [dot] rr.com> From: "Ramon Moore" To: Cc: "interval" References: +ADw-LOBBLFLDLCOABDBDAMICAEJFEBAA.berz+AEA-msu.edu+AD4- Subject: Re: Taylor models - how to get valid comparisons Date: Fri, 8 Mar 2002 12:06:18 -0500 MIME-Version: 1.0 Content-Type: text/plain; charset="utf-7" Content-Transfer-Encoding: 7bit X-Priority: 3 X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook Express 5.50.4133.2400 X-MimeOLE: Produced By Microsoft MimeOLE V5.50.4133.2400 Sender: owner-reliable_computing [at] interval [dot] louisiana.edu Precedence: bulk +AD4- P.S.: While I was typing this, I saw the posting of Nataraj+ADs- however, I couldn't open the .ps files, Ghostview 4 stopped somewhere early on in both papers and I couldn't print them either - can others open them? Can Nataraj perhaps post a pdf? +AD4- I had the same problem, and I have the same request. Ramon Moore From owner-reliable_computing [at] interval [dot] louisiana.edu Fri Mar 8 11:28:12 2002 Received: (from daemon@localhost) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) id g28HSC819272 for reliable_computing-outgoing; Fri, 8 Mar 2002 11:28:12 -0600 (CST) Received: from mailgate.rz.uni-karlsruhe.de (mailgate.rz.uni-karlsruhe.de [129.13.64.97]) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) with ESMTP id g28HS6319268 for ; Fri, 8 Mar 2002 11:28:07 -0600 (CST) Received: from there (rz37@rzm-lohner.rz.uni-karlsruhe.de [172.21.99.37]) by mailgate.rz.uni-karlsruhe.de with smtp (Exim 3.33 #1) id 16jO9k-0006PK-00; Fri, 08 Mar 2002 18:27:48 +0100 Content-Type: text/plain; charset="iso-8859-1" From: Rudolf Lohner Reply-To: Rudolf.Lohner [at] rz [dot] uni-karlsruhe.de Organization: Universitaet Karlsruhe (TH), Rechenzentrum To: Subject: RE: Taylor models - how to get valid comparisons Date: Fri, 8 Mar 2002 18:27:47 +0100 X-Mailer: KMail [version 1.3.2] MIME-Version: 1.0 Content-Transfer-Encoding: 8bit Message-Id: Sender: owner-reliable_computing [at] interval [dot] louisiana.edu Precedence: bulk Edit the .ps files in your favorite text editor and remove the printer job control commands. The files should start with the line %!PS-Adobe-3.0 and end with the line %%EOF Remove everything before and after. Then you should be able to open the files with gv and print them. Rudolf > +AD4- P.S.: While I was typing this, I saw the posting of Nataraj+ADs- > however, I > couldn't open the .ps files, Ghostview 4 stopped somewhere early on in both > papers and I couldn't print them either - can others open them? Can Nataraj > perhaps post a pdf? > +AD4- > > I had the same problem, and I have the same request. -- Rudolf Lohner --- Universitaet Karlsruhe (TH) --- Rechenzentrum Zirkel 2, D-76128 Karlsruhe, phone/fax: +49 721 {608-6958 | 32550} www: http://www.uni-karlsruhe.de/~Rudolf.Lohner email: Rudolf.Lohner [at] rz [dot] uni-karlsruhe.de From owner-reliable_computing [at] interval [dot] louisiana.edu Fri Mar 8 11:40:31 2002 Received: (from daemon@localhost) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) id g28HeVo19368 for reliable_computing-outgoing; Fri, 8 Mar 2002 11:40:31 -0600 (CST) Received: from mail.epost.de ([64.39.38.76]) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) with ESMTP id g28HeP319364 for ; Fri, 8 Mar 2002 11:40:25 -0600 (CST) Received: from TP570Berz (12.245.210.11) by mail.epost.de (5.5.052) (authenticated as martin.berz [at] epost [dot] de) id 3C6B20EC0047B9B2; Fri, 8 Mar 2002 18:39:56 +0100 Reply-To: From: "Martin Berz" To: "Arnold Neumaier" , "P. S. V. Nataraj" Cc: Subject: RE: Taylor models - how to get valid comparisons Date: Fri, 8 Mar 2002 12:39:50 -0500 Message-ID: MIME-Version: 1.0 Content-Type: text/plain; charset="utf-7" X-Priority: 3 (Normal) X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook IMO, Build 9.0.2416 (9.0.2911.0) Importance: Normal In-Reply-To: <3C88DCAE.9BD1F39D [at] univie [dot] ac.at> X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2600.0000 Content-Transfer-Encoding: 8bit X-MIME-Autoconverted: from quoted-printable to 8bit by interval.louisiana.edu id g28HeR319365 Sender: owner-reliable_computing [at] interval [dot] louisiana.edu Precedence: bulk Follow up to my previous message, which was written before receiving this one... the speed and efficiency of the internet keeps you on your toes+ACE- From what Arnold says, it seems these papers contain another way of bounding the polynomial part that shows that higher order accuracies are possible not only for the remainder bound, but even for the polyomial part. That confirms my statement, and good to hear that independently. However, the statement +AD4- the Taylor model only has the quadratic approximation property, in my opinion is not quite right. It should read something like: +ACI-The Taylor model method provides an asymptotic accuracy equal to min(n+-1,p), where n is the order of the Taylor polynomial, and p the order of the scheme used for bounding the Taylor polynomial. If mere interval evaluation of the polynomial is used, then p+AD0-2, as for all naive interval bounding. But various more efficient polynomial bounding schemes exist with p higher than 2 exist. +ACI- I hope we can agree on this, ok? Regarding the collection of optimization examples that Arnold mentions, and similar such things by Baker, George, and others: would it make sense or of interest to provide a FORTRAN code for our nasty invariant defect function? Martin +AD4- -----Original Message----- +AD4- From: owner-reliable+AF8-computing+AEA-interval.louisiana.edu +AD4- +AFs-mailto:owner-reliable+AF8-computing+AEA-interval.louisiana.edu+AF0-On Behalf Of +AD4- Arnold Neumaier +AD4- Sent: Friday, March 08, 2002 10:46 AM +AD4- To: P. S. V. Nataraj +AD4- Cc: reliable+AF8-computing+AEA-interval.louisiana.edu +AD4- Subject: Re: Taylor models - how to get valid comparisons +AD4- +AD4- +AD4- +ACI-P. S. V. Nataraj+ACI- wrote: +AD4- +AD4- +AD4- For a new inclusion function form having higher order convergence, along +AD4- +AD4- with a convergence study of this form vs. that of Berz's Taylor model, +AD4- +AD4- www.ee.iitb.ac.in/+AH4-nataraj/Super+AF8-TB+AF8-ps.zip +AD4- +AD4- www.ee.iitb.ac.in/+AH4-nataraj/GOTB+AF8-Final+AF8-PS.zip +AD4- +AD4- Thanks for the papers. Very nice results+ACE- It confirms my suspicion that +AD4- the Taylor model only has the quadratic approximation property, +AD4- and it shows a way how to modify it to get truly high order +AD4- with good efficiency in low dimensions. +AD4- +AD4- One correction: +AD4- You state that the Bernstein form gives exact results for sufficiently +AD4- small boxes. But this is true only if the extremal value is attained in +AD4- a box+ADs- it is not the case, e.g., for f(x)+AD0-x+AF4-3-6x in +AFs-0,2+AF0- since the +AD4- minimum +AD4- is irrational. Thus you need slight modifications in your GO algorithm: +AD4- you must use the computed upper bound for min p(x) instead of min p(x), +AD4- and probably you must also adjust your criterion for stopping Bernstein +AD4- subdivision. I'd also recommend that you mention the leading order of +AD4- the +AD4- cost of the transformation to Bernstein form, since this may explain the +AD4- limitations of your method in higher dimensions. +AD4- +AD4- +AD4- In all our studies, we badly needed a good collection of +AD4- challenge problems +AD4- +AD4- to study the convergence behavior of the various forms. +AD4- +AD4- +AD4- +AD4- Can somebody suggest a good collection of such problems - +AD4- preferably with a +AD4- +AD4- structure more general than polynomial, and dimensions varying +AD4- from 1 to (at +AD4- +AD4- least) 6 ? +AD4- +AD4- http://www.mat.univie.ac.at/+AH4-neum/glopt/test.html +AD4- contains collections of test problems for global optimization. +AD4- Both the Dixon-Szego test set and the More test set with bounds by Gay +AD4- are low-dimiensional (up to dimension 12, but some have varying +AD4- dimensions). +AD4- Instead of global optimization, you can also find the ranges, +AD4- and shrink the domains to see how the size of the box +AD4- affects the overestimation of the range. +AD4- +AD4- You can generate arbitrarily challenging test problems with the +AD4- rational test problem generator described in +AD4- http://www.mat.univie.ac.at/+AH4-neum/papers.html+ACM-ratglob +AD4- +AD4- Best wishes, +AD4- +AD4- Arnold +AD4- +AD4- PS. Do you have a WWW page woth your publications? +AD4- From owner-reliable_computing [at] interval [dot] louisiana.edu Fri Mar 8 12:11:07 2002 Received: (from daemon@localhost) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) id g28IB7519497 for reliable_computing-outgoing; Fri, 8 Mar 2002 12:11:07 -0600 (CST) Received: from mailbox.univie.ac.at (mailbox.univie.ac.at [131.130.1.27]) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) with ESMTP id g28IB1319493 for ; Fri, 8 Mar 2002 12:11:02 -0600 (CST) Received: from univie.ac.at (hektor.mat.univie.ac.at [131.130.16.21]) by mailbox.univie.ac.at (8.12.2/8.12.2) with ESMTP id g28IAs0E076068; Fri, 8 Mar 2002 19:10:55 +0100 Message-ID: <3C88FEAE.2336F990 [at] univie [dot] ac.at> Date: Fri, 08 Mar 2002 19:10:54 +0100 From: Arnold Neumaier Organization: University of Vienna X-Mailer: Mozilla 4.78 [en] (X11; U; Linux 2.4.9-21 i686) X-Accept-Language: en, de MIME-Version: 1.0 To: berz [at] msu [dot] edu, reliable_computing [at] interval [dot] louisiana.edu Subject: Re: Taylor models - how to get valid comparisons References: Content-Type: text/plain; charset=UTF-7 Content-Transfer-Encoding: 7bit Sender: owner-reliable_computing [at] interval [dot] louisiana.edu Precedence: bulk Martin Berz wrote: > the 6D example is "industrial strength", a composition of three polynomials of order 10 in six variables. That makes it impossible to list in detail, since each of these polynomials has many thousands of coefficients. However, we are happy to provide you or anyone else who asks the set of coefficients to try whatever method they want on it, which would certainly be interesting. I'd be happy with a problem description in machine readable form. There is no need to have the polynomials represented in power form; any form that produces the right function values is ok. AMPL input would be most desirable for our group in Vienna, but a Fortran or C program that defines the polynomials as a sequence of simple assignments, with loops allowed, would be ok, too. Providing this on your web site would make the problem accessible to everyone. > we evaluated the system with linear Taylor models which [...] is still > far from sufficient to crack this problem. I suppose when Kyoko is back > from her meeting, she may possibly be able to dig out some detailed > numbers and provide them. Yes, please! > If you know how to do it with NEOS, in particular how to make the > interface to evaluate the function in this complicated case, > I think this would be very worthwhile to pursue. The page on the NEOS server I quoted gives the instructions; namely, essentially provide f95 evaluation routines for f and df. But maybe it GLOBMIN also suffers from digestion. In any case, reporting such information is important to gauge the quality of a new approach. > so far we usually don't even get to the point of explaining > sophisticated polynomial bounding because just getting across > the basic Taylor model idea takes all the time. But the basic model gets across in *one* publication, and then the others should go into more detail! And the dissertation by Makino has *no* sophisticated polynomial bounding apart from Horner; nothing to suggest that the overestimation can be kept at high order. > And again, in most cases the enclosure of the polynomial turns out to offer > not so much additional gain for the overall accuracy, once you add > the minimum level of sophistication of using an intrinsic square operation > for even orders. Some representative examples of this are given, > for example, in Makino's dissertation > http://bt.nscl.msu.edu/cgi-bin/display.pl?name=makinophd p. 119-123. But this means that you are stuck with O(h^2) accuracy in dimension > 1, against what you claimed. Since you nowhere compute the exact range, one cannot see the overestimation in your tables. > Keep in mind too that you never have to range bound any > intermediate results, you leave them in the Taylor model form. > That's one of the tricks that helps so much in the verified integrators. This is what Hansen does in his recent Computing paper, but only with quadratic term. It is what I guess is the *only* trick that makes the Taylor model form in the form you propose useful, but I found this nowhere spelled out as such. > we know how to range bound general quadratic forms How? I do not know any efficient way of bounding quadratic forms to higher than O(h^2) accuracy, except through subdivision algorithms. > in case of univariate Taylor models This is known; I agree. > what's left to bound is some kind of higher order part; > and even if you use just plain intervals for that, > the overestimation of the true range is usually in the range of > a few percent only. I agree here; but this sounds quite different from your high order claims - it does not go down to a few millionth if you increase the order! > However, if you REALLY want to, and for the sake of the theoretical > argument, you CAN ALSO BOUND THE POLYNOMIAL of the Taylor model with > an overestimation that truly scales with order (n+-1) like the remainder. You say it now, but never in your publications, that this needs additional techniques (and you don't provide these techniques!). What you wrote in print always sounded like your method solved these problems already, and the interested reader (and more readers than you think are sophisticated enough that they don't need to get the same starter every time, and don't notice that they are never served the real meat) is left to wonder what precisely is the degree of reliability of your statements. (It is better to claim less than to raise expectations that cannot be satisfied. In the past, the image of interval methods suffered severely from initial overenthusiasm that turned interested people away, or even hostile, when they noticed that the claims were not substantiated.) > it ONLY works for Taylor polynomials where the contributions of the > higher orders fall off quickly, which however is what we have in > Taylor models. Only for high order terms! Nothing guarantees that for a general function, the quadratic contribution is already small. If I give you a homogeneous quadratic form in 6 variables to be enclosed in [-0.05,0.05], the Taylor form of order n=5 is the original function, and I want to see your method that provides an overestimation that truly scales with order n+-1. > The first method is outlined in Makino's dissertation p. 128-130, but only in the univariate case, where Cornelius and Lohner solved the problem already in 1984. The case important for most applications is that of dimensions 2-6 (or even higher). > Sure, we are always motivated ;-) > To this end, we'd be more than happy to work with anybody > who wants to try his/her favorite tool on the normal form > defect problem and see what happens. But this requires that you put the analytic form of sample examples on the Web. I'll have a look at the application in your book... > I couldn't open the .ps files, Ghostview 4 stopped somewhere early on > in both papers and I couldn't print them either - can others open them? > Can Nataraj perhaps post a pdf? On my system there is ps2pdf, and it translates both papers to pdf, which can be printed with acroread. > "The Taylor model method provides an asymptotic accuracy equal to > min(n+-1,p), where n is the order of the Taylor polynomial, > and p the order of the scheme used for bounding the Taylor polynomial. > If mere interval evaluation of the polynomial is used, then p=2, > as for all naive interval bounding. But various more efficient > polynomial bounding schemes exist with p higher than 2 exist. " In this form, your statement is correct. But my earlier statement, > the Taylor model only has the quadratic approximation property, was correct, too, for any of the versions of the Taylor model you presented (for dimension >1) in any of your publications, including Makino's thesis. Nataraj's achievement consists in making the Taylor model truly of higher order. > Regarding the collection of optimization examples that > Arnold mentions, and similar such things by Baker, George, > and others: would it make sense or of interest to provide > a FORTRAN code for our nasty invariant defect function? Yes, very much! Best wishes, Arnold From owner-reliable_computing [at] interval [dot] louisiana.edu Sat Mar 9 01:25:15 2002 Received: (from daemon@localhost) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) id g297PFs20872 for reliable_computing-outgoing; Sat, 9 Mar 2002 01:25:15 -0600 (CST) Received: from mailhost.iitb.ac.in (garbo.iitb.ac.in [203.197.74.149] (may be forged)) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) with SMTP id g297P6320868 for ; Sat, 9 Mar 2002 01:25:07 -0600 (CST) Received: (qmail 13610 invoked from network); 9 Mar 2002 07:01:46 -0000 Received: from mailscan1.iitb.ac.in (HELO thisdomain) (144.16.108.201) by mailhost.iitb.ac.in with SMTP; 9 Mar 2002 07:01:46 -0000 Received: from nataraj by iitb.ac.in ; Sat, 09 Mar 2002 12:55:01 +0530 Date: Sat, 09 Mar 2002 12:55:01 +0530 X-Originating-IP: 144.16.100.71 X-Auth-User: nataraj [at] ee [dot] iitb.ernet.in From: "P. S. V. Nataraj" To: "interval" Cc: "Ramon Moore" , Subject: PDF files of Super-conv form and Taylor model papers Date: Sat, 9 Mar 2002 12:59:51 +0530 Message-ID: MIME-Version: 1.0 Content-Type: text/plain; charset="utf-7" X-Priority: 3 (Normal) X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook IMO, Build 9.0.2416 (9.0.2910.0) In-Reply-To: <000e01c1c6c3$90d9cdc0$66ae1841 [at] columbus [dot] rr.com> Importance: Normal X-MimeOLE: Produced By Microsoft MimeOLE V5.50.4807.1700 Content-Transfer-Encoding: 8bit X-MIME-Autoconverted: from quoted-printable to 8bit by interval.louisiana.edu id g297P8320869 Sender: owner-reliable_computing [at] interval [dot] louisiana.edu Precedence: bulk Hi, I am sorry for the trouble you have had with my zipped postscript files. I have made a PDF version of these files: www.ee.iitb.ac.in/+AH4-nataraj/GOTB+AF8-Final+AF8-PDF.zip www.ee.iitb.ac.in/+AH4-nataraj/Super+AF8-TB+AF8-PDF.zip (Sorry, I do not have a WWW page). Regards, Nataraj .............................................. Prof. P. S. V. Nataraj Systems and Control Engineering Group Department of Electrical Engineering Indian Institute of Technology Bombay 400 076 India Ph: +-91-022-5723757 Fax: +-91-022-5726263 Email: nataraj+AEA-ee.iitb.ernet.in -----Original Message----- From: owner-reliable+AF8-computing+AEA-interval.louisiana.edu +AFs-mailto:owner-reliable+AF8-computing+AEA-interval.louisiana.edu+AF0-On Behalf Of Ramon Moore Sent: Friday, March 08, 2002 10:44 PM To: berz+AEA-msu.edu Cc: interval Subject: Re: Taylor models - how to get valid comparisons +AD4- P.S.: While I was typing this, I saw the posting of Nataraj+ADs- however, I couldn't open the .ps files, Ghostview 4 stopped somewhere early on in both papers and I couldn't print them either - can others open them? Can Nataraj perhaps post a pdf? +AD4- I had the same problem, and I have the same request. Ramon Moore --- Incoming mail is certified Virus Free. Checked by AVG anti-virus system (http://www.grisoft.com). Version: 6.0.325 / Virus Database: 182 - Release Date: 2/19/02 --- Outgoing mail is certified Virus Free. Checked by AVG anti-virus system (http://www.grisoft.com). Version: 6.0.325 / Virus Database: 182 - Release Date: 2/19/02 From owner-reliable_computing [at] interval [dot] louisiana.edu Mon Mar 11 01:33:44 2002 Received: (from daemon@localhost) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) id g2B7Xhc04565 for reliable_computing-outgoing; Mon, 11 Mar 2002 01:33:43 -0600 (CST) Received: from mailhost.iitb.ac.in (garbo.iitb.ac.in [203.197.74.149] (may be forged)) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) with SMTP id g2B7XZ304560 for ; Mon, 11 Mar 2002 01:33:36 -0600 (CST) Received: (qmail 6004 invoked from network); 11 Mar 2002 07:10:22 -0000 Received: from mailscan1.iitb.ac.in (HELO thisdomain) (144.16.108.201) by mailhost.iitb.ac.in with SMTP; 11 Mar 2002 07:10:22 -0000 Received: from nataraj by iitb.ac.in ; Mon, 11 Mar 2002 13:03:34 +0530 Date: Mon, 11 Mar 2002 13:03:34 +0530 X-Originating-IP: 144.16.100.71 X-Auth-User: nataraj [at] ee [dot] iitb.ernet.in From: "P. S. V. Nataraj" To: "Arnold Neumaier" Cc: Subject: RE: Taylor models - how to get valid comparisons Date: Mon, 11 Mar 2002 13:08:56 +0530 Message-ID: MIME-Version: 1.0 Content-Type: text/plain; charset="utf-7" X-Priority: 3 (Normal) X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook IMO, Build 9.0.2416 (9.0.2910.0) Importance: Normal In-Reply-To: <3C88DCAE.9BD1F39D [at] univie [dot] ac.at> X-MimeOLE: Produced By Microsoft MimeOLE V5.50.4807.1700 Content-Transfer-Encoding: 8bit X-MIME-Autoconverted: from quoted-printable to 8bit by interval.louisiana.edu id g2B7Xc304562 Sender: owner-reliable_computing [at] interval [dot] louisiana.edu Precedence: bulk Prof. Neumaier, Thank you very much for your kind comments on our papers and the correction. Sorry, I do not have a WWW page. We will investigate the performance of the various forms on the collections of test problems you suggested. Regards, Nataraj .............................................. Prof. P. S. V. Nataraj Systems and Control Engineering Group Department of Electrical Engineering Indian Institute of Technology Bombay 400 076 India Ph: +-91-022-5723757 Fax: +-91-022-5726263 Email: nataraj+AEA-ee.iitb.ernet.in -----Original Message----- From: neum+AEA-mailbox.univie.ac.at +AFs-mailto:neum+AEA-mailbox.univie.ac.at+AF0-On Behalf Of Arnold Neumaier Sent: Friday, March 08, 2002 9:16 PM To: P. S. V. Nataraj Cc: reliable+AF8-computing+AEA-interval.louisiana.edu Subject: Re: Taylor models - how to get valid comparisons +ACI-P. S. V. Nataraj+ACI- wrote: +AD4- For a new inclusion function form having higher order convergence, along +AD4- with a convergence study of this form vs. that of Berz's Taylor model, +AD4- www.ee.iitb.ac.in/+AH4-nataraj/Super+AF8-TB+AF8-ps.zip +AD4- www.ee.iitb.ac.in/+AH4-nataraj/GOTB+AF8-Final+AF8-PS.zip Thanks for the papers. Very nice results+ACE- It confirms my suspicion that the Taylor model only has the quadratic approximation property, and it shows a way how to modify it to get truly high order with good efficiency in low dimensions. One correction: You state that the Bernstein form gives exact results for sufficiently small boxes. But this is true only if the extremal value is attained in a box+ADs- it is not the case, e.g., for f(x)+AD0-x+AF4-3-6x in +AFs-0,2+AF0- since the minimum is irrational. Thus you need slight modifications in your GO algorithm: you must use the computed upper bound for min p(x) instead of min p(x), and probably you must also adjust your criterion for stopping Bernstein subdivision. I'd also recommend that you mention the leading order of the cost of the transformation to Bernstein form, since this may explain the limitations of your method in higher dimensions. +AD4- In all our studies, we badly needed a good collection of challenge problems +AD4- to study the convergence behavior of the various forms. +AD4- +AD4- Can somebody suggest a good collection of such problems - preferably with a +AD4- structure more general than polynomial, and dimensions varying from 1 to (at +AD4- least) 6 ? http://www.mat.univie.ac.at/+AH4-neum/glopt/test.html contains collections of test problems for global optimization. Both the Dixon-Szego test set and the More test set with bounds by Gay are low-dimiensional (up to dimension 12, but some have varying dimensions). Instead of global optimization, you can also find the ranges, and shrink the domains to see how the size of the box affects the overestimation of the range. You can generate arbitrarily challenging test problems with the rational test problem generator described in http://www.mat.univie.ac.at/+AH4-neum/papers.html+ACM-ratglob Best wishes, Arnold PS. Do you have a WWW page woth your publications? --- Incoming mail is certified Virus Free. Checked by AVG anti-virus system (http://www.grisoft.com). Version: 6.0.325 / Virus Database: 182 - Release Date: 2/19/02 --- Outgoing mail is certified Virus Free. Checked by AVG anti-virus system (http://www.grisoft.com). Version: 6.0.325 / Virus Database: 182 - Release Date: 2/19/02 From owner-reliable_computing [at] interval [dot] louisiana.edu Tue Mar 12 22:59:28 2002 Received: (from daemon@localhost) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) id g2D4xRB09607 for reliable_computing-outgoing; Tue, 12 Mar 2002 22:59:27 -0600 (CST) Received: from mail.epost.de ([64.39.38.70]) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) with ESMTP id g2D4x6309603 for ; Tue, 12 Mar 2002 22:59:10 -0600 (CST) Received: from TP570Berz (12.245.210.11) by mail.epost.de (5.5.052) (authenticated as martin.berz [at] epost [dot] de) id 3C6B1E3900087A38 for reliable_computing [at] interval [dot] louisiana.edu; Wed, 13 Mar 2002 05:58:29 +0100 Reply-To: From: "Martin Berz" To: "Reliable+AF8-Computing+AEA-Interval. Louisiana. Edu" Subject: FW: Taylor models - how to get valid comparisons Date: Tue, 12 Mar 2002 23:58:44 -0500 Message-ID: MIME-Version: 1.0 Content-Type: text/plain; charset="utf-7" X-Priority: 3 (Normal) X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook IMO, Build 9.0.2416 (9.0.2911.0) Importance: Normal X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2600.0000 Content-Transfer-Encoding: 8bit X-MIME-Autoconverted: from quoted-printable to 8bit by interval.louisiana.edu id g2D4xL309604 Sender: owner-reliable_computing [at] interval [dot] louisiana.edu Precedence: bulk Hm, I sent this message yesterday, but I am not sure it worked ... it went to reliable+AF8-computing+AEA-interval.usl.edu, but it seems the right address now is reliable+AF8-computing+AEA-interval.louisiana.edu. So here it is again. Best, Martin -----Original Message----- From: Martin Berz +AFs-mailto:berz+AEA-msu.edu+AF0- Sent: Monday, March 11, 2002 5:50 PM To: Arnold Neumaier+ADs- Reliable Computing Subject: RE: Taylor models - how to get valid comparisons Dear Arnold et al., well, first of all it looks like there is still some hot air, and I propose to try to cool this down as carefully as we can+ADs- if we can do this, I believe it will help towards really effective communication and limit unsubstantiated claims. Also, along these lines, at the end of this message I will propose a few simple computations that we can do with various methods to check their performance, and encourage others to propose more of them. These examples will also help prevent just talking in vacuo, or more importantly, prevent from spreading ideas that aren't really true. I hope everybody can be agreeable to that. +AD4- +AD4- I'd be happy with a problem description in machine readable form. +AD4- Ok, we will do that. +AD4- +AD4- we evaluated the system with linear Taylor models which +AFs-...+AF0- is still +AD4- +AD4- far from sufficient to crack this problem. I suppose when Kyoko is back +AD4- +AD4- from her meeting, she may possibly be able to dig out some detailed +AD4- +AD4- numbers and provide them. +AD4- +AD4- Yes, please+ACE- Ok, we will do that too. +AD4- +AD4- +AD4- If you know how to do it with NEOS, in particular how to make the +AD4- +AD4- interface to evaluate the function in this complicated case, +AD4- +AD4- I think this would be very worthwhile to pursue. +AD4- +AD4- The page on the NEOS server I quoted gives the instructions+ADs- namely, +AD4- essentially provide f95 evaluation routines for f and df. +AD4- But maybe it GLOBMIN also suffers from digestion. In any case, reporting +AD4- such information is important to gauge the quality of a new approach. +AD4- >From the limited information I have about NEOS and a few discussions I had with Jorge More about this problem, I believe that NEOS has the digestion problem too, unless dramatic things have happened in the last two or so years that I don't know about yet. +AD4- +AD4- so far we usually don't even get to the point of explaining +AD4- +AD4- sophisticated polynomial bounding because just getting across +AD4- +AD4- the basic Taylor model idea takes all the time. +AD4- +AD4- But the basic model gets across in +ACo-one+ACo- publication, Well, to the really careful reader, yes. Yet we are still frequently and repeatedly being asked many of the basic questions, like +ACI-how and to what extent does it control dependency+ACI-, +ACI-why does the sharpness of the remainder bound decrease with order (n+-1)+ACI-, +ACI-why are higher orders beyond say two useful at all+ACI-. All these are important but perhaps not completely universally clear+ADs- to anybody to whom they are clear, I apologize. Actually one of the questions that is still borderline basic, yet very important here, is +ACI-what role does the sophistication of the polynomial bounding play+ACI-. +AD4- and then the others should go into more detail+ACE- For those to whom these points are clear, we have written some advanced papers on details about the dependency problem, quadrature, ODE integration, DAEs, near earth asteroids, stability theory, verified inversion, etc. Most of these, except when asked for by referees etc, don't include the basic theory again. In most of these applications, the range bounding question is not important because either 1) values at the endpoints are plugged in (quadrature or ODE integration), or 2) the problem is dominated by a complicated functional dependency situation (ODE integration, stability theory etc). For those topics that are relevant that we haven't yet addressed, sorry, we are humans, and there is life outside intervals too ... Altogether, from what we see so far, the strength of the Taylor models lies in these listed areas+ADs- and as a general rule, one should keep all operations in Taylor model format from beginning to end, without intermediate representation via intervals, i.e. not do intermediate range bounding. For plain range bounding itself, Taylor models may be useful if other methods fail because of the dependency problem+ADs- this is addressed to the extent it fits into the page limit in the paper we were talking about last time. In this case, and almost only there, does the question of how to compute the range of the polynomial of the Taylor model matter. But even there, for practical problems the question is not necessarily to squeeze out the last bit of sharpness. If you want to do that, however, you need high order range bounding of the polynomial. +AD4- And the dissertation by Makino has +AD4- +ACo-no+ACo- sophisticated polynomial bounding apart from Horner+ADs- nothing to +AD4- suggest that the overestimation can be kept at high order. +AD4- I don't agree (besides the above point that it's often not needed). Kyoko's dissertation contains the method of the linear dominated bounder which under the assumption of regularity of the linear part (and mild limitations on the floating point error) can get an inclusion with a sharpness that is aymptocially of the size of the remainder bound, order (n+-1), and which works in high dimensions, is robust and fast. (Additionally, should the linear part not be regular, the method usually reduces the dimensionality of the problem by the rank of the linear part, and then you can attack it perhaps with other means.) Whether or not this is +ACI-sophisticated+ACI- lies in the eye of the beholder, but it gets the job done in almost all cases. (In fact in her diss. she illustrates it only in one variable and for bounds of order three, but it is clear how this generalizes - should this not be the case, I'd be happy to say more next time, but in any case what she wrote should +ACI-suggest+ACI- that this is possible). In addition, she also describes a multivariate quadratic form bounder which allows to get a third order inclusion under rather mild conditions, but it doesn't scale so favorably to higher dimensions+ADs- and there is the one dimensional fifth order bounder. +AD4- +AD4- But this means that you are stuck with O(h+AF4-2) accuracy in dimension +AD4- 1, +AD4- against what you claimed. we are not stuck with the range bounders mentioned above. But nevertheless, let's still look at this in more detail. In a case where you really would want to use Taylor models, i.e. big overestimation, if it works you win because Taylor models can to some extent control dependencies. This effect is much more important than the ability to range bound most efficiently. +AD4- Since you nowhere compute the exact range, one cannot see the overestimation in your tables. +AD4- Well, if we knew the exact range, we wouldn't need intervals +ADs--) But besides this ironic comment, it's not quite true that we +ACI-nowhere+ACI- give the exact range. Many of the examples give the actual Taylor model, and from a rough estimate of the range of the polynomial (for example by rastering), you can get a rather reasonable estimate of the overestimation you have. +AD4- +AD4- Keep in mind too that you never have to range bound any +AD4- +AD4- intermediate results, you leave them in the Taylor model form. +AD4- +AD4- That's one of the tricks that helps so much in the verified integrators. +AD4- +AD4- This is what Hansen does in his recent Computing paper, but only with +AD4- quadratic term. I would be interested to see the final version of Hansen's paper. I had the pleasure to roughly discuss the matter with him about a year ago and much enjoyed it, but haven't seen the final product. From what I recall the method seemed a little similar to linear Taylor models. But in any case, if we have details we can also include it in the list of things to try. +AD4- It is what I guess is the +ACo-only+ACo- trick that makes +AD4- the Taylor model form in the form you propose useful, +AD4- but I found this nowhere spelled out as such. You don't find this spelled out because it is NOT the +ACo-only+ACo- trick that makes the Taylor model form useful. To begin with, the higher orders also matter+ACE- You can see this clearly with the Gritton problem and in many other cases+ADs- I hope in the course of this discussion, we will be able to get this point across fully clearly. And for all of the non-range bounding topics mentioned above, you really like the high orders because they give you more rapid convergence. +AD4- +AD4- +AD4- what's left to bound is some kind of higher order part+ADs- +AD4- +AD4- and even if you use just plain intervals for that, +AD4- +AD4- the overestimation of the true range is usually in the range of +AD4- +AD4- a few percent only. +AD4- +AD4- I agree here+ADs- but this sounds quite different from your high order claims +AD4- - it does not go down to a few millionth if you increase the order+ACE- YES, IT DOES go down to a few millionths+ACE- The problem of the normal form problem is that of any typical dependency problem. By the time you have gotten the final Taylor model, all the bad cancellations HAVE ALREADY HAPPENED in the Taylor polynomials, WHERE THEY DIDN'T HURT in the process. And in the end, the Taylor polynomial has REASONABLY SMALL coefficients, reflecting the fact that in fact the function is reasonably small compared to its interval evaluatoin. But in the intermediate steps the coefficients were all very large, they just cancel each other in the end. This is precisely what happens in any of the blow-up based problems where Taylor models are useful. In fact, with the 6D normal form / invariant defect problem in particular, it REALLY doesn't matter how you bound your polynomial in the end, you will get the few millionths. +AD4- +AD4- +AD4- However, if you REALLY want to, and for the sake of the theoretical +AD4- +AD4- argument, you CAN ALSO BOUND THE POLYNOMIAL of the Taylor model with +AD4- +AD4- an overestimation that truly scales with order (n+-1) like the remainder. +AD4- +AD4- You say it now, but never in your publications, that this needs +AD4- additional techniques (and you don't provide these techniques+ACE-). +AD4- What you wrote in print always sounded like your method solved these +AD4- problems already, Well, what we were saying in the publications is something like +ACI-the remainder bound scales with order (n+-1)+ACI-, or +ACI-the approximation by the polynomial scales with order (n+-1)+ACI-, which is certainly true. For the sub-problem of range bounding, a little more thought (see above) is needed. +AD4- and the interested reader (and more readers than you +AD4- think are sophisticated enough that they don't need to get the same +AD4- starter every time, Again, sorry if anybody was reading the same thing repeatedly +AD4- and don't notice that they are never served the real meat) +AD4- is left to wonder what precisely is the degree of reliability of your +AD4- statements. (It is better to claim less than to raise expectations +AD4- that cannot be satisfied. In the past, the image of interval methods +AD4- suffered severely from initial overenthusiasm that turned interested +AD4- people away, or even hostile, when they noticed that the claims +AD4- were not substantiated.) +AD4- Yes, I agree one has to be careful about not raising expectations+ADs- yet I also hope that the above paragraph didn't try to express a fundamental doubt about the reliability of my statements. Anyway, if at the end of this long and at times frustrating but hopefully ultimately useful discussion, it turns out that there was something we severely oversold, then I'll apologize and make my journey to Canossa ... ok? But what I claim, and many others agree now, and George said it from the beginning of this recent discussion: the polynomial bounding ISN't the real meat of the Taylor model approach+ACE- The ability to control the dependency problem is the key issue at hand here, and that happens via the use of floating point coefficients long before you get to the bounding of the final range. Furthermore, most of the time you can get order (n+-1) convergence not only in the enclosure of the function through its Taylor model, but also for predicted range bounds. +AD4- +AD4- it ONLY works for Taylor polynomials where the contributions of the +AD4- +AD4- higher orders fall off quickly, which however is what we have in +AD4- +AD4- Taylor models. +AD4- +AD4- Only for high order terms+ACE- Nothing guarantees that for a general +AD4- function, the quadratic contribution is already small. If I give you +AD4- a homogeneous quadratic form in 6 variables to be enclosed in +AD4- +AFs--0.05,0.05+AF0-, +AD4- the Taylor form of order n+AD0-5 is the original function, and I want to see +AD4- your method that provides an overestimation that truly scales with order +AD4- n+-1. +AD4- Perhaps from the statements above it is clear what I mean now. +AD4- +AD4- +AD4- +ACI-The Taylor model method provides an asymptotic accuracy equal to +AD4- +AD4- min(n+-1,p), where n is the order of the Taylor polynomial, +AD4- +AD4- and p the order of the scheme used for bounding the Taylor polynomial. +AD4- +AD4- If mere interval evaluation of the polynomial is used, then p+AD0-2, +AD4- +AD4- as for all naive interval bounding. But various more efficient +AD4- +AD4- polynomial bounding schemes exist with p higher than 2 exist. +ACI- +AD4- +AD4- In this form, your statement is correct. But my earlier statement, +AD4- +AD4- +AD4- the Taylor model only has the quadratic approximation property, +AD4- +AD4- was correct, too, for any of the versions of the Taylor model you +AD4- presented (for dimension +AD4-1) in any of your publications, +AD4- including Makino's thesis. Again, perhaps the above statements clarify what is meant. If not, anybody please feel free to ask again and we'll take another round at it. +AD4- +AD4- Nataraj's achievement consists in making the Taylor model truly of +AD4- higher order. I have received pdf's of Prof. Nataraj's paper, but haven't had a chance to read it (as said above, even if it may sound hard to believe, there is life outside the interval world that also needs to be tended to +ADs--) ). However, if it's truly a general and efficient polynomial bounder, then it's definitely a useful asset to have. Now, in order to focus this discussion back as much as possible to the science and away from the hot air, may I propose the following. Let us start with a simple example that has overestimation but that everybody who wants can handle easily+ADs- perhaps Gritton G(x). Next, we can proceed to a more complicated example, perhaps multidimensional Gritton G(ax+-by) etc., or if we want to, the normal form invariant defect problem - any other good suggestions? Everybody can use it to show their methods. Kyoko and I do it with Taylor models+ADs- if somebody (Arnold?) has code readily available, (s)he could run it with centered forms, affine arithmetic, Hansens's method, and whatever else can be done. If nobody has code for it, we here or Kyoko can probably try to put it together for the simpler schemes, but it may take a short time to do so. For the Taylor models, we can use different bounding schemes, beginning from trivial interval evaluation, then the default bounder, then the high-order linear dominated bounder, and perhaps also the bounders of Prof. Nataraj, with his help. We will start with this as soon as I have contact with Makino again (I believe she is back from her trip now) and keep you posted. Best wishes to all, Martin From owner-reliable_computing [at] interval [dot] louisiana.edu Wed Mar 13 14:55:30 2002 Received: (from daemon@localhost) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) id g2DKtTs11472 for reliable_computing-outgoing; Wed, 13 Mar 2002 14:55:29 -0600 (CST) Received: from harpia.petrobras.com.br (harpia.petrobras.com.br [200.255.28.1]) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) with ESMTP id g2DKtM311468 for ; Wed, 13 Mar 2002 14:55:23 -0600 (CST) Received: by harpia.petrobras.com.br; (8.8.8/1.3/10May95) id RAA16337; Wed, 13 Mar 2002 17:53:04 -0300 (GMT-0300) From: Subject: Interval Linear Systems of Equations - Overestimation - Help !! To: "interval" X-Mailer: Lotus Notes Release 5.0.2c (Intl)8 Fevereiro 2000 Message-ID: Date: Wed, 13 Mar 2002 17:50:42 -0300 X-MIMETrack: Serialize by Router on CENPESLN01/CENPES/C/Petrobras(Release 5.07a |May 14, 2001) at 13/03/2002 17:50:42 MIME-Version: 1.0 Content-type: text/plain; charset=us-ascii Sender: owner-reliable_computing [at] interval [dot] louisiana.edu Precedence: bulk Dear colleagues, I am trying to solve a 1D heat conduction problem by FInite Elements Method ( Ax=b). My stiffness matrix (A) is 51 x 51 ( row, column). This example accessories my Phd thesis. I am using the thermal conductivity (k) as interval number, with a little variation in k I get a large overestimation in the results. I have already tried the Preconditioning Gauss Elimination and Gauss-Seidel Methods, the Combinatorial and the Powell Optimization Method. Do anybody know how to treat and avoid this so large overestimation ? Papers, routines, books an examples are welcome. Best Regards, Thank you very much. Sebastiao Pereira Petrobras - Brazilian Oil Company http://www.petrobras.com.br phone: 55 21 38656436 fax: 55 21 3865 6441 From owner-reliable_computing [at] interval [dot] louisiana.edu Wed Mar 13 22:17:35 2002 Received: (from daemon@localhost) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) id g2E4HYc12270 for reliable_computing-outgoing; Wed, 13 Mar 2002 22:17:34 -0600 (CST) Received: from gcet_int.gcet.ac.in ([210.212.139.36]) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) with ESMTP id g2E4HN312266 for ; Wed, 13 Mar 2002 22:17:25 -0600 (CST) Received: from KK ([10.10.10.12]) by gcet_int.gcet.ac.in with SMTP (Microsoft Exchange Internet Mail Service Version 5.5.2653.13) id GY45N2K6; Wed, 13 Mar 2002 13:41:59 +0530 Message-ID: <003201c1ca68$66127620$0c0a0a0a@kk> From: "Mr. Ketan Kotecha" To: References: +ADw-NEBBJIKHDMLBDKBEGMNEGEOLCCAA.nataraj+AEA-ee.iitb.ernet.in+AD4- Subject: Fortran 90/95-timings Date: Wed, 13 Mar 2002 13:53:41 +0530 MIME-Version: 1.0 Content-Type: text/plain; charset="utf-7" Content-Transfer-Encoding: 7bit X-Priority: 3 X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook Express 5.00.2919.6600 X-MimeOLE: Produced By Microsoft MimeOLE V5.00.2919.6600 Sender: owner-reliable_computing [at] interval [dot] louisiana.edu Precedence: bulk Dear all: Can any one tell me:, is that possible to get program execution timings in milliseconds, or even in micro seconds, using microsoft fortran 90/Compaq fortran/fortefortran 95? I am getting it atmost accurately upto 1E-2 seconds. Thanks in advance, Ketan Kotecha Research Scholar IIT Bombay. From owner-reliable_computing [at] interval [dot] louisiana.edu Thu Mar 14 06:50:06 2002 Received: (from daemon@localhost) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) id g2ECo5X13478 for reliable_computing-outgoing; Thu, 14 Mar 2002 06:50:05 -0600 (CST) Received: from sunshine.math.utah.edu (IDENT:mk/dftbfX/M76F3t467afYOXViMYngxt [at] sunshine [dot] math.utah.edu [128.110.198.2]) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) with ESMTP id g2ECnw313474 for ; Thu, 14 Mar 2002 06:49:59 -0600 (CST) Received: from suncore.math.utah.edu (IDENT:Zr1cJyJJN/d0V3B1G5m0ARxJkAuvR0KI [at] suncore0 [dot] math.utah.edu [128.110.198.5]) by sunshine.math.utah.edu (8.9.3/8.9.3) with ESMTP id FAA21272; Thu, 14 Mar 2002 05:49:56 -0700 (MST) Received: (from beebe@localhost) by suncore.math.utah.edu (8.9.3/8.9.3) id FAA15054; Thu, 14 Mar 2002 05:49:55 -0700 (MST) Date: Thu, 14 Mar 2002 05:49:55 -0700 (MST) From: "Nelson H. F. Beebe" To: reliable_computing [at] interval [dot] louisiana.edu Cc: beebe [at] math [dot] utah.edu, "Mr. Ketan Kotecha" X-US-Mail: "Center for Scientific Computing, Department of Mathematics, 110 LCB, University of Utah, 155 S 1400 E RM 233, Salt Lake City, UT 84112-0090, USA" X-Telephone: +1 801 581 5254 X-FAX: +1 801 585 1640, +1 801 581 4148 X-URL: http://www.math.utah.edu/~beebe Subject: Re: Fortran 90/95-timings In-Reply-To: Your message of Wed, 13 Mar 2002 13:53:41 +0530 Message-ID: Sender: owner-reliable_computing [at] interval [dot] louisiana.edu Precedence: bulk "Ketan Kotecha" asks on Wed, 13 Mar 2002 13:53:41 +0530 about getting program execution timings accurate to milliseconds or better. Sadly, on the vast majority of operating systems today, with clock speeds sometimes exceeding 1GHz, process timers remain extremely crude, with 60 to 100 ticks per second. This is true no matter what programming language you use. Some processors (DEC Alpha, IBM PowerPC, Intel Pentium, Sun UltraSPARC) actually have timer instructions that can produce a resolution of less than a microsec, but these tend to require kernel privileges to execute, and remain unavailable to user processes, even UNIX root (super-user) processes. Cray supercomputers have been about the only ones I've encountered where accurate timing is available. The workaround is to wrap benchmark codes with outer loops that repeat the calculation long enough to get reliable timings, or use input data that makes them run long enough. Of course, this doesn't work if you are trying to accurately time various parts of real code. If you delve into C header files, look for the symbols CLK_TCK, CLOCKS_PER_SEC, HZ, and AHZ, and consult "man sysconf"; Sun Solaris systems get CLK_TCK from the function call _sysconf(3). On a GNU/Linux Red Hat 6.2 system on Intel x86, I find: /* ISO/IEC 9899:1990 7.12.1: The macro `CLOCKS_PER_SEC' is the number per second of the value returned by the `clock' function. CAE XSH, Issue 4, Version 2: The value of CLOCKS_PER_SEC is required to be 1 million on all XSI-conformant systems. */ # define CLOCKS_PER_SEC 1000000 /* Even though CLOCKS_PER_SEC has such a strange value CLK_TCK presents the real value for clock ticks per second for the system. */ # define CLK_TCK 100 Thus, CLOCKS_PER_SEC may lie about the true timer resolution. You also need to be aware of the effect of cache vs memory: the first iteration your program runs, data may be in memory, but on subsequent iterations, it may be in cache. The effect on timing can be dramatic. Typically, register access takes 1 cycle, level-1 cache 2 to 5 cycles, level-2 cache 8 to 15 cycles, and memory access 10 to 100 cycles. For more details, see my February 2001 talk on microprocessors, starting about slide 55, available at http://www.chpc.utah.edu/cc/talks/beebe/index.htm http://www.chpc.utah.edu/cc/talks/beebe/handout-color.pdf Here is a copy of a file in which I recorded notes on 7-Oct-1996 about timers on various systems when I implemented a profiling facility in the awk programming language: ------------------------------------------------------------------------ These notes record my attempts to try to get access to a cheap high-resolution clock for profiling on several systems. ------------------------------------------------------------------------ On Sun UltraSPARC, modified assembly code from timercheck.c in function foo() ! 1 #include ! 2 ! 3 typedef long long LONG; ! 4 ! 5 LONG foo(void) ! 6 { ! 7 return (1000000L + sizeof(LONG)); sethi %hi(.L_cseg0),%l0 or %l0,%lo(.L_cseg0),%l0 ldd [%l0+0],%l0 std %l0,[%fp-8] rd %tick,%l0 !<== inserted read of 63-bit cycle clock ba .L76 nop ! block 2 .L76: ldd [%fp-8],%l0 mov %l1,%i1 mov %l0,%i0 jmp %i7+8 restore Compilation was cc -c -xarch=v8plus timercheck.s cc -o timercheck -xarch=v8plus timercheck.s and running the job produces privilege failure, both as me, and as root: signal ILL (privileged opcode) in foo at 0x107e4 It is aggravating that such a useful instruction is locked out by the operating system for `security reasons'. Older SPARC architectures (pre-V9) lack any kind of a high-resolution cycle timer. ------------------------------------------------------------------------ On the DEC Alpha, the rscc (read system cycle counter) is potentially available in unprivileged VMS PALcode, but alas, not in OSF/1 PALcode. ------------------------------------------------------------------------ On the IBM RS/6000 PowerPC (see ``The PowerPC Architecture'', p. 354), the 64-bit Time Base register can be read by a 5-instruction loop to get a 64-bit value in two adjacent registers (r3 and r4 are needed for a long long function return value). I modified the output assembly code of timertest.c like this, to just read the lower 32 bits: .foo: # 0x00000000 (H.10.NO_SYMBOL) .file "timercheck.c" cau r0,r0,0x0002 ai r3,r0,-31068 mftb r3 # <== inserted bcr BO_ALWAYS,CR0_LT This will not compile in the Power or Power2 architectures (-qarch=pwr or -qarch=pwr2). It compiles in the PowerPC architecture (-qarch=ppc), but on the Power and Power2 systems that I ran the test on, it dies with Illegal instruction (core dumped) I don't have access to a PowerPC system running UNIX to make further tests. ------------------------------------------------------------------------ The MIPS architecture does not appear to have any documented facility for reading a high-precision clock, though on DEC ULTRIX, DEC OSF/1, and SGI IRIX operating systems, the pixie/pixstats utilities appear to be able to get exact cycle counts for instructions. ------------------------------------------------------------------------ I don't have online documentation of the HP-UX architecture to investigate further on our HP workstations. ------------------------------------------------------------------------ ------------------------------------------------------------------------------- - Nelson H. F. Beebe Tel: +1 801 581 5254 - - Center for Scientific Computing FAX: +1 801 585 1640, +1 801 581 4148 - - University of Utah Internet e-mail: beebe [at] math [dot] utah.edu - - Department of Mathematics, 110 LCB beebe [at] acm [dot] org beebe [at] computer [dot] org - - 155 S 1400 E RM 233 beebe [at] ieee [dot] org - - Salt Lake City, UT 84112-0090, USA URL: http://www.math.utah.edu/~beebe - ------------------------------------------------------------------------------- From owner-reliable_computing [at] interval [dot] louisiana.edu Thu Mar 14 17:47:09 2002 Received: (from daemon@localhost) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) id g2ENl8S14666 for reliable_computing-outgoing; Thu, 14 Mar 2002 17:47:08 -0600 (CST) Received: from patan.sun.com (patan.Sun.COM [192.18.98.43]) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) with ESMTP id g2ENkw314662 for ; Thu, 14 Mar 2002 17:46:59 -0600 (CST) Received: from vic.Aus.Sun.COM ([129.158.87.25]) by patan.sun.com (8.9.3+Sun/8.9.3) with ESMTP id QAA17574 for ; Thu, 14 Mar 2002 16:46:55 -0700 (MST) Received: from retreat (retreat [129.158.87.155]) by vic.Aus.Sun.COM (8.10.2+Sun/8.10.2/ENSMAIL,v2.2) with SMTP id g2ENkrh27825 for ; Fri, 15 Mar 2002 10:46:53 +1100 (EST) Message-Id: <200203142346.g2ENkrh27825 [at] vic [dot] Aus.Sun.COM> Date: Fri, 15 Mar 2002 10:47:44 +1100 (EST) From: Richard Smith - Systems Engineer - Melbourne Reply-To: Richard Smith - Systems Engineer - Melbourne Subject: Re: Fortran 90/95-timings To: reliable_computing [at] interval [dot] louisiana.edu MIME-Version: 1.0 Content-Type: TEXT/plain; charset=us-ascii Content-MD5: jSDuc3Aonvn1JFX0DEK5MQ== X-Mailer: dtmail 1.3.0 @(#)CDE Version 1.4.2 SunOS 5.8 sun4u sparc Sender: owner-reliable_computing [at] interval [dot] louisiana.edu Precedence: bulk Following up on Nelson Beebe's comments, SPARC does indeed have a high resolution clock that increments every cpu clock cycle. In the SPARC architecture rdtick causes a privileged_action exception if PSTATE.PRIV = 0 and TICK.NPT = 1. From Solaris 7 onwards the default setting of the bits permits rdtick in user code. An alternative is to use the high resolution timer routine gethrtime(3C). Admittedly it has higher overhead than rdtick but will work when threads get migrated across multiple cpus. Depending on which method you use and the specific cpu you're running on overhead is in the region of 10-100 cycles. At a minimum rdtick stalls the cpu pipeline. I tend to use inline(1) assembler code fragments to instrument code. Infact it is possible to use a fragment to replace gethrtime with rdtick. Note the different argument conventions required between v8 and v9: gethrtimev8plus.il: .inline gethrtime,0 rd %tick,%o0 sllx %o0,32,%o1 srlx %o0,32,%o0 srlx %o1,32,%o1 .end gethrtimev9.il: .inline gethrtime,0 rd %tick,%o0 .end cc -fast gethrtimev8plus.il junk.c ... Fortran code should use a pragma to specify that the "function" has a C interface: !$pragma C(gethrtime) Related tricks can be used to exploit the hardware performance counters. More information can be found in "Techniques For Optimizing Applications" by Garg and Sharapov. For Intel architecture I suggest consulting "Michael Abrash's Graphics Programming Black Book" Ch 3 Zen Timer http://www.ddj.com/documents/s=865/ddj0165f/ ============================================================================ ,-_|\ Richard Smith - SE Melbourne / \ Sun Microsystems Australia Phone : +61 3 9869 6200 richard.smith [at] Sun [dot] COM Direct : +61 3 9869 6224 \_,-._/ 476 St Kilda Road Fax : +61 3 9869 6290 v Melbourne Vic 3004 Australia =========================================================================== From owner-reliable_computing [at] interval [dot] louisiana.edu Fri Mar 15 18:33:50 2002 Received: (from daemon@localhost) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) id g2G0Xnm17382 for reliable_computing-outgoing; Fri, 15 Mar 2002 18:33:49 -0600 (CST) Received: from cs.utep.edu (mail.cs.utep.edu [129.108.5.3]) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) with ESMTP id g2G0Xi317378 for ; Fri, 15 Mar 2002 18:33:45 -0600 (CST) Received: from aragorn (aragorn [129.108.5.35]) by cs.utep.edu (8.11.3/8.11.3) with SMTP id g2G0XdK24345; Fri, 15 Mar 2002 17:33:39 -0700 (MST) Message-Id: <200203160033.g2G0XdK24345 [at] cs [dot] utep.edu> Date: Fri, 15 Mar 2002 17:33:37 -0700 (MST) From: Vladik Kreinovich Reply-To: Vladik Kreinovich Subject: conference in Bulgaria To: reliable_computing [at] interval [dot] louisiana.edu Cc: krat [at] bas [dot] bg MIME-Version: 1.0 Content-Type: TEXT/plain; charset=us-ascii Content-MD5: GYXAeYUX0hlhzmu5bbDsnA== X-Mailer: dtmail 1.3.0 @(#)CDE Version 1.4 SunOS 5.8 sun4u sparc Sender: owner-reliable_computing [at] interval [dot] louisiana.edu Precedence: bulk FYI: Krassimir Atanassov is organizing a session on intuitionistic fuzzy sets at the IEEE International Symposium 'Intelligent Systems' Methodology, Models, Applications in Emerging Technologies Hotel Palace, Sunny Day, Bulgaria, September 10-12, 2002 For this session, deadline was extended until the end of March. For details of submission, check the conference's webpage http://www.iinf.bas.bg/is/ Right after the IEEE Conference, on September 12-13, there will be an annual Conference on Intuitionistic Fuzzy Sets; the deadline for this conference is May 31, 2002. Please contact Krassimir at krat [at] bas [dot] bg for details. Vladik From owner-reliable_computing [at] interval [dot] louisiana.edu Wed Mar 20 10:19:20 2002 Received: (from daemon@localhost) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) id g2KGJKQ00764 for reliable_computing-outgoing; Wed, 20 Mar 2002 10:19:20 -0600 (CST) Received: from cs.utep.edu (mail.cs.utep.edu [129.108.5.3]) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) with ESMTP id g2KGJGr00760 for ; Wed, 20 Mar 2002 10:19:16 -0600 (CST) Received: from aragorn (aragorn [129.108.5.35]) by cs.utep.edu (8.11.3/8.11.3) with SMTP id g2KGJBo18599; Wed, 20 Mar 2002 09:19:11 -0700 (MST) Message-Id: <200203201619.g2KGJBo18599 [at] cs [dot] utep.edu> Date: Wed, 20 Mar 2002 09:19:10 -0700 (MST) From: Vladik Kreinovich Reply-To: Vladik Kreinovich Subject: Second call for participation To: reliable_computing [at] interval [dot] louisiana.edu, interval [at] cs [dot] utep.edu MIME-Version: 1.0 Content-Type: TEXT/plain; charset=us-ascii Content-MD5: QPy0mxYcg20UJFBk8lO6Tg== X-Mailer: dtmail 1.3.0 @(#)CDE Version 1.4 SunOS 5.8 sun4u sparc Sender: owner-reliable_computing [at] interval [dot] louisiana.edu Precedence: bulk deadline passed but they are still accepting abstracts ------------- Begin Forwarded Message ------------- To: jaf21 [at] pastor [dot] pdmi.ras.ru From: "Alexei V. Pastor" Date: Wed, 20 Mar 2002 14:38:55 +0300 (MSK) Subject: Second call for participation MIME-Version: 1.0 Content-Transfer-Encoding: 7bit Second call for participation International meeting "21th DAYS OF WEAK ARITHMETICS" June 7-9, 2002 St.Petersburg, Russia MAIN TOPICS For a detailed description of what are "Weak Arithmetics" visit http://www.univ-paris12.fr/lacl/jaf/html/wa.html The meeting will cover traditional topics of the "Days" such as: Provability in weak arithmetics Definability in weak arithmetics Weak arithmetics and model theory Decidability/undecidability of weak logical theories Modelling computations in the frameworks of weak arithmetics Weak arithmetics and computational comlexity Computer science applications of weak arithmetics Besides them a special session will be organized on: Coding and processing information in biological systems PERMANENT COMMITTEE of the DAYS: Patrick CEGIELSKI Jean-Pierre RESSAYRE Denis RICHARD University Paris-12 CNRS, Jussieu University of Auvergne Fontainebleau IUT Clermont IUT, LLAIC1 PROGRAM COMMITTEE Paola d'AQUINO (Italy) Anatoly BELTIUKOV (Russia) Patrick CEGIELSKI (France) Gregory KUCHEROV (France) Krzysztof LORYS (Poland) Yuri MATIYASSEVICH (Russia), the chairman Jean-Pierre RESSAYRE (France) Denis RICHARD (France) Maxim VSEMIRNOV (Russia) Local ORGANIZING COMMITTEE : Dmitri KARPOV Elena NOVIKOVA Vladimir OREVKOV Alexei PASTOR Yuri MATIYASEVICH (Head) Maxim VSEMIRNOV Working LANGUAGE: English SUBMISSION of papers To this goal the facilities of Atlas Mathematical Conference Abstracts will be used. If you wish to present a paper, please, submit an (extended) abstract via http://at.yorku.ca/cgi-bin/amca/submit/cail-01 following the instruction on this site. The deadline for submission passed on March 15, 2002, if you want to participate, submit your abstract as soon as possible. REGISTRATION Please, do it via http://logic.pdmi.ras.ru/jaf21/registration Participation FEE: The fee will cover common meals and coffee breaks (no return for possibly non-taken meals). The fee equivalent to 120USD can be paid on arrival. PLACE of the meeting: The "21th DAYS" will be held at the Euler International Mathematical Institute (which now is a part of St.Petersburg Department of Steklov Institute of Mathematics of the Russian Academy of Sciences). The building of the EIMI is situated at 10, Pesochnaya embankment. ACCOMODATION It is suggested that all participants stay at the nearby hotel of the Palace of Youth (this is a standard place of living for visitors to the EIMI). Current prices (slight increase is possible): Single-room apartment for one persons: equivalent to 60 USD; Single-room apartment for two persons: equivalent to 70 USD; If you prefer to stay at another hotel (more or less expensive, or closer to the center of the city), the Organizing Committee will help you to find such accomodation. Please, take into account that June is high tourist season in St.Petersburg so rooms should be booked as soon as possible. You can do it directly or we can do it for you (please, inform us if you are going to share the room with particular person). VISAS Please, take into account that participants from the majority of countries need visas to enrty Russia. After receiving your registration form, the Organizing Committee will issue a proper invitation. Getting the visa will take some time, so apply as soon as possible. CONTACTS Meeting's sites (mirrored): http://logic.pdmi.ras.ru/jaf21 http://llaic3.u-clermont1.fr/~yumat/jaf21 E-mail: arithmet [at] logic [dot] pdmi.ras.ru FAX: 7 (812) 310 53 77 (Program Committee) 7 (812) 234 58 19 (Organizing Committee) Relevant LINKS : All previous "Days": http://www.univ-paris12.fr/lacl/jaf/html/issues.html http://llaic3.u-clermont1.fr/E/jaf.htm "14th Days" held also in St.Petersburg: http://logic.pdmi.ras.ru/jaf14 Euler International Mathematical Institute: http://www.pdmi.ras.ru/EIMI Steklov Institute of Mathematics at St.Petersburg: http://www.pdmi.ras.ru Life in St.Petersburg: http://www.spb.ru -- Alexei Pastor E-mail: pastor [at] pdmi [dot] ras.ru ------------- End Forwarded Message ------------- From owner-reliable_computing [at] interval [dot] louisiana.edu Wed Mar 20 13:55:45 2002 Received: (from daemon@localhost) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) id g2KJtiF01196 for reliable_computing-outgoing; Wed, 20 Mar 2002 13:55:44 -0600 (CST) Received: from zeus.polsl.gliwice.pl (root [at] zeus [dot] polsl.gliwice.pl [157.158.1.3]) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) with ESMTP id g2KJtbr01192 for ; Wed, 20 Mar 2002 13:55:38 -0600 (CST) Received: from andrzej1 (a18.asystent.polsl.gliwice.pl [157.158.187.18]) by zeus.polsl.gliwice.pl (8.9.3 (PHNE_25183)/8.9.3) with SMTP id UAA00859; Wed, 20 Mar 2002 20:55:23 +0100 (MET) Message-ID: <01d001c1d049$30752c60$0200a8c0@andrzej1> From: "Andrzej Pownuk" To: , Cc: "Andrzej Pownuk" Subject: Re: Interval Linear Systems of Equations - Overestimation - Help !! Date: Wed, 20 Mar 2002 20:55:24 +0100 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_01CD_01C1D051.8ECE07C0" X-Priority: 3 X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook Express 6.00.2600.0000 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2600.0000 Sender: owner-reliable_computing [at] interval [dot] louisiana.edu Precedence: bulk This is a multi-part message in MIME format. ------=_NextPart_000_01CD_01C1D051.8ECE07C0 Content-Type: text/plain; charset="iso-8859-2" Content-Transfer-Encoding: quoted-printable I have worked on this problem since 1995. Now I can solve system of 1000 equations with dependent interval = parameters. (I have solved system of 2300 equations=20 with two intervals Young's modulus.) This algorithm can be downloaded directly from the internet. http://157.158.187.18/papers/Interval_FEM.pdf This algorithm was presented on the following conferences: 1) Pownuk A., Modelling of structures with uncertain parameters. XLI Sympsion on Modeling in Mechanics, 18-22.02.2002, Wis=B3a, Poland (Theoretical background) 2) Pownuk A., Numerical solutions of fuzzy partial differential equation = and its application in computational mechanics. Assessment and New Directions for Research. FUZZY PARTIAL DIFFERENTIAL EQUATIONS, FUZZY RELATIONAL EQUATIONS, FUZZY DIFFERENCE EQUATIONS. March 15-17, 2002 University of California-Berkeley , California 94720 - USA (Applications) Regards, Andrzej Pownuk ............................... Andrzej Pownuk Chair of Theoretical Mechanics Faculty of Civil Engineering Silesian University of Technology E-mail: pownuk [at] zeus [dot] polsl.gliwice.pl URL: http://zeus.polsl.gliwice.pl/~pownuk/ ............................... >=20 > Dear colleagues, >=20 > I am trying to solve a 1D heat conduction problem by FInite Elements > Method ( Ax=3Db). My stiffness matrix (A) is 51 x 51 ( row, column). = This > example accessories my Phd thesis. > I am using the thermal conductivity (k) as interval number, with a = little > variation in k I get a large overestimation in the results. > I have already tried the Preconditioning Gauss Elimination and = Gauss-Seidel > Methods, the Combinatorial and the Powell Optimization Method. > Do anybody know how to treat and avoid this so large overestimation ? > Papers, routines, books an examples are welcome. >=20 > Best Regards, >=20 > Thank you very much. >=20 > Sebastiao Pereira > Petrobras - Brazilian Oil Company > http://www.petrobras.com.br > phone: 55 21 38656436 > fax: 55 21 3865 6441 >=20 > ------=_NextPart_000_01CD_01C1D051.8ECE07C0 Content-Type: text/html; charset="iso-8859-2" Content-Transfer-Encoding: quoted-printable
I have worked on this problem since = 1995.
Now I=20 can solve system of 1000 equations with dependent interval = parameters.
(I=20 have solved system of 2300 equations
with two intervals = Young’s=20 modulus.)
 
This algorithm can be downloaded = directly from the=20 internet.
 
http://157.158.187= .18/papers/Interval_FEM.pdf
 

This algorithm was presented on the following = conferences:
 
1) Pownuk A., Modelling of structures with uncertain = parameters.
XLI=20 Sympsion on Modeling in Mechanics,
18-22.02.2002, Wis=B3a,=20 Poland
(Theoretical background)
 

2) Pownuk A., Numerical solutions of fuzzy partial differential = equation
and its application in computational = mechanics.
Assessment and=20 New Directions for Research.
FUZZY PARTIAL DIFFERENTIAL = EQUATIONS,
FUZZY=20 RELATIONAL EQUATIONS,
FUZZY DIFFERENCE EQUATIONS.
March 15-17,=20 2002
University of California-Berkeley ,
California 94720 -=20 USA
(Applications)
 
         Regards,
 
           &n= bsp; =20 Andrzej Pownuk
 
...............................
Andrzej Pownuk
Chair of = Theoretical=20 Mechanics
Faculty of Civil Engineering
Silesian University of=20 Technology
E-mail: pownuk [at] zeus [dot] polsl.gliwice.pl=
URL:=20 http://zeus.polsl.gliwice.= pl/~pownuk/
...............................
 
 
 
>
> Dear colleagues,
>
> I am trying to = solve a 1D=20 heat conduction problem by  FInite Elements
> Method ( = Ax=3Db). My=20 stiffness matrix (A) is 51 x 51 ( row, column). This
> example = accessories=20 my Phd thesis.
> I am using the thermal conductivity (k) as = interval=20 number, with a little
> variation in k I get a large = overestimation in the=20 results.
> I have already tried the Preconditioning Gauss = Elimination and=20 Gauss-Seidel
> Methods,  the Combinatorial and the Powell=20 Optimization Method.
> Do anybody know how to treat and avoid this = so=20 large overestimation ?
> Papers, routines, books an examples are=20 welcome.
>
> Best Regards,
>
> Thank  you = very=20 much.
>
> Sebastiao Pereira
> Petrobras - Brazilian = Oil=20 Company
> http://www.petrobras.com.br
&= gt;=20 phone: 55 21 38656436
> fax: 55 21 3865 6441
>=20
>
------=_NextPart_000_01CD_01C1D051.8ECE07C0-- From owner-reliable_computing [at] interval [dot] louisiana.edu Wed Mar 20 14:53:38 2002 Received: (from daemon@localhost) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) id g2KKrcf01351 for reliable_computing-outgoing; Wed, 20 Mar 2002 14:53:38 -0600 (CST) Received: from clmboh1-smtp3.columbus.rr.com (clmboh1-smtp3.columbus.rr.com [65.24.0.112]) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) with ESMTP id g2KKrWr01346 for ; Wed, 20 Mar 2002 14:53:33 -0600 (CST) Received: from oemcomputer (dhcp065-024-174-102.columbus.rr.com [65.24.174.102]) by clmboh1-smtp3.columbus.rr.com (8.11.2/8.11.2) with SMTP id g2KKrJB07560; Wed, 20 Mar 2002 15:53:19 -0500 (EST) Message-ID: <003701c1d050$d30a1920$66ae1841 [at] columbus [dot] rr.com> From: "Ramon Moore" To: "Andrzej Pownuk" Cc: "interval" References: <01d001c1d049$30752c60$0200a8c0@andrzej1> Subject: Re: Interval Linear Systems of Equations - Overestimation - Help !! Date: Wed, 20 Mar 2002 15:50:09 -0500 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0034_01C1D026.E9E024C0" X-Priority: 3 X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook Express 5.50.4133.2400 X-MimeOLE: Produced By Microsoft MimeOLE V5.50.4133.2400 Sender: owner-reliable_computing [at] interval [dot] louisiana.edu Precedence: bulk This is a multi-part message in MIME format. ------=_NextPart_000_0034_01C1D026.E9E024C0 Content-Type: text/plain; charset="iso-8859-2" Content-Transfer-Encoding: quoted-printable Dear Andrzej, I am very interested in your work and would very much like to read the = publications you refer to below, however I am unable to open the = download from the URL you give below. Perhaps there is another version as a pdf file that I could open and = read? [Can anyone else in the reliable_computing group help with this?] best wishes, Ramon Moore ----- Original Message -----=20 From: Andrzej Pownuk=20 To: reliable_computing [at] interval [dot] louisiana.edu ; = sebastiaoc [at] cenpes [dot] petrobras.com.br=20 Cc: Andrzej Pownuk=20 Sent: Wednesday, March 20, 2002 2:55 PM Subject: Re: Interval Linear Systems of Equations - Overestimation - = Help !! I have worked on this problem since 1995. Now I can solve system of 1000 equations with dependent interval = parameters. (I have solved system of 2300 equations=20 with two intervals Young's modulus.) This algorithm can be downloaded directly from the internet. http://157.158.187.18/papers/Interval_FEM.pdf This algorithm was presented on the following conferences: 1) Pownuk A., Modelling of structures with uncertain parameters. XLI Sympsion on Modeling in Mechanics, 18-22.02.2002, Wis=B3a, Poland (Theoretical background) 2) Pownuk A., Numerical solutions of fuzzy partial differential = equation=20 and its application in computational mechanics. Assessment and New Directions for Research. FUZZY PARTIAL DIFFERENTIAL EQUATIONS, FUZZY RELATIONAL EQUATIONS, FUZZY DIFFERENCE EQUATIONS. March 15-17, 2002 University of California-Berkeley , California 94720 - USA (Applications) Regards, Andrzej Pownuk ............................... Andrzej Pownuk Chair of Theoretical Mechanics Faculty of Civil Engineering Silesian University of Technology E-mail: pownuk [at] zeus [dot] polsl.gliwice.pl URL: http://zeus.polsl.gliwice.pl/~pownuk/ ............................... >=20 > Dear colleagues, >=20 > I am trying to solve a 1D heat conduction problem by FInite = Elements > Method ( Ax=3Db). My stiffness matrix (A) is 51 x 51 ( row, column). = This > example accessories my Phd thesis. > I am using the thermal conductivity (k) as interval number, with a = little > variation in k I get a large overestimation in the results. > I have already tried the Preconditioning Gauss Elimination and = Gauss-Seidel > Methods, the Combinatorial and the Powell Optimization Method. > Do anybody know how to treat and avoid this so large overestimation = ? > Papers, routines, books an examples are welcome. >=20 > Best Regards, >=20 > Thank you very much. >=20 > Sebastiao Pereira > Petrobras - Brazilian Oil Company > http://www.petrobras.com.br > phone: 55 21 38656436 > fax: 55 21 3865 6441 >=20 > ------=_NextPart_000_0034_01C1D026.E9E024C0 Content-Type: text/html; charset="iso-8859-2" Content-Transfer-Encoding: quoted-printable
Dear Andrzej,
 
I am very interested in your work and would very = much like=20 to read the publications you refer to below, however I am unable to open = the=20 download from the URL you give below.
Perhaps there is another version as a pdf file = that I=20 could open and read?
 
[Can anyone else in the reliable_computing group = help with=20 this?]
 
best wishes,
 
Ramon Moore
----- Original Message -----
From:=20 Andrzej Pownuk
To: reliable_comput= ing [at] interval [dot] louisiana.edu=20 ; sebastiaoc [at] cenpes [dot] petr= obras.com.br=20
Sent: Wednesday, March 20, 2002 = 2:55=20 PM
Subject: Re: Interval Linear = Systems of=20 Equations - Overestimation - Help !!

I have worked on this problem since = 1995.
Now=20 I can solve system of 1000 equations with dependent interval = parameters.
(I=20 have solved system of 2300 equations
with two intervals = Young’s=20 modulus.)
 
This algorithm can be downloaded = directly from=20 the internet.
 
http://157.158.187= .18/papers/Interval_FEM.pdf
 

This algorithm was presented on the following = conferences:
 
1) Pownuk A., Modelling of structures with uncertain = parameters.
XLI=20 Sympsion on Modeling in Mechanics,
18-22.02.2002, Wis=B3a,=20 Poland
(Theoretical background)
 

2) Pownuk A., Numerical solutions of fuzzy partial = differential=20 equation
and its application in computational = mechanics.
Assessment and=20 New Directions for Research.
FUZZY PARTIAL DIFFERENTIAL = EQUATIONS,
FUZZY=20 RELATIONAL EQUATIONS,
FUZZY DIFFERENCE EQUATIONS.
March 15-17,=20 2002
University of California-Berkeley ,
California 94720 -=20 USA
(Applications)
 
         Regards,
 
=
           &n= bsp; =20 Andrzej Pownuk
 
...............................
Andrzej Pownuk
Chair of = Theoretical=20 Mechanics
Faculty of Civil Engineering
Silesian University of=20 Technology
E-mail: pownuk [at] zeus [dot] polsl.gliwice.pl=
URL:=20 http://zeus.polsl.gliwice.= pl/~pownuk/
...............................
 
 
 
>
> Dear colleagues,
>
> I am trying to = solve a 1D=20 heat conduction problem by  FInite Elements
> Method ( = Ax=3Db). My=20 stiffness matrix (A) is 51 x 51 ( row, column). This
> example=20 accessories my Phd thesis.
> I am using the thermal conductivity = (k) as=20 interval number, with a little
> variation in k I get a large=20 overestimation in the results.
> I have already tried the=20 Preconditioning Gauss Elimination and Gauss-Seidel
> = Methods,  the=20 Combinatorial and the Powell Optimization Method.
> Do anybody = know how=20 to treat and avoid this so large overestimation ?
> Papers, = routines,=20 books an examples are welcome.
>
> Best Regards,
> =
>=20 Thank  you very much.
>
> Sebastiao Pereira
>=20 Petrobras - Brazilian Oil Company
> http://www.petrobras.com.br
&= gt;=20 phone: 55 21 38656436
> fax: 55 21 3865 6441
>=20
>
------=_NextPart_000_0034_01C1D026.E9E024C0-- From owner-reliable_computing [at] interval [dot] louisiana.edu Wed Mar 20 15:48:58 2002 Received: (from daemon@localhost) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) id g2KLmv301504 for reliable_computing-outgoing; Wed, 20 Mar 2002 15:48:57 -0600 (CST) Received: from clmboh1-smtp3.columbus.rr.com (clmboh1-smtp3.columbus.rr.com [65.24.0.112]) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) with ESMTP id g2KLmqr01500 for ; Wed, 20 Mar 2002 15:48:52 -0600 (CST) Received: from oemcomputer (dhcp065-024-174-102.columbus.rr.com [65.24.174.102]) by clmboh1-smtp3.columbus.rr.com (8.11.2/8.11.2) with SMTP id g2KLmpB14662 for ; Wed, 20 Mar 2002 16:48:51 -0500 (EST) Message-ID: <009901c1d058$91a825a0$66ae1841 [at] columbus [dot] rr.com> From: "Ramon Moore" To: "interval" Subject: sorry, my problem Date: Wed, 20 Mar 2002 16:45:35 -0500 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0096_01C1D02E.A89FC300" X-Priority: 3 X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook Express 5.50.4133.2400 X-MimeOLE: Produced By Microsoft MimeOLE V5.50.4133.2400 Sender: owner-reliable_computing [at] interval [dot] louisiana.edu Precedence: bulk This is a multi-part message in MIME format. ------=_NextPart_000_0096_01C1D02E.A89FC300 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable Dear People, Sorry for the trouble. The problem was at my end. I have the paper = Interval_FEM.pdf now. Thanks for your help Ramon Moore ------=_NextPart_000_0096_01C1D02E.A89FC300 Content-Type: text/html; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable
Dear People,
 
Sorry for the trouble. The problem was at my = end. I have=20 the paper Interval_FEM.pdf
now.
 
Thanks for your help
 
Ramon Moore
------=_NextPart_000_0096_01C1D02E.A89FC300-- From owner-reliable_computing [at] interval [dot] louisiana.edu Thu Mar 21 06:16:47 2002 Received: (from daemon@localhost) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) id g2LCGku05980 for reliable_computing-outgoing; Thu, 21 Mar 2002 06:16:46 -0600 (CST) Received: from mailbox.univie.ac.at (mailbox.univie.ac.at [131.130.1.27]) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) with ESMTP id g2LCGer05976 for ; Thu, 21 Mar 2002 06:16:41 -0600 (CST) Received: from univie.ac.at (hektor.mat.univie.ac.at [131.130.16.21]) by mailbox.univie.ac.at (8.12.2/8.12.2) with ESMTP id g2LCDqZW152222; Thu, 21 Mar 2002 13:13:53 +0100 Message-ID: <3C99CE80.8B9DA49B [at] univie [dot] ac.at> Date: Thu, 21 Mar 2002 13:13:52 +0100 From: Arnold Neumaier Organization: University of Vienna X-Mailer: Mozilla 4.78 [en] (X11; U; Linux 2.4.9-21 i686) X-Accept-Language: en, de MIME-Version: 1.0 To: reliable_computing [at] interval [dot] louisiana.edu CC: sebastiaoc [at] cenpes [dot] petrobras.com.br Subject: Re: Interval Linear Systems of Equations References: <01d001c1d049$30752c60$0200a8c0@andrzej1> Content-Type: text/plain; charset=UTF-7 Content-Transfer-Encoding: 7bit Sender: owner-reliable_computing [at] interval [dot] louisiana.edu Precedence: bulk > Andrzej Pownuk wrote: > > I have worked on this problem since 1995. > Now I can solve system of 1000 equations with dependent interval > parameters. > (I have solved system of 2300 equations > with two intervals Young?s modulus.) > > This algorithm can be downloaded directly from the internet. > > http://157.158.187.18/papers/Interval_FEM.pdf However, your algorithm depends on unverified assumptions (after (3)), that are unlikely to be satisfied if the intervals are wide. To make this a valid algorithm you need to provide verifiable sufficient conditions. Moreover, the statement on p.3 is simply wrong. You should have been suspicious about your method and proof, since it would show polynomiality of a NP-hard problem - thus you would have won one of the Millenium prizes (for P=NP) with such a simple argument! I think Jansson and Rohn wrote papers on linear interval systems with linear dependence relations and narrow intervals, also based on monotony arguments, that are on a sounder basis, and give *reliable* results. Maybe someone can give precise references. Arnold Neumaier From owner-reliable_computing [at] interval [dot] louisiana.edu Thu Mar 21 07:02:37 2002 Received: (from daemon@localhost) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) id g2LD2ag06191 for reliable_computing-outgoing; Thu, 21 Mar 2002 07:02:36 -0600 (CST) Received: from rztsun.rz.tu-harburg.de (rztsun.rz.tu-harburg.de [134.28.200.14]) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) with ESMTP id g2LD2Wr06187 for ; Thu, 21 Mar 2002 07:02:32 -0600 (CST) Received: from hp0.mat.tu-harburg.de (hp0.mat.tu-harburg.de [134.28.61.10]) by rztsun.rz.tu-harburg.de (8.9.0/8.8.8) with ESMTP id NAA10919; Thu, 21 Mar 2002 13:55:44 +0100 (MET) Received: from zemke.mat.tu-harburg.de (zemke.mat.tu-harburg.de [134.28.61.33]) by hp0.mat.tu-harburg.de with ESMTP (8.9.3 (PHNE_24419)/8.7.1) id NAA25959; Thu, 21 Mar 2002 13:55:44 +0100 (MET) Date: Thu, 21 Mar 2002 13:54:53 +0100 (MET) From: Jens Zemke X-X-Sender: To: Arnold Neumaier cc: , Subject: Re: Interval Linear Systems of Equations In-Reply-To: <3C99CE80.8B9DA49B [at] univie [dot] ac.at> Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: owner-reliable_computing [at] interval [dot] louisiana.edu Precedence: bulk -----BEGIN PGP SIGNED MESSAGE----- Hash: SHA1 The precise reference to the Jansson/Rohn paper is C. Jansson and J. Rohn. An Algorithm for Checking Regularity of Interval Matrices. SIAM J. Matrix Anal. Appl., 20(3):756-776, 1999. It is a new version of C. Jansson and J. Rohn. An Algorithm for Checking Regularity of Interval Matrices. Technical Report 96.4, Berichte des Forschungsschwerpunktes Informations- und Kommunikationstechnik, TUHH, 1996. Also of interest might be C. Jansson. Calculation of Exact Bounds for the Solution Set of Linear Interval Systems. Linear Algebra and its Applications 251, 251:321-340, 1997. Concerning dependent data the article S.M. Rump. Verification Methods for Dense and Sparse Systems of Equations. In J. Herzberger, editor, Topics in Validated Computations --- Studies in Computational Mathematics, pages 63-136, Elsevier, Amsterdam, 1994 might be of interest, especially p. 106 on data dependencies in the input data. Jens Zemke - -- Jens-Peter M. Zemke -- zemke@tu-harburg.de AB Mathematik / Mathematics Department Technische Universitaet Hamburg -- Harburg -----BEGIN PGP SIGNATURE----- Version: GnuPG v1.0.6 (GNU/Linux) Comment: Weitere Infos: siehe http://www.gnupg.org iD8DBQE8mdgl04CU2LVV1ggRAtUTAJ9l3sZymUXuKPS2zS8ikSM/8XkbDgCfYqbE gFrhMXC5zbzGOTbLWDof2bQ= =8afB -----END PGP SIGNATURE----- From owner-reliable_computing [at] interval [dot] louisiana.edu Thu Mar 21 09:00:27 2002 Received: (from daemon@localhost) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) id g2LF0Qc06474 for reliable_computing-outgoing; Thu, 21 Mar 2002 09:00:26 -0600 (CST) Received: from zeus.polsl.gliwice.pl (root [at] zeus [dot] polsl.gliwice.pl [157.158.1.3]) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) with ESMTP id g2LF0Kr06470 for ; Thu, 21 Mar 2002 09:00:21 -0600 (CST) Received: from andrzej1 (a18.asystent.polsl.gliwice.pl [157.158.187.18]) by zeus.polsl.gliwice.pl (8.9.3 (PHNE_25183)/8.9.3) with SMTP id PAA22896; Thu, 21 Mar 2002 15:59:05 +0100 (MET) Message-ID: <003501c1d0e8$f17c4d90$0200a8c0@andrzej1> From: "Andrzej Pownuk" To: , Cc: "Andrzej Pownuk" References: +ADw-01d001c1d049+ACQ-30752c60+ACQ-0200a8c0+AEA-andrzej1+AD4- +ADw-3C99CE80.8B9DA49B+AEA-univie.ac.at+AD4- Subject: Re: Interval Linear Systems of Equations Date: Thu, 21 Mar 2002 15:59:02 +0100 MIME-Version: 1.0 Content-Type: text/plain; charset="UTF-7" Content-Transfer-Encoding: 7bit X-Priority: 3 X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook Express 6.00.2600.0000 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2600.0000 Sender: owner-reliable_computing [at] interval [dot] louisiana.edu Precedence: bulk +AD4- +AD4- Andrzej Pownuk wrote: +AD4- +AD4- +AD4- +AD4- I have worked on this problem since 1995. +AD4- +AD4- Now I can solve system of 1000 equations with dependent interval +AD4- +AD4- parameters. +AD4- +AD4- (I have solved system of 2300 equations +AD4- +AD4- with two intervals Young?s modulus.) +AD4- +AD4- +AD4- +AD4- This algorithm can be downloaded directly from the internet. +AD4- +AD4- +AD4- +AD4- http://157.158.187.18/papers/Interval+AF8-FEM.pdf +AD4- +AD4- However, your algorithm depends on unverified assumptions (after (3)), +AD4- that are unlikely to be satisfied if the intervals are wide. +AD4- To make this a valid algorithm you need to provide verifiable +AD4- sufficient conditions. After point (3) I assume that the intervals are narrow. (+IBw-in technical applications the intervals are usually narrow+IB0-) Additionally for small intervals we can assume that the solution is monotone and depend only on the endpoints of the intervals. There are a lot of papers, which are based on these assumptions. Akpan U.O., Koko T.S., Orisamolu I.R., Gallant B.K., Practical fuzzy finite element analysis of structures, Finite Elements in Analysis and Design, 38 (2000) 93-111 McWilliam S., Anti-optimization of uncertain structures using interval analysis, Computers and Structures, 79 (2000) 421-430 Noor A.K., Starnes J.H., Peters J.M., Uncertainty analysis of composite structures, Computer methods in applied mechanics and engineering, 79 (2000) 413-232 Valliappan S., Pham T.D., Elasto-Plastic Finite Element Analysis with Fuzzy Parameters, International Journal for Numerical Methods in Engineering, 38 (1995) 531-548 etc. +AD4- +AD4- Moreover, the statement on p.3 is simply wrong. +AD4- You should have been suspicious about your method and proof, +AD4- since it would show polynomiality of a NP-hard problem - thus you +AD4- would have won one of the Millenium prizes (for P+AD0-NP) with such a +AD4- simple argument+ACE- Unfortunately, you are right. I apologize for this error but I still think that this algorithm can give good solution in the case of narrow intervals. Regards, Andrzej Pownuk From owner-reliable_computing [at] interval [dot] louisiana.edu Thu Mar 21 09:43:55 2002 Received: (from daemon@localhost) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) id g2LFhta06636 for reliable_computing-outgoing; Thu, 21 Mar 2002 09:43:55 -0600 (CST) Received: from mailbox.univie.ac.at (mailbox.univie.ac.at [131.130.1.27]) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) with ESMTP id g2LFhnr06632 for ; Thu, 21 Mar 2002 09:43:50 -0600 (CST) Received: from univie.ac.at (hektor.mat.univie.ac.at [131.130.16.21]) by mailbox.univie.ac.at (8.12.2/8.12.2) with ESMTP id g2LFf6JO032940; Thu, 21 Mar 2002 16:41:07 +0100 Message-ID: <3C99FF12.5E6987EB [at] univie [dot] ac.at> Date: Thu, 21 Mar 2002 16:41:06 +0100 From: Arnold Neumaier Organization: University of Vienna X-Mailer: Mozilla 4.78 [en] (X11; U; Linux 2.4.9-21 i686) X-Accept-Language: en, de MIME-Version: 1.0 To: Andrzej Pownuk CC: sebastiaoc [at] cenpes [dot] petrobras.com.br, reliable_computing [at] interval [dot] louisiana.edu Subject: Re: Interval Linear Systems of Equations References: +ADw-01d001c1d049+ACQ-30752c60+ACQ-0200a8c0+AEA-andrzej1+AD4- +ADw-3C99CE80.8B9DA49B+AEA-univie.ac.at+AD4- <003501c1d0e8$f17c4d90$0200a8c0@andrzej1> Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Sender: owner-reliable_computing [at] interval [dot] louisiana.edu Precedence: bulk Andrzej Pownuk wrote: > After point (3) I assume that the intervals are narrow. > (in technical applications the intervals are usually narrow) > Additionally for small intervals we can assume > that the solution is monotone and depend only on > the endpoints of the intervals. > > There are a lot of papers, which are based on these assumptions. People in applications may well be satisfied with results based on uncertain assumptions. But in *reliable computing*, on whose mailing list you are posting, the common denominator is that *all* assumptions going into the calculations are *verified* (including the effects of rounding errors, which are usually deemed irrelevant in practical applications, too). If one uses a method that works if the intervals are narrow enough one still needs to check whether the *particular* intervals in the application are *narrow enough* for the method to work; one might be well within the valid domain, or just outside. The reliability of the result depends on the ability to perform such a check. You are well advised to study the work of Jansson and of Rohn to learn about techniques how to do such checking. If one cannot check the assumptions, there is no need to use verified methods at all; using interval techniques to get approximate answers is wasted effort. My recommendation to those satsified with a reasonable approximation of a range is to do traditional approximate sensitivity analysis, which is cheap and gives adequate results for all kinds of problems in most applications with narrow intervals. > I still think that this algorithm can give > good solution in the case of narrow intervals. You should call it 'good approximation', not 'good solution'. Arnold Neumaier From owner-reliable_computing [at] interval [dot] louisiana.edu Thu Mar 21 09:46:36 2002 Received: (from daemon@localhost) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) id g2LFka506717 for reliable_computing-outgoing; Thu, 21 Mar 2002 09:46:36 -0600 (CST) Received: from clmboh1-smtp3.columbus.rr.com (clmboh1-smtp3.columbus.rr.com [65.24.0.112]) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) with ESMTP id g2LFkUr06713 for ; Thu, 21 Mar 2002 09:46:31 -0600 (CST) Received: from oemcomputer (dhcp065-024-174-102.columbus.rr.com [65.24.174.102]) by clmboh1-smtp3.columbus.rr.com (8.11.2/8.11.2) with SMTP id g2LFkQn02109; Thu, 21 Mar 2002 10:46:26 -0500 (EST) Message-ID: <003801c1d0ef$1fea6260$66ae1841 [at] columbus [dot] rr.com> From: "Ramon Moore" To: "Andrzej Pownuk" Cc: "interval" References: +ADw-01d001c1d049+ACQ-30752c60+ACQ-0200a8c0+AEA-andrzej1+AD4- +ADw-3C99CE80.8B9DA49B+AEA-univie.ac.at+AD4- +ADw-003501c1d0e8+ACQ-f17c4d90+ACQ-0200a8c0+AEA-andrzej1+AD4- Subject: Re: Interval Linear Systems of Equations Date: Thu, 21 Mar 2002 10:43:18 -0500 MIME-Version: 1.0 Content-Type: text/plain; charset="UTF-7" Content-Transfer-Encoding: 7bit X-Priority: 3 X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook Express 5.50.4133.2400 X-MimeOLE: Produced By Microsoft MimeOLE V5.50.4133.2400 Sender: owner-reliable_computing [at] interval [dot] louisiana.edu Precedence: bulk Dear Colleagues, Andrzej Pownuk makes an important point in reminding us that in many practical applications intervals are narrow. Ramon Moore ----- Original Message ----- From: +ACI-Andrzej Pownuk+ACI- +ADw-pownuk+AEA-zeus.polsl.gliwice.pl+AD4- To: +ADw-sebastiaoc+AEA-cenpes.petrobras.com.br+AD4AOw- +ADw-reliable+AF8-computing+AEA-interval.louisiana.edu+AD4- Cc: +ACI-Andrzej Pownuk+ACI- +ADw-pownuk+AEA-zeus.polsl.gliwice.pl+AD4- Sent: Thursday, March 21, 2002 9:59 AM Subject: Re: Interval Linear Systems of Equations +AD4- +AD4- +AD4- Andrzej Pownuk wrote: +AD4- +AD4- +AD4- +AD4- +AD4- +AD4- I have worked on this problem since 1995. +AD4- +AD4- +AD4- Now I can solve system of 1000 equations with dependent interval +AD4- +AD4- +AD4- parameters. +AD4- +AD4- +AD4- (I have solved system of 2300 equations +AD4- +AD4- +AD4- with two intervals Young?s modulus.) +AD4- +AD4- +AD4- +AD4- +AD4- +AD4- This algorithm can be downloaded directly from the internet. +AD4- +AD4- +AD4- +AD4- +AD4- +AD4- http://157.158.187.18/papers/Interval+AF8-FEM.pdf +AD4- +AD4- +AD4- +AD4- However, your algorithm depends on unverified assumptions (after (3)), +AD4- +AD4- that are unlikely to be satisfied if the intervals are wide. +AD4- +AD4- To make this a valid algorithm you need to provide verifiable +AD4- +AD4- sufficient conditions. +AD4- +AD4- After point (3) I assume that the intervals are narrow. +AD4- (+IBw-in technical applications the intervals are usually narrow+IB0-) +AD4- Additionally for small intervals we can assume +AD4- that the solution is monotone and depend only on +AD4- the endpoints of the intervals. +AD4- +AD4- There are a lot of papers, which are based on these assumptions. +AD4- +AD4- Akpan U.O., Koko T.S., Orisamolu I.R., Gallant B.K., +AD4- Practical fuzzy finite element analysis of structures, +AD4- Finite Elements in Analysis and Design, 38 (2000) 93-111 +AD4- +AD4- McWilliam S., Anti-optimization of uncertain structures +AD4- using interval analysis, +AD4- Computers and Structures, 79 (2000) 421-430 +AD4- +AD4- Noor A.K., Starnes J.H., Peters J.M., +AD4- Uncertainty analysis of composite structures, +AD4- Computer methods in applied mechanics and engineering, +AD4- 79 (2000) 413-232 +AD4- +AD4- Valliappan S., Pham T.D., Elasto-Plastic Finite Element Analysis +AD4- with Fuzzy Parameters, International Journal +AD4- for Numerical Methods in Engineering, 38 (1995) 531-548 +AD4- +AD4- etc. +AD4- +AD4- +AD4- +AD4- +AD4- Moreover, the statement on p.3 is simply wrong. +AD4- +AD4- You should have been suspicious about your method and proof, +AD4- +AD4- since it would show polynomiality of a NP-hard problem - thus you +AD4- +AD4- would have won one of the Millenium prizes (for P+AD0-NP) with such a +AD4- +AD4- simple argument+ACE- +AD4- +AD4- Unfortunately, you are right. +AD4- +AD4- I apologize for this error +AD4- but I still think that this algorithm can give +AD4- good solution in the case of narrow intervals. +AD4- +AD4- Regards, +AD4- +AD4- Andrzej Pownuk +AD4- +AD4- +AD4- From owner-reliable_computing [at] interval [dot] louisiana.edu Thu Mar 21 10:29:47 2002 Received: (from daemon@localhost) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) id g2LGTk606888 for reliable_computing-outgoing; Thu, 21 Mar 2002 10:29:46 -0600 (CST) Received: from imf09bis.bellsouth.net (mail309.mail.bellsouth.net [205.152.58.169]) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) with ESMTP id g2LGTfr06884 for ; Thu, 21 Mar 2002 10:29:42 -0600 (CST) Received: from u8174 ([66.20.82.156]) by imf09bis.bellsouth.net (InterMail vM.5.01.04.05 201-253-122-122-105-20011231) with SMTP id <20020321163055.SQXO26876.imf09bis.bellsouth.net@u8174>; Thu, 21 Mar 2002 11:30:55 -0500 Message-Id: <2.2.32.20020321162923.00995c44 [at] pop [dot] louisiana.edu> X-Sender: rbk5287 [at] pop [dot] louisiana.edu X-Mailer: Windows Eudora Pro Version 2.2 (32) Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" Date: Thu, 21 Mar 2002 10:29:23 -0600 To: "Ramon Moore" , "Andrzej Pownuk" From: "R. Baker Kearfott" Subject: Re: Interval Linear Systems of Equations Cc: "interval" Sender: owner-reliable_computing [at] interval [dot] louisiana.edu Precedence: bulk Colleagues, Yes, indeed. And narrow intervals occur in diverse practical contexts, when rigorous error bounds on a floating-point approximate solution are desired. Depending on how the mathematics is done and how the algorithms are arranged, working "in the small" can lead to much nicer computational complexity. Of course, it's easy for me to make such statements, but we need to provide solid details :-) Best regards, Baker At 10:43 AM 3/21/02 -0500, Ramon Moore wrote: >Dear Colleagues, > >Andrzej Pownuk makes an important point in reminding us that in many >practical applications >intervals are narrow. > >Ramon Moore > --------------------------------------------------------------- R. Baker Kearfott, rbk [at] louisiana [dot] edu (337) 482-5346 (fax) (337) 482-5270 (work) (337) 981-9744 (home) URL: http://interval.louisiana.edu/kearfott.html Department of Mathematics, University of Louisiana at Lafayette Box 4-1010, Lafayette, LA 70504-1010, USA --------------------------------------------------------------- From owner-reliable_computing [at] interval [dot] louisiana.edu Thu Mar 21 10:58:03 2002 Received: (from daemon@localhost) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) id g2LGw3S07013 for reliable_computing-outgoing; Thu, 21 Mar 2002 10:58:03 -0600 (CST) Received: from iph.bio.bas.bg (IDENT:0@bas-bio.lines.bas.bg [195.96.252.58]) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) with ESMTP id g2LGvvr07009 for ; Thu, 21 Mar 2002 10:57:57 -0600 (CST) Received: from biomath2 (biomath2.bio.bas.bg [195.96.247.143]) by iph.bio.bas.bg (8.11.6/8.9.3) with ESMTP id g2LGxBr19062 for ; Thu, 21 Mar 2002 18:59:11 +0200 From: "Evgenija D. Popova" Organization: Institute of Mathematics & Informatics, BAS To: reliable_computing [at] interval [dot] louisiana.edu Date: Thu, 21 Mar 2002 19:02:49 +0100 MIME-Version: 1.0 Content-type: text/plain; charset=US-ASCII Content-transfer-encoding: 7BIT Subject: Re: Interval Linear Systems of Equations Reply-to: epopova [at] iph [dot] bio.bas.bg Message-ID: <3C9A2E59.2127.1D30854@localhost> In-reply-to: <3C99CE80.8B9DA49B [at] univie [dot] ac.at> X-mailer: Pegasus Mail for Win32 (v3.12c) Sender: owner-reliable_computing [at] interval [dot] louisiana.edu Precedence: bulk The following paper: E. Popova, Quality of the Solution Sets of Parameter-Dependent Interval Linear Systems, to appear in ZAMM. http://www.math.bas.bg/~epopova/papers/ZAMMPopova.pdf contains some verifiable sufficient conditions for exact finite characterisation of the bounds of the solution set of a parametrised interval linear system. In: E.Popova, On the Solution of Parametrised Linear Systems. W. Kraemer, J. Wolff von Gudenberg (Eds.): Scientific Computing, Validated Numerics, Interval Methods. Kluwer Acad. Publishers, 2001, pp. 127-138. http://www.math.bas.bg/~epopova/papers/SCAN2000Popova.ps on a practical example, we have presented our experience in using Rump's algorithm and a way to sharpen the enclosing intervals whenever it is necessary. Evgenija D. Popova --------------------------------------------------------- Institute of Mathematics & Informatics Bulgarian Academy of Sciences Acad. G. Bonchev str., block 8 BG-1113 Sofia, Bulgaria Phone: (+359 2) 979-3704 Fax: (+359 2) 971-3649 http://www.math.bas.bg/~epopova --------------------------------------------------------- From owner-reliable_computing [at] interval [dot] louisiana.edu Thu Mar 21 11:13:47 2002 Received: (from daemon@localhost) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) id g2LHDlw07122 for reliable_computing-outgoing; Thu, 21 Mar 2002 11:13:47 -0600 (CST) Received: from kathmandu.sun.com (kathmandu.sun.com [192.18.98.36]) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) with ESMTP id g2LHDfr07118 for ; Thu, 21 Mar 2002 11:13:42 -0600 (CST) Received: from engmail1.Eng.Sun.COM ([129.146.1.13]) by kathmandu.sun.com (8.9.3+Sun/8.9.3) with ESMTP id KAA11002; Thu, 21 Mar 2002 10:13:36 -0700 (MST) Received: from phys-mpkmaila (phys-mpkmaila.Eng.Sun.COM [129.146.18.131]) by engmail1.Eng.Sun.COM (8.9.3+Sun/8.9.3/ENSMAIL,v2.1p1) with ESMTP id JAA00770; Thu, 21 Mar 2002 09:13:35 -0800 (PST) Received: from nubbins (nubbins.Eng.Sun.COM [129.146.79.58]) by mpkmail.eng.sun.com (iPlanet Messaging Server 5.2 (built Jan 13 2002)) with SMTP id <0GTC00HN32JQOQ [at] mpkmail [dot] eng.sun.com>; Thu, 21 Mar 2002 09:14:15 -0800 (PST) Date: Thu, 21 Mar 2002 09:13:35 -0800 (PST) From: Marty Itzkowitz Subject: Re: Fortran 90/95-timings To: beebe [at] math [dot] utah.edu, reliable_computing [at] interval [dot] louisiana.edu, kk [at] gcet [dot] ac.in Cc: Martin.Itzkowitz [at] Eng [dot] Sun.COM, gvt [at] mpkmail [dot] eng.sun.com Reply-to: Marty Itzkowitz Message-id: <0GTC00HN42JROQ [at] mpkmail [dot] eng.sun.com> MIME-version: 1.0 X-Mailer: dtmail 1.3.0 @(#)CDE Version 1.4 SunOS 5.8 sun4u sparc Content-type: TEXT/plain; charset=us-ascii Content-transfer-encoding: 7BIT Content-MD5: ktf/lMCIpdpLCvqmHYePsg== Sender: owner-reliable_computing [at] interval [dot] louisiana.edu Precedence: bulk This email was forwarded to me by Gregory Tarsy. There is a gethrtime(3C) routine available on Solaris that will give very fast high-resolution timing. On my Ultra-60, the gethrtime call takes approximately 300 nanoseconds. Its man page reads as follows: Standard C Library Functions gethrtime(3C) NAME gethrtime, gethrvtime - get high resolution time SYNOPSIS #include hrtime_t gethrtime(void); hrtime_t gethrvtime(void); DESCRIPTION The gethrtime() function returns the current high-resolution real time. Time is expressed as nanoseconds since some arbi- trary time in the past; it is not correlated in any way to the time of day, and thus is not subject to resetting or drifting by way of adjtime(2) or settimeofday(3C). The hi- res timer is ideally suited to performance measurement tasks, where cheap, accurate interval timing is required. The gethrvtime() function returns the current high- resolution LWP virtual time, expressed as total nanoseconds of execution time. This function requires that micro state accounting be enabled with the ptime utility (see proc(1)). The gethrtime() and gethrvtime() functions both return an hrtime_t, which is a 64-bit (long long) signed integer. EXAMPLES The following code fragment measures the average cost of getpid(2): hrtime_t start, end; int i, iters = 100; start = gethrtime(); for (i = 0; i < iters; i++) getpid(); end = gethrtime(); printf("Avg getpid() time = %lld nsec\n", (end - start) / iters); ATTRIBUTES See attributes(5) for descriptions of the following attri- butes: ____________________________________________________________ | ATTRIBUTE TYPE | ATTRIBUTE VALUE | |_____________________________|_____________________________| | MT-Level | MT-Safe | |_____________________________|_____________________________| SunOS 5.8 Last change: 10 Apr 1997 1 Standard C Library Functions gethrtime(3C) SEE ALSO proc(1), adjtime(2), gettimeofday(3C), settimeofday(3C), attributes(5) NOTES Although the units of hi-res time are always the same (nanoseconds), the actual resolution is hardware dependent. Hi-res time is guaranteed to be monotonic (it won't go back- ward, it won't periodically wrap) and linear (it won't occa- sionally speed up or slow down for adjustment, like the time of day can), but not necessarily unique: two sufficiently proximate calls may return the same value. SunOS 5.8 Last change: 10 Apr 1997 2 Marty Itzkowitz Project Lead, Forte Developer Performance Tools ------------- Begin Forwarded Message ------------- Date: Thu, 14 Mar 2002 12:25:38 -0800 (PST) From: "Gregory V. Tarsy" Subject: Re: Fortran 90/95-timings To: martyi [at] susila [dot] eng.sun.com A query that you could answer ... ------------- Begin Forwarded Message ------------- Date: Thu, 14 Mar 2002 05:49:55 -0700 (MST) From: "Nelson H. F. Beebe" To: reliable_computing [at] interval [dot] louisiana.edu Cc: beebe [at] math [dot] utah.edu, "Mr. Ketan Kotecha" X-US-Mail: "Center for Scientific Computing, Department of Mathematics, 110 LCB, University of Utah, 155 S 1400 E RM 233, Salt Lake City, UT 84112-0090, USA" X-Telephone: +1 801 581 5254 X-FAX: +1 801 585 1640, +1 801 581 4148 X-URL: http://www.math.utah.edu/~beebe Subject: Re: Fortran 90/95-timings "Ketan Kotecha" asks on Wed, 13 Mar 2002 13:53:41 +0530 about getting program execution timings accurate to milliseconds or better. Sadly, on the vast majority of operating systems today, with clock speeds sometimes exceeding 1GHz, process timers remain extremely crude, with 60 to 100 ticks per second. This is true no matter what programming language you use. Some processors (DEC Alpha, IBM PowerPC, Intel Pentium, Sun UltraSPARC) actually have timer instructions that can produce a resolution of less than a microsec, but these tend to require kernel privileges to execute, and remain unavailable to user processes, even UNIX root (super-user) processes. Cray supercomputers have been about the only ones I've encountered where accurate timing is available. The workaround is to wrap benchmark codes with outer loops that repeat the calculation long enough to get reliable timings, or use input data that makes them run long enough. Of course, this doesn't work if you are trying to accurately time various parts of real code. If you delve into C header files, look for the symbols CLK_TCK, CLOCKS_PER_SEC, HZ, and AHZ, and consult "man sysconf"; Sun Solaris systems get CLK_TCK from the function call _sysconf(3). On a GNU/Linux Red Hat 6.2 system on Intel x86, I find: /* ISO/IEC 9899:1990 7.12.1: The macro `CLOCKS_PER_SEC' is the number per second of the value returned by the `clock' function. CAE XSH, Issue 4, Version 2: The value of CLOCKS_PER_SEC is required to be 1 million on all XSI-conformant systems. */ # define CLOCKS_PER_SEC 1000000 /* Even though CLOCKS_PER_SEC has such a strange value CLK_TCK presents the real value for clock ticks per second for the system. */ # define CLK_TCK 100 Thus, CLOCKS_PER_SEC may lie about the true timer resolution. You also need to be aware of the effect of cache vs memory: the first iteration your program runs, data may be in memory, but on subsequent iterations, it may be in cache. The effect on timing can be dramatic. Typically, register access takes 1 cycle, level-1 cache 2 to 5 cycles, level-2 cache 8 to 15 cycles, and memory access 10 to 100 cycles. For more details, see my February 2001 talk on microprocessors, starting about slide 55, available at http://www.chpc.utah.edu/cc/talks/beebe/index.htm http://www.chpc.utah.edu/cc/talks/beebe/handout-color.pdf Here is a copy of a file in which I recorded notes on 7-Oct-1996 about timers on various systems when I implemented a profiling facility in the awk programming language: ------------------------------------------------------------------------ These notes record my attempts to try to get access to a cheap high-resolution clock for profiling on several systems. ------------------------------------------------------------------------ On Sun UltraSPARC, modified assembly code from timercheck.c in function foo() ! 1 #include ! 2 ! 3 typedef long long LONG; ! 4 ! 5 LONG foo(void) ! 6 { ! 7 return (1000000L + sizeof(LONG)); sethi %hi(.L_cseg0),%l0 or %l0,%lo(.L_cseg0),%l0 ldd [%l0+0],%l0 std %l0,[%fp-8] rd %tick,%l0 !<== inserted read of 63-bit cycle clock ba .L76 nop ! block 2 .L76: ldd [%fp-8],%l0 mov %l1,%i1 mov %l0,%i0 jmp %i7+8 restore Compilation was cc -c -xarch=v8plus timercheck.s cc -o timercheck -xarch=v8plus timercheck.s and running the job produces privilege failure, both as me, and as root: signal ILL (privileged opcode) in foo at 0x107e4 It is aggravating that such a useful instruction is locked out by the operating system for `security reasons'. Older SPARC architectures (pre-V9) lack any kind of a high-resolution cycle timer. ------------------------------------------------------------------------ On the DEC Alpha, the rscc (read system cycle counter) is potentially available in unprivileged VMS PALcode, but alas, not in OSF/1 PALcode. ------------------------------------------------------------------------ On the IBM RS/6000 PowerPC (see ``The PowerPC Architecture'', p. 354), the 64-bit Time Base register can be read by a 5-instruction loop to get a 64-bit value in two adjacent registers (r3 and r4 are needed for a long long function return value). I modified the output assembly code of timertest.c like this, to just read the lower 32 bits: .foo: # 0x00000000 (H.10.NO_SYMBOL) .file "timercheck.c" cau r0,r0,0x0002 ai r3,r0,-31068 mftb r3 # <== inserted bcr BO_ALWAYS,CR0_LT This will not compile in the Power or Power2 architectures (-qarch=pwr or -qarch=pwr2). It compiles in the PowerPC architecture (-qarch=ppc), but on the Power and Power2 systems that I ran the test on, it dies with Illegal instruction (core dumped) I don't have access to a PowerPC system running UNIX to make further tests. ------------------------------------------------------------------------ The MIPS architecture does not appear to have any documented facility for reading a high-precision clock, though on DEC ULTRIX, DEC OSF/1, and SGI IRIX operating systems, the pixie/pixstats utilities appear to be able to get exact cycle counts for instructions. ------------------------------------------------------------------------ I don't have online documentation of the HP-UX architecture to investigate further on our HP workstations. ------------------------------------------------------------------------ ------------------------------------------------------------------------------- - Nelson H. F. Beebe Tel: +1 801 581 5254 - - Center for Scientific Computing FAX: +1 801 585 1640, +1 801 581 4148 - - University of Utah Internet e-mail: beebe [at] math [dot] utah.edu - - Department of Mathematics, 110 LCB beebe [at] acm [dot] org beebe [at] computer [dot] org - - 155 S 1400 E RM 233 beebe [at] ieee [dot] org - - Salt Lake City, UT 84112-0090, USA URL: http://www.math.utah.edu/~beebe - ------------------------------------------------------------------------------- ------------- End Forwarded Message ------------- ------------- End Forwarded Message ------------- From owner-reliable_computing [at] interval [dot] louisiana.edu Thu Mar 21 14:17:43 2002 Received: (from daemon@localhost) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) id g2LKHhm07495 for reliable_computing-outgoing; Thu, 21 Mar 2002 14:17:43 -0600 (CST) Received: from smtp.sunflower.com (smtp.sunflower.com [24.124.0.128]) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) with ESMTP id g2LKHcr07491 for ; Thu, 21 Mar 2002 14:17:39 -0600 (CST) Received: from tirazezhk1exbw (dv137s77.lawrence.ks.us [24.124.77.137]) by smtp.sunflower.com (8.11.6/8.11.6) with SMTP id g2LKHbv26353 for ; Thu, 21 Mar 2002 14:17:37 -0600 Message-ID: <001b01c1d114$81125d20$6401a8c0@tirazezhk1exbw> From: "tiraz birdie" To: References: <3C9A2E59.2127.1D30854@localhost> Subject: Re: Interval Linear Systems of Equations Date: Thu, 21 Mar 2002 14:10:48 -0600 MIME-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: 7bit X-Priority: 3 X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook Express 6.00.2600.0000 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2600.0000 Sender: owner-reliable_computing [at] interval [dot] louisiana.edu Precedence: bulk If my memory serves me right, then the heat conduction equation represents a convex problem in the (conductivity) parameter space. Therefore, you can obtain the bounds by simply using the upper and lower limits of the parameter. Tiraz Birdie From owner-reliable_computing [at] interval [dot] louisiana.edu Thu Mar 21 14:53:51 2002 Received: (from daemon@localhost) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) id g2LKrod07621 for reliable_computing-outgoing; Thu, 21 Mar 2002 14:53:50 -0600 (CST) Received: from kathmandu.sun.com (kathmandu.sun.com [192.18.98.36]) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) with ESMTP id g2LKrjr07617 for ; Thu, 21 Mar 2002 14:53:46 -0600 (CST) Received: from engmail3.Eng.Sun.COM ([129.144.170.5]) by kathmandu.sun.com (8.9.3+Sun/8.9.3) with ESMTP id NAA12081; Thu, 21 Mar 2002 13:53:39 -0700 (MST) Received: from phys-mpkmaila (phys-mpkmaila.Eng.Sun.COM [129.146.18.131]) by engmail3.Eng.Sun.COM (8.9.3+Sun/8.9.3/ENSMAIL,v2.1p1) with ESMTP id MAA28694; Thu, 21 Mar 2002 12:53:39 -0800 (PST) Received: from gww (gww.Eng.Sun.COM [129.146.78.116]) by mpkmail.eng.sun.com (iPlanet Messaging Server 5.2 (built Jan 13 2002)) with SMTP id <0GTC00HZ1CQIOQ [at] mpkmail [dot] eng.sun.com>; Thu, 21 Mar 2002 12:54:18 -0800 (PST) Date: Thu, 21 Mar 2002 12:53:38 -0800 (PST) From: William Walster Subject: Re: Interval Linear Systems of Equations To: rmoore17 [at] columbus [dot] rr.com, pownuk [at] zeus [dot] polsl.gliwice.pl, rbk [at] louisiana [dot] edu Cc: reliable_computing [at] interval [dot] louisiana.edu Reply-to: William Walster Message-id: <0GTC00HZ2CQIOQ [at] mpkmail [dot] eng.sun.com> MIME-version: 1.0 X-Mailer: dtmail 1.3.0 @(#)CDE Version 1.4.2 SunOS 5.8 sun4u sparc Content-type: TEXT/plain; charset=us-ascii Content-transfer-encoding: 7BIT Content-MD5: Xy/rWcBDhiCc10Bs7i2fNw== Sender: owner-reliable_computing [at] interval [dot] louisiana.edu Precedence: bulk Would it be possible to "cheaply" but rigorously verify if the necessary assumptions have been met? This might be less work than our normal procedures. With a contracting map, once the necessary conditions have been satisfied, the tests should not be necessary. Bill >Date: Thu, 21 Mar 2002 10:29:23 -0600 >From: "R. Baker Kearfott" >Subject: Re: Interval Linear Systems of Equations >X-Sender: rbk5287 [at] pop [dot] louisiana.edu >To: Ramon Moore , Andrzej Pownuk >Cc: interval >MIME-version: 1.0 > >Colleagues, > >Yes, indeed. And narrow intervals occur in diverse practical contexts, >when rigorous error bounds on a floating-point approximate solution >are desired. Depending on how the mathematics is done and how the >algorithms are arranged, working "in the small" can lead to much >nicer computational complexity. > >Of course, it's easy for me to make such statements, but we need >to provide solid details :-) > >Best regards, > >Baker > >At 10:43 AM 3/21/02 -0500, Ramon Moore wrote: >>Dear Colleagues, >> >>Andrzej Pownuk makes an important point in reminding us that in many >>practical applications >>intervals are narrow. >> >>Ramon Moore >> > >--------------------------------------------------------------- >R. Baker Kearfott, rbk [at] louisiana [dot] edu (337) 482-5346 (fax) >(337) 482-5270 (work) (337) 981-9744 (home) >URL: http://interval.louisiana.edu/kearfott.html >Department of Mathematics, University of Louisiana at Lafayette >Box 4-1010, Lafayette, LA 70504-1010, USA >--------------------------------------------------------------- > From owner-reliable_computing [at] interval [dot] louisiana.edu Thu Mar 21 15:55:15 2002 Received: (from daemon@localhost) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) id g2LLtFV07816 for reliable_computing-outgoing; Thu, 21 Mar 2002 15:55:15 -0600 (CST) Received: from mailbox.univie.ac.at (mailbox.univie.ac.at [131.130.1.27]) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) with ESMTP id g2LLt9r07812 for ; Thu, 21 Mar 2002 15:55:10 -0600 (CST) Received: from univie.ac.at (hektor.mat.univie.ac.at [131.130.16.21]) by mailbox.univie.ac.at (8.12.2/8.12.2) with ESMTP id g2LLsuSi065322; Thu, 21 Mar 2002 22:54:57 +0100 Message-ID: <3C9A56B0.C7CA176A [at] univie [dot] ac.at> Date: Thu, 21 Mar 2002 22:54:56 +0100 From: Arnold Neumaier Organization: University of Vienna X-Mailer: Mozilla 4.78 [en] (X11; U; Linux 2.4.9-21 i686) X-Accept-Language: en, de MIME-Version: 1.0 To: William Walster CC: rmoore17 [at] columbus [dot] rr.com, pownuk [at] zeus [dot] polsl.gliwice.pl, rbk [at] louisiana [dot] edu, reliable_computing [at] interval [dot] louisiana.edu Subject: Re: Interval Linear Systems of Equations References: <0GTC00HZ2CQIOQ [at] mpkmail [dot] eng.sun.com> Content-Type: text/plain; charset=UTF-7 Content-Transfer-Encoding: 7bit Sender: owner-reliable_computing [at] interval [dot] louisiana.edu Precedence: bulk William Walster wrote: > > Would it be possible to "cheaply" but rigorously verify if the > necessary assumptions have been met? This might be less work than > our normal procedures. With a contracting map, once the necessary > conditions have been satisfied, the tests should not be necessary. Hansen has an *old* paper in which he used *verified* monotony of some components of an interval linear system to fix these components at a bound and showed that in the 'inverse stable' case, where all signs are fixed (which holds with high probability if all intervals are narrow), the total work is reduced to O(n^4); when the signs have special patterns (which is the case only in special situations, even for thin interval matrices) the work is even O(n^3) only. These cases correspond precisely to the situation discussed by Pownuk on his faulty p.3. Which sign patterns must be checked in the general case is stated precisely in Corollary 6.2.4 of my book, following work of Rohn. The *verification* of signs is based in both cases on the ability to first compute an enclosure of the inverse of the interval coefficient matrix, which is cheap, O(n^3), but limited to strongly regular matrices. This limitation is removed to some extent by Jansson and Rohn, I believe. Pownuk discusses (in the correct pages 1 and 2) also the case of dependent interval coefficients (which was not considered by Hansen). Probably his results are covered for linear dependences by Jansson and Rohn, who give (for whatever results they actually have) *verifiable* conditions. Since I have the papers not readily available, I can't check at the moment what they actually did; maybe the authors should summarize their main results here to make them more generally known. What is probably new about Pownuk's results is the generalization to nonlinear parameter dependence, and the sign verification in this case can probably be done in analogy to what has been done in the linear case. If done properly and written up with the same care as done by Hansen, Rohn and Jansson, this will be a useful addition to the tool set for linear interval equations. However, we should not fall below already established standards of quality. Best wishes, Arnold From owner-reliable_computing [at] interval [dot] louisiana.edu Thu Mar 21 16:18:06 2002 Received: (from daemon@localhost) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) id g2LMI5p07934 for reliable_computing-outgoing; Thu, 21 Mar 2002 16:18:05 -0600 (CST) Received: from kathmandu.sun.com (kathmandu.sun.com [192.18.98.36]) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) with ESMTP id g2LMI0r07930 for ; Thu, 21 Mar 2002 16:18:01 -0600 (CST) Received: from engmail1.Eng.Sun.COM ([129.146.1.13]) by kathmandu.sun.com (8.9.3+Sun/8.9.3) with ESMTP id PAA25666; Thu, 21 Mar 2002 15:17:40 -0700 (MST) Received: from phys-mpkmaila (phys-mpkmaila.Eng.Sun.COM [129.146.18.131]) by engmail1.Eng.Sun.COM (8.9.3+Sun/8.9.3/ENSMAIL,v2.1p1) with ESMTP id OAA27460; Thu, 21 Mar 2002 14:17:39 -0800 (PST) Received: from gww (gww.Eng.Sun.COM [129.146.78.116]) by mpkmail.eng.sun.com (iPlanet Messaging Server 5.2 (built Jan 13 2002)) with SMTP id <0GTC00EQXGMIMP [at] mpkmail [dot] eng.sun.com>; Thu, 21 Mar 2002 14:18:18 -0800 (PST) Date: Thu, 21 Mar 2002 14:17:38 -0800 (PST) From: William Walster Subject: Re: Interval Linear Systems of Equations To: Bill.Walster [at] Eng [dot] Sun.COM, Arnold.Neumaier [at] univie [dot] ac.at Cc: rmoore17 [at] columbus [dot] rr.com, pownuk [at] zeus [dot] polsl.gliwice.pl, rbk [at] louisiana [dot] edu, reliable_computing [at] interval [dot] louisiana.edu Reply-to: William Walster Message-id: <0GTC00EQYGMIMP [at] mpkmail [dot] eng.sun.com> MIME-version: 1.0 X-Mailer: dtmail 1.3.0 @(#)CDE Version 1.4.2 SunOS 5.8 sun4u sparc Content-type: TEXT/plain; charset=us-ascii Content-transfer-encoding: 7BIT Content-MD5: i1c9i2NM4G8ZseM6Hm02OA== Sender: owner-reliable_computing [at] interval [dot] louisiana.edu Precedence: bulk Great. Thanks, Arnold. Of course, I *absolutely* agree with your last sentence :) Cheers, Bill >Date: Thu, 21 Mar 2002 22:54:56 +0100 >From: Arnold Neumaier >Subject: Re: Interval Linear Systems of Equations >To: William Walster >Cc: rmoore17 [at] columbus [dot] rr.com, pownuk [at] zeus [dot] polsl.gliwice.pl, rbk [at] louisiana [dot] edu, reliable_computing [at] interval [dot] louisiana.edu >MIME-version: 1.0 >Content-transfer-encoding: 7bit >X-Accept-Language: en, de > >William Walster wrote: >> >> Would it be possible to "cheaply" but rigorously verify if the >> necessary assumptions have been met? This might be less work than >> our normal procedures. With a contracting map, once the necessary >> conditions have been satisfied, the tests should not be necessary. > >Hansen has an *old* paper in which he used *verified* monotony of some >components of an interval linear system to fix these components at a >bound >and showed that in the 'inverse stable' case, where all signs are fixed >(which holds with high probability if all intervals are narrow), >the total work is reduced to O(n^4); when the signs have special >patterns (which is the case only in special situations, even for thin >interval matrices) the work is even O(n^3) only. These cases correspond >precisely to the situation discussed by Pownuk on his faulty p.3. > >Which sign patterns must be checked in the general case is stated >precisely in Corollary 6.2.4 of my book, following work of Rohn. >The *verification* of signs is based in both cases on the ability to >first compute an enclosure of the inverse of the interval coefficient >matrix, which is cheap, O(n^3), but limited to strongly regular >matrices. >This limitation is removed to some extent by Jansson and Rohn, I >believe. > >Pownuk discusses (in the correct pages 1 and 2) also the case of >dependent interval coefficients (which was not considered by Hansen). >Probably his results are covered for linear dependences >by Jansson and Rohn, who give (for whatever results they actually have) >*verifiable* conditions. Since I have the papers not readily available, >I can't check at the moment what they actually did; maybe the authors >should summarize their main results here to make them more generally >known. > >What is probably new about Pownuk's results is the generalization to >nonlinear parameter dependence, and the sign verification in this case >can >probably be done in analogy to what has been done in the linear case. >If done properly and written up with the same care as done by >Hansen, Rohn and Jansson, this will be a useful addition to the tool >set for linear interval equations. However, we should not fall below >already established standards of quality. > > >Best wishes, > >Arnold From owner-reliable_computing [at] interval [dot] louisiana.edu Fri Mar 22 05:56:29 2002 Received: (from daemon@localhost) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) id g2MBuSi09594 for reliable_computing-outgoing; Fri, 22 Mar 2002 05:56:28 -0600 (CST) Received: from sgu.ssu.runnet.ru (sgu.ssu.runnet.ru [212.193.32.14]) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) with ESMTP id g2MBoTr09581 for ; Fri, 22 Mar 2002 05:50:47 -0600 (CST) Received: from info.sgu.ru (info.sgu.ru [212.193.32.6]) by sgu.ssu.runnet.ru (8.12.2/8.12.2) with ESMTP id g2MBlvrl065545 for ; Fri, 22 Mar 2002 14:47:57 +0300 (MSK) Received: from info.sgu.ru ([212.193.39.42]) by info.sgu.ru (Netscape Messaging Server 3.6) with SMTP id AAA2880 for ; Fri, 22 Mar 2002 14:35:40 +0300 Status: U Received: from sgu.ssu.runnet.ru ([212.193.32.14]) by info.sgu.ru (Netscape Messaging Server 3.6) with ESMTP id AAA53C3; Thu, 21 Mar 2002 19:52:46 +0300 Received: from interval.louisiana.edu (interval.louisiana.edu [130.70.132.145]) by sgu.ssu.runnet.ru (8.12.2/8.12.2) with ESMTP id g2LH3Irl034950; Thu, 21 Mar 2002 20:03:47 +0300 (MSK) Received: from localhost (daemon@localhost) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) with SMTP id g2LH1Ga07092; Thu, 21 Mar 2002 11:01:17 -0600 (CST) Received: (from daemon@localhost) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) id g2LGw3S07013 for reliable_computing-outgoing; Thu, 21 Mar 2002 10:58:03 -0600 (CST) Received: from iph.bio.bas.bg (IDENT:0@bas-bio.lines.bas.bg [195.96.252.58]) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) with ESMTP id g2LGvvr07009 for ; Thu, 21 Mar 2002 10:57:57 -0600 (CST) Received: from biomath2 (biomath2.bio.bas.bg [195.96.247.143]) by iph.bio.bas.bg (8.11.6/8.9.3) with ESMTP id g2LGxBr19062 for ; Thu, 21 Mar 2002 18:59:11 +0200 From: "Evgenija D. Popova" Organization: Institute of Mathematics & Informatics, BAS To: reliable_computing [at] interval [dot] louisiana.edu Date: Thu, 21 Mar 2002 19:02:49 +0100 MIME-Version: 1.0 Content-type: text/plain; charset=US-ASCII Content-transfer-encoding: 7BIT Subject: Re: Interval Linear Systems of Equations Reply-to: epopova [at] iph [dot] bio.bas.bg Message-ID: <3C9A2E59.2127.1D30854@localhost> In-reply-to: <3C99CE80.8B9DA49B [at] univie [dot] ac.at> X-mailer: Pegasus Mail for Win32 (v3.12c) Sender: owner-reliable_computing [at] interval [dot] louisiana.edu Precedence: bulk The following paper: E. Popova, Quality of the Solution Sets of Parameter-Dependent Interval Linear Systems, to appear in ZAMM. http://www.math.bas.bg/~epopova/papers/ZAMMPopova.pdf contains some verifiable sufficient conditions for exact finite characterisation of the bounds of the solution set of a parametrised interval linear system. In: E.Popova, On the Solution of Parametrised Linear Systems. W. Kraemer, J. Wolff von Gudenberg (Eds.): Scientific Computing, Validated Numerics, Interval Methods. Kluwer Acad. Publishers, 2001, pp. 127-138. http://www.math.bas.bg/~epopova/papers/SCAN2000Popova.ps on a practical example, we have presented our experience in using Rump's algorithm and a way to sharpen the enclosing intervals whenever it is necessary. Evgenija D. Popova --------------------------------------------------------- Institute of Mathematics & Informatics Bulgarian Academy of Sciences Acad. G. Bonchev str., block 8 BG-1113 Sofia, Bulgaria Phone: (+359 2) 979-3704 Fax: (+359 2) 971-3649 http://www.math.bas.bg/~epopova --------------------------------------------------------- From owner-reliable_computing [at] interval [dot] louisiana.edu Fri Mar 22 06:38:54 2002 Received: (from daemon@localhost) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) id g2MCcr209767 for reliable_computing-outgoing; Fri, 22 Mar 2002 06:38:53 -0600 (CST) Received: from rztsun.rz.tu-harburg.de (rztsun.rz.tu-harburg.de [134.28.200.14]) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) with ESMTP id g2MCcmr09763 for ; Fri, 22 Mar 2002 06:38:49 -0600 (CST) Received: from gamma.ti3.tu-harburg.de (gamma.ti3.tu-harburg.de [134.28.20.71]) by rztsun.rz.tu-harburg.de (8.9.0/8.8.8) with ESMTP id NAA23543 for ; Fri, 22 Mar 2002 13:38:47 +0100 (MET) Received: (from jansson@localhost) by gamma.ti3.tu-harburg.de (AIX4.2/UCB 8.7/8.7) id NAA30206 for reliable_computing [at] interval [dot] louisiana.edu; Fri, 22 Mar 2002 13:43:36 +0100 (NFT) Date: Fri, 22 Mar 2002 13:43:36 +0100 (NFT) From: "PD Dr. Christian Jansson" Message-Id: <200203221243.NAA30206 [at] gamma [dot] ti3.tu-harburg.de> To: reliable_computing [at] interval [dot] louisiana.edu Subject: Interval Linear Systems Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-MD5: DOfwjgT6GWSa8f0nu6yjRA== Content-Transfer-Encoding: 8bit X-MIME-Autoconverted: from quoted-printable to 8bit by interval.louisiana.edu id g2MCcnr09764 Sender: owner-reliable_computing [at] interval [dot] louisiana.edu Precedence: bulk Dear Colleagues, Arnold Neumaier asked to summarize some of the results of the joint paper with Jiri Rohn: C. Jansson and J. Rohn. An Algorithm for Checking Regularity of Interval Matrices. SIAM J. Matrix Anal. Appl., 20(3):756-776, 1999. Checking regularity and singularity of interval matrices is a known NP-hard problem. This paper presents an algorithm for checking regularity/singularity of interval matrices. There are no assumptions on the diameter of the intervals, i.e. it is an exact method which checks in each case regularity or singularity. The problem of checking regularity naturally arises in solving linear interval systems, but it is also important in applications since some frequently used properties of interval matrices (such as positive defineteness, stability, or the P-matrix property) can be verified via checking regularity. Because of the NP-hardness the algorithm behaves in the worst case exponentially. However, in many cases the computational work is reasonable. In our numerical experiments we have considered linear interval systems of large diameter up to size n=50. A paper which is more related to the computation of exact bounds for linear interval systems is: C. Jansson. Calculation of Exact Bounds for the Solution Set of Linear Interval Systems. Linear Algebra and its Applications 251, 251:321-340, 1997. This paper presents some topological and graphtheoretical properties of the solution set of linear algebraic systems with interval coefficients. Based on these properties, an exact method is given which, in a finite number of steps, either calculates exact bounds for each component of the solution set, or finds a singular matrix within the interval coefficients. The calculation of exact bounds of the solution set is known to be NP-hard. This method needs p calls of a polynomial-time algorithm, where p is the number of nonempty intersections of the solution set with the orthants. Frequently, due to physical or economical requirements, many variables do not change the sign. In those cases p is small, and the method works efficiently. In both papers it is assumed that the input data vary independently between given upper and lower bounds, and data dependencies are not considered. Concerning dependent input data the following two articles may be of interest: C. Jansson. Interval Linear Systems with Symmetric Matrices, Skew-Symmetric Matrices, and Dependencies in the Right Hand Side. Computing 47, 265-274, 1991 This article considers an algorithm for computing bounds of the solution set corresponding to the linear interval system with the above mentioned special data dependencies. The bounds are not necessarily exact or optimal. However, The quality of the bounds can be reliable estimated by computing also so-called inner inclusions. The general case of affine-linear data dependencies is presented in S.M. Rump. Verification Methods for Dense and Sparse Systems of Equations. In J. Herzberger, editor, Topics in Validated Computations --- Studies in Computational Mathematics, pages 63-136, Elsevier, Amsterdam, 1994 The solution set of linear interval systems has a complicated structure. There are some very interesting papers of Alefeld, Kreinovich, and Mayer which consider these structures in the case of data dependencies. Best wishes, Christian From owner-reliable_computing [at] interval [dot] louisiana.edu Fri Mar 22 08:45:20 2002 Received: (from daemon@localhost) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) id g2MEjJl10077 for reliable_computing-outgoing; Fri, 22 Mar 2002 08:45:19 -0600 (CST) Received: from rztsun.rz.tu-harburg.de (rztsun.rz.tu-harburg.de [134.28.200.14]) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) with ESMTP id g2MEjEr10073 for ; Fri, 22 Mar 2002 08:45:15 -0600 (CST) Received: from gamma.ti3.tu-harburg.de (gamma.ti3.tu-harburg.de [134.28.20.71]) by rztsun.rz.tu-harburg.de (8.9.0/8.8.8) with ESMTP id PAA02586 for ; Fri, 22 Mar 2002 15:45:08 +0100 (MET) Received: (from jansson@localhost) by gamma.ti3.tu-harburg.de (AIX4.2/UCB 8.7/8.7) id PAA29256 for reliable_computing [at] interval [dot] louisiana.edu; Fri, 22 Mar 2002 15:49:58 +0100 (NFT) Date: Fri, 22 Mar 2002 15:49:58 +0100 (NFT) From: Jansson Message-Id: <200203221449.PAA29256 [at] gamma [dot] ti3.tu-harburg.de> To: reliable_computing [at] interval [dot] louisiana.edu Subject: Interval Linear Systems Mime-Version: 1.0 Content-Type: text/plain; charset=US-ASCII Content-Transfer-Encoding: 7bit Content-MD5: wdgrrO2y5av+Wm9dV6GuaA== Sender: owner-reliable_computing [at] interval [dot] louisiana.edu Precedence: bulk Dear Colleagues, one additional remark to my previous e-mail. In the case where the intervals are narrow and we have nonlinear interval data dependencies, these dependencies can be enclosed very well by using a mean value or centered form. This leads to a linear interval system with affine-linear dependencies and the results of the previously mentioned paper of Rump S.M. Rump. Verification Methods for Dense and Sparse Systems of Equations. In J. Herzberger, editor, Topics in Validated Computations --- Studies in Computational Mathematics, pages 63-136, Elsevier, Amsterdam, 1994 can be applied. Then the error bounds are rigorous. Best wishes, Christian From owner-reliable_computing [at] interval [dot] louisiana.edu Fri Mar 22 10:18:12 2002 Received: (from daemon@localhost) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) id g2MGIBf10310 for reliable_computing-outgoing; Fri, 22 Mar 2002 10:18:11 -0600 (CST) Received: from mailbox.univie.ac.at (mailbox.univie.ac.at [131.130.1.27]) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) with ESMTP id g2MGI1r10306 for ; Fri, 22 Mar 2002 10:18:02 -0600 (CST) Received: from univie.ac.at (hektor.mat.univie.ac.at [131.130.16.21]) by mailbox.univie.ac.at (8.12.2/8.12.2) with ESMTP id g2MGHs1r027160; Fri, 22 Mar 2002 17:17:55 +0100 Message-ID: <3C9B5932.6C2B937A [at] univie [dot] ac.at> Date: Fri, 22 Mar 2002 17:17:54 +0100 From: Arnold Neumaier Organization: University of Vienna X-Mailer: Mozilla 4.78 [en] (X11; U; Linux 2.4.9-21 i686) X-Accept-Language: en, de MIME-Version: 1.0 To: "PD Dr. Christian Jansson" CC: reliable_computing [at] interval [dot] louisiana.edu Subject: Re: Interval Linear Systems References: <200203221243.NAA30206 [at] gamma [dot] ti3.tu-harburg.de> Content-Type: text/plain; charset=UTF-7 Content-Transfer-Encoding: 7bit Sender: owner-reliable_computing [at] interval [dot] louisiana.edu Precedence: bulk Jansson wrote: > In the case where the intervals are narrow and we have nonlinear > interval data dependencies, these dependencies can be enclosed > very well by using a mean value or centered form. This leads to > a linear interval system with affine-linear dependencies and the > results of the previously mentioned paper of Rump > > S.M. Rump. Verification Methods for > Dense and Sparse Systems of Equations. > In J. Herzberger, editor, Topics in > Validated Computations --- Studies in > Computational Mathematics, pages 63-136, > Elsevier, Amsterdam, 1994 > > can be applied. Then the error bounds are rigorous. Moreover, if one uses centered forms with centers at all points determined by Pownuk's semirigorous approach, and slopes in place of derivatives, and intersects all intervals obtained in this way, then one gets not only rigorous error bounds but also bounds that are optimal or close to optimal in the case of narrow input. And from Theorem 2.3.3 of my book, or from interval evaluations at these points, one can also obtain inner enclosures, so that the overestimation can be assessed with high quality. I hope Andrzej Pownuk will write a nice paper along these lines, with interesting examples from finite element analysis. Arnold Neumaier From owner-reliable_computing [at] interval [dot] louisiana.edu Fri Mar 22 12:10:16 2002 Received: (from daemon@localhost) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) id g2MIAGj10586 for reliable_computing-outgoing; Fri, 22 Mar 2002 12:10:16 -0600 (CST) Received: from zeus.polsl.gliwice.pl (root [at] zeus [dot] polsl.gliwice.pl [157.158.1.3]) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) with ESMTP id g2MIAAr10581 for ; Fri, 22 Mar 2002 12:10:10 -0600 (CST) Received: from andrzej1 (a18.asystent.polsl.gliwice.pl [157.158.187.18]) by zeus.polsl.gliwice.pl (8.9.3 (PHNE_25183)/8.9.3) with SMTP id TAA03253; Fri, 22 Mar 2002 19:10:06 +0100 (MET) Message-ID: <011b01c1d1cc$d073ad20$0200a8c0@andrzej1> From: "Andrzej Pownuk" To: , "Arnold Neumaier" Cc: "Andrzej Pownuk" References: +ADw-200203221243.NAA30206+AEA-gamma.ti3.tu-harburg.de+AD4- +ADw-3C9B5932.6C2B937A+AEA-univie.ac.at+AD4- Subject: Re: Interval Linear Systems Date: Fri, 22 Mar 2002 19:10:12 +0100 MIME-Version: 1.0 Content-Type: text/plain; charset="UTF-7" Content-Transfer-Encoding: 7bit X-Priority: 3 X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook Express 6.00.2600.0000 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2600.0000 Sender: owner-reliable_computing [at] interval [dot] louisiana.edu Precedence: bulk Thank you very much for very useful information. I apologize once again for my error. I try to apply all these comments in my future works. I work for the Department of Civil Engineering. I am interested in modeling of structures with uncertain parameters. In my work I usually use finite element method (FEM) (boundary element method (BEM), finite difference method (FDM) etc.). to obtain the solution of partial differential equations. These method lead to the very large systems of linear equations (thousands or even million degree of freedom). To modeling of uncertain parameters we can apply interval numbers (or interval valued random variables). In this situations we have system of thousands equations with interval parameters in the following form: K(h)u+AD0-Q(h) where h belong to +AFs-h+AF0- where h is a vector of uncertain parameters. These are usually parameter dependent system of equations. (I don't know (direct) practical applications of system of linear interval equations with independent coefficients in computational mechanics) There are many methods of solution of the system of linear interval equations (with independent coefficients). Unfortunately due to overestimation problem they cannot be apply in practical applications. In may country we have a small team which work on these topics. (http://www.ippt.gov.pl/+AH4-zkulpa/quaphys/QAGroup.html) I know the work of Mr. Jansson, but I don+IBk-t know any example of application in computational mechanics which is connected with his method. I know the work of Mr Roump and Ms. Popova. In my country we tested this method. Unfortunately there are two problems: 1) The dependency of the parameters in my problems is more complicated than linear combination of interval parameters. In the simplest cases we have the following coefficient of the stiffness matrix: K(i,j)+AD0-2+ACo-p1+ACo-p2+ACo-p3+-8+ACo-p1+ACo-p3+-5+ACo-p1+ACo-p2+-...+-4+ACo-p3+ACo-p4 where K(i,j) is a coefficient of the matrix K, p1,p2,p3,p4 are interval parameters. We can apply linear combination of interval parameters only in very special cases. 2) I don't know the efficiency of Rump+IBk-s ( and Popova) method in high dimensional problems. (1000, 10 000 equations) In my country Ms Iwona Skalna have tested only examples with approximately 20 degree of freedom. Mr Muhanna in the paper Muhanna R.L., Mullen R.L., Uncertainty in Mechanics Problems - Interval - Based Approach. Journal of Engineering Mechanics, Vol.127, No.6, 2001, 557-556, have solved problems with about 20 degree of freedom and one interval parameter. What happened when we will have thousand equations? I think that this topic need more investigation. ( or I have to read something (????) ) I know that my method or other method which are based on sensitivity analysis gives only approximate solution but they has three advantages: (in my Ph.D. dissertation I investigated some other method.) 1) Can be applied to very high dimensional problems. 2) Unsung this method we can take into account very complicated cases of dependency of the interval parameters. 3) These methods can be applied for nonlinear problems (even dynamics). Additionally as far as I know most of practical method of computation gives only approximate results. (but I understood that this is not the topic of this mailing list.) The problem is how accurate results we can calculate. I met a lot of person which are very interested in practical (commercial) applications of the problems, which I described above. Thank you very much for all comments, Regards, Andrzej Pownuk +AD4- +AD4- Jansson wrote: +AD4- +AD4- +AD4- In the case where the intervals are narrow and we have nonlinear +AD4- +AD4- interval data dependencies, these dependencies can be enclosed +AD4- +AD4- very well by using a mean value or centered form. This leads to +AD4- +AD4- a linear interval system with affine-linear dependencies and the +AD4- +AD4- results of the previously mentioned paper of Rump +AD4- +AD4- +AD4- +AD4- S.M. Rump. Verification Methods for +AD4- +AD4- Dense and Sparse Systems of Equations. +AD4- +AD4- In J. Herzberger, editor, Topics in +AD4- +AD4- Validated Computations --- Studies in +AD4- +AD4- Computational Mathematics, pages 63-136, +AD4- +AD4- Elsevier, Amsterdam, 1994 +AD4- +AD4- +AD4- +AD4- can be applied. Then the error bounds are rigorous. +AD4- +AD4- Moreover, if one uses centered forms with centers at all points +AD4- determined by Pownuk's semirigorous approach, and slopes in +AD4- place of derivatives, and intersects all intervals obtained in +AD4- this way, then one gets not only rigorous error bounds but +AD4- also bounds that are optimal or close to optimal in the +AD4- case of narrow input. +AD4- +AD4- And from Theorem 2.3.3 of my book, or from interval evaluations +AD4- at these points, one can also obtain inner enclosures, so that +AD4- the overestimation can be assessed with high quality. +AD4- +AD4- I hope Andrzej Pownuk will write a nice paper along these lines, +AD4- with interesting examples from finite element analysis. +AD4- +AD4- +AD4- Arnold Neumaier +AD4- From owner-reliable_computing [at] interval [dot] louisiana.edu Sat Mar 23 03:00:42 2002 Received: (from daemon@localhost) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) id g2N90fe12123 for reliable_computing-outgoing; Sat, 23 Mar 2002 03:00:41 -0600 (CST) Received: from suntana.fh-konstanz.de (suntana.fh-konstanz.de [141.37.9.230]) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) with ESMTP id g2N90Tr12119 for ; Sat, 23 Mar 2002 03:00:29 -0600 (CST) Received: from fh-konstanz.de (dialup60.fh-konstanz.de [141.37.217.60]) by suntana.fh-konstanz.de (8.9.3+Sun/8.9.3) with ESMTP id JAA03259 for ; Sat, 23 Mar 2002 09:58:01 +0100 (MET) Message-ID: <3C9C470A.64CE4ED1@fh-konstanz.de> Date: Sat, 23 Mar 2002 10:12:42 +0100 From: Garloff [at] suntana [dot] fh-konstanz.de, =?iso-8859-1?Q?J=FCrgen?= Organization: Fachhochschule Konstanz X-Mailer: Mozilla 4.51 [en] (Win98; I) X-Accept-Language: en MIME-Version: 1.0 To: reliable_computing [at] interval [dot] louisiana.edu Subject: Taylor models - how to get valid comparisons Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Sender: owner-reliable_computing [at] interval [dot] louisiana.edu Precedence: bulk Dear Arnold, I am late but I want to comment on your remarks: > One correction: > You state that the Bernstein form gives exact results for sufficiently > small boxes. But this is true only if the extremal value is attained in > a box; it is not the case, e.g., for f(x)=x^3-6x in [0,2] since the > minimum > is irrational. The precise statement (for simplicity stated here in the 1D case) is that Bernstein expansion over a fixed interval, [a,b] say, gives the exact range on [a, a + eps] and [b - eps,b] for eps sufficiently small. This result can be found in the dissertation by Volker Stahl (Interval Methods for Bounding the Range of Polynomials and Solving Systems of Nonlinear Eqs., RISC, University of Linz, 1995) which contains also a lot of experimental comparisons in the univariate case of the Bernstein form with other forms, e.g., slope interpolation form and parabolic boundary value form. I'd also recommend that you mention the leading order of > the > cost of the transformation to Bernstein form, since this may explain the > limitations of your method in higher dimensions. > This transformation can be accomplished e.g. by the Horner scheme (in the multivariate case possibly in parallel)- alternatively by de Casteljau's algorithm. If subdivision is applied, this transformation has to be performed only at the beginning. Best wishes, Juergen From owner-reliable_computing [at] interval [dot] louisiana.edu Sat Mar 23 06:33:44 2002 Received: (from daemon@localhost) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) id g2NCXhT12795 for reliable_computing-outgoing; Sat, 23 Mar 2002 06:33:43 -0600 (CST) Received: from mail.comset.net (mail.comset.net [213.172.0.3]) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) with ESMTP id g2NCXbr12791 for ; Sat, 23 Mar 2002 06:33:38 -0600 (CST) Received: from 9-136.dialup.comset.net ([213.172.9.136] helo=e0gumi46) by mail.comset.net with smtp (Exim 3.33 #1) id 16okjF-0004hc-00 for reliable_computing [at] interval [dot] louisiana.edu; Sat, 23 Mar 2002 15:34:38 +0300 Message-ID: <00df01c1d266$6027c200$710bfea9 [at] wplus [dot] net> From: "Vyacheslav Nesterov" To: "RC mailing list" Subject: Email address of Reliable Computing will be changed tomorrow Date: Sat, 23 Mar 2002 15:28:24 +0300 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_00DC_01C1D27F.5F493F80" X-Priority: 3 X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook Express 5.00.2615.200 X-MimeOLE: Produced By Microsoft MimeOLE V5.00.2615.200 Sender: owner-reliable_computing [at] interval [dot] louisiana.edu Precedence: bulk This is a multi-part message in MIME format. ------=_NextPart_000_00DC_01C1D27F.5F493F80 Content-Type: text/plain; charset="koi8-r" Content-Transfer-Encoding: quoted-printable Dear Colleagues, This message is to inform you that my address and=20 official address of Reliable Computing journal changes tomorrow from slavanest [at] yahoo [dot] com to=20 slnest [at] comset [dot] net First address will work properly during at least one month and hopefully = much more, however please use the new address for all kind of communications with = me. Best regards, Slava Nesterov ------=_NextPart_000_00DC_01C1D27F.5F493F80 Content-Type: text/html; charset="koi8-r" Content-Transfer-Encoding: quoted-printable
Dear Colleagues,
 
This message is to inform you = that my address=20 and
official address of Reliable = Computing=20 journal changes tomorrow
from
 
slavanest [at] yahoo [dot] com
 
to
 
slnest [at] comset [dot] net
 
First address will work properly = during at=20 least one month and hopefully much more,
however please use the new = address for all=20 kind of communications with me.
 
Best regards,
Slava = Nesterov
------=_NextPart_000_00DC_01C1D27F.5F493F80-- From owner-reliable_computing [at] interval [dot] louisiana.edu Sat Mar 23 17:58:28 2002 Received: (from daemon@localhost) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) id g2NNwRG13913 for reliable_computing-outgoing; Sat, 23 Mar 2002 17:58:27 -0600 (CST) Received: from bologna.vision.caltech.edu (bologna.vision.caltech.edu [131.215.134.19]) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) with ESMTP id g2NNwMr13909 for ; Sat, 23 Mar 2002 17:58:23 -0600 (CST) Received: from peking.vision.caltech.edu (peking [131.215.134.18]) by bologna.vision.caltech.edu (8.9.3/8.8.7) with ESMTP id PAA12537 for ; Sat, 23 Mar 2002 15:58:22 -0800 Received: (from arrigo@localhost) by peking.vision.caltech.edu (8.9.3+Sun/8.9.1) id PAA20200; Sat, 23 Mar 2002 15:57:46 -0800 (PST) X-Authentication-Warning: peking.vision.caltech.edu: arrigo set sender to arrigo [at] vision [dot] caltech.edu using -f To: reliable_computing [at] interval [dot] louisiana.edu Subject: Re: Taylor models - how to get valid comparisons References: <3C9C470A.64CE4ED1@fh-konstanz.de> From: Arrigo Benedetti Date: 23 Mar 2002 15:57:46 -0800 In-Reply-To: <3C9C470A.64CE4ED1@fh-konstanz.de> Message-ID: Lines: 34 User-Agent: Gnus/5.0808 (Gnus v5.8.8) Emacs/20.4 MIME-Version: 1.0 Content-Type: text/plain; charset=iso-8859-1 Content-Transfer-Encoding: 8bit X-MIME-Autoconverted: from quoted-printable to 8bit by interval.louisiana.edu id g2NNwNr13910 Sender: owner-reliable_computing [at] interval [dot] louisiana.edu Precedence: bulk Garloff [at] suntana [dot] fh-konstanz.de, Jürgen writes: > Dear Arnold, > I am late but I want to comment on your remarks: > > > > One correction: > > You state that the Bernstein form gives exact results for sufficiently > > small boxes. But this is true only if the extremal value is attained in > > a box; it is not the case, e.g., for f(x)=x^3-6x in [0,2] since the > > minimum > > is irrational. > > The precise statement (for simplicity stated here in the 1D case) is > that Bernstein expansion over a fixed interval, [a,b] say, gives the > exact range on [a, a + eps] and [b - eps,b] for eps sufficiently small. > This result can be found in the dissertation by Volker Stahl (Interval > Methods for Bounding the Range of Polynomials and Solving Systems of > Nonlinear Eqs., RISC, University of Linz, 1995) which contains also a > lot of experimental comparisons in the univariate case of the Bernstein > form with other forms, e.g., slope interpolation form and parabolic > boundary value form. > Just in case anyone is interested, this thesis is available at: ftp://ftp.risc.uni-linz.ac.at/pub/private/vstahl/thesis.ps Best, -Arrigo -- Dr. Arrigo Benedetti e-mail: arrigo [at] vision [dot] caltech.edu Caltech, MS 136-93 phone: (626) 644-3757 Pasadena, CA 91125 fax: (626) 795-8649 From owner-reliable_computing [at] interval [dot] louisiana.edu Mon Mar 25 06:40:44 2002 Received: (from daemon@localhost) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) id g2PCei918093 for reliable_computing-outgoing; Mon, 25 Mar 2002 06:40:44 -0600 (CST) Received: from mailbox.univie.ac.at (mailbox.univie.ac.at [131.130.1.27]) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) with ESMTP id g2PCebr18089 for ; Mon, 25 Mar 2002 06:40:38 -0600 (CST) Received: from univie.ac.at (hektor.mat.univie.ac.at [131.130.16.21]) by mailbox.univie.ac.at (8.12.2/8.12.2) with ESMTP id g2PCeTsg062294; Mon, 25 Mar 2002 13:40:30 +0100 Message-ID: <3C9F1ABD.5D290ADE [at] univie [dot] ac.at> Date: Mon, 25 Mar 2002 13:40:29 +0100 From: Arnold Neumaier Organization: University of Vienna X-Mailer: Mozilla 4.78 [en] (X11; U; Linux 2.4.9-31 i686) X-Accept-Language: en, de MIME-Version: 1.0 To: Andrzej Pownuk CC: reliable_computing [at] interval [dot] louisiana.edu Subject: On rigor in reliable computing References: +ADw-200203221243.NAA30206+AEA-gamma.ti3.tu-harburg.de+AD4- +ADw-3C9B5932.6C2B937A+AEA-univie.ac.at+AD4- <011b01c1d1cc$d073ad20$0200a8c0@andrzej1> Content-Type: text/plain; charset=UTF-7 Content-Transfer-Encoding: 7bit Sender: owner-reliable_computing [at] interval [dot] louisiana.edu Precedence: bulk Andrzej Pownuk wrote: > Additionally as far as I know > most of practical method of computation gives only approximate results. > (but I understood that this is not the topic of this mailing list.) > The problem is how accurate results we can calculate. I think it is ok to discuss on the list *all* problems with intervals, and also propose approximation algorithms that are not rigorous, for problems that currently defy rigorous methods. Indeed, less rigorous methods may contain the germs for more rigorous ones, and problems thast can be solved now by less rigorous methods pose challenges for more rigorous attacks. But the degree of rigor used should be clearly visible (and mentioned in papers at prominent places - abstract, introduction, conclusion). There are four increasingly demanding stages of rigor: 1. With uncontrolled approximations. For problems involving functions, this is the realm of traditional numerical analysis, where asymptotic estimates are all that is available to justify a method. 2. Without uncontrolled approximations but with unverified assumptions. This is the case e.g. for methods that assume that the intervals are sufficiently narrow, without giving precise criteria for how narrow is enough. 3. Without uncontrolled approximations or unverified assumptions, but without error control. This would be rigorous in exact arithmetic, but is not in floating point arithmetic. An example is Gaussian elimination for exact data. For bounding interval linear systems, Rohn's sign accord algorithms belongs to this category, since so far no one has proposed a safe strategy for making the algorithm behave correctly in floating points. 4. Without uncontrolled approximations or unverified assumptions, and with rounding error control. These are the only algorithms that deserve the designation 'rigorous' or 'exact'. Progress consists in bringing new problems into stage 1, or bringing older or larger problems than before one stage higher. Arnold Neumaier From owner-reliable_computing [at] interval [dot] louisiana.edu Mon Mar 25 07:57:57 2002 Received: (from daemon@localhost) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) id g2PDvuA18299 for reliable_computing-outgoing; Mon, 25 Mar 2002 07:57:56 -0600 (CST) Received: from marnier.ucs.louisiana.edu (root [at] marnier [dot] ucs.louisiana.edu [130.70.132.233]) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) with ESMTP id g2PDvpr18295 for ; Mon, 25 Mar 2002 07:57:52 -0600 (CST) Received: from liberty (liberty.louisiana.edu [130.70.46.171]) by marnier.ucs.louisiana.edu (8.11.3/8.11.3/ull-ucs-mx-host_1.6) with SMTP id g2PDvT402815; Mon, 25 Mar 2002 07:57:33 -0600 (CST) Message-Id: <2.2.32.20020325140521.01536cf4 [at] 130 [dot] 70.132.231> X-Sender: rbk5287 [at] 130 [dot] 70.132.231 X-Mailer: Windows Eudora Pro Version 2.2 (32) Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" Date: Mon, 25 Mar 2002 08:05:21 -0600 To: Arnold Neumaier , Andrzej Pownuk From: "R. Baker Kearfott" Subject: Re: On rigor in reliable computing Cc: reliable_computing [at] interval [dot] louisiana.edu Sender: owner-reliable_computing [at] interval [dot] louisiana.edu Precedence: bulk Well said, Arnold! Best regards, Baker At 01:40 PM 3/25/02 +0100, Arnold Neumaier wrote: >Andrzej Pownuk wrote: > >> Additionally as far as I know >> most of practical method of computation gives only approximate results. >> (but I understood that this is not the topic of this mailing list.) >> The problem is how accurate results we can calculate. > >I think it is ok to discuss on the list *all* problems with >intervals, and also propose approximation algorithms that are not >rigorous, for problems that currently defy rigorous methods. >Indeed, less rigorous methods may contain the germs for more rigorous >ones, and problems thast can be solved now by less rigorous methods >pose challenges for more rigorous attacks. > >But the degree of rigor used should be clearly visible >(and mentioned in papers at prominent places - abstract, >introduction, conclusion). > >There are four increasingly demanding stages of rigor: > >1. With uncontrolled approximations. For problems involving functions, >this is the realm of traditional numerical analysis, where asymptotic >estimates are all that is available to justify a method. > >2. Without uncontrolled approximations but with unverified assumptions. >This is the case e.g. for methods that assume that the intervals are >sufficiently narrow, without giving precise criteria for how narrow is >enough. > >3. Without uncontrolled approximations or unverified assumptions, >but without error control. This would be rigorous in >exact arithmetic, but is not in floating point arithmetic. >An example is Gaussian elimination for exact data. >For bounding interval linear systems, Rohn's sign accord >algorithms belongs to this category, since so far no one has >proposed a safe strategy for making the algorithm behave >correctly in floating points. > >4. Without uncontrolled approximations or unverified assumptions, >and with rounding error control. These are the only algorithms >that deserve the designation 'rigorous' or 'exact'. > >Progress consists in bringing new problems into stage 1, or bringing >older or larger problems than before one stage higher. > >Arnold Neumaier > > --------------------------------------------------------------- R. Baker Kearfott, rbk [at] louisiana [dot] edu (337) 482-5346 (fax) (337) 482-5270 (work) (337) 981-9744 (home) URL: http://interval.louisiana.edu/kearfott.html Department of Mathematics, University of Louisiana at Lafayette Box 4-1010, Lafayette, LA 70504-1010, USA --------------------------------------------------------------- From owner-reliable_computing [at] interval [dot] louisiana.edu Mon Mar 25 08:06:40 2002 Received: (from daemon@localhost) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) id g2PE6eW18398 for reliable_computing-outgoing; Mon, 25 Mar 2002 08:06:40 -0600 (CST) Received: from alfik.ms.mff.cuni.cz (alfik.ms.mff.cuni.cz [195.113.19.71]) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) with ESMTP id g2PE6Wr18394 for ; Mon, 25 Mar 2002 08:06:32 -0600 (CST) Received: from helena.ms.mff.cuni.cz [195.113.19.218] by alfik.ms.mff.cuni.cz (8.9.3/2.1/20011017.1454) with ESMTP id PAA17036; Mon, 25 Mar 2002 15:05:48 +0100 (MET) Received: from HELENA/SpoolDir by helena.ms.mff.cuni.cz (Mercury 1.40); 25 Mar 102 15:11:25 +0100 Received: from SpoolDir by HELENA (Mercury 1.40); 25 Mar 102 15:11:13 +0100 Received: from j (195.113.18.85) by helena.ms.mff.cuni.cz (Mercury 1.40); 25 Mar 102 15:11:03 +0100 Reply-To: From: "Jiri Rohn" To: Subject: Interval linear eqs. with dependent coeffs. Date: Mon, 25 Mar 2002 15:05:27 +0100 Message-ID: <000901c1d406$1e84da80$551271c3 [at] rohn [dot] ms.mff.cuni.cz> MIME-Version: 1.0 Content-Type: text/plain; charset="iso-8859-2" X-Priority: 3 (Normal) X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook CWS, Build 9.0.2416 (9.0.2910.0) Importance: Normal X-MimeOLE: Produced By Microsoft MimeOLE V5.00.2314.1300 X-MIME-Autoconverted: from 8bit to quoted-printable by alfik.ms.mff.cuni.cz id PAA17036 Content-Transfer-Encoding: 8bit X-MIME-Autoconverted: from quoted-printable to 8bit by interval.louisiana.edu id g2PE6Zr18395 Sender: owner-reliable_computing [at] interval [dot] louisiana.edu Precedence: bulk Dear colleagues, As my name was mentioned several times during last week's discussion, I feel I should add a few words on the topic. But let me start with a quotation. In 1992 I met Prof. Babuska, a leading numerical analyst, and I told him briefly about interval linear systems and results achieved there. I remember that he grasped immediately the essence of the matter and responded with words (I quote from memory): "Unless you are able to handle dependent data, you will never gain interest of the engineers". Now, 10 years later, his remark is even more relevant and timely as the foregoing discussion suggests. Yet there is a simple idea, originated by E. Hansen in the 1960's, of exploiting the signs of partial derivatives of the solution components to take the dependences into account. On April 21, 1999 I wrote a personal letter to a friend (whose name I do not wish to disclose here) where I described my ideas. But they were never elaborated and remained unpublished. Today I would write it in a more simplified form, but I think even in the "old" form it can serve its purpose. Those of you who are interested can find this four-page letter (which describes a general approach to systems with linearly dependent coefficients) at www.ms.mff.cuni.cz/~rohn/letter/letter.ps (warning: there may be occasional downloading problems because of reconstruction works in our building) Finally, I would like to clarify that our joint paper with Christian Jansson, mentioned in the discussion, is devoted to checking regularity of interval matrices only and does not handle interval linear systems or dependent data. With best Easter wishes to all of you, Jiri Rohn --- Odchozí zpráva neobsahuje viry. Zkontrolováno antivirovým systémem AVG (http://www.grisoft.cz). Verze: 6.0.314 / Virová báze: 175 - datum vydání: 11.1.2002 From owner-reliable_computing [at] interval [dot] louisiana.edu Mon Mar 25 09:50:19 2002 Received: (from daemon@localhost) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) id g2PFoJg18708 for reliable_computing-outgoing; Mon, 25 Mar 2002 09:50:19 -0600 (CST) Received: from pheriche.sun.com (pheriche.sun.com [192.18.98.34]) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) with ESMTP id g2PFoDr18704 for ; Mon, 25 Mar 2002 09:50:14 -0600 (CST) Received: from engmail1.Eng.Sun.COM ([129.146.1.13]) by pheriche.sun.com (8.9.3+Sun/8.9.3) with ESMTP id IAA26859 for ; Mon, 25 Mar 2002 08:50:12 -0700 (MST) Received: from phys-mpkmaila (phys-mpkmaila.Eng.Sun.COM [129.146.18.131]) by engmail1.Eng.Sun.COM (8.9.3+Sun/8.9.3/ENSMAIL,v2.1p1) with ESMTP id HAA05624 for ; Mon, 25 Mar 2002 07:50:11 -0800 (PST) Received: from gww (gww.Eng.Sun.COM [129.146.78.116]) by mpkmail.eng.sun.com (iPlanet Messaging Server 5.2 (built Jan 13 2002)) with SMTP id <0GTJ00L5FDCVAH [at] mpkmail [dot] eng.sun.com> for reliable_computing [at] interval [dot] louisiana.edu; Mon, 25 Mar 2002 07:50:55 -0800 (PST) Date: Mon, 25 Mar 2002 07:50:11 -0800 (PST) From: William Walster Subject: Optimization Prize for Young Researchers To: reliable_computing [at] interval [dot] louisiana.edu Reply-to: William Walster Message-id: <0GTJ00L5GDCVAH [at] mpkmail [dot] eng.sun.com> MIME-version: 1.0 X-Mailer: dtmail 1.3.0 @(#)CDE Version 1.4.2 SunOS 5.8 sun4u sparc Content-type: TEXT/plain; charset=us-ascii Content-transfer-encoding: 7BIT Content-MD5: 4OH+NJCdfLKCzsT6YxDu4A== Sender: owner-reliable_computing [at] interval [dot] louisiana.edu Precedence: bulk ------------- Begin Forwarded Message ------------- From: Renato Monteiro Date: Tue, 19 Mar 2002 16:28:18 -0500 (EST) Subject: Optimization Prize for Young Researchers CALL FOR NOMINATIONS Optimization Prize for Young Researchers PRINCIPAL GUIDELINE. The Optimization Prize for Young Researchers, established in 1998 and administered by the Optimization Section (OS) within the Institute for Operations Research and Management Science (INFORMS), is awarded annually at the INFORMS Fall National Meeting to one (or more) young researchers for the most outstanding paper in optimization that is submitted to or published in a refereed professional journal. The prize serves as an esteemed recognition of promising colleagues who are at the beginning of their academic or industrial career. DESCRIPTION OF THE AWARD. The optimization award includes a cash amount of US$1,000 and a citation certificate. The award winners will be invited to give a fifteen minute presentation of the winning paper at the Optimization Section Business Meeting held during the INFORMS Fall National Meeting in the year the award is made. It is expected that the winners will be responsible for the travel expenses to present the paper at the INFORMS meeting. ELIGIBILITY. The authors and paper must satisfy the following three conditions to be eligible for the prize: (a) the paper must either be published in a refereed professional journal no more than three years before the closing date of nomination, or be submitted to and received by a refereed professional journal no more than three years before the closing date of nomination; (b) all authors must have been awarded their terminal degree within five years of the closing date of nomination; (c) the topic of the paper must belong to the field of optimization in its broadest sense. NOMINATION. A letter of nomination should be sent (preferably by email) on or before this year's closing date of July 1, 2002, to: Renato D.C. Monteiro monteiro [at] isye [dot] gatech.edu Georgia Tech School of ISyE Atlanta GA 30332-0205 PAST AWARDEES. The past winners of the Optimization Prize for Young Researchers are: Year Prize Winner 1999 Francois Oustry 2000 Kevin Wayne 2001 Kamal Jain ------------- End Forwarded Message ------------- From owner-reliable_computing [at] interval [dot] louisiana.edu Mon Mar 25 10:04:58 2002 Received: (from daemon@localhost) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) id g2PG4vH18824 for reliable_computing-outgoing; Mon, 25 Mar 2002 10:04:57 -0600 (CST) Received: from zeus.polsl.gliwice.pl (root [at] zeus [dot] polsl.gliwice.pl [157.158.1.3]) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) with ESMTP id g2PG4qr18820 for ; Mon, 25 Mar 2002 10:04:52 -0600 (CST) Received: from andrzej1 (a18.asystent.polsl.gliwice.pl [157.158.187.18]) by zeus.polsl.gliwice.pl (8.9.3 (PHNE_25183)/8.9.3) with SMTP id RAA09969; Mon, 25 Mar 2002 17:04:49 +0100 (MET) Message-ID: <005501c1d416$c975bcb0$0200a8c0@andrzej1> From: "Andrzej Pownuk" To: Cc: "Andrzej Pownuk" References: <000901c1d406$1e84da80$551271c3 [at] rohn [dot] ms.mff.cuni.cz> Subject: Re: Interval linear eqs. with dependent coeffs. Date: Mon, 25 Mar 2002 17:04:46 +0100 MIME-Version: 1.0 Content-Type: text/plain; charset="iso-8859-2" Content-Transfer-Encoding: 7bit X-Priority: 3 X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook Express 6.00.2600.0000 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2600.0000 Sender: owner-reliable_computing [at] interval [dot] louisiana.edu Precedence: bulk > Dear colleagues, > > As my name was mentioned several times during last week's > discussion, I feel I should add a few words on the topic. > But let me start with a quotation. In 1992 I met Prof. Babuska, > a leading numerical analyst, and I told him briefly about > interval linear systems and results achieved there. I remember > that he grasped immediately the essence of the matter and > responded with words (I quote from memory): "Unless you are > able to handle dependent data, you will never gain interest > of the engineers". Now, 10 years later, his remark is even > more relevant and timely as the foregoing discussion suggests. > > Yet there is a simple idea, originated by E. Hansen in > the 1960's, of exploiting the signs of partial derivatives > of the solution components to take the dependences into account. > On April 21, 1999 I wrote a personal letter to a friend > (whose name I do not wish to disclose here) where I described > my ideas. But they were never elaborated and remained unpublished. > Today I would write it in a more simplified form, but I think > even in the "old" form it can serve its purpose. Those of you > who are interested can find this four-page letter (which describes > a general approach to systems with linearly dependent > coefficients) at www.ms.mff.cuni.cz/~rohn/letter/letter.ps > (warning: there may be occasional downloading problems because > of reconstruction works in our building). In my paper (in the paragraph 5) I used very similar method. This method can be also applied for nonlinear equations. Jasinski M., Pownuk A., Modelling of heat transfer in biological tissue by interval FEM, Computer Assisted Mechanics and Engineering Sciences, vol. 7, No. 4, 2000, pp.551-558 http://157.158.187.18/publications/IntervalFEM.pdf When I worked on this paper I met the following problems: 1) The sign of the interval extension of the derivative usually is not constant. In this case I have to divide the intervals and repeat the whole procedure. If the intervals are sufficiently small then the sign of the derivative is constant and the method can be applied. Unfortunately this is very time-consuming process. 2) In some cases the interval extension of the matrix A (in my paper K) is singular. >From my experience this method is inefficient even in low dimensional problems. (but I don't know the efficiency of the presented method i.e www.ms.mff.cuni.cz/~rohn/letter/letter.ps) This method gives results with guaranteed accuracy. This is the main advantage of this algorithm. Regards, Andrzej Pownuk From owner-reliable_computing [at] interval [dot] louisiana.edu Mon Mar 25 21:23:25 2002 Received: (from daemon@localhost) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) id g2Q3NO420081 for reliable_computing-outgoing; Mon, 25 Mar 2002 21:23:24 -0600 (CST) Received: from espresso.cafe.net (espresso.cafe.net [204.244.119.1]) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) with ESMTP id g2Q3NJr20077 for ; Mon, 25 Mar 2002 21:23:20 -0600 (CST) Received: from [192.168.1.2] (CPE00045a0ac3bf.cpe.net.cable.rogers.com [24.156.35.65]) by espresso.cafe.net (8.9.3/8.9.3) with ESMTP id VAA13955 for ; Mon, 25 Mar 2002 21:31:24 -0800 Mime-Version: 1.0 X-Sender: tupper [at] www [dot] peda.com Message-Id: In-Reply-To: <3C9F1ABD.5D290ADE [at] univie [dot] ac.at> References: <3C9F1ABD.5D290ADE [at] univie [dot] ac.at> Date: Mon, 25 Mar 2002 22:22:44 -0500 To: reliable_computing [at] interval [dot] louisiana.edu From: Jeff Tupper / Pedagoguery Software Inc Subject: Re: On rigor in reliable computing Content-Type: text/plain; charset="us-ascii" ; format="flowed" Sender: owner-reliable_computing [at] interval [dot] louisiana.edu Precedence: bulk As usual, Prof. Neumaier gives much good advice. At 7:40 AM -0500 3/25/02, Arnold Neumaier wrote: >There are four increasingly demanding stages of rigor: >... >4. Without uncontrolled approximations or unverified assumptions, >and with rounding error control. These are the only algorithms >that deserve the designation 'rigorous' or 'exact'. Personally, I would use the term "exact" only when there is perfect agreement between a computed result and the true mathematical result. I would characterize algorithms in stage as 4 as being "rigorous" but not necessarily as always returning "exact" results. I too would also like authors to clearly state the assumptions they make. Many interval papers I read implicitly assume that computed quantities are well-defined mathematically and do not address division by zero, taking the square root of a negative quantity, etc. At 4:54 PM -0500 3/21/02, Arnold Neumaier wrote: >However, we should not fall below already established standards of quality. I have been going through some interval papers recently and I would like to commend reviewers for the care they have given to many papers. But I have been disappointed by the appearance of minor mistakes in some papers, so I would urge reviewers to remain ever-vigilant. Errors in published papers will tarnish the reputation of interval methods and may cause some to question if such methods are capable of mathematical rigor. Best wishes, Jeff -- From owner-reliable_computing [at] interval [dot] louisiana.edu Tue Mar 26 06:28:16 2002 Received: (from daemon@localhost) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) id g2QCSGm21296 for reliable_computing-outgoing; Tue, 26 Mar 2002 06:28:16 -0600 (CST) Received: from imf04bis.bellsouth.net (mail104.mail.bellsouth.net [205.152.58.44]) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) with ESMTP id g2QCSBr21292 for ; Tue, 26 Mar 2002 06:28:11 -0600 (CST) Received: from u8174 ([65.81.242.25]) by imf04bis.bellsouth.net (InterMail vM.5.01.04.05 201-253-122-122-105-20011231) with SMTP id <20020326122926.VZDH11676.imf04bis.bellsouth.net@u8174> for ; Tue, 26 Mar 2002 07:29:26 -0500 Message-Id: <2.2.32.20020326115331.009c84f0 [at] pop [dot] louisiana.edu> X-Sender: rbk5287 [at] pop [dot] louisiana.edu X-Mailer: Windows Eudora Pro Version 2.2 (32) Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" Date: Tue, 26 Mar 2002 05:53:31 -0600 To: reliable_computing [at] interval [dot] louisiana.edu From: "R. Baker Kearfott" Subject: Validated Computing 2002: Program now on the web Sender: owner-reliable_computing [at] interval [dot] louisiana.edu Precedence: bulk Dear colleagues: A detailed preliminary program for Validated Computing 2002 is now on the web at: http://www.cs.utep.edu/interval-comp/interval.02/program.html This program includes a schedule of all the sessions, with authors and titles that are linked to abstracts in *.ps and *.pdf format. There is also an author index, with links to sessions. Vladik Kreinovich did a quick and impressive job of creating this program in the SIAM format. Who says Spring Break is just for students to drink beer? :-) I remind you that complete information for the conference is available from the page http://interval.louisiana.edu/conferences/Validated_computing_2002/html_notice.html This includes links to on-line registration and hotel information, as well as a call for proceedings papers. If you have received our reviewer reports, if your paper has been accepted, and if you have not yet submitted a revised abstract, please submit it immediately, so we may include your abstract on the program and in the printed copies we supply to participants. Best regards, R. Baker Kearfott --------------------------------------------------------------- R. Baker Kearfott, rbk [at] louisiana [dot] edu (337) 482-5346 (fax) (337) 482-5270 (work) (337) 981-9744 (home) URL: http://interval.louisiana.edu/kearfott.html Department of Mathematics, University of Louisiana at Lafayette Box 4-1010, Lafayette, LA 70504-1010, USA --------------------------------------------------------------- From owner-reliable_computing [at] interval [dot] louisiana.edu Tue Mar 26 15:34:15 2002 Received: (from daemon@localhost) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) id g2QLYEV22350 for reliable_computing-outgoing; Tue, 26 Mar 2002 15:34:14 -0600 (CST) Received: from hemispheres.gtrep.gatech.edu (hemispheres.gtrep.gatech.edu [168.20.168.31]) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) with ESMTP id g2QLY9r22346 for ; Tue, 26 Mar 2002 15:34:10 -0600 (CST) Received: from rmuhannalptp (rmuhanna-lptp.gtrep.gatech.edu [168.20.168.90]) by hemispheres.gtrep.gatech.edu (8.11.6/8.11.0) with ESMTP id g2QLWFu10005; Tue, 26 Mar 2002 16:32:15 -0500 Message-ID: <001e01c1d50d$810ae2f0$5aa814a8@rmuhannalptp> From: "Rafi Muhanna" To: Cc: "Rafi Muhanna" Subject: Interval Linear Systems Date: Tue, 26 Mar 2002 16:30:50 -0500 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_001B_01C1D4E3.975131B0" X-Priority: 3 X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook Express 5.00.2919.6700 X-MimeOLE: Produced By Microsoft MimeOLE V5.00.2919.6700 Sender: owner-reliable_computing [at] interval [dot] louisiana.edu Precedence: bulk This is a multi-part message in MIME format. ------=_NextPart_000_001B_01C1D4E3.975131B0 Content-Type: text/plain; charset="Windows-1252" Content-Transfer-Encoding: quoted-printable Dear Colleagues,=20 Robert Mullen and I have been following your interesting and valuable = discussion for the last week, and we would like to highlight some = aspects of the short history of Interval Finite Element and the latest = developments in this fast growing area. To make this e-mail brief, you = can download the short review from the following link: www.gtrep.gatech.edu/~rmuhanna/IFEM_review.pdf We do not claim that our review is inclusive. It is, however, complete = to our knowledge and we would appreciate list members to send us = additional information that we have missed. We would also like to = include links to the text of each paper where allowed by copyright law. We would like to add the following comments to the discussion on = interval finite element methods. While intervals are narrow in many = practical applications, we believe that interval treatment in finite = elements requires the use of wide intervals. For example, the live = loads that a structure has to be designed for often vary between zero = and the maximum design load. Material strengths are often uncertain by = at least 10%. In problems associated with computational mechanics we = might face in the same problem narrow and wide intervals, i.e., modulus = of elasticity (narrow) and load (wide). We feel that restricting = solutions of finite element problems to narrow intervals has only = limited usefulness. More important, successful application of interval methods to finite = element problems requires that the issues of parameter dependency be = included in the analysis. Failure to consider dependency results in = interval solutions that are much too wide (see paper by Rao et al.)=20 While it is well known that the general problem of solving interval = system of equations is NP-hard, the system of equations associated with = finite element problem can have different additional properties = depending on the weak form used as a basis of the finite element = formulation. For example, for a class of structural problems, using a = mixed finite element formulation, one can construct a multiplicative = decomposition of the system of equations that diagonalizes the matrix of = engineering (interval) parameters. Solution of such a problem is not = NP-hard. As we are striving to use intervals to represent uncertainty in our = practical applications (analysis and design), consequently we would = expect to enhance the reliability of final products, and we believe that = cannot be done without rigorous verified methods. Some times, = approximate or less rigorous method (unless they are conservative) might = lead to narrower solutions in comparison with the exact ones. Such = cases can have a catastrophic impact in design problems. In addition, we would like to comment on the remarks of Andrzej Pownuk. = In his last e-mail he wrote: "Mr. Muhanna in the paper Muhanna R.L., = Mullen R.L., Uncertainty in Mechanics Problems - Interval - Based = Approach. Journal of Engineering Mechanics, Vol.127, No.6, 2001, = 557-556, have solved problems with about 20 degree of freedom and one = interval parameter. What happened when we will have thousand equations?" In this paper we present solutions that can be compared to = combinatorial solutions to demonstrate sharp interval bounds. We cannot = calculate combinatorial solutions to large problems and therefore did = not included larger models in the paper. In other conference papers, we = have solved (using similar algorithms) problems up to 1600 elements. It = should also be noted that we allow for a different stiffness for each = elements (not one parameter but one parameter per element).=20 www.gtrep.gatech.edu/~rmuhanna/IFEM_EBE_paper.pdf Finally, we are going to host a website dedicated to "Interval Finite = Element Methods", and this website will be linked to the main interval = website, i.e., http://www.cs.utep.edu/interval-comp/main.html. It will = include the latest development in the relevant development in the area = of IFEM and related publications. We welcome any contribution from = researchers in this field. Moreover, we will try to design the site = with Java-based interval computational capabilities. We found discussions on the interval list and private communications = with Arnold Neumaier very useful during our investigation of Interval = Finite Element and hope that this level of communications continues. Rafi Muhanna and Robert Mullen ________________________________________________ Rafi L. Muhanna =20 Department of Civil Engineering Regional Engineering Program Georgia Institute of Technology =20 6001 Chatham Center Dr., Suite 350 Savannah, GA 31405 USA =20 Email: rafi.muhanna [at] gtrep [dot] gatech.edu Phone: (912) 651-7547 Fax: (912) 651-7279 ------=_NextPart_000_001B_01C1D4E3.975131B0 Content-Type: text/html; charset="Windows-1252" Content-Transfer-Encoding: quoted-printable
Dear = Colleagues,=20

Robert=20 Mullen and I have been following your interesting and valuable = discussion for=20 the last week, and we would like to highlight some aspects of the short = history=20 of Interval Finite Element and the latest developments in this fast = growing=20 area.  To make this e-mail = brief,=20 you can download the short review from the following = link:

 www.gtrep.= gatech.edu/~rmuhanna/IFEM_review.pdf

We do not=20 claim that our review is inclusive. =20 It is, however, complete to our knowledge and we would appreciate = list=20 members to send us additional information that we have missed.  We would also like to include = links to=20 the text of each paper where allowed by copyright = law.

We would=20 like to add the following comments to the discussion on interval finite = element=20 methods. While intervals are narrow in many practical applications, we = believe=20 that interval treatment in finite elements requires the use of wide=20 intervals.  For example, = the live=20 loads that a structure has to be designed for often vary between zero = and the=20 maximum design load. Material strengths are often uncertain by at least=20 10%.  In problems = associated with=20 computational mechanics we might face in the same problem narrow and = wide=20 intervals, i.e., modulus of elasticity (narrow) and load (wide).  We feel that restricting = solutions of=20 finite element problems to narrow intervals has only limited=20 usefulness.

More=20 important, successful application of interval methods to finite element = problems=20 requires that the issues of parameter dependency be included in the=20 analysis.  Failure to = consider=20 dependency results in interval solutions that are much too wide (see = paper by=20 Rao et al.)

While it is=20 well known that the general problem of solving interval system of = equations is=20 NP-hard, the system of equations associated with finite element problem = can have=20 different additional properties depending on the weak form used as a = basis of=20 the finite element formulation.  = For=20 example, for a class of structural problems, using a mixed finite = element=20 formulation, one can construct a multiplicative decomposition of the = system of=20 equations that diagonalizes the matrix of engineering (interval)=20 parameters.  Solution of = such a=20 problem is not NP-hard.

 As we are striving to use = intervals=20 to represent uncertainty in our practical applications (analysis and = design),=20 consequently we would expect to enhance the reliability of final = products, and=20 we believe that cannot be done without rigorous verified methods.  Some times, approximate or = less rigorous=20 method (unless they are conservative) might lead to narrower solutions = in=20 comparison with the exact ones. =20 Such cases can have a catastrophic impact in design=20 problems.

 In addition, we would like = to=20 comment on the remarks of Andrzej Pownuk. =20 In his last e-mail he wrote: "Mr. Muhanna in the paper Muhanna = R.L.,=20 Mullen R.L., Uncertainty in Mechanics Problems - Interval - Based = Approach.=20 Journal of Engineering Mechanics, Vol.127, No.6, 2001, 557-556, have = solved=20 problems with about 20 degree of freedom and one interval parameter. = What=20 happened when we will have thousand equations?"

 In this paper we present = solutions=20 that can be compared to combinatorial solutions to demonstrate sharp = interval=20 bounds.  We cannot = calculate=20 combinatorial solutions to large problems and therefore did not included = larger=20 models in the paper.  In = other=20 conference papers, we have solved (using similar algorithms) problems up = to 1600=20 elements. It should also be noted that we allow for a different = stiffness for=20 each elements (not one parameter but one parameter per element).=20

 www.gtr= ep.gatech.edu/~rmuhanna/IFEM_EBE_paper.pdf

 Finally, we are going to = host a=20 website dedicated to "Interval Finite Element Methods", and this website = will be=20 linked to the main interval website, i.e., http://www.cs.ute= p.edu/interval-comp/main.html.  It will include the latest = development=20 in the relevant development in the area of IFEM and related=20 publications.  We welcome = any=20 contribution from researchers in this field.  Moreover, we will try to = design the site=20 with Java-based interval computational = capabilities.

We found discussions on the interval list and = private=20 communications with Arnold Neumaier very useful during our investigation = of=20 Interval Finite Element and hope that this level of communications=20 continues.

Rafi Muhanna and Robert Mullen

________________________________________________
 
Rafi L. Muhanna
 
Department = of Civil=20 Engineering
Regional Engineering Program
Georgia Institute of=20 Technology
 
6001 Chatham Center Dr., Suite 350
Savannah, = GA=20 31405
USA
 
Email:       = ; =20 rafi.muhanna [at] gtrep [dot] gatech.e= du
Phone:      =20 (912)=20 651-7547
Fax:         &nb= sp;=20 (912) 651-7279
------=_NextPart_000_001B_01C1D4E3.975131B0-- From owner-reliable_computing [at] interval [dot] louisiana.edu Tue Mar 26 16:21:36 2002 Received: (from daemon@localhost) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) id g2QMLZE22552 for reliable_computing-outgoing; Tue, 26 Mar 2002 16:21:35 -0600 (CST) Received: from clmboh1-smtp3.columbus.rr.com (clmboh1-smtp3.columbus.rr.com [65.24.0.112]) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) with ESMTP id g2QMLUr22548 for ; Tue, 26 Mar 2002 16:21:31 -0600 (CST) Received: from oemcomputer (dhcp065-024-174-102.columbus.rr.com [65.24.174.102]) by clmboh1-smtp3.columbus.rr.com (8.11.2/8.11.2) with SMTP id g2QMLRB05016; Tue, 26 Mar 2002 17:21:27 -0500 (EST) Message-ID: <001001c1d514$1d529c60$66ae1841 [at] columbus [dot] rr.com> From: "Ramon Moore" To: "Rafi Muhanna" Cc: "interval" References: <001e01c1d50d$810ae2f0$5aa814a8@rmuhannalptp> Subject: Re: Interval Linear Systems Date: Tue, 26 Mar 2002 17:18:09 -0500 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_000D_01C1D4EA.33887060" X-Priority: 3 X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook Express 5.50.4133.2400 X-MimeOLE: Produced By Microsoft MimeOLE V5.50.4133.2400 Sender: owner-reliable_computing [at] interval [dot] louisiana.edu Precedence: bulk This is a multi-part message in MIME format. ------=_NextPart_000_000D_01C1D4EA.33887060 Content-Type: text/plain; charset="Windows-1252" Content-Transfer-Encoding: quoted-printable Dear Rafi and Robert, Brilliant work ! It opens the doors for many important applications of = interval methods to problems in mechanics.=20 My sincere congratulations and best wishes. Ramon Moore ----- Original Message -----=20 From: Rafi Muhanna=20 To: reliable_computing [at] interval [dot] louisiana.edu=20 Cc: Rafi Muhanna=20 Sent: Tuesday, March 26, 2002 4:30 PM Subject: Interval Linear Systems Dear Colleagues,=20 Robert Mullen and I have been following your interesting and valuable = discussion for the last week, and we would like to highlight some = aspects of the short history of Interval Finite Element and the latest = developments in this fast growing area. To make this e-mail brief, you = can download the short review from the following link: www.gtrep.gatech.edu/~rmuhanna/IFEM_review.pdf We do not claim that our review is inclusive. It is, however, = complete to our knowledge and we would appreciate list members to send = us additional information that we have missed. We would also like to = include links to the text of each paper where allowed by copyright law. We would like to add the following comments to the discussion on = interval finite element methods. While intervals are narrow in many = practical applications, we believe that interval treatment in finite = elements requires the use of wide intervals. For example, the live = loads that a structure has to be designed for often vary between zero = and the maximum design load. Material strengths are often uncertain by = at least 10%. In problems associated with computational mechanics we = might face in the same problem narrow and wide intervals, i.e., modulus = of elasticity (narrow) and load (wide). We feel that restricting = solutions of finite element problems to narrow intervals has only = limited usefulness. More important, successful application of interval methods to finite = element problems requires that the issues of parameter dependency be = included in the analysis. Failure to consider dependency results in = interval solutions that are much too wide (see paper by Rao et al.)=20 While it is well known that the general problem of solving interval = system of equations is NP-hard, the system of equations associated with = finite element problem can have different additional properties = depending on the weak form used as a basis of the finite element = formulation. For example, for a class of structural problems, using a = mixed finite element formulation, one can construct a multiplicative = decomposition of the system of equations that diagonalizes the matrix of = engineering (interval) parameters. Solution of such a problem is not = NP-hard. As we are striving to use intervals to represent uncertainty in our = practical applications (analysis and design), consequently we would = expect to enhance the reliability of final products, and we believe that = cannot be done without rigorous verified methods. Some times, = approximate or less rigorous method (unless they are conservative) might = lead to narrower solutions in comparison with the exact ones. Such = cases can have a catastrophic impact in design problems. In addition, we would like to comment on the remarks of Andrzej = Pownuk. In his last e-mail he wrote: "Mr. Muhanna in the paper Muhanna = R.L., Mullen R.L., Uncertainty in Mechanics Problems - Interval - Based = Approach. Journal of Engineering Mechanics, Vol.127, No.6, 2001, = 557-556, have solved problems with about 20 degree of freedom and one = interval parameter. What happened when we will have thousand equations?" In this paper we present solutions that can be compared to = combinatorial solutions to demonstrate sharp interval bounds. We cannot = calculate combinatorial solutions to large problems and therefore did = not included larger models in the paper. In other conference papers, we = have solved (using similar algorithms) problems up to 1600 elements. It = should also be noted that we allow for a different stiffness for each = elements (not one parameter but one parameter per element).=20 www.gtrep.gatech.edu/~rmuhanna/IFEM_EBE_paper.pdf Finally, we are going to host a website dedicated to "Interval Finite = Element Methods", and this website will be linked to the main interval = website, i.e., http://www.cs.utep.edu/interval-comp/main.html. It will = include the latest development in the relevant development in the area = of IFEM and related publications. We welcome any contribution from = researchers in this field. Moreover, we will try to design the site = with Java-based interval computational capabilities. We found discussions on the interval list and private communications = with Arnold Neumaier very useful during our investigation of Interval = Finite Element and hope that this level of communications continues. Rafi Muhanna and Robert Mullen ________________________________________________ Rafi L. Muhanna =20 Department of Civil Engineering Regional Engineering Program Georgia Institute of Technology =20 6001 Chatham Center Dr., Suite 350 Savannah, GA 31405 USA =20 Email: rafi.muhanna [at] gtrep [dot] gatech.edu Phone: (912) 651-7547 Fax: (912) 651-7279 ------=_NextPart_000_000D_01C1D4EA.33887060 Content-Type: text/html; charset="Windows-1252" Content-Transfer-Encoding: quoted-printable
Dear Rafi and Robert,
 
Brilliant  work ! It opens the doors = for many=20 important applications of interval methods to problems in mechanics.=20
 
My sincere congratulations and best = wishes.
 
Ramon Moore
----- Original Message -----
From:=20 Rafi Muhanna
To: reliable_comput= ing [at] interval [dot] louisiana.edu=20
Cc: Rafi Muhanna
Sent: Tuesday, March 26, 2002 = 4:30=20 PM
Subject: Interval Linear = Systems

Dear = Colleagues,=20

Robert=20 Mullen and I have been following your interesting and valuable = discussion for=20 the last week, and we would like to highlight some aspects of the = short=20 history of Interval Finite Element and the latest developments in this = fast=20 growing area.  To make = this e-mail=20 brief, you can download the short review from the following=20 link:

 www.gtrep.= gatech.edu/~rmuhanna/IFEM_review.pdf

We do not=20 claim that our review is inclusive. =20 It is, however, complete to our knowledge and we would = appreciate list=20 members to send us additional information that we have missed.  We would also like to = include links to=20 the text of each paper where allowed by copyright = law.

We would=20 like to add the following comments to the discussion on interval = finite=20 element methods. While intervals are narrow in many practical = applications, we=20 believe that interval treatment in finite elements requires the use of = wide=20 intervals.  For example, = the live=20 loads that a structure has to be designed for often vary between zero = and the=20 maximum design load. Material strengths are often uncertain by at = least=20 10%.  In problems = associated with=20 computational mechanics we might face in the same problem narrow and = wide=20 intervals, i.e., modulus of elasticity (narrow) and load (wide).  We feel that restricting = solutions of=20 finite element problems to narrow intervals has only limited=20 usefulness.

More=20 important, successful application of interval methods to finite = element=20 problems requires that the issues of parameter dependency be included = in the=20 analysis.  Failure to = consider=20 dependency results in interval solutions that are much too wide (see = paper by=20 Rao et al.)

While it=20 is well known that the general problem of solving interval system of = equations=20 is NP-hard, the system of equations associated with finite element = problem can=20 have different additional properties depending on the weak form used = as a=20 basis of the finite element formulation. =20 For example, for a class of structural problems, using a mixed = finite=20 element formulation, one can construct a multiplicative decomposition = of the=20 system of equations that diagonalizes the matrix of engineering = (interval)=20 parameters.  Solution of = such a=20 problem is not NP-hard.

 As we are striving to = use=20 intervals to represent uncertainty in our practical applications = (analysis and=20 design), consequently we would expect to enhance the reliability of = final=20 products, and we believe that cannot be done without rigorous verified = methods.  Some times, = approximate=20 or less rigorous method (unless they are conservative) might lead to = narrower=20 solutions in comparison with the exact ones.  Such cases can have a = catastrophic=20 impact in design problems.

 In addition, we would = like to=20 comment on the remarks of Andrzej Pownuk.  In his last e-mail he wrote: = "Mr.=20 Muhanna in the paper Muhanna R.L., Mullen R.L., Uncertainty in = Mechanics=20 Problems - Interval - Based Approach. Journal of Engineering = Mechanics,=20 Vol.127, No.6, 2001, 557-556, have solved problems with about 20 = degree of=20 freedom and one interval parameter. What happened when we will have = thousand=20 equations?"

 In this paper we present = solutions=20 that can be compared to combinatorial solutions to demonstrate sharp = interval=20 bounds.  We cannot = calculate=20 combinatorial solutions to large problems and therefore did not = included=20 larger models in the paper.  = In=20 other conference papers, we have solved (using similar algorithms) = problems up=20 to 1600 elements. It should also be noted that we allow for a = different=20 stiffness for each elements (not one parameter but one parameter per = element).=20

 www.gtr= ep.gatech.edu/~rmuhanna/IFEM_EBE_paper.pdf

 Finally, we are going to = host a=20 website dedicated to "Interval Finite Element Methods", and this = website will=20 be linked to the main interval website, i.e., http://www.cs.ute= p.edu/interval-comp/main.html.  It will include the latest = development=20 in the relevant development in the area of IFEM and related=20 publications.  We = welcome any=20 contribution from researchers in this field.  Moreover, we will try to = design the=20 site with Java-based interval computational=20 capabilities.

We found discussions on the interval list and = private=20 communications with Arnold Neumaier very useful during our = investigation of=20 Interval Finite Element and hope that this level of communications=20 continues.

Rafi Muhanna and Robert Mullen

________________________________________________
 
Rafi L. = Muhanna
 
Department of Civil=20 Engineering
Regional Engineering Program
Georgia Institute of=20 Technology
 
6001 Chatham Center Dr., Suite = 350
Savannah, GA=20 = 31405
USA
 
Email:       = ; =20 rafi.muhanna [at] gtrep [dot] gatech.e= du
Phone:      =20 (912)=20 = 651-7547
Fax:         &nb= sp;=20 (912) 651-7279
------=_NextPart_000_000D_01C1D4EA.33887060-- From owner-reliable_computing [at] interval [dot] louisiana.edu Wed Mar 27 10:25:36 2002 Received: (from daemon@localhost) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) id g2RGPau00749 for reliable_computing-outgoing; Wed, 27 Mar 2002 10:25:36 -0600 (CST) Received: from mailbox.univie.ac.at (mailbox.univie.ac.at [131.130.1.27]) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) with ESMTP id g2RGPUW00745 for ; Wed, 27 Mar 2002 10:25:31 -0600 (CST) Received: from univie.ac.at (hektor.mat.univie.ac.at [131.130.16.21]) by mailbox.univie.ac.at (8.12.2/8.12.2) with ESMTP id g2RGPLfg170442; Wed, 27 Mar 2002 17:25:23 +0100 Message-ID: <3CA1F271.83A836D8 [at] univie [dot] ac.at> Date: Wed, 27 Mar 2002 17:25:21 +0100 From: Arnold Neumaier Organization: University of Vienna X-Mailer: Mozilla 4.78 [en] (X11; U; Linux 2.4.9-31 i686) X-Accept-Language: en, de MIME-Version: 1.0 To: rohn [at] helena [dot] ms.mff.cuni.cz CC: reliable_computing [at] interval [dot] louisiana.edu Subject: Re: Interval linear eqs. with dependent coeffs. References: <000901c1d406$1e84da80$551271c3 [at] rohn [dot] ms.mff.cuni.cz> Content-Type: text/plain; charset=UTF-7 Content-Transfer-Encoding: 7bit Sender: owner-reliable_computing [at] interval [dot] louisiana.edu Precedence: bulk Jiri Rohn wrote: > In 1992 I met Prof. Babuska, > a leading numerical analyst, and I told him briefly about > interval linear systems and results achieved there. I remember > that he grasped immediately the essence of the matter and > responded with words (I quote from memory): "Unless you are > able to handle dependent data, you will never gain interest > of the engineers". For narrow intervals, dependence is a real obstacle only in problems where the defining formulas have much cancellation (where Taylor expansion techniques might offer a remedy, but only for small problems). If not much cancellation occurs, as is the case in typical finite element calculations (whether linear or nonlinear), linearization techniqes are sufficient to cope with dependence. In particular, Corollary 4.2 of my paper A. Neumaier, Rigorous sensitivity analysis for parameter-dependent systems of equations, J. Math. Anal. Appl. 144 (1989), 16-25. contains enclosures for the range of all solutions of a nonlinear system F(x,t)=0 [dim F = dim x] near a given solution (x_0,t_0), where t is a parameter vector varying in a box that is not too large, together with computable bounds on the overestimation, which has second order in the parameter width and hence is quite accurate for narrow intervals. Thus it takes proper account of dependence and nonlinearities. Applied to finite element problems in the element by element formulation of R.L. Muhanna and R.L Mullen, Uncertainty in Mechanics Problems - Interval-Based Approach, J. Engin. Mech. 127 (2001), 557-566. see also the powerpoint presentation http://ecivwww.cwru.edu/civil/rlm/emae2002_files/v3_document.htm this should give good results as long as the interval coefficient matrix is strongly regular, which is guaranteed for narrow intervals but (depending on the formulation) may hold over wider domains. Results are less good than (rigorous) results based on monotony arguments if these apply, but they are much cheaper, and valid without any monotony assumption. Note that although the results are phrased in terms of a slope defined by an integral, they are true for any slope; in particular, the automatic differentiation-like slope arithmetic in Intlab. may be used to compute the required interval matrices A and B. Arnold Neumaier From owner-reliable_computing [at] interval [dot] louisiana.edu Wed Mar 27 14:03:55 2002 Received: (from daemon@localhost) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) id g2RK3sN01169 for reliable_computing-outgoing; Wed, 27 Mar 2002 14:03:54 -0600 (CST) Received: from cs.utep.edu (mail.cs.utep.edu [129.108.5.3]) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) with ESMTP id g2RK3oW01165 for ; Wed, 27 Mar 2002 14:03:50 -0600 (CST) Received: from aragorn (aragorn [129.108.5.35]) by cs.utep.edu (8.11.3/8.11.3) with SMTP id g2RK3fU08349; Wed, 27 Mar 2002 13:03:41 -0700 (MST) Message-Id: <200203272003.g2RK3fU08349 [at] cs [dot] utep.edu> Date: Wed, 27 Mar 2002 13:03:41 -0700 (MST) From: Vladik Kreinovich Reply-To: Vladik Kreinovich Subject: session of potential interest to interval researchers To: reliable_computing [at] interval [dot] louisiana.edu, interval [at] cs [dot] utep.edu MIME-Version: 1.0 Content-Type: TEXT/plain; charset=us-ascii Content-MD5: aAUbKyviMdzuUXeSwDogXg== X-Mailer: dtmail 1.3.0 @(#)CDE Version 1.4 SunOS 5.8 sun4u sparc Sender: owner-reliable_computing [at] interval [dot] louisiana.edu Precedence: bulk March 27, 2002 Dear Friends, T. Y. Lin, the Founding Chair of the Granular Computing Special Interest Group, is organizing a session on "Granular Computing and Data Mining" at the Third International Conference on Rough Sets and Current Trends in Computing RSCTC2002 (http://www.gv.psu.edu/Conferences/RSCTC2002/) at Penn State (near Philadelphia) on October 14-16, 2002. Since interval computations are an important part of granular computing, interval-related papers are welcome. If you are interested in participation, please submit your paper by the April 15, 2002 deadline. Please indicate, in your paper, that this paper is for a session organized by T.-Y. Lin. Detailed formatting instructions are give on the conference website. For more information about the planned session, feel free to contact me or Dr. T. Y. Lin at tylin [at] cs [dot] sjsu.edu. Yours Vladik From owner-reliable_computing [at] interval [dot] louisiana.edu Wed Mar 27 17:38:14 2002 Received: (from daemon@localhost) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) id g2RNcE501583 for reliable_computing-outgoing; Wed, 27 Mar 2002 17:38:14 -0600 (CST) Received: from cs.utep.edu (mail.cs.utep.edu [129.108.5.3]) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) with ESMTP id g2RNc9W01579 for ; Wed, 27 Mar 2002 17:38:10 -0600 (CST) Received: from aragorn (aragorn [129.108.5.35]) by cs.utep.edu (8.11.3/8.11.3) with SMTP id g2RNc4W10463; Wed, 27 Mar 2002 16:38:04 -0700 (MST) Message-Id: <200203272338.g2RNc4W10463 [at] cs [dot] utep.edu> Date: Wed, 27 Mar 2002 16:38:04 -0700 (MST) From: Vladik Kreinovich Reply-To: Vladik Kreinovich Subject: P.S. Granular Computing and Data Mining To: reliable_computing [at] interval [dot] louisiana.edu, interval [at] cs [dot] utep.edu MIME-Version: 1.0 Content-Type: TEXT/plain; charset=us-ascii Content-MD5: 513Wvbjc+6tfxx0+CVev7w== X-Mailer: dtmail 1.3.0 @(#)CDE Version 1.4 SunOS 5.8 sun4u sparc Sender: owner-reliable_computing [at] interval [dot] louisiana.edu Precedence: bulk Bill Walster correctly noticed that the term "granular computing" may not be very familiar to most interval folks (thanks a lot for noticing!). This is an informal term actively used by many AI researchers to indicate cases when instead of individual points, we consider "granules" of points: could be intervals, could be fuzzy sets, could be distributions, etc. BTW, a "rough set" is also a notion related to intervals. Crudely speaking, we do not know the exact set X, we know two sets instead X- and X+, one inside, the second containing the unknown set. In other words, it is a set analogue of an interval. From owner-reliable_computing [at] interval [dot] louisiana.edu Wed Mar 27 19:24:52 2002 Received: (from daemon@localhost) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) id g2S1OpR01842 for reliable_computing-outgoing; Wed, 27 Mar 2002 19:24:51 -0600 (CST) Received: from clmboh1-smtp3.columbus.rr.com (clmboh1-smtp3.columbus.rr.com [65.24.0.112]) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) with ESMTP id g2S1OkW01838 for ; Wed, 27 Mar 2002 19:24:47 -0600 (CST) Received: from oemcomputer (dhcp065-024-174-102.columbus.rr.com [65.24.174.102]) by clmboh1-smtp3.columbus.rr.com (8.11.2/8.11.2) with SMTP id g2S1OhB22771; Wed, 27 Mar 2002 20:24:43 -0500 (EST) Message-ID: <000601c1d5f6$cff81e20$66ae1841 [at] columbus [dot] rr.com> From: "Ramon Moore" To: "Vladik Kreinovich" Cc: "interval" References: <200203272338.g2RNc4W10463 [at] cs [dot] utep.edu> Subject: Re: P.S. Granular Computing and Data Mining Date: Wed, 27 Mar 2002 20:20:55 -0500 MIME-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: 7bit X-Priority: 3 X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook Express 5.50.4133.2400 X-MimeOLE: Produced By Microsoft MimeOLE V5.50.4133.2400 Sender: owner-reliable_computing [at] interval [dot] louisiana.edu Precedence: bulk Vladik wrote: > Bill Walster correctly noticed that the term "granular computing" may not be > very familiar to most interval folks (thanks a lot for noticing!). > > This is an informal term actively used by many AI researchers to indicate cases > when instead of individual points, we consider "granules" of points: could be > intervals, > could be fuzzy sets, could be distributions, etc. > > BTW, a "rough set" is also a notion related to intervals. Crudely speaking, we > do not know the exact set X, we know two sets instead X- and X+, one inside, > the second containing the unknown set. In other words, it is a set analogue of > an interval. Yes, years ago Karl Nickel pointed out to me, in connection with Birkhoff's lattice theory, that we can consider intervals in any partially ordered set. An interesting example is the set of subsets of a given set with the partial order relation of inclusion. Thus, we can consider intervals of subsets [S, T], with S and T subsets of some given set U, with [S, T] defined as the set of all V that are subsets of U for which S is contained in V and V is contained in T. In this way we can talk about intervals of sets. A partial order relation is just a relation that is transitive. Set inclusion is transitive. Ramon Moore From owner-reliable_computing [at] interval [dot] louisiana.edu Thu Mar 28 13:24:53 2002 Received: (from daemon@localhost) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) id g2SJOq704969 for reliable_computing-outgoing; Thu, 28 Mar 2002 13:24:52 -0600 (CST) Received: from imf07bis.bellsouth.net (mail307.mail.bellsouth.net [205.152.58.167]) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) with ESMTP id g2SJOlW04965 for ; Thu, 28 Mar 2002 13:24:47 -0600 (CST) Received: from u8174 ([216.78.183.171]) by imf07bis.bellsouth.net (InterMail vM.5.01.04.05 201-253-122-122-105-20011231) with SMTP id <20020328192602.ODPH1184.imf07bis.bellsouth.net@u8174>; Thu, 28 Mar 2002 14:26:02 -0500 Message-Id: <2.2.32.20020328192439.009d8b64 [at] pop [dot] louisiana.edu> X-Sender: rbk5287 [at] pop [dot] louisiana.edu X-Mailer: Windows Eudora Pro Version 2.2 (32) Mime-Version: 1.0 Content-Type: multipart/mixed; boundary="=====================_1017365079==_" Date: Thu, 28 Mar 2002 13:24:39 -0600 To: "P. S. V. Nataraj" , From: "R. Baker Kearfott" Subject: RE: Taylor models - how to get valid comparisons X-Attachments: C:\GlobSol\examples\Gritton\gritton2.RO1; Sender: owner-reliable_computing [at] interval [dot] louisiana.edu Precedence: bulk --=====================_1017365079==_ Content-Type: text/plain; charset="us-ascii" Prof. Nataraj, That is impressive work overall. It is a laudable contribution, and the experiments make their point well. I wish to point out minor discrepancies in the experiments on page 10 of the work you make available below, as follows: 1. You define Gritton's second problem as finding the minimum of a particular degree 18 univariate polynomial. However, the original problem that I got from Prof. Seader at the University of Utah was to find all of the ROOTS of that polynomial in [-12,8], not find the minimum in [1,2]. Readers should take this into account when comparing the results you report to the results in previous papers. 2. You mention that "Kearfott and Arazyan report in [the AD 2000 proceedings] that GlobSol had some difficulty in tackling this problem." That is only true in a certain sense, as follows: (a) We did not report results on the minimization problem, only on the root-finding problem. Using "find_roots" from an early version of GlobSol, all 18 roots were found without too much problem. I attach the output file (b) I just tried GlobSol with default configuration on the "minimization" version of Gritton's problem, and GlobSol found the minimum with NO problems on [-12,8], in particular, with only three bisections. (It reported .06 seconds on a 450 MHZ machine; however, due to OS inadequacies, this is total elapsed time, and I was simultaneously processing a stereo MP3 stream through the CPU and a software-based network connection.) On your interval [1,2], the problem was indeed somewhat harder, with the default configuration and not using any Taylor arithmetic: GlobSol made 516 bisections before obtaining the result, and took roughly 2 seconds of total elapsed time. However, this is much less than the failure after an hour you reported with the raw Moore-Skelboe algorithm, regardless of timing questions. (c) GlobSol uses alternate techniques, such as finding approximate solutions followed by epsilon inflation and interval Newton methods, to make things efficient. However, GlobSol is configurable. The purpose of our experiments in the AD 2000 proceedings was to test the usefulness of the Taylor extensions defined by Berz; we turned off some of GlobSol's capabilities to simplify the environment for our experiments. We did find that the Taylor extensions of Berz et al were very useful in certain circumstances. Of course, your experiments show that your Bernstein-Taylor extensions are even better, and MUCH better at that, in certain circumstances. In any case, I'm impressed that you have found a practical way of doing higher-order inclusions for multivariate functions. I had been thinking about that for some time, but had not devised a scheme :-) Best regards, Baker At 05:06 PM 3/8/02 +0530, P. S. V. Nataraj wrote: >Dear All, > >For a new inclusion function form having higher order convergence, along >with a convergence study of this form vs. that of Berz's Taylor model, pl. >see our paper (under review): >www.ee.iitb.ac.in/~nataraj/Super_TB_ps.zip >In the new form, Bernstein polynomial techniques were used to bound the >range of the Taylor polynomial part. We studied in this paper six problems, >of dimensions varying from 1 to 6. > >The new form (along with Taylor model) has also been used in a modified >Moore-Skelboe Global Optimization algorithm in our paper (to appear in Jl. >Global Optimization) >www.ee.iitb.ac.in/~nataraj/GOTB_Final_PS.zip > >In all our studies, we badly needed a good collection of challenge problems >to study the convergence behavior of the various forms. > >Can somebody suggest a good collection of such problems - preferably with a >structure more general than polynomial, and dimensions varying from 1 to (at >least) 6 ? > >Regards, >Nataraj > >.............................................. >Prof. P. S. V. Nataraj >Systems and Control Engineering Group >Department of Electrical Engineering >Indian Institute of Technology >Bombay 400 076 India >Ph: +91-022-5723757 Fax: +91-022-5726263 >Email: nataraj [at] ee [dot] iitb.ernet.in > >-----Original Message----- >From: owner-reliable_computing [at] interval [dot] louisiana.edu >[mailto:owner-reliable_computing [at] interval [dot] louisiana.edu]On Behalf Of >Arnold Neumaier >Sent: Thursday, March 07, 2002 1:12 AM >To: berz [at] msu [dot] edu; reliable_computing [at] interval [dot] louisiana.edu >Subject: Re: Taylor models - how to get valid comparisons > > >Arnold Neumaier wrote: > >> you'd have to compare it at least to techniques that use centered forms >> with slopes which are available for easy use in Rump's INTLAB >> (and perhaps to Bernstein polynomial techniques, and to affine >> arithmetic), > >Another relevant comparison would be to a modified Moore-Skelboe >algorithm, >applied twice to get the lower and upper bound. This is a little >inefficient since part of the wirk is duplicated, but it has the >advantage that it can be done in a simple way since the NEOS server >allows one to solve global optimization problems on a box online >via the WWW. See >http://www-neos.mcs.anl.gov/neos/solvers/GO:GLOBMIN/ >--- >Incoming mail is certified Virus Free. >Checked by AVG anti-virus system (http://www.grisoft.com). >Version: 6.0.325 / Virus Database: 182 - Release Date: 2/19/02 > >--- >Outgoing mail is certified Virus Free. >Checked by AVG anti-virus system (http://www.grisoft.com). >Version: 6.0.325 / Virus Database: 182 - Release Date: 2/19/02 > > > --=====================_1017365079==_ Content-Type: text/plain; charset="us-ascii" Content-Disposition: attachment; filename="gritton2.RO1" Output from FIND_ROOTS on 05/22/1999 at 19:31:25. Version for the system is: May 19, 1999 Codelist file name is: gritton2.CDL Box data file name is: gritton2.DT1 Heuristic parameter alpha: 0.5000D+00 Singular expansion factor: 0.3162D+05 Configuration settings: print_roots_delete: 0 print_inflate: 0 use_interval_newton: T slope_not_jacobi: T use_second_order: T refinement_in_verification: F VERY_GOOD_INITIAL_GUESS: F USE_USER_SUBSIT: F Initial box coordinates: [ -0.1200D+02, 0.8000D+01 ] EPS_DOMAIN 0.1000D-08 EPS_CHECK 0.5000D-01 Default point solver was used. (BOX_MODIFIED_POINT_NEWTON) Solver statistics Number of dense Jacobi matrix evaluations: 975 Number of evaluations of a row of the expanded Jacobi matrix: 36075 Number of evaluations of a row of the point expanded Jacobi matrix: 1253486 Number of bisections: 2 Number of dense interval residual evaluations: 12322 Total number of boxes pushed on the list: 1837 Number of orig. system inverse midpoint preconditioner rows: 246 Number of orig. system C-LP preconditioner rows: 2866 Total number of forward_substitutions: 25140 Number of Gauss--Seidel steps on the dense system: 3112 Number point dense residual evaluations: 34824 Number of point dense jacobi matrix evaluations: 33878 Total number of iterations of one of the point Newton methods: 19018 Total number of dense slope matrix evaluations: 4985 Total number second-order interval evaluations of the original function: 1873 Number of times a box was rejected in the interval Newton method due to an empty intersection: 947 Number of times Epsilon-inflation was attempted: 18 Number of times the approximate solver was called: 957 Number times a box was rejected due to 0 not in the second order interval function evaluation: 888 CPU time: 0.5110D+01 Time in approximate solver: 0.3490D+01 Time in point function: 0.0000D+00 Time in SUBSIT: 0.0000D+00 The following boxes have been verified to contain unique roots: Box no.: 1 Box coordinates: [ 0.2741D+01, 0.2741D+01 ] Level: 0 Box contains the following approximate root: 0.2741D+01 Small box in which the root must lie: [ 0.2741D+01, 0.2741D+01 ] Interval residuals over the small box: [ -0.8197D-06, 0.8209D-06 ] Unknown = F Contains_root = T Changed coordinates: T ------------------------------------------------- Box no.: 2 Box coordinates: [ 0.2286D+01, 0.2286D+01 ] Level: 0 Box contains the following approximate root: 0.2286D+01 Small box in which the root must lie: [ 0.2286D+01, 0.2286D+01 ] Interval residuals over the small box: [ -0.3773D-07, 0.3766D-07 ] Unknown = F Contains_root = T Changed coordinates: T ------------------------------------------------- Box no.: 3 Box coordinates: [ 0.6791D+01, 0.7124D+01 ] Level: 0 Box contains the following approximate root: 0.6958D+01 Small box in which the root must lie: [ 0.6958D+01, 0.6958D+01 ] Interval residuals over the small box: [ -0.4099D+02, 0.4099D+02 ] Unknown = F Contains_root = T Changed coordinates: T ------------------------------------------------- Box no.: 4 Box coordinates: [ 0.3541D+01, 0.3542D+01 ] Level: 0 Box contains the following approximate root: 0.3541D+01 Small box in which the root must lie: [ 0.3541D+01, 0.3541D+01 ] Interval residuals over the small box: [ -0.8782D-04, 0.8783D-04 ] Unknown = F Contains_root = T Changed coordinates: T ------------------------------------------------- Box no.: 5 Box coordinates: [ 0.4863D+01, 0.4897D+01 ] Level: 0 Box contains the following approximate root: 0.4880D+01 Small box in which the root must lie: [ 0.4880D+01, 0.4880D+01 ] Interval residuals over the small box: [ -0.3397D-01, 0.3397D-01 ] Unknown = F Contains_root = T Changed coordinates: T ------------------------------------------------- Box no.: 6 Box coordinates: [ -0.3779D+01, -0.3603D+01 ] Level: 0 Box contains the following approximate root: -0.3691D+01 Small box in which the root must lie: [ -0.3691D+01, -0.3691D+01 ] Interval residuals over the small box: [ -0.1015D+00, 0.1015D+00 ] Unknown = F Contains_root = T Changed coordinates: T ------------------------------------------------- Box no.: 7 Box coordinates: [ -0.6392D+01, -0.6093D+01 ] Level: 0 Box contains the following approximate root: -0.6242D+01 Small box in which the root must lie: [ -0.6242D+01, -0.6242D+01 ] Interval residuals over the small box: [ -0.3937D+02, 0.3937D+02 ] Unknown = F Contains_root = T Changed coordinates: T ------------------------------------------------- Box no.: 8 Box coordinates: [ 0.8425D+00, 0.8494D+00 ] Level: 0 Box contains the following approximate root: 0.8460D+00 Small box in which the root must lie: [ 0.8460D+00, 0.8460D+00 ] Interval residuals over the small box: [ -0.4398D-07, 0.4398D-07 ] Unknown = F Contains_root = T Changed coordinates: T ------------------------------------------------- Box no.: 9 Box coordinates: [ -0.5110D+01, -0.4871D+01 ] Level: 0 Box contains the following approximate root: -0.4990D+01 Small box in which the root must lie: [ -0.4990D+01, -0.4990D+01 ] Interval residuals over the small box: [ -0.1585D+01, 0.1585D+01 ] Unknown = F Contains_root = T Changed coordinates: T ------------------------------------------------- Box no.: 10 Box coordinates: [ -0.3014D+00, -0.2535D+00 ] Level: 0 Box contains the following approximate root: -0.2775D+00 Small box in which the root must lie: [ -0.2775D+00, -0.2775D+00 ] Interval residuals over the small box: [ -0.3322D-06, 0.3322D-06 ] Unknown = F Contains_root = T Changed coordinates: T ------------------------------------------------- Box no.: 11 Box coordinates: [ -0.1200D+02, -0.9076D+01 ] Level: 1 Box contains the following approximate root: -0.1109D+02 Small box in which the root must lie: [ -0.1109D+02, -0.1109D+02 ] Interval residuals over the small box: [ -0.9163D+06, 0.9163D+06 ] Unknown = F Contains_root = T Changed coordinates: T ------------------------------------------------- Box no.: 12 Box coordinates: [ -0.3563D+00, -0.3084D+00 ] Level: 1 Box contains the following approximate root: -0.3324D+00 Small box in which the root must lie: [ -0.3324D+00, -0.3324D+00 ] Interval residuals over the small box: [ -0.4004D-06, 0.4004D-06 ] Unknown = F Contains_root = T Changed coordinates: T ------------------------------------------------- Box no.: 13 Box coordinates: [ -0.2296D+01, -0.1589D+01 ] Level: 0 Box contains the following approximate root: -0.1943D+01 Small box in which the root must lie: [ -0.1943D+01, -0.1943D+01 ] Interval residuals over the small box: [ -0.1327D-02, 0.1327D-02 ] Unknown = F Contains_root = T Changed coordinates: T ------------------------------------------------- The following boxes have been not been verified to contain roots, but are small, and may contain roots: Box no.: 1 Box coordinates: [ 0.1966D+01, 0.1967D+01 ] Level: 0 Box contains the following approximate root: 0.1967D+01 Interval residuals over the box: [ -0.5488D+03, 0.5491D+03 ] Unknown = T Contains_root = F Changed coordinates: T ------------------------------------------------- Box no.: 2 Box coordinates: [ 0.1752D+01, 0.1752D+01 ] Level: 0 Box contains the following approximate root: 0.1752D+01 Interval residuals over the box: [ -0.2624D+03, 0.2626D+03 ] Unknown = T Contains_root = F Changed coordinates: T ------------------------------------------------- Box no.: 3 Box coordinates: [ 0.1593D+01, 0.1594D+01 ] Level: 0 Box contains the following approximate root: 0.1594D+01 Interval residuals over the box: [ -0.1507D+03, 0.1508D+03 ] Unknown = T Contains_root = F Changed coordinates: T ------------------------------------------------- Box no.: 4 Box coordinates: [ 0.1479D+01, 0.1480D+01 ] Level: 0 Box contains the following approximate root: 0.1480D+01 Interval residuals over the box: [ -0.1003D+03, 0.1004D+03 ] Unknown = T Contains_root = F Changed coordinates: T ------------------------------------------------- Box no.: 5 Box coordinates: [ 0.1381D+01, 0.1382D+01 ] Level: 0 Box contains the following approximate root: 0.1381D+01 Interval residuals over the box: [ -0.7003D+02, 0.7007D+02 ] Unknown = T Contains_root = F Changed coordinates: T ------------------------------------------------- --=====================_1017365079==_ Content-Type: text/plain; charset="us-ascii" --------------------------------------------------------------- R. Baker Kearfott, rbk [at] louisiana [dot] edu (337) 482-5346 (fax) (337) 482-5270 (work) (337) 981-9744 (home) URL: http://interval.louisiana.edu/kearfott.html Department of Mathematics, University of Louisiana at Lafayette Box 4-1010, Lafayette, LA 70504-1010, USA --------------------------------------------------------------- --=====================_1017365079==_-- From owner-reliable_computing [at] interval [dot] louisiana.edu Thu Mar 28 16:15:19 2002 Received: (from daemon@localhost) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) id g2SMFJT05289 for reliable_computing-outgoing; Thu, 28 Mar 2002 16:15:19 -0600 (CST) Received: from mtiwmhc23.worldnet.att.net (mtiwmhc23.worldnet.att.net [204.127.131.48]) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) with ESMTP id g2SMFEW05285 for ; Thu, 28 Mar 2002 16:15:14 -0600 (CST) Received: from nd.edu ([12.84.17.137]) by mtiwmhc23.worldnet.att.net (InterMail vM.4.01.03.27 201-229-121-127-20010626) with ESMTP id <20020328221506.CYR8815.mtiwmhc23.worldnet.att.net [at] nd [dot] edu>; Thu, 28 Mar 2002 22:15:06 +0000 Message-ID: <3CA395E7.9A2BB97B [at] nd [dot] edu> Date: Thu, 28 Mar 2002 17:15:03 -0500 From: Mark Stadtherr Reply-To: markst [at] nd [dot] edu Organization: University of Notre Dame X-Mailer: Mozilla 4.75 [en] (Win95; U) X-Accept-Language: en MIME-Version: 1.0 To: reliable_computing [at] interval [dot] louisiana.edu CC: "R. Baker Kearfott" , "P. S. V. Nataraj" Subject: Re: Taylor models - how to get valid comparisons References: <2.2.32.20020328192439.009d8b64 [at] pop [dot] louisiana.edu> Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Sender: owner-reliable_computing [at] interval [dot] louisiana.edu Precedence: bulk An aside to the discussion of "Gritton's second problem"-- While it may be interesting from a computational standpoint to find (rigorously) all roots in [-12,8], or the minimum in [1,2], it should be noted that the variable in this problem is a mole fraction, which by definition must be in [0,1]. Thus, from an application (chemical engineering) standpoint, it is pointless (and a waste of computational effort) to consider any interval other than [0,1]. Incidently, there is (exactly) one root in [0,1]. Mark -- Mark A. Stadtherr Professor Department of Chemical Engineering University of Notre Dame Notre Dame, IN 46556 USA Telephone: (574) 631-9318 Fax: (574) 631-8366 E-mail: markst [at] nd [dot] edu WWW: http://www.nd.edu/~markst ======================================================================== "R. Baker Kearfott" wrote: > > Prof. Nataraj, > > That is impressive work overall. It is a laudable contribution, and > the experiments make their point well. > > I wish to point out minor discrepancies in the experiments on page 10 of > the work you make available below, as follows: > > 1. You define Gritton's second problem as finding the minimum of > a particular degree 18 univariate polynomial. However, the original > problem that I got from Prof. Seader at the University of Utah > was to find all of the ROOTS of that polynomial in [-12,8], not find the > minimum in [1,2]. Readers should take this into account when comparing > the results you report to the results in previous papers. > > 2. You mention that "Kearfott and Arazyan report in [the AD 2000 > proceedings] that GlobSol had some difficulty in tackling this > problem." That is only true in a certain sense, as follows: > > (a) We did not report results on the minimization problem, only > on the root-finding problem. Using "find_roots" from > an early version of GlobSol, all 18 roots were found without > too much problem. I attach the output file > > (b) I just tried GlobSol with default configuration on the > "minimization" version of Gritton's problem, and GlobSol > found the minimum with NO problems on [-12,8], in particular, > with only three bisections. > (It reported .06 seconds on a 450 MHZ machine; however, > due to OS inadequacies, this is total elapsed time, and > I was simultaneously processing a stereo MP3 stream > through the CPU and a software-based network connection.) > On your interval [1,2], the problem was indeed somewhat > harder, with the default configuration and not using > any Taylor arithmetic: GlobSol made 516 bisections before > obtaining the result, and took roughly 2 seconds of total > elapsed time. However, this is much less than the failure > after an hour you reported with the raw Moore-Skelboe algorithm, > regardless of timing questions. > > (c) GlobSol uses alternate techniques, such as finding approximate > solutions followed by epsilon inflation and interval Newton > methods, to make things efficient. > However, GlobSol is configurable. The purpose of our experiments > in the AD 2000 proceedings was to test the usefulness of the > Taylor extensions defined by Berz; we turned off some of GlobSol's > capabilities to simplify the environment for our experiments. > We did find that the Taylor extensions of Berz et al were very > useful in certain circumstances. Of course, your experiments > show that your Bernstein-Taylor extensions are even better, > and MUCH better at that, in certain circumstances. > > In any case, I'm impressed that you have found a practical way of doing > higher-order inclusions for multivariate functions. I had been thinking > about that for some time, but had not devised a scheme :-) > > Best regards, > > Baker > > At 05:06 PM 3/8/02 +0530, P. S. V. Nataraj wrote: > >Dear All, > > > >For a new inclusion function form having higher order convergence, along > >with a convergence study of this form vs. that of Berz's Taylor model, pl. > >see our paper (under review): > >www.ee.iitb.ac.in/~nataraj/Super_TB_ps.zip > >In the new form, Bernstein polynomial techniques were used to bound the > >range of the Taylor polynomial part. We studied in this paper six problems, > >of dimensions varying from 1 to 6. > > > >The new form (along with Taylor model) has also been used in a modified > >Moore-Skelboe Global Optimization algorithm in our paper (to appear in Jl. > >Global Optimization) > >www.ee.iitb.ac.in/~nataraj/GOTB_Final_PS.zip > > > >In all our studies, we badly needed a good collection of challenge problems > >to study the convergence behavior of the various forms. > > > >Can somebody suggest a good collection of such problems - preferably with a > >structure more general than polynomial, and dimensions varying from 1 to (at > >least) 6 ? > > > >Regards, > >Nataraj > > > >.............................................. > >Prof. P. S. V. Nataraj > >Systems and Control Engineering Group > >Department of Electrical Engineering > >Indian Institute of Technology > >Bombay 400 076 India > >Ph: +91-022-5723757 Fax: +91-022-5726263 > >Email: nataraj [at] ee [dot] iitb.ernet.in > > > >-----Original Message----- > >From: owner-reliable_computing [at] interval [dot] louisiana.edu > >[mailto:owner-reliable_computing [at] interval [dot] louisiana.edu]On Behalf Of > >Arnold Neumaier > >Sent: Thursday, March 07, 2002 1:12 AM > >To: berz [at] msu [dot] edu; reliable_computing [at] interval [dot] louisiana.edu > >Subject: Re: Taylor models - how to get valid comparisons > > > > > >Arnold Neumaier wrote: > > > >> you'd have to compare it at least to techniques that use centered forms > >> with slopes which are available for easy use in Rump's INTLAB > >> (and perhaps to Bernstein polynomial techniques, and to affine > >> arithmetic), > > > >Another relevant comparison would be to a modified Moore-Skelboe > >algorithm, > >applied twice to get the lower and upper bound. This is a little > >inefficient since part of the wirk is duplicated, but it has the > >advantage that it can be done in a simple way since the NEOS server > >allows one to solve global optimization problems on a box online > >via the WWW. See > >http://www-neos.mcs.anl.gov/neos/solvers/GO:GLOBMIN/ > >--- > >Incoming mail is certified Virus Free. > >Checked by AVG anti-virus system (http://www.grisoft.com). > >Version: 6.0.325 / Virus Database: 182 - Release Date: 2/19/02 > > > >--- > >Outgoing mail is certified Virus Free. > >Checked by AVG anti-virus system (http://www.grisoft.com). > >Version: 6.0.325 / Virus Database: 182 - Release Date: 2/19/02 > > > > > > > > ------------------------------------------------------------------------ > > gritton2.RO1Name: gritton2.RO1 > Type: Plain Text (text/plain) > > ------------------------------------------------------------------------ > > --------------------------------------------------------------- > R. Baker Kearfott, rbk [at] louisiana [dot] edu (337) 482-5346 (fax) > (337) 482-5270 (work) (337) 981-9744 (home) > URL: http://interval.louisiana.edu/kearfott.html > Department of Mathematics, University of Louisiana at Lafayette > Box 4-1010, Lafayette, LA 70504-1010, USA > --------------------------------------------------------------- -- Mark A. Stadtherr Professor Department of Chemical Engineering University of Notre Dame Notre Dame, IN 46556 USA Telephone: (219) 631-9318 Fax: (219) 631-8366 E-mail: markst [at] nd [dot] edu WWW: http://www.nd.edu/~markst From owner-reliable_computing [at] interval [dot] louisiana.edu Thu Mar 28 17:06:07 2002 Received: (from daemon@localhost) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) id g2SN67R05466 for reliable_computing-outgoing; Thu, 28 Mar 2002 17:06:07 -0600 (CST) Received: from mail8.wi.rr.com (mkc-162-176.kc.rr.com [24.94.162.176]) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) with ESMTP id g2SN62W05462 for ; Thu, 28 Mar 2002 17:06:03 -0600 (CST) Received: from Marquette.edu ([65.29.171.35]) by mail8.wi.rr.com with Microsoft SMTPSVC(5.5.1877.537.53); Thu, 28 Mar 2002 17:06:43 -0600 Message-ID: <3CA3A1D1.3020803 [at] Marquette [dot] edu> Date: Thu, 28 Mar 2002 17:05:53 -0600 From: "Dr. George Corliss" Reply-To: George.Corliss [at] Marquette [dot] edu Organization: Marquette University, EECE User-Agent: Mozilla/5.0 (Windows; U; Windows NT 5.1; en-US; rv:0.9.4) Gecko/20011128 Netscape6/6.2.1 X-Accept-Language: en-us MIME-Version: 1.0 To: markst [at] nd [dot] edu CC: reliable_computing [at] interval [dot] louisiana.edu, "R. Baker Kearfott" , "P. S. V. Nataraj" Subject: Re: Taylor models - how to get valid comparisons References: <2.2.32.20020328192439.009d8b64 [at] pop [dot] louisiana.edu> <3CA395E7.9A2BB97B [at] nd [dot] edu> Content-Type: text/plain; charset=us-ascii; format=flowed Content-Transfer-Encoding: 7bit Sender: owner-reliable_computing [at] interval [dot] louisiana.edu Precedence: bulk Excellent. Once again illustrating the Design Pattern: A little applications insight is faster than one HELL of a big computer. (Sun's championing of the cause not withstanding.) or a room full of smart numerical analysts. Thanks, Mark. Dr. George F. Corliss Electrical and Computer Engineering Haggerty Engineering 296 Marquette University P.O. Box 1881 Milwaukee, WI 53201-1881 USA George.Corliss [at] Marquette [dot] edu Office: 414-288-6599; Dept: 288-6820; Fax: 288-5579 Mark Stadtherr wrote: > An aside to the discussion of "Gritton's second problem"-- > While it may be interesting from a computational standpoint to > find (rigorously) all roots in [-12,8], or the minimum > in [1,2], it should be noted that the variable in this problem > is a mole fraction, which by definition must be in [0,1]. > Thus, from an application (chemical engineering) standpoint, > it is pointless (and a waste of computational effort) to > consider any interval other than [0,1]. Incidently, there > is (exactly) one root in [0,1]. > > Mark > > From owner-reliable_computing [at] interval [dot] louisiana.edu Thu Mar 28 17:13:19 2002 Received: (from daemon@localhost) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) id g2SNDJf05561 for reliable_computing-outgoing; Thu, 28 Mar 2002 17:13:19 -0600 (CST) Received: from clmboh1-smtp3.columbus.rr.com (clmboh1-smtp3.columbus.rr.com [65.24.0.112]) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) with ESMTP id g2SNDEW05557 for ; Thu, 28 Mar 2002 17:13:14 -0600 (CST) Received: from oemcomputer (dhcp065-024-174-102.columbus.rr.com [65.24.174.102]) by clmboh1-smtp3.columbus.rr.com (8.11.2/8.11.2) with SMTP id g2SNDBB08066; Thu, 28 Mar 2002 18:13:11 -0500 (EST) Message-ID: <009e01c1d6ad$9b410360$66ae1841 [at] columbus [dot] rr.com> From: "Ramon Moore" To: Cc: "interval" References: <2.2.32.20020328192439.009d8b64 [at] pop [dot] louisiana.edu> <3CA395E7.9A2BB97B [at] nd [dot] edu> <3CA3A1D1.3020803 [at] Marquette [dot] edu> Subject: Re: Taylor models - how to get valid comparisons Date: Thu, 28 Mar 2002 18:09:25 -0500 MIME-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: 7bit X-Priority: 3 X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook Express 5.50.4133.2400 X-MimeOLE: Produced By Microsoft MimeOLE V5.50.4133.2400 Sender: owner-reliable_computing [at] interval [dot] louisiana.edu Precedence: bulk Absolutely true, George, and well put. Ray (Ramon Moore) ----- Original Message ----- From: "Dr. George Corliss" To: Cc: ; "R. Baker Kearfott" ; "P. S. V. Nataraj" Sent: Thursday, March 28, 2002 6:05 PM Subject: Re: Taylor models - how to get valid comparisons > Excellent. Once again illustrating the Design Pattern: > > A little applications insight is faster > than one HELL of a big computer. > > (Sun's championing of the cause not withstanding.) > > or a room full of smart numerical analysts. > > > Thanks, Mark. > > Dr. George F. Corliss > Electrical and Computer Engineering > Haggerty Engineering 296 > Marquette University > P.O. Box 1881 > Milwaukee, WI 53201-1881 USA > George.Corliss [at] Marquette [dot] edu > Office: 414-288-6599; Dept: 288-6820; Fax: 288-5579 > > Mark Stadtherr wrote: > > > An aside to the discussion of "Gritton's second problem"-- > > While it may be interesting from a computational standpoint to > > find (rigorously) all roots in [-12,8], or the minimum > > in [1,2], it should be noted that the variable in this problem > > is a mole fraction, which by definition must be in [0,1]. > > Thus, from an application (chemical engineering) standpoint, > > it is pointless (and a waste of computational effort) to > > consider any interval other than [0,1]. Incidently, there > > is (exactly) one root in [0,1]. > > > > Mark > > > > > > > From owner-reliable_computing [at] interval [dot] louisiana.edu Fri Mar 29 17:04:32 2002 Received: (from daemon@localhost) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) id g2TN4WF08683 for reliable_computing-outgoing; Fri, 29 Mar 2002 17:04:32 -0600 (CST) Received: from lcyoung.math.wisc.edu (lcyoung.math.wisc.edu [144.92.166.90]) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) with ESMTP id g2TN4KW08679 for ; Fri, 29 Mar 2002 17:04:20 -0600 (CST) Received: from ultra8.math.wisc.edu (ultra8.math.wisc.edu [144.92.166.178]) by lcyoung.math.wisc.edu (8.11.6/8.11.6) with ESMTP id g2TN4Il10697; Fri, 29 Mar 2002 17:04:18 -0600 (CST) Received: from localhost (hans@localhost) by ultra8.math.wisc.edu (8.11.6+Sun/8.11.6) with ESMTP id g2TN4DU00953; Fri, 29 Mar 2002 17:04:14 -0600 (CST) Date: Fri, 29 Mar 2002 17:04:13 -0600 (CST) From: Hans Schneider To: NETS -- at-net , "Hershkowitz, Danny -- Hershkowitz Daniel" , Danny Hershkowitz , E-LETTER , "na.digest" , ipnet-digest [at] math [dot] msu.edu, wim@bell-labs.com, hjt [at] eos [dot] ncsu.edu, vkm [at] eedsp [dot] gatech.edu, reliable_computing [at] interval [dot] louisiana.edu Subject: LAA contents Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: owner-reliable_computing [at] interval [dot] louisiana.edu Precedence: bulk ------------------------------------------------------------------------------ Hans Schneider hans [at] math [dot] wisc.edu. Department of Mathematics 608-262-1402 (Work) Van Vleck Hall 608-271-7252 (Home) 480 Lincoln Drive 608-263-8891 (Work FAX) University of Wisconsin-Madison No Home FAX at present Madison WI 53706 USA http://www.math.wisc.edu/~hans (URL) ------------------------------------------------------------------------------ Dear Net Organizer: Please circulate the attached LAA contents over your net. Thanks hans ************************************************************************ Journal: Linear Algebra and its Applications ISSN : 0024-3795 Volume : 346 Issue : 1-3 Date : 01-May-2002 Visit the journal at http://www.elsevier.nl/locate/jnlnr/07738 An algorithmic version of the theorem by Latimer and MacDuffee for 2x2 integral matrices A. Behn, A.B. Van der Merwe pp 1-14 Full text via ScienceDirect : http://www.sciencedirect.com/science?_ob=GatewayURL&_origin=CONTENTS&_method=citationSearch&_piikey=S0024379501005183&_version=1&md5=c22f5109d7cc3bb88b6e35e05f550407 On the nonlinear matrix equation X+A^*F(X)A=Q: solutions and perturbation theory A.C.M. Ran, M.C.B. Reurings pp 15-26 Full text via ScienceDirect : http://www.sciencedirect.com/science?_ob=GatewayURL&_origin=CONTENTS&_method=citationSearch&_piikey=S0024379501005080&_version=1&md5=896c5f92c42dc5692aec04fd95740889 Global reduction to the Kronecker canonical form of a C^r-family of time-invariant linear systems X. Puerta, F. Puerta, J. Ferrer pp 27-45 Full text via ScienceDirect : http://www.sciencedirect.com/science?_ob=GatewayURL&_origin=CONTENTS&_method=citationSearch&_piikey=S0024379501005250&_version=1&md5=6237e01d86a4a3d3ec6df924e3525566 Convexity and the separability problem of quantum mechanical density matrices A.O. Pittenger, M.H. Rubin pp 47-71 Full text via ScienceDirect : http://www.sciencedirect.com/science?_ob=GatewayURL&_origin=CONTENTS&_method=citationSearch&_piikey=S0024379501005249&_version=1&md5=24040a6b489a2e160472963aeea4332e More on matrix semigroup homomorphisms D. Kokol-Bukovsek pp 73-95 Full text via ScienceDirect : http://www.sciencedirect.com/science?_ob=GatewayURL&_origin=CONTENTS&_method=citationSearch&_piikey=S0024379501005031&_version=1&md5=f9d78076eab244980b0c00e393e5dd61 Pole-shifting for linear systems over commutative rings M. Carriegos, J.A. Hermida-Alonso, T. Sanchez-Giralda pp 97-107 Full text via ScienceDirect : http://www.sciencedirect.com/science?_ob=GatewayURL&_origin=CONTENTS&_method=citationSearch&_piikey=S002437950100502X&_version=1&md5=f084c7eae9c4121f998e4203817aedc5 Maximal graphs and graphs with maximal spectral radius D.D. Olesky, A. Roy, P. van den Driessche pp 109-130 Full text via ScienceDirect : http://www.sciencedirect.com/science?_ob=GatewayURL&_origin=CONTENTS&_method=citationSearch&_piikey=S0024379501005043&_version=1&md5=361480aba1653247cabe026703e3c952 Obtaining simultaneous solutions of linear subsystems of inequalities and duals E. Castillo, F. Jubete, R.E. Pruneda, C. Solares pp 131-154 Full text via ScienceDirect : http://www.sciencedirect.com/science?_ob=GatewayURL&_origin=CONTENTS&_method=citationSearch&_piikey=S0024379501005006&_version=1&md5=3fd20a21a4d8761fdb50febc49a53ca9 On matrix differential equations and abstract FG algorithm M. Przybylska pp 155-175 Full text via ScienceDirect : http://www.sciencedirect.com/science?_ob=GatewayURL&_origin=CONTENTS&_method=citationSearch&_piikey=S0024379501005110&_version=1&md5=8b149d644160910d28f6e7434465b946 A polynomial fit preconditioner for band Toeplitz matrices in image reconstruction P. Favati, G. Lotti, O. Menchi pp 177-197 Full text via ScienceDirect : http://www.sciencedirect.com/science?_ob=GatewayURL&_origin=CONTENTS&_method=citationSearch&_piikey=S0024379501005134&_version=1&md5=4b1ee92092902acf612cab38bc2602e1 Coherence invariant mappings on block triangular matrix spaces W.L. Chooi, M.H. Lim pp 199-238 Full text via ScienceDirect : http://www.sciencedirect.com/science?_ob=GatewayURL&_origin=CONTENTS&_method=citationSearch&_piikey=S0024379501005109&_version=1&md5=bbcb4f6d2439e33c2d810025d115af09 A boundary Nevanlinna-Pick problem for a class of analytic matrix-valued functions in the unit ball V. Bolotnikov pp 239-260 Full text via ScienceDirect : http://www.sciencedirect.com/science?_ob=GatewayURL&_origin=CONTENTS&_method=citationSearch&_piikey=S0024379501005171&_version=1&md5=d6b00fda6be4dcb8904f711cdd0cbba8 Irreducible, pattern k-potent ray pattern matrices J.L. Stuart, L. Beasley, B. Shader pp 261-271 Full text via ScienceDirect : http://www.sciencedirect.com/science?_ob=GatewayURL&_origin=CONTENTS&_method=citationSearch&_piikey=S0024379501005419&_version=1&md5=c87d43d6e4a3ab1ef287a47af6ab9d91 On block completion problems for Arov-normalized j"q"q-J"q-elementary factors B. Fritzsche, B. Kirstein, M. Mosch pp 273-291 Full text via ScienceDirect : http://www.sciencedirect.com/science?_ob=GatewayURL&_origin=CONTENTS&_method=citationSearch&_piikey=S0024379501005237&_version=1&md5=db018353b97d13cc08aef4b3c8970e16 Author index pp 293 Full text via ScienceDirect : http://www.sciencedirect.com/science?_ob=GatewayURL&_origin=CONTENTS&_method=citationSearch&_piikey=S0024379502002860&_version=1&md5=8d2b828474cf7ba61e76e4015ca96371 From owner-reliable_computing [at] interval [dot] louisiana.edu Fri Mar 29 17:07:11 2002 Received: (from daemon@localhost) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) id g2TN7A408756 for reliable_computing-outgoing; Fri, 29 Mar 2002 17:07:10 -0600 (CST) Received: from lcyoung.math.wisc.edu (lcyoung.math.wisc.edu [144.92.166.90]) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) with ESMTP id g2TN73W08748 for ; Fri, 29 Mar 2002 17:07:03 -0600 (CST) Received: from ultra8.math.wisc.edu (ultra8.math.wisc.edu [144.92.166.178]) by lcyoung.math.wisc.edu (8.11.6/8.11.6) with ESMTP id g2TN72l10718; Fri, 29 Mar 2002 17:07:02 -0600 (CST) Received: from localhost (hans@localhost) by ultra8.math.wisc.edu (8.11.6+Sun/8.11.6) with ESMTP id g2TN71S00957; Fri, 29 Mar 2002 17:07:01 -0600 (CST) Date: Fri, 29 Mar 2002 17:07:01 -0600 (CST) From: Hans Schneider To: NETS -- at-net , "Hershkowitz, Danny -- Hershkowitz Daniel" , Danny Hershkowitz , E-LETTER , "na.digest" , ipnet-digest [at] math [dot] msu.edu, wim@bell-labs.com, hjt [at] eos [dot] ncsu.edu, vkm [at] eedsp [dot] gatech.edu, reliable_computing [at] interval [dot] louisiana.edu Subject: LAA contents, alternate version Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: owner-reliable_computing [at] interval [dot] louisiana.edu Precedence: bulk ------------------------------------------------------------------------------ Hans Schneider hans [at] math [dot] wisc.edu. Department of Mathematics 608-262-1402 (Work) Van Vleck Hall 608-271-7252 (Home) 480 Lincoln Drive 608-263-8891 (Work FAX) University of Wisconsin-Madison No Home FAX at present Madison WI 53706 USA http://www.math.wisc.edu/~hans (URL) ------------------------------------------------------------------------------ Dear Net Organizer: I have made an alternate version of these LAA contents without the precise URL reference for each article. Please let me know if you wish me to continue to do so. best hans ************************************************************************ Journal: Linear Algebra and its Applications ISSN : 0024-3795 Volume : 346 Issue : 1-3 Date : 01-May-2002 Visit the journal at http://www.elsevier.nl/locate/jnlnr/07738 An algorithmic version of the theorem by Latimer and MacDuffee for 2x2 integral matrices A. Behn, A.B. Van der Merwe pp 1-14 On the nonlinear matrix equation X+A^*F(X)A=Q: solutions and perturbation theory A.C.M. Ran, M.C.B. Reurings pp 15-26 Global reduction to the Kronecker canonical form of a C^r-family of time-invariant linear systems X. Puerta, F. Puerta, J. Ferrer pp 27-45 Convexity and the separability problem of quantum mechanical density matrices A.O. Pittenger, M.H. Rubin pp 47-71 More on matrix semigroup homomorphisms D. Kokol-Bukovsek pp 73-95 Pole-shifting for linear systems over commutative rings M. Carriegos, J.A. Hermida-Alonso, T. Sanchez-Giralda pp 97-107 Maximal graphs and graphs with maximal spectral radius D.D. Olesky, A. Roy, P. van den Driessche pp 109-130 Obtaining simultaneous solutions of linear subsystems of inequalities and duals E. Castillo, F. Jubete, R.E. Pruneda, C. Solares pp 131-154 On matrix differential equations and abstract FG algorithm M. Przybylska pp 155-175 A polynomial fit preconditioner for band Toeplitz matrices in image reconstruction P. Favati, G. Lotti, O. Menchi pp 177-197 Coherence invariant mappings on block triangular matrix spaces W.L. Chooi, M.H. Lim pp 199-238 A boundary Nevanlinna-Pick problem for a class of analytic matrix-valued functions in the unit ball V. Bolotnikov pp 239-260 Irreducible, pattern k-potent ray pattern matrices J.L. Stuart, L. Beasley, B. Shader pp 261-271 On block completion problems for Arov-normalized j"q"q-J"q-elementary factors B. Fritzsche, B. Kirstein, M. Mosch pp 273-291 Author index pp 293 From owner-reliable_computing [at] interval [dot] louisiana.edu Sat Mar 30 15:25:11 2002 Received: (from daemon@localhost) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) id g2ULPBL11308 for reliable_computing-outgoing; Sat, 30 Mar 2002 15:25:11 -0600 (CST) Received: from imf14bis.bellsouth.net (mail114.mail.bellsouth.net [205.152.58.54]) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) with ESMTP id g2ULP5W11304 for ; Sat, 30 Mar 2002 15:25:05 -0600 (CST) Received: from u8174 ([66.20.80.183]) by imf14bis.bellsouth.net (InterMail vM.5.01.04.05 201-253-122-122-105-20011231) with SMTP id <20020330212619.HDID25581.imf14bis.bellsouth.net@u8174>; Sat, 30 Mar 2002 16:26:19 -0500 Message-Id: <2.2.32.20020330212237.009efe18 [at] pop [dot] louisiana.edu> X-Sender: rbk5287 [at] pop [dot] louisiana.edu X-Mailer: Windows Eudora Pro Version 2.2 (32) Mime-Version: 1.0 Content-Type: multipart/mixed; boundary="=====================_1017544957==_" Date: Sat, 30 Mar 2002 15:22:37 -0600 To: markst [at] nd [dot] edu, reliable_computing [at] interval [dot] louisiana.edu From: "R. Baker Kearfott" Subject: Re: Taylor models - how to get valid comparisons Cc: "P. S. V. Nataraj" X-Attachments: C:\GlobSol\examples\Gritton\new_nonlinear_system\gritton2 nls.OT1; Sender: owner-reliable_computing [at] interval [dot] louisiana.edu Precedence: bulk --=====================_1017544957==_ Content-Type: text/plain; charset="us-ascii" Mark, Thank you for the enlightenment. It's nice to run realistic test problems, even when we're just testing optimization packages :-) I'm sure J. D. Seader knew about Gritton's problem; he must have just given it to me over [-12,8] as a challenge or a test problem. In any case, I've solved the nonlinear system over [0,1], and attached the output from the present version of GlobSol (just for fun). Apparently, the problem over [0,1] is significantly easier :-) You can probably do even better than GlobSol with your group's codes. Best regards, Baker P.S. My tardy reply was due to air conditioning problems where the university's incoming mail server resides. It is definitely spring :-) At 05:15 PM 3/28/02 -0500, Mark Stadtherr wrote: > >An aside to the discussion of "Gritton's second problem"-- >While it may be interesting from a computational standpoint to >find (rigorously) all roots in [-12,8], or the minimum >in [1,2], it should be noted that the variable in this problem >is a mole fraction, which by definition must be in [0,1]. >Thus, from an application (chemical engineering) standpoint, >it is pointless (and a waste of computational effort) to >consider any interval other than [0,1]. Incidently, there >is (exactly) one root in [0,1]. > >Mark > >-- >Mark A. Stadtherr >Professor >Department of Chemical Engineering >University of Notre Dame >Notre Dame, IN 46556 USA >Telephone: (574) 631-9318 >Fax: (574) 631-8366 >E-mail: markst [at] nd [dot] edu >WWW: http://www.nd.edu/~markst > >======================================================================== > >"R. Baker Kearfott" wrote: >> >> Prof. Nataraj, >> >> That is impressive work overall. It is a laudable contribution, and >> the experiments make their point well. >> >> I wish to point out minor discrepancies in the experiments on page 10 of >> the work you make available below, as follows: >> >> 1. You define Gritton's second problem as finding the minimum of >> a particular degree 18 univariate polynomial. However, the original >> problem that I got from Prof. Seader at the University of Utah >> was to find all of the ROOTS of that polynomial in [-12,8], not find the >> minimum in [1,2]. Readers should take this into account when comparing >> the results you report to the results in previous papers. >> >> 2. You mention that "Kearfott and Arazyan report in [the AD 2000 >> proceedings] that GlobSol had some difficulty in tackling this >> problem." That is only true in a certain sense, as follows: >> >> (a) We did not report results on the minimization problem, only >> on the root-finding problem. Using "find_roots" from >> an early version of GlobSol, all 18 roots were found without >> too much problem. I attach the output file >> >> (b) I just tried GlobSol with default configuration on the >> "minimization" version of Gritton's problem, and GlobSol >> found the minimum with NO problems on [-12,8], in particular, >> with only three bisections. >> (It reported .06 seconds on a 450 MHZ machine; however, >> due to OS inadequacies, this is total elapsed time, and >> I was simultaneously processing a stereo MP3 stream >> through the CPU and a software-based network connection.) >> On your interval [1,2], the problem was indeed somewhat >> harder, with the default configuration and not using >> any Taylor arithmetic: GlobSol made 516 bisections before >> obtaining the result, and took roughly 2 seconds of total >> elapsed time. However, this is much less than the failure >> after an hour you reported with the raw Moore-Skelboe algorithm, >> regardless of timing questions. >> >> (c) GlobSol uses alternate techniques, such as finding approximate >> solutions followed by epsilon inflation and interval Newton >> methods, to make things efficient. >> However, GlobSol is configurable. The purpose of our experiments >> in the AD 2000 proceedings was to test the usefulness of the >> Taylor extensions defined by Berz; we turned off some of GlobSol's >> capabilities to simplify the environment for our experiments. >> We did find that the Taylor extensions of Berz et al were very >> useful in certain circumstances. Of course, your experiments >> show that your Bernstein-Taylor extensions are even better, >> and MUCH better at that, in certain circumstances. >> >> In any case, I'm impressed that you have found a practical way of doing >> higher-order inclusions for multivariate functions. I had been thinking >> about that for some time, but had not devised a scheme :-) >> >> Best regards, >> >> Baker >> >> At 05:06 PM 3/8/02 +0530, P. S. V. Nataraj wrote: >> >Dear All, >> > >> >For a new inclusion function form having higher order convergence, along >> >with a convergence study of this form vs. that of Berz's Taylor model, pl. >> >see our paper (under review): >> >www.ee.iitb.ac.in/~nataraj/Super_TB_ps.zip >> >In the new form, Bernstein polynomial techniques were used to bound the >> >range of the Taylor polynomial part. We studied in this paper six problems, >> >of dimensions varying from 1 to 6. >> > >> >The new form (along with Taylor model) has also been used in a modified >> >Moore-Skelboe Global Optimization algorithm in our paper (to appear in Jl. >> >Global Optimization) >> >www.ee.iitb.ac.in/~nataraj/GOTB_Final_PS.zip >> > >> >In all our studies, we badly needed a good collection of challenge problems >> >to study the convergence behavior of the various forms. >> > >> >Can somebody suggest a good collection of such problems - preferably with a >> >structure more general than polynomial, and dimensions varying from 1 to (at >> >least) 6 ? >> > >> >Regards, >> >Nataraj >> > >> >.............................................. >> >Prof. P. S. V. Nataraj >> >Systems and Control Engineering Group >> >Department of Electrical Engineering >> >Indian Institute of Technology >> >Bombay 400 076 India >> >Ph: +91-022-5723757 Fax: +91-022-5726263 >> >Email: nataraj [at] ee [dot] iitb.ernet.in >> > >> >-----Original Message----- >> >From: owner-reliable_computing [at] interval [dot] louisiana.edu >> >[mailto:owner-reliable_computing [at] interval [dot] louisiana.edu]On Behalf Of >> >Arnold Neumaier >> >Sent: Thursday, March 07, 2002 1:12 AM >> >To: berz [at] msu [dot] edu; reliable_computing [at] interval [dot] louisiana.edu >> >Subject: Re: Taylor models - how to get valid comparisons >> > >> > >> >Arnold Neumaier wrote: >> > >> >> you'd have to compare it at least to techniques that use centered forms >> >> with slopes which are available for easy use in Rump's INTLAB >> >> (and perhaps to Bernstein polynomial techniques, and to affine >> >> arithmetic), >> > >> >Another relevant comparison would be to a modified Moore-Skelboe >> >algorithm, >> >applied twice to get the lower and upper bound. This is a little >> >inefficient since part of the wirk is duplicated, but it has the >> >advantage that it can be done in a simple way since the NEOS server >> >allows one to solve global optimization problems on a box online >> >via the WWW. See >> >http://www-neos.mcs.anl.gov/neos/solvers/GO:GLOBMIN/ >> >--- >> >Incoming mail is certified Virus Free. >> >Checked by AVG anti-virus system (http://www.grisoft.com). >> >Version: 6.0.325 / Virus Database: 182 - Release Date: 2/19/02 >> > >> >--- >> >Outgoing mail is certified Virus Free. >> >Checked by AVG anti-virus system (http://www.grisoft.com). >> >Version: 6.0.325 / Virus Database: 182 - Release Date: 2/19/02 >> > >> > >> > >> >> ------------------------------------------------------------------------ >> >> gritton2.RO1Name: gritton2.RO1 >> Type: Plain Text (text/plain) >> >> ------------------------------------------------------------------------ >> >> --------------------------------------------------------------- >> R. Baker Kearfott, rbk [at] louisiana [dot] edu (337) 482-5346 (fax) >> (337) 482-5270 (work) (337) 981-9744 (home) >> URL: http://interval.louisiana.edu/kearfott.html >> Department of Mathematics, University of Louisiana at Lafayette >> Box 4-1010, Lafayette, LA 70504-1010, USA >> --------------------------------------------------------------- > >-- >Mark A. Stadtherr >Professor >Department of Chemical Engineering >University of Notre Dame >Notre Dame, IN 46556 USA >Telephone: (219) 631-9318 >Fax: (219) 631-8366 >E-mail: markst [at] nd [dot] edu >WWW: http://www.nd.edu/~markst > > --=====================_1017544957==_ Content-Type: text/plain; charset="us-ascii" Content-Disposition: attachment; filename="gritton2nls.OT1" Output from FIND_GLOBAL_MIN on 03/30/2002 at 15:14:49. Version for the system is: October 10, 2000 Codelist file name is: gritton2nlsG.CDL Box data file name is: gritton2nls.DT1 Initial box: [ 0.0000D+00, 0.1000D+01 ] BOUND_CONSTRAINT: F F --------------------------------------- CONFIGURATION VALUES: EPS_DOMAIN: 0.1000D-08 MAXITR: 20000 MAX_CPU_SECONDS: 0.3600D+04 DO_INTERVAL_NEWTON: T QUADRATIC: T FULL_SPACE: F VERY_GOOD_INITIAL_GUESS: F USE_SUBSIT: T OUTPUT UNIT: 7 PRINT_LENGTH: 1 PHI_MUST_CONVERGE: T EQ_CNS_MUST_CONVERGE: T INEQ_CNS_MUST_CONVERGE: T PHI_THICKNESS_FACTOR: 0.5000D+00 EQ_CNS_THICKNESS_FACTOR: 0.5000D+00 INEQ_CNS_THICKNESS_FACTOR: 0.5000D+00 PHI_MUST_CONVERGE: T EQ_CNS_MUST_CONVERGE: T INEQ_CNS_MUST_CONVERGE: T PHI_CONVERGENCE_FACTOR: 0.1000D-13 EQ_CNS_CONVERGENCE_FACTOR: 0.1000D-13 INEQ_CNS_CONVERGENCE_FACTOR: 0.1000D-13 CONTINUITY_ACROSS_BRANCHES: F SINGULAR_EXPANSION_FACTOR: 0.1000D+02 HEURISTIC PARAMETER ALPHA: 0.5000D+00 APPROX_OPT_BEFORE_BISECTION: F USE_TAYLOR_EQUALITY_CONSTRAINTS F USE_TAYLOR_INEQ_CONSTRAINTS F USE_TAYLOR_OBJECTIVE F USE_TAYLOR_EQ_CNS_GRD F USE_TAYLOR_GRAD F USE_TAYLOR_INEQ_CNS_GRD F USE_TAYLOR_REDUCED_INEWTON F COSY_POLYNOMIAL_ORDER 5 LEAST_SQUARES_FUNCTIONS: F NONLINEAR_SYSTEM: T Default point optimizer was used. THERE WERE NO BOXES IN THE LIST OF SMALL BOXES. LIST OF BOXES CONTAINING VERIFIED FEASIBLE POINTS: Box no.: 1 Box coordinates: [ 0.8456D+00, 0.8463D+00 ] PHI: [ 0.0000D+00, 0.1827D+00 ] B%LIUI(1,*): F B%LIUI(2,*): F B%SIDE(*): F B%PEEL(*): F Level: 0 Box contains the following approximate root: 0.8460D+00 OBJECTIVE ENCLOSURE AT APPROXIMATE ROOT: [ 0.0000D+00, 0.2889D-21 ] Unknown = T Contains_root = T Changed coordinates: F U0: [ 0.0000D+00, 0.1000D+01 ] V: [ -0.1000D+01, 0.1000D+01 ] ------------------------------------------------- ALGORITHM COMPLETED WITH LESS THAN THE MAXIMUM NUMBER, 20000 OF BOXES. Number of bisections: 36 No. dense interval residual evaluations -- gradient code list: 380 Number of orig. system inverse midpoint preconditioner rows: 1 Number of orig. system C-LP preconditioner rows: 78 Number of solutions for a component in the expanded system: 10808 Total number of forward_substitutions: 799 Number of Gauss--Seidel steps on the dense system: 79 Number of gradient evaluations from a gradient code list: 110 Total number of dense slope matrix evaluations: 76 Total number second-order interval evaluations of the original function: 76 Total number dense interval constraint evaluations: 489 Total number dense interval constraint gradient component evaluations: 225 Total number dense point constraint gradient component evaluations: 8 Number of times a box was rejected in the interval Newton method due to an empty intersection: 34 Number of times the interval Newton method made a coordinate interval smaller: 8 Number of times a box was rejected because of a large lower bound on the objective function: 3 Number of times the approximate solver was called: 1 Number of times SUBSIT decreased one or more coordinate widths: 1 Number of times SUBSIT rejected a box: 1 Total number of boxes processed in loop: 74 BEST_ESTIMATE: 0.0000D+00 Overall CPU time: 0.1700D+00 CPU time in PEEL_BOUNDARY: 0.0000D+00 CPU time in REDUCED_INTERVAL_NEWTON: 0.0000D+00 --=====================_1017544957==_ Content-Type: text/plain; charset="us-ascii" --------------------------------------------------------------- R. Baker Kearfott, rbk [at] louisiana [dot] edu (337) 482-5346 (fax) (337) 482-5270 (work) (337) 981-9744 (home) URL: http://interval.louisiana.edu/kearfott.html Department of Mathematics, University of Louisiana at Lafayette Box 4-1010, Lafayette, LA 70504-1010, USA --------------------------------------------------------------- --=====================_1017544957==_-- From owner-reliable_computing [at] interval [dot] louisiana.edu Sun Mar 31 08:13:43 2002 Received: (from daemon@localhost) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) id g2VEDhi02361 for reliable_computing-outgoing; Sun, 31 Mar 2002 08:13:43 -0600 (CST) Received: from mail.comset.net (mail.comset.net [213.172.0.3]) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) with ESMTP id g2VEDb302357 for ; Sun, 31 Mar 2002 08:13:37 -0600 (CST) Received: from 9-242.dialup.comset.net ([213.172.9.242] helo=e0gumi46) by mail.comset.net with smtp (Exim 3.33 #1) id 16rbkr-0007U5-00 for reliable_computing [at] interval [dot] louisiana.edu; Sun, 31 Mar 2002 13:36:06 +0400 Message-ID: <004e01c1d896$baad5420$710bfea9 [at] wplus [dot] net> From: "Vyacheslav Nesterov" To: "RC mailing list" Subject: Reliable Computing, Volume 8, Issue 3, 2002 Date: Sun, 31 Mar 2002 13:19:56 +0400 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0012_01C1D8B6.C01F2A80" X-Priority: 3 X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook Express 5.00.2615.200 X-MimeOLE: Produced By Microsoft MimeOLE V5.00.2615.200 Sender: owner-reliable_computing [at] interval [dot] louisiana.edu Precedence: bulk This is a multi-part message in MIME format. ------=_NextPart_000_0012_01C1D8B6.C01F2A80 Content-Type: text/plain; charset="koi8-r" Content-Transfer-Encoding: quoted-printable 3-2002 Reliable Computing Volume 8, issue 3, 2002 Mathematical Research Bounding Perturbations in Zeros of Nonlinear Systems Michael A. Wolfe 177-188 Formal Solution to Systems of Interval Linear or Non-Linear Equations Miguel A. Sainz, Ernest Gardenyes, Lambert Jorba 189-211 In Case of Interval (or More General) Uncertainty, No Algorithm Can Choose the Simplest Representative Gerhard Heindl, Vladik Kreinovich, Maria Rifqi 213-227 An Approach to Overcome Division by Zero in the Interval Gauss Algorithm Jan Mayer 229-237 Intervals of Inverse M-matrices Charles R. Johnson, Ronald L. Smith 239-243 Short communication Rump's Example Revisited Eugene Loh, G. William Walster 245-248 ------=_NextPart_000_0012_01C1D8B6.C01F2A80 Content-Type: text/html; charset="koi8-r" Content-Transfer-Encoding: quoted-printable
          &nbs= p;         =20 3-2002
          &nb= sp;    =20 Reliable=20 Computing
          =   =20 Volume 8, issue 3, 2002
 
          &nbs= p; =20 Mathematical Research
 
Bounding Perturbations in Zeros = of Nonlinear=20 Systems
Michael A. Wolfe
177-188
 
Formal Solution to Systems of = Interval=20 Linear  or Non-Linear Equations
Miguel A. Sainz, Ernest = Gardenyes,=20 Lambert Jorba
189-211
 
In Case of Interval (or More = General)=20 Uncertainty,  No Algorithm Can
Choose the Simplest=20 Representative
Gerhard Heindl, Vladik Kreinovich, Maria=20 Rifqi
213-227
 
An Approach to Overcome Division = by Zero in=20 the Interval Gauss Algorithm
Jan Mayer
229-237
 
Intervals of Inverse = M-matrices
Charles R.=20 Johnson, Ronald L. Smith
239-243
 
          &nbs= p; =20 Short communication
 
Rump's Example = Revisited
Eugene Loh, G.=20 William Walster
245-248
------=_NextPart_000_0012_01C1D8B6.C01F2A80-- From owner-reliable_computing [at] interval [dot] louisiana.edu Sun Mar 31 22:36:38 2002 Received: (from daemon@localhost) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) id g314aat03870 for reliable_computing-outgoing; Sun, 31 Mar 2002 22:36:36 -0600 (CST) Received: from cs.utep.edu (mail.cs.utep.edu [129.108.5.3]) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) with ESMTP id g314aR303866 for ; Sun, 31 Mar 2002 22:36:28 -0600 (CST) Received: from aragorn (aragorn [129.108.5.35]) by cs.utep.edu (8.11.3/8.11.3) with SMTP id g314aOI04946 for ; Sun, 31 Mar 2002 21:36:24 -0700 (MST) Message-Id: <200204010436.g314aOI04946 [at] cs [dot] utep.edu> Date: Sun, 31 Mar 2002 21:36:24 -0700 (MST) From: Vladik Kreinovich Reply-To: Vladik Kreinovich Subject: ECCAD 2002 - Second Announcement To: reliable_computing [at] interval [dot] louisiana.edu MIME-Version: 1.0 Content-Type: TEXT/plain; charset=us-ascii Content-MD5: qH5OqnVhnkDZRxSI5HGnHg== X-Mailer: dtmail 1.3.0 @(#)CDE Version 1.4 SunOS 5.8 sun4u sparc Sender: owner-reliable_computing [at] interval [dot] louisiana.edu Precedence: bulk a conference of potential interest to at least some interval researchers ------------- Begin Forwarded Message ------------- Date: Thu, 28 Mar 2002 19:19:06 -0500 From: William Sit To: ECCAD 2002 Subject: ECCAD 2002 - Second Announcement Mime-Version: 1.0 Content-Disposition: inline ---------------------------------------------------------------------- EAST COAST COMPUTER ALGEBRA DAY 2002 SECOND ANNOUNCEMENT AND CALL FOR PARTICIPATION ---------------------------------------------------------------------- The 9th annual East Coast Computer Algebra Day (ECCAD'2002) will be held on Saturday, May 18, 2002. It will be hosted at LaGuardia Community College The City University of New York Long Island City, New York, USA You can pre-register and get more information by going to our World Wide Web site at http://www.lagcc.cuny.edu/Mathematics/ECCAD2002 Alternatively, send e-mail with your name, affiliation (and if you are a student, the name of your advisor), and any other relevant information to one of the organizers listed below. There is no registration fee for the conference, but pre-registration is encouraged to help with organization. Vendors (publishers, software companies with products in symbolic computation) are encouraged to apply for a free table to display their ware of interest to ECCAD participants. Please contact the organizers directly. Tables are reserved on a first come first serve basis. TIME AND LOCATION Saturday, May 18, 2002, 8:15 am - 5:00 pm Little Theater, LaGuardia Community College, The City University of New York Long Island City, New York, USA THEMES * Algebraic Algorithms * Hybrid Symbolic-Numeric Computation * Computer Algebra Systems and Generic Programming * Mathematical Communication * Complexity of Algebraic Problems INVITED PRESENTATIONS * Dr. Gert-Martin Greuel, Professor, and Head of Center for Computer Algebra Universitat Kaiserslautern, Fachbereich Mathematik; Director, Mathematical Research Institute at Oberwolfach. Title: The system SINGULAR and applications of computer algebra * Dr. Richard D. Jenks and Sam Dooley, IBM Research, Hawthorne, New York. Title: The IBM MathML Expression Editor * Professor Doron Zeilberger, Department of Mathematics (Hill Center), Rutgers University. Title: Computerized deconstruction Abstracts of the above are available at http://www.lagcc.cuny.edu/Mathematics/ECCAD2002 POSTER SESSIONS In keeping with tradition, there will be two poster sessions offering all participants an opportunity to present timely research in an informal environment. If you wish to submit a poster, please send a title and abstract to William Sit at wyscc [at] cunyvm [dot] cuny.edu. If you would like to set up a demo at the conference, please send space and other requirements to Jerry G. Ianni at iannije [at] lagcc [dot] cuny.edu. All submissions are due by Deadline: April 30, 2002. Recent post-docs and graduate students are particularly encouraged to submit posters and arrange to set up a demo of their work. ACCOMMODATIONS New York has plenty of inexpensive accommodations (as well as fancy and expensive ones of course). We have arranged with Pan American Hotel, which is very close to LaGuardia Community College to reserve up to 30 single/double rooms at $89 per room per night for the period May 17 through May 19. Each room can be shared by up to 4 persons at no additional cost. You can rent a cot for $6 per night if desired. Address: 79-00 Queens Blvd, New York, New York. Please call 1-800-937-7374 (toll-free reservation) or 1-718-446-7676 (direct) and mention ECCAD to get the discount rate. Their fax number is 1-718-446-7991. Email: reservations [at] panamhotel [dot] com. For details about the rooms, visit their web site at http://www.panamhotel.com or http://www.newyorkviews.com/newyork/panamericanhotel.htm All reservations must be made by May 1, 2002 and must mention ECCAD. You will need a credit card to hold the reservation. TRAVEL SUPPORT Subject to successful funding, a limited number of participants will be supported to attend ECCAD 2002. We have requested support for US participants who may not be originating from the East Coast, as well as those from the East Coast. Support may cover only partially your travel expenses and lodging for up to two nights. There will be no stipends. If you plan to attend and apply for support, please send your inquiry with an itemized estimate of your expenses to William Sit at wyscc [at] cunyvm [dot] cuny.edu. Please note that if your application is approved and if funding is available, you may not get your reimbursements until after the meeting. The organizers expect to know about the funding situation a few weeks before the meeting. You are, however, encouraged to submit requests for funding as soon as you have decided to attend. All applications for support are due by Deadline: April 30, 2002. ORGANIZATION Advisory Committee: * Bruce Char, Drexel University * David Saunders, University of Delaware Organizing Committee: * William Y. Sit (Chair), City College of New York, * Jerry G. Ianni, LaGuardia Community College, Program Committee (Poster/Demo): * William Y. Sit (Chair), City College of New York, * Robert Lewis, Fordham University Local Arrangement: * Jerry G. Ianni (Chair), LaGuardia Community College, * Elvin Escano, LaGuardia Community College * Luis A. Gonzalez, LaGuardia Community College OTHER INFORMATION (1) Up-to-date information regarding accommodations, directions, area restaurants, conference dinner, reception, registration, etc. will be posted at the official web site: http://www.lagcc.cuny.edu/Mathematics/ECCAD2002 (2) The Kolchin Seminar in Differential Algebra will meet on May 19, at 10 am. Place to be announced. Professor Gert-Martin Greuel will give a two-hour talk on: Using Computer Algebra in Mathematical Research. see URL: http://math0.sci.ccny.cuny.edu/DASC for abstract. If you are interested in attending, please drop the organizers a note so we can plan on the most convenient location. PLEASE HELP COMMUNICATE THIS ANNOUNCEMENT TO ALL INTERESTED PARTIES. ------------- End Forwarded Message -------------