From owner-reliable_computing Fri Dec 8 07:08:37 1995 Received: by interval.usl.edu id AA29865 (5.65c/IDA-1.4.4 for reliable_computing-outgoing); Fri, 8 Dec 1995 15:08:49 -0600 Received: from mailhost.lanl.gov by interval.usl.edu with SMTP id AA29859 (5.65c/IDA-1.4.4 for ); Fri, 8 Dec 1995 15:08:43 -0600 Received: from xdiv.lanl.gov by mailhost.lanl.gov (8.6.12/1.2) id OAA22635; Fri, 8 Dec 1995 14:08:40 -0700 Received: from xdiv.lanl.gov.xdiv (angus.lanl.gov [128.165.123.30]) by xdiv.lanl.gov (8.6.12/8.6.12) with SMTP id OAA02519 for ; Fri, 8 Dec 1995 14:08:37 -0700 Date: Fri, 8 Dec 1995 14:08:37 -0700 From: "Rajendra B. Patil" Message-Id: <199512082108.OAA02519 [at] xdiv [dot] lanl.gov> Received: by xdiv.lanl.gov.xdiv (4.1/SMI-4.1) id AA04777; Fri, 8 Dec 95 14:08:20 MST To: reliable_computing [at] interval [dot] usl.edu Subject: Applications info.... Reply-To: Raj Patil Sender: owner-reliable_computing Precedence: bulk Content-Length: 1531 X-Lines: 43 Status: O Hi: I am looking for help in collecting following information to convince a group of people in using Interval Arithmetic for verified computing in problems where safety issues are important in the design of a physical system. 1) Very simple examples to demonstrate the concept of verification for people without much mathematical and computer background. 2) ANY industrial application areas with a need for such verification properties. 3) What are the benefits of verification in an application a. economic impact, e.g. saving over unverified systems (estimates are ok), b. reducing risk (due to verification), c. improved working conditions, d. improved environment, e. whatever improvements you think are possible that will help convince administration for using this approach. As you may notice all of the above advantages can be addresses on the basis of safety that can be guaranteed by the verified computing paradigm. I am looking for some concrete explanation and convincing examples. I will post the summary, IF i receive any replies.... Thanks in advance, ------------------------------------------------------------- Raj Patil email: rbp1 [at] lanl [dot] gov phone: (505)-665-0581 MS - B 258 fax : (505)-665-4479 XCM, Los Alamos National Laboratory Los Alamos, NM 87545 ------------- alternate address ----------------------------- P.O Box 1145 Los Alamos, NM 88003 ------------------------------------------------------------- From owner-reliable_computing Sun Dec 10 13:37:06 1995 Received: by interval.usl.edu id AA00874 (5.65c/IDA-1.4.4 for reliable_computing-outgoing); Sun, 10 Dec 1995 21:37:12 -0600 Received: from cs.utep.edu by interval.usl.edu with SMTP id AA00868 (5.65c/IDA-1.4.4 for ); Sun, 10 Dec 1995 21:37:08 -0600 Received: from earth.cs.utep.edu by cs.utep.edu (4.1/SMI-4.1) id AA11322; Sun, 10 Dec 95 20:37:06 MST Date: Sun, 10 Dec 95 20:37:06 MST From: vladik [at] cs [dot] utep.edu (Vladik Kreinovich) Message-Id: <9512110337.AA11322 [at] cs [dot] utep.edu> To: reliable_computing [at] interval [dot] usl.edu Subject: Reviews on Applications of Interval Computations Sender: owner-reliable_computing Precedence: bulk Content-Length: 2257 X-Lines: 51 Status: O Reviews on Applications of Interval Computations It has been proposed to start a new section in the journal "Reliable Computing" that will contain short reviews of papers and books on applications of (broadly understood) interval computations. The experience of the 1995 Workshop on Applications of Interval Computations has shown that there are many unexpected application areas and results that are not widely known in the interval community. It is very difficult to trace such papers because papers of applications of interval computations are published not only in mathematical journals (that are usually covered by MR, ZBlatt, etc.), but also in the journals of the corresponding application areas, that are, as a rule, not covered by traditional mathematical review journals. Moreover, reviews published in MR, Zblatt, etc, may only describe the result without explaining that interval methods have been actually used. In view of this difficulty, we decided to provide the readers of "Reliable Computing" with short reviews of application papers (something like an ongoing annotated bibliography). We strongly believe that the information about the current applications is of interest to this community. For the reasons expressed above, we are not sure that we are covering all current application papers. To increase the coverage, we need your help. If you know of any recent papers devoted to applications of interval computations and related techniques, please send the references to Vladik Kreinovich at vladik [at] cs [dot] utep.edu, or by regular mail to Vladik Kreinovich Department of Computer Science University of Texas at El Paso El Paso, TX 79968, USA If you have written your own reviews, or if you would like to write reviews, please contact Vladik as well. Authors, please send information and/or copies of your own application papers (papers in Russian and German are also welcome). Reviews should be in LaTeX, but ASCII is also acceptable. This is a new section, and we want the reader's input about how to make it better. For example, when we have more reviews, it may make sense to divide this section into subsection devoted to different application areas. Any suggestions and recommendations will be highly welcome. From owner-reliable_computing Tue Dec 12 04:19:39 1995 Received: by interval.usl.edu id AA01844 (5.65c/IDA-1.4.4 for reliable_computing-outgoing); Tue, 12 Dec 1995 10:19:41 -0600 Received: by interval.usl.edu id AA01834 (5.65c/IDA-1.4.4 for reliable_computing); Tue, 12 Dec 1995 10:19:39 -0600 Date: Tue, 12 Dec 1995 10:19:39 -0600 From: "Kearfott R. Baker" Message-Id: <199512121619.AA01834 [at] interval [dot] usl.edu> To: reliable_computing Subject: Re: information on applications Content-Type: X-sun-attachment Sender: owner-reliable_computing Precedence: bulk X-Lines: 84 Status: O Content-Length: 4537 ---------- X-Sun-Data-Type: text X-Sun-Data-Description: text X-Sun-Data-Name: text X-Sun-Content-Lines: 11 X-Sun-Content-Length: 607 I would like to encourage people to reply, as specifically as possible, to Rajendra Patil's request for information on actual applications of interval computations. For convenience, I have attached his original message. --------------------------------------------------------------- R. Baker Kearfott, rbk [at] usl [dot] edu (318) 482-5346 (fax) (318) 482-5270 (work) (318) 981-9744 (home) URL: ftp://interval.usl.edu/pub/interval_math/www/kearfott.html Department of Mathematics, University of Southwestern Louisiana --------------------------------------------------------------- ---------- X-Sun-Data-Type: sun-deskset-message X-Sun-Data-Name: sun-deskset-message X-Sun-Encoding-Info: uuencode X-Sun-Content-Lines: 61 X-Sun-Content-Length: 3628 begin 600 sun-deskset-message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ed, 13 Dec 1995 11:01:27 -0600 Received: from plot79.math.utah.edu (beebe [at] plot79 [dot] math.utah.edu [128.110.198.3]) by csc-sun.math.utah.edu (8.7.1/8.7.1) with ESMTP id KAA11868; Wed, 13 Dec 1995 10:01:10 -0700 (MST) From: "Nelson H. F. Beebe" Received: (from beebe@localhost) by plot79.math.utah.edu (8.7.1/8.7.1) id KAA19113; Wed, 13 Dec 1995 10:01:06 -0700 (MST) Date: Wed, 13 Dec 1995 10:01:06 -0700 (MST) To: reliable_computing [at] interval [dot] usl.edu Cc: beebe [at] math [dot] utah.edu, bibnet [at] math [dot] utah.edu, ehg [at] research [dot] att.com X-Us-Mail: "Center for Scientific Computing, University of Utah, Salt Lake City, UT 84112, USA" X-Telephone: +1 801 581 5254 X-Fax: +1 801 581 4148 X-Url: http://www.math.utah.edu/~beebe Subject: Seeking bibliographies of interval arithmetic Message-Id: Sender: owner-reliable_computing Precedence: bulk Content-Length: 1475 X-Lines: 33 Status: O Are any of you aware of Internet-accessible bibliographies of papers and books on interval arithmetic? As one of the editors of the BibNet Project, I think that it would be useful to have such a bibliography in the BibNet archives. The master archive for the Project resides at ftp://ftp.math.utah.edu/pub/bibnet, and is mirrored nightly to Netlib hosts at AT&T and Oak Ridge National Laboratory. The WorldWideWeb URL http://www.netlib.org/bibnet/ this morning reports: >> ... >> bibnet >> There have been 25,956 accesses to this library. >> (Count updated 12/12/95 at 03:08:00) >> ... and the ftp logs for ftp.math.utah.edu show 7478 accesses as of 7-Dec-1995, so evidently, people are finding the collection useful. I have a summer 1994 snapshot of 632 bibliographies from the huge Computer Science bibliography archive in Karlsruhe (http://www.ira.uka.de/), and no single file in that collection concerns itself exclusively with interval arithmetic, though I'm confident that the collection could be mined to start such a bibliography. ======================================================================== Nelson H. F. Beebe Tel: +1 801 581 5254 Center for Scientific Computing FAX: +1 801 581 4148 Department of Mathematics, 105 JWB Internet: beebe [at] math [dot] utah.edu University of Utah URL: http://www.math.utah.edu/~beebe Salt Lake City, UT 84112, USA ======================================================================== From owner-reliable_computing Fri Dec 15 08:35:53 1995 Received: by interval.usl.edu id AA04263 (5.65c/IDA-1.4.4 for reliable_computing-outgoing); Fri, 15 Dec 1995 00:42:15 -0600 Received: from ixgate02.dfnrelay.d400.de by interval.usl.edu with SMTP id AA04257 (5.65c/IDA-1.4.4 for ); Fri, 15 Dec 1995 00:41:54 -0600 X400-Received: by mta d400relay in /PRMD=dfnrelay/ADMD=d400/C=de/; Relayed; Fri, 15 Dec 1995 07:41:00 +0100 X400-Received: by /PRMD=UNI-LEIPZIG/ADMD=D400/C=DE/; Relayed; Fri, 15 Dec 1995 07:35:53 +0100 Date: Fri, 15 Dec 1995 07:35:53 +0100 X400-Originator: rex [at] mathematik [dot] uni-leipzig.d400.de X400-Recipients: non-disclosure:; X400-Mts-Identifier: [/PRMD=UNI-LEIPZIG/ADMD=D400/C=DE/;ULMTA10389951215073553-W9K] X400-Content-Type: P2-1984 (2) From: Kearfott "R." Baker Message-Id: <951215073552*/S=rex/OU=mathematik/PRMD=UNI-LEIPZIG/ADMD=D400/C=DE/@MHS> To: RELIABLE_COMPUTING [at] INTERVAL [dot] USL.edu (Receipt Notification Requested) (Non Receipt Notification Requested) Cc: rbk [at] usl [dot] edu Subject: Re: information on applications Sender: owner-reliable_computing Precedence: bulk Content-Length: 5198 X-Lines: 99 Status: O ------------------------------ Start of body part 1 ----- Additional Unix mail Header Lines ----- Received: from mihp710.mathematik.uni-leipzig.de by server1.rz.uni-leipzig.de with SMTP (1.37.109.16/16.2) id AA066387126; Tue, 12 Dec 1995 20:38:46 +0100 Received: from interval.usl.edu by mihp710.mathematik.uni-leipzig.de with SMTP (16.8/15.6) id AA06118; Tue, 12 Dec 95 20:34:03 +0100 Received: from localhost by interval.usl.edu with SMTP id AA01870 (5.65c/IDA-1.4.4); Tue, 12 Dec 1995 10:21:07 -0600 Received: by interval.usl.edu id AA01844 (5.65c/IDA-1.4.4 for reliable_computing-outgoing); Tue, 12 Dec 1995 10:19:41 -0600 Received: by interval.usl.edu id AA01834 (5.65c/IDA-1.4.4 for reliable_computing); Tue, 12 Dec 1995 10:19:39 -0600 ------------------------------ Start of body part 2 ---------- I would like to encourage people to reply, as specifically as possible, to Rajendra Patil's request for information on actual applications of interval computations. For convenience, I have attached his original message. --------------------------------------------------------------- R. Baker Kearfott, rbk [at] usl [dot] edu (318) 482-5346 (fax) (318) 482-5270 (work) (318) 981-9744 (home) URL: ftp://interval.usl.edu/pub/interval_math/www/kearfott.html Department of Mathematics, University of Southwestern Louisiana --------------------------------------------------------------- Dear Baker, Thank you for this message, but unfortunately I can not read this text. Georg begin 600 sun-deskset-message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ri, 15 Dec 1995 13:59:23 -0600 Received: from xdiv.lanl.gov by mailhost.lanl.gov (8.6.12/1.2) id MAA26564; Fri, 15 Dec 1995 12:59:20 -0700 Received: from xdiv.lanl.gov.xdiv (angus.lanl.gov [128.165.123.30]) by xdiv.lanl.gov (8.6.12/8.6.12) with SMTP id MAA22684 for ; Fri, 15 Dec 1995 12:59:12 -0700 Date: Fri, 15 Dec 1995 12:59:12 -0700 From: "Rajendra B. Patil" Message-Id: <199512151959.MAA22684 [at] xdiv [dot] lanl.gov> Received: by xdiv.lanl.gov.xdiv (4.1/SMI-4.1) id AA02371; Fri, 15 Dec 95 12:59:12 MST To: reliable_computing [at] interval [dot] usl.edu Subject: Applications... Reply-To: Raj Patil Sender: owner-reliable_computing Precedence: bulk Content-Length: 764 X-Lines: 23 Status: O Hi: Regarding my previous message, Dr. Birdie suggested I change the wording as that might provoke many responses. Until now I have only received 3 messages. Interval arithmetic/analysis appears to provide elegant tools for dealing with system uncertainty. I am looking for examples of these in the design and development of real systems. Thanks, ------------------------------------------------------------- Raj Patil email: rbp1 [at] lanl [dot] gov phone: (505)-665-0581 MS - F 645 fax : (505)-665-4479 XCM, Los Alamos National Laboratory Los Alamos, NM 87545 ------------- alternate address ----------------------------- P.O Box 1145 Los Alamos, NM 88003 ------------------------------------------------------------- From owner-reliable_computing Fri Dec 15 08:02:34 1995 Received: by interval.usl.edu id AA04880 (5.65c/IDA-1.4.4 for reliable_computing-outgoing); Fri, 15 Dec 1995 14:02:35 -0600 Received: by interval.usl.edu id AA04870 (5.65c/IDA-1.4.4 for reliable_computing); Fri, 15 Dec 1995 14:02:34 -0600 Date: Fri, 15 Dec 1995 14:02:34 -0600 From: "Kearfott R. Baker" Message-Id: <199512152002.AA04870 [at] interval [dot] usl.edu> To: reliable_computing Subject: Request for applications -- another try and an apology Sender: owner-reliable_computing Precedence: bulk Content-Length: 2657 X-Lines: 72 Status: O Colleagues: I apologize that the attachment on my previous message may have been unreadable. Sorry for the inconvenience. I'm learning! I repeat: I would like to encourage people to reply, as specifically as possible, to Rajendra Patil's request for information on actual applications of interval computations. For convenience, I have appended his original message (in a hopefully readable form this time). --------------------------------------------------------------- R. Baker Kearfott, rbk [at] usl [dot] edu (318) 482-5346 (fax) (318) 482-5270 (work) (318) 981-9744 (home) URL: ftp://interval.usl.edu/pub/interval_math/www/kearfott.html Department of Mathematics, University of Southwestern Louisiana --------------------------------------------------------------- From: "Rajendra B. Patil" Message-Id: <199512082108.OAA02519 [at] xdiv [dot] lanl.gov> Received: by xdiv.lanl.gov.xdiv (4.1/SMI-4.1) id AA04777; Fri, 8 Dec 95 14:08:20 MST To: reliable_computing [at] interval [dot] usl.edu Subject: Applications info.... Reply-To: Raj Patil Sender: owner-reliable_computing Hi: I am looking for help in collecting following information to convince a group of people in using Interval Arithmetic for verified computing in problems where safety issues are important in the design of a physical system. 1) Very simple examples to demonstrate the concept of verification for people without much mathematical and computer background. 2) ANY industrial application areas with a need for such verification properties. 3) What are the benefits of verification in an application a. economic impact, e.g. saving over unverified systems (estimates are ok), b. reducing risk (due to verification), c. improved working conditions, d. improved environment, e. whatever improvements you think are possible that will help convince administration for using this approach. As you may notice all of the above advantages can be addresses on the basis of safety that can be guaranteed by the verified computing paradigm. I am looking for some concrete explanation and convincing examples. I will post the summary, IF i receive any replies.... Thanks in advance, ------------------------------------------------------------- Raj Patil email: rbp1 [at] lanl [dot] gov phone: (505)-665-0581 MS - B 258 fax : (505)-665-4479 XCM, Los Alamos National Laboratory Los Alamos, NM 87545 ------------- alternate address ----------------------------- P.O Box 1145 Los Alamos, NM 88003 ------------------------------------------------------------- From owner-reliable_computing Tue Dec 19 11:26:53 1995 Received: by interval.usl.edu id AA07119 (5.65c/IDA-1.4.4 for reliable_computing-outgoing); Tue, 19 Dec 1995 03:27:47 -0600 Received: from paradoxe.irisa.fr by interval.usl.edu with SMTP id AA07113 (5.65c/IDA-1.4.4 for ); Tue, 19 Dec 1995 03:27:30 -0600 Received: from paraffine.irisa.fr (paraffine.irisa.fr [131.254.20.8]) by paradoxe.irisa.fr (8.6.12/8.6.9) with ESMTP id KAA00476; Tue, 19 Dec 1995 10:26:54 +0100 From: Olivier Beaumont Date: Tue, 19 Dec 1995 10:26:53 +0100 Message-Id: <199512190926.KAA06030 [at] paraffine [dot] irisa.fr> To: reliable_computing [at] interval [dot] usl.edu Subject: Polynomials... Sender: owner-reliable_computing Precedence: bulk _________________________________________________________________________ I'd like to know methods in order to enclose surely the image set of a polynomial in one or several variables. The aim is not to compute exactly the image set but just to enclose it as precisely as possible. Any reference would be helpful. Thank you for your help, Olivier Beaumont (33) 99 84 75 05 obeaumon [at] irisa [dot] fr Institut de Recherche en Informatique et Sytemes aleatoires. Rennes. France _________________________________________________________________________ From owner-reliable_computing Thu Dec 21 12:04:04 1995 Received: by interval.usl.edu id AA13722 (5.65c/IDA-1.4.4 for reliable_computing-outgoing); Thu, 21 Dec 1995 20:04:13 -0600 Received: from cs.utep.edu by interval.usl.edu with SMTP id AA13716 (5.65c/IDA-1.4.4 for ); Thu, 21 Dec 1995 20:04:08 -0600 Received: from earth.cs.utep.edu by cs.utep.edu (4.1/SMI-4.1) id AA13180; Thu, 21 Dec 95 19:04:04 MST Date: Thu, 21 Dec 95 19:04:04 MST From: vladik [at] cs [dot] utep.edu (Vladik Kreinovich) Message-Id: <9512220204.AA13180 [at] cs [dot] utep.edu> To: reliable_computing [at] interval [dot] usl.edu Subject: Double bubble paper on-line Sender: owner-reliable_computing Precedence: bulk The paper by J. Hass et al. in which interval computations have been used to solve a long-standing geometric problem is on-line (for details on the problem, see my previous announcement). It can be accessed from the homepage of the first author: URL http://www.math.ucdavis.edu/~hass This homepage can also be accessed from the Personalia and Applications sections of the Interval Computations homepage: URL http://cs.utep.edu/interval-comp/main.html From owner-reliable_computing Fri Dec 22 09:33:00 1995 Received: by interval.usl.edu id AA14507 (5.65c/IDA-1.4.4 for reliable_computing-outgoing); Fri, 22 Dec 1995 15:34:14 -0600 Received: from boris.mscs.mu.edu by interval.usl.edu with SMTP id AA14501 (5.65c/IDA-1.4.4 for ); Fri, 22 Dec 1995 15:34:05 -0600 Received: by boris.mscs.mu.edu (Smail3.1.28.1 #9) id m0tTF5s-000083C; Fri, 22 Dec 95 15:33 CST Message-Id: Date: Fri, 22 Dec 95 15:33 CST From: georgec [at] boris [dot] mscs.mu.edu (Dr. George F. Corliss MU MSCS) To: reliable_computing [at] interval [dot] usl.edu Subject: Applications of intervals (long) X-Sun-Charset: US-ASCII Sender: owner-reliable_computing Precedence: bulk Raj, > I am looking for help in collecting following information to convince > a group of people in using Interval Arithmetic for verified computing > in problems where safety issues are important in the design of a > physical system. Here are some stories I know. Interval techniques are no "silver bullet", but they CAN provide insight. 1. arctan Some years back, a major oil company was running a portion of an oil reservoir simulation code. They were running identical codes on identical IBM 3090 computers with identical software configurations at two different sites. One machine consistently gave non-physical negative concentrations. After MUCH investication, the difference was traced to different arctan routines in the vendor-supplied libraries. The engineers, of course, standardized on the library that led to positive concentrations, ALTHOUGH THAT WAS THE ARCTAN THAT YIELDED _WRONG_ RESULTS. The engineers went away satisfied, but the computer support people were MOST dismayed, and called me as an interval consultant. When I arrived, though, they preferred that I look at different problems. I do not know the outcome of the engineers continuing intentionally to use the incorrect arctan. 2. sqrt (negative) The same oil company DID ask me to look at a crack propagation submodel in their reservoir simulation model. The crack propagation model was a nonlinear PDE, which they solve by time stepping. At each time step, they solved a nonlinear system by Newton's method. Computations proceeded smoothly for about 100 time steps, when suddenly, computation halted with a sqrt (negative). We used interval techniques to validate the existing approximating computations and to discover that the problem was a need for a more accurate starting guess for the Newton iteration. That modification was made to the approximate code, and the simulation ran for more than 200 time steps. 3. Singular linear systems. The same oil company gave me a linear system they were trying to solve by a conjugate gradient algorithm. The algorithm was not converging as they expected. I coded it up using the accurate linear solver in ACRITH, and ACRITH computed 1 ULP enclosures of an answer on an IBM 3090 while we ate lunch. They hadn't solved the problem, but I had. Great! Actually, the long accumulator had more to do with the success than interval arithmetic, but the interval techniques DID allow us to validate our results. Next, I wanted to explore the sensitivity of the result, so I inclated their matrix coefficients by a relative width of about 10^{-3}. ACRITH said the interval matrix contained a singular matrix. After further experiments, when I inflated ANY element of their original point-valued matrix to an interval 1 UPL wide, ACRITH again said that the matrix was singular. So I went to the engineers and triumphantly proclaimed, "The reason you were having trouble is that you are trying to solve a singular system, and interval techniques have validated that!" One engineer looked puzzled. "Oh, didn't we tell you? ALL our matrices are singular." So I learned something, even if they knew it already. These stories are described in more detail in the attached paper. 4. Bounded solutions of linear systems. A biomodeler cam to me with a problem of blood flow in a capillary bed. He had a pressure in a single incoming artery and the pressure in a single outgoing vein. In between was a complicated network flow problem. He had formulated the pressure model as a linear system, which he had solved, but he was concerned about the accuracy and robustness of his results. After all, the problem was very sensitive, since a restriction in one vessel in the bed causes BIG changes in the flows of nearby vesslels. We set up an interval-valued system to reflect the uncertainties in his data. This time, when we passed the system to an interval solver, we were surprised to find everything perfectly well behaved. There was no evidence (in the pressures) of any sensitivity at all. My client thought for a moment and exclaimed, "Of course! The pressures _must_ be bounded by the input and the output pressures, so we have no fear of unbounded solutions!" The application of interval techniques allowed him to concentrate on his model, without fears of numerical difficulties. 5. In recent talks, Professor Kulisch has described a turbine application. You should get details from him. 6. A book of a couple years ago edited by Adams and Kulisch contains many articles describing applications. 7. I headr a wonderful talk by Wolfram Klein at SCAN 95 in Wuppertal about using interval techniques and automatic differentiation for a circuit simulation problem. 8. Ernst Adams and others have applied interval techniques to the study of chaotic dynamical systems. % File: INT_TEC2.TEX 03-JAN-1990 % % Industrial Applications of Interval Techniques % Paper for Basel conference % Based on technical report 301 % How Can You Apply Interval Techniques in an Industrial Setting? % No page numbers? % Line spacing? \documentstyle [11pt]{article} \topmargin -0.2 true in %\textheight 8.5 true in \textheight 232 mm %\textwidth 6 true in \textwidth 160 mm \hoffset 0.5 true in \begin {document} \renewcommand {\vector}[1]{{\bf #1}} \newcommand {\beginsection}[1]{{\vspace {15pt} \message{#1} \leftline{\bf #1} \nobreak \smallskip \vskip-\parskip}} \newcommand {\subsect}[1]{{\bigskip \leftline{\bf #1} \nobreak \smallskip \vskip-\parskip}} \newcommand {\subsubsub}[1]{ \vspace {20 pt} \noindent {\large \em #1} \vspace {10 pt} } \newcommand {\recommend}[1]{ \vspace {20 pt} \noindent {\bf #1} \vspace {20 pt} } \newcommand {\real}{\bf R} \large \pagestyle {empty} \baselineskip 8 mm \begin {center} {\Large INDUSTRIAL APPLICATIONS OF INTERVAL TECHNIQUES } \vspace {10 pt} \\ George F. Corliss \\ Marquette University \\ Milwaukee, WI 53233 USA \vspace {10 pt} \end {center} \noindent {\bf Abstract.} Tools for achieving high accuracy and guaranteed inclusions in scientific computations have been promoted by the research community for many years. Commercial products have made those tools widely available. We apply the tools of ACRITH to a least squares problem, a problem involving matrix products and inverses, the conjugate gradient algorithm, and a partial differential equation with a moving boundary. We discuss the lessons learned from these experiences about how interval techniques and high accuracy computations can be used cost-effectively for industrial problems. \beginsection {1. Introduction} Since Moore's {\sl Interval Analysis\/} \cite{Moor66a} appeared in 1966, the academic research community has been promoting the use of interval techniques to achieve guaranteed bounds for the results of scientific computations. Conventional algorithms, or {\em point\/} algorithms, compute an estimate for an answer and perhaps an error estimate. The user cannot tell how accurate the estimated answer may be without extensive, expensive error analysis. The quality of scientific software packages has improved, but there are still problems for which catastrophically incorrect answers are returned to the user with no warning. Interval techniques compute an interval in which the correct answer is guaranteed to lie. If interval techniques compute an answer $ [1.23456, 1.23457] $, then 6 decimal places are {\em known\/} to be correct. If an algorithm yields the interval $ [-10^{30}, \ +10^{30} ] $, then you {\em know\/} that you do not know the answer. Hence, interval results carry with them an assurance of their quality. In the early days, it was hard for a scientist or engineer to apply interval techniques because suitable software was not widely available. Now there are at least four commercially available systems which support interval arithmetic and the Kulisch-Miranker accurate scalar product \cite{Kuli81a}: \begin {description} \item [Pascal-SC] - Variant of Pascal for scientific computation. Runs on popular microcomputers. Available from Wiley-Teubner. \cite{Kuli87a} \item [ACRITH] - Fortran callable subroutine library. Runs on IBM 370 architecture. Available from IBM. \cite{IBM86a} Fortran-SC \cite{Metz87a} makes ACRITH subroutine calls easier to program. \item [ARITHMOS] - Fortran callable subroutine library. Runs on Siemens BS 2000. Available from Siemens. \cite{Siem86a} \item [NAG] - Ada programming language library. Requires VAX/VMS Ada compiler. Available from Numerical Algorithms Group. \cite{Delv87a} \end {description} Hence, a scientist or engineer now has ready access to these tools on a variety of commonly used computers. This paper addresses ways in which tools for interval methods can be applied to industrial problems drawn from the author's consulting experience. We illustrate with a least squares problem, a problem involving matrix products and inverses, the conjugate gradient algorithm, and a partial differential equation with a moving boundary. We consider when interval tools should be used and how to use them. \beginsection {2. High Accuracy} Interval techniques compute an interval in which the correct answer is validated to lie. On the other hand, techniques for high accuracy compute an answer which is accurate to as many places as possible. Interval and high accuracy techniques are related, because we want the widths of interval inclusions to be as narrow as possible. However, techniques for high accuracy computation are very useful in their own right, with no reference to intervals. This section describes the support provided by ACRITH for high accuracy computations and how those tools have are applied to some actual problems. \subsect {2.1. Accurate scalar product} The fundamental tool provided by ACRITH for high accuracy computation is the accurate scalar product. Its use often prevents the error from growing with problem size. Let $ \vector {w} = (w_1, w_2, \ldots , w_n) $ and $ \vector {v} = (v_1, v_2, \ldots , v_n) $ be vectors in $ \real^n $. Then ACRITH's scalar product computes \begin {equation} \vector {w} \bullet \vector {v} = \sum_{i=0}^n w_i \cdot v_i \label {Scalar_Product} \end {equation} with only {\em one rounding error.\/} The ACRITH scalar product is more accurate than the corresponding Fortran DO loop. On some hardware, it is also about 15% faster. A Fortran DO loop for evaluating Equation (\ref{Scalar_Product}) commits $ 2 n $ rounding errors, while the ACRITH program commits only one rounding error {\em independent\/} of the size of $ n $. This is important since industrial problems tend to be large. Since the ACRITH program commits only one rounding error at the conclusion of the calculation, it is not subject to catastrophic cancellation errors or to intermediate overflow. Conventional wisdom in numerical analysis suggests that it is often useful to accumulate scalar products in a higher precision. If a Fortran DO loop accumulates the sum in double precision, the $ 2 n $ rounding errors are of the order of double precision machine roundoff. The result is more accurate one computed by the single precision program, but it is not as accurate as the results from ACRITH. Further, the double precision accumulation of scalar products is still sensitive to catastrophic cancellation and overflow of intermediate results. \subsect {2.2. How does it work?} A detailed discussion of the algorithms used by ACRITH is beyond the scope of this paper (see \cite{IBM86a}, pp. 57--61 for details). ACRITH uses a long accumulator with a capacity for 328 hexadecimal digits. The accumulation of the scalar product is done in fixed-point arithmetic with partial products being added into the accumulator in the appropriate digit positions. When the summation is completed, the result is rounded once for storing back into a single or double precision memory location. \subsect {2.3. An example -- Least squares} Let $ A $ be an $ m \times n $ matrix, with $ m > n $. We wish to solve \begin {equation} A \vector {x} = \vector {b} \label {Least_Squares} \end {equation} in the least squares sense. This problem arises often in many industrial applications, and it is often very large. The least squares solution to Equation (\ref{Least_Squares}) is the same as the solution to the square system \begin {equation} A^T A \vector {x} = A^T \vector {b}, \label {Normal_Equations} \end {equation} which are called the normal equations. It is well known that the normal equations are very poorly conditioned. The conventional wisdom in numerical analysis is that one can almost never solve a least squares problem using the normal equations. Methods such as Gram-Schmidt orthonormalization, Q-R factorization, or singular valued decomposition were developed to overcome the numerical difficulties of ill conditioned systems. The accurate scalar product calls this conventional wisdom into question. The reason the ill conditioning of the normal equations is a problem is that with conventional arithmetic, roundoff errors accumulate so fast that the computed solution is worthless. With ACRITH's scalar product, the accumulation of roundoff errors is much less severe, and it is independent of the size of the problem. The program in Listing 1 solves a modest least squares problem. This program uses ACRITH routines to form $ A^T A $, $ A^T b $, and to solve the normal equations. \begin {verbatim} PARAMETER (M = 228, N = 178) DOUBLE PRECISION A (M, N), AT(N, M), APROD(N, N), B(M), + BT(N), XL(N), XU(N) C Get matrix A C AT = A transpose C APROD = A transpose * A CALL DMAMN (N, M, N, AT, A, APROD, IER) C BT = A transpose * B CALL DMAMN (N, M, 1, AT, B, BT, IER) C Solve A trans A x = A trans b CALL DLIN (N, APROD, BT, BT, XL, XU, IER) \end{verbatim} \begin {center} Listing 1. Solve the normal equations. \end {center} \vspace {10 pt} For the data we were given, ACRITH was able to solve the normal equations with an absolute residual error of 4.02E-14. For comparison, Listing 2 is code using IMSL routines to form and solve the normal equations. \begin {verbatim} C APROD = A transpose * A CALL DMXTXF (M, N, A, M, N, APROD, N) C BT = A transpose * B CALL DMXTYF (M, N, A, M, M, 1, B, M, N, 1, BT, N) C Solve A trans A x = A trans b CALL DL2ARG (N, APROD, N, BT, 1, XL, FAC, IPVT, WK) \end{verbatim} \begin {center} Listing 2. Solve the normal equations using IMSL routines. \end {center} \vspace {10 pt} Since the difficulty of using tools for high accuracy is an issue, we remark that changing from the IMSL subroutines to the ACRITH subroutines requires only the replacing of one set of subroutine calls by another. The two programs are of the same complexity. The IMSL library provides a routine L2BRR for solving linear least squares problems using a Q-R decomposition with optimal column pivoting. This routine was unable to solve this problem; it fails because of an underflow. In Section 4, we will see that L2BRR fails because the problem is very close to a singular problem. If the ACRITH linear equation solving routine DLIN is unable to solve the normal equations because of excess width arising from the explicit formation of $ A^T A $ and $ A^T b $, then the normal equations are recast as an $ (2 m + n) \times (2 m + n) $ block system with no matrix products: \begin {equation} \left ( \begin {array}{ccc} A & -I \\ & A^T \\ && I \end {array} \right ) \left ( \begin {array}{c} x \\ y \\ z \end {array} \right ) = \left ( \begin {array}{c} 0 \\ 0 \\ b \end {array} \right ). \label {Large_System} \end {equation} Then DLIN is called to solve Equation (\ref{Large_System}). % derivation: % A^T A x = A^T b % % A x = y % A^T y = A^T b % % A^T y - A^T z = 0 % z = b % % x y z % A -I 0 m % A^T -A^T 0 n % I b m % n m m \subsect {2.4. An example - Conjugate gradient} A second approach to solving Problem (\ref{Least_Squares}) is the conjugate gradient algorithm \cite{Hest52a}, which involves many scalar products. We suspected that an accumulation of roundoff errors was forcing a conventional program to take several times as many iterations than should be necessary. We replaced the existing scalar products with ACRITH calls to improve the accuracies. We present an outline of the analysis and our experiences as an example of how to apply the tools of ACRITH. Suppose that $ A $ is an $ m \times n $ matrix. The conjugate gradient algorithm for solving the least squares problem $$ A \vector {x} = \vector {b} $$ is: \begin {tabbing} \hspace {1.3 in} \= Choose $ \vector {x}_0 $; \\ \> Put \= $ \vector {s}_0 = \vector {b} - A \vector {x}_0 $; \\ \> \> $ \vector {r}_0 = \vector {p}_0 = A^T (\vector {b} - A \vector {x}_0) $; \\ \> \> $ \vector {q}_0 = A \vector {p}_0 $; \\ \> For $ k = 0, 1, \ldots $ \\ \> \> $ \alpha_{k+1} = (\vector {r}_k, \vector {r}_k) / (\vector {q}_k, \vector {q}_k) $; \\ \> \> $ \vector {x}_{k+1} = \vector {x}_k + \alpha_{k+1} \vector {p}_k $; \\ \> \> $ \vector {s}_{k+1} = \vector {s}_k - \alpha_{k+1} \vector {q}_k $; \\ \> \> $ \vector {r}_{k+1} = A^T \vector {s}_{k+1} $; \\ \> \> $ \beta_{k+1} = (\vector {r}_{k+1}, \vector {r}_{k+1}) / (\vector {r}_k, \vector {r}_k) $; \\ \> \> $ \vector {p}_{k+1} = \vector {r}_{k+1} + \beta_{k+1} \vector {p}_k $; \\ \> \> $ \vector {q}_{k+1} = A \vector {p}_{k+1} $; \end {tabbing} If arithmetic is done with infinite precision, the conjugate gradient algorithm converges in $ n $ iterations. In practice, the accumulation of roundoff errors requires extra iterations. We observed 4 -- 5 $ \times \ n $ iterations being required for satisfactory convergence. Initially, we replaced each of the scalar products $ ( \cdot , \cdot ) $ by a call to the scalar product from ACRITH. Calculations such as $ \vector {b} - A \vector {x} $ or $ \vector {x} + \alpha \vector {p} $ are potentially subject to catastrophic cancellation, so each was reformulated to use the ACRITH scalar product. It is critical to evaluate the expression \begin {eqnarray} \vector {x}_{k+1} = \vector {x}_k + \alpha_{k+1} \vector {p}_k = \vector {x}_0 + \sum_{j=0}^k \alpha_{j+1} \vector {p}_j. \label {x_k+1} \end {eqnarray} accurately. By storing $ \vector {p}_0, \ldots , \vector {p}_k $, each component of $ \vector {x}_{k+1} $ can be computed with a single rounding error. Similarly, each component of $ \vector {s}_{k+1} $ can be computed with a single rounding error. This requires storing all previous iterates and negates one of the advantages of the conjugate gradient algorithm. We developed a compromise algorithm which retained about 10 previous iterates. It is common to design algorithms to achieve high accuracy by reformulating a problem as a residual correction. Suppose we wish to compute the double precision value $ y = \sum_{j=1}^n u_j v_j $ as for each component of $ \vector {x}_{k+1} $ in Equation (\ref{x_k+1}). The value of $ y $ is computed with only one rounding error, but suppose that still introduces too much error into succeeding calculations. If so, we compute $ y_1 = \sum_{j=1}^n u_j v_j $ and then $ y_2 = - y_1 + \sum_{j=1}^n u_j v_j $ as a single scalar product. The effect is that $ y_1 + y_2 $ is a quadruple precision value of $ y $. Later computations use $ y_1 + y_2 $ in place of $ y $. We applied this trick to the computation of $ \vector {x}_{k+1} $ and $ \vector {s}_{k+1} $. We had eliminated several different sources of accumulation of rounding error from the original program, so we ran the modified program with great anticipation. The resulting vectors $ \vector {x}_k $ agreed to several figures with those calculated previously. That is, there was no significant improvement. Close inspection of the intermediate results showed that while the vectors $ \vector {x}_k $ agreed with those calculated previously, the search directions $ \vector {p}_k $ and the residual vectors $ \vector {r}_k $ were completely different after the first few iterations. When we checked the orthogonalities $$ (\vector {r}_j, A \vector {r}_{k+1}) = 0 \mbox { for } j = 0, \ldots , k $$ predicted by the theory, the ``accurate'' residuals were only slightly more nearly orthogonal than the original residuals. This suggests that to improve the accuracy of the algorithm, we must do a partial reorthonormalization. What did we learn from this attempt to apply the ACRITH scalar product to the conjugate gradient algorithm? First, we were able to replace calls to conventional scalar product routines by calls to the ACRITH scalar product routine. Second, we learned about the behavior of the algorithm. We now understand that the source of the inaccuracies is a gradual deterioration in the orthonormality of the residual vectors. Finally, we are reminded that hard problems are hard. The scalar product from ACRITH is a useful tool to achieve increased accuracies, but it is not a magic wand. To use the tool effectively, we must understand the algorithm to which it is being applied. A mechanical replacement of one scalar product by another is not necessarily enough. Since the human time required to use these methods is an issue, we remark that these results have required about 3 man-days of an expert on ACRITH who was unfamiliar with the conjugate gradient algorithm. \subsect {2.5. An example - Hat matrices} Another example using ACRITH's high accuracy arises in computing Allen's PRESS criterion for selecting variables in regression \cite{Alle74a}. Let $ A $ be an $ m \times n $ matrix of derivatives of a model function with respect to $ n $ model parameters at $ m $ values of the independent variable. Problems of interest are about $ 100 \times 5 $. The PRESS criterion requires the diagonal elements of the $ m \times m $ ``hat'' matrix \begin {equation} H = A (A^T A)^{-1} A^T. \label {Hat} \end {equation} The sum of the diagonal elements of $ H $ is $ n $, and $ H $ is idempotent ($ H \cdot H = H $). These two properties can be used to check the computed values. Hoaglin and Welsch \cite{Hoag78a} suggest using Householder transformations or singular value decompositions to minimize roundoff errors. If possible, we preferred a simpler implementation based on the definition for ease of coding and maintenance. Neither single nor double precision routines from the Harwell or IMSL libraries were sufficiently accurate, even for systems as small as $ 10 \times 4 $. When the calls to Harwell or IMSL library routines are replaced by calls to double precision ACRITH routines, the sum of the diagonal elements is $ n $, and $ H $ is idempotent to full machine accuracy. The human and machine time required is an issue. The engineer who did this work remarks, ``In my application, the extra time required by ACRITH routines is insignificant in comparison to the time it would have required to program more complex methods and to maintain the program as changes in FORTRAN, etc. occur. It certainly is less than the clock and CPU time I would have wasted by now in trying to make sense of less accurate computed results.'' If the accumulation of errors is still too great using ACRITH routines to evaluate $ H $, Equation (\ref{Hat}) can be recast as a block system in $ m \times m + n \times m $ unknowns \begin {equation} \left ( \begin {array}{cc} I & -A \\ A^T \end {array} \right ) \left ( \begin {array}{c} H \\ V \end {array} \right ) = \left ( \begin {array}{c} 0 \\ A^T \end {array} \right ). \label {HAT} \end {equation} Then DLIN is called to solve Equation (\ref{HAT}). % derivation: % H = A (A^T A)^{-1} A^T % % H = A V H is m \times m; V is n \times m % V = (A^T A)^{-1} A^T % (A^T A) V = A^T % % A V = W implies W = H % A^T W = A^T H = A^T % % H V % I -A 0 m % A^T A^T n % m n % % The unknowns are the elements of H and V. That is, this % matrix equation should be viewed in terms of the column vectors % of H and V. \subsect {2.6. How should you use techniques for high accuracy?} Scientists and engineers can begin at once to apply tools for high accuracy computation to their industrial problems. \subsubsub {Use accurate scalar product in new or existing code.} The accurate scalar product can be profitably used whenever a scalar or dot product is required. Both authors of new development codes and those maintaining existing codes can look for code segments which sum vectors or compute scalar products. Replacing conventional code with a call to the appropriate ACRITH scalar product routine improves the accuracy. Depending on the host hardware, the execution speed may also be improved. This strategy is similar to that of seeking to recognize scalar products in algorithms being implemented on vector or parallel machines. In both cases, the strategy works because the scalar product is treated as an elementary operation by the machine. \subsubsub {Use high accuracy general-purpose problem solving routines.} ACRITH provides several powerful, general-purpose problem solving routines which can serve as major building blocks. Examples include solving linear equations, eigenvalue/eigenvector analysis, polynomial rootfinding, and solving nonlinear equations. These routines can be used modularly in the same manner as routines from the NAG or IMSL libraries. Hence, a scientist or engineer can include the ACRITH library among the libraries searched for existing software to help solve applications problems. \subsubsub {Use simpler methods before more complicated ones.} The accurate scalar product from ACRITH can be used to make algorithms which have historically been discarded as numerically unstable behave satisfactorily. Examples include solving a least squares approximation problem using the normal equations or by Gram-Schmidt orthonormalization. Such methods are poorly conditioned using conventional arithmetic. However, they may be well behaved when carefully programmed using the accurate scalar product. The simpler methods may work using accurate arithmetic, and they are much easier to program. Of course, some bad methods are still bad methods, even with the accurate scalar product. \subsect {2.7. What are the costs?} The primary cost of using the ACRITH tools for high accuracy is the human cost of learning to use them. Those costs are highly variable and difficult to quantify. Learning costs may be minimal, since one form of a scalar product is being replaced with another. Learning costs may be modest, since problem solving routines from the ACRITH library replace similar routines from other libraries. Learning costs also may be large, since new programming tools are replacing familiar ones. Further, a fundamental analysis of the algorithm being used is sometimes required to achieve the potential benefits of the accurate scalar product. \beginsection {3. When should interval techniques be considered?} Now we turn our attention to our main focus: interval techniques for computing validated inclusions. We address a scientist or engineer who is acquainted with conventional point numerical algorithms and who is interested in appropriate uses of interval algorithms for their problems. Interval techniques should be considered in two circumstances: \begin {enumerate} \item Accuracy of the computed answer is in doubt. \item Input data or parameters in the problem are uncertain. \end {enumerate} We consider each of these circumstances separately. \subsect {3.1. Accuracy of the computed answer is in doubt.} Interval techniques should be considered whenever to provide some assurance about how accurately the answer is computed. An interval answer $ [1.23456, 1.23457] $ provides assurance that the answer is known correctly to 6 decimal places, while a point answer 1.23456 may have the wrong sign. The accuracy of many scientific and engineering calculations is in doubt. Sections 4 and 5 give two examples. \subsect {3.2. Uncertain data or parameters.} The second circumstance in which interval techniques should be considered are problems whose input data or parameters are uncertain. Often, data are known to be accurate to only 3 or 4 decimal places. How does the uncertainty in the data contribute to uncertainty in the answer? This question is especially important when the problem is known to be ill conditioned. The question may be ignored, or it may be addressed by hard analysis, by simulations, or by appealing to experience. The analysis is often difficult or impossible; simulations are expensive, especially in many dimensions; and experience may be lacking or misleading. In contrast, interval techniques provide guarantees, and they may be applied almost automatically. If we start with intervals in which the input data or parameters must lie, then computations using interval arithmetic give intervals in which the answers must lie. \beginsection {4. An example - Least squares} Interval techniques should be considered when the accuracy of the computed solution is in doubt. For example, we asserted in Section 2.4 that the program in Listing 1 solved a least squares problem accurately. We gave as evidence a residual of 4.02E-14. However, a small residual does not assure a small relative error in the solution, and the conventional wisdom suggests that it is not possible to solve the normal equations as accurately as we claim to have done. Hence, the accuracy of our answer is in doubt. The ACRITH routine DLIN in Listing 1 computes an interval [XL(J), XU(J)] containing each component of the solution: \begin {verbatim} J [XL(J), XU(J)] 1 ( 0.1044068767878713D+02, 0.1044068767878714D+02 ) 2 ( 0.1182954682872744D+02, 0.1182954682872745D+02 ) . . . 177 ( -0.1808366597477822D+02, -0.1808366597477821D+02 ) 178 ( 0.6148360648647853D-01, 0.6148360648647854D-01 ) \end{verbatim} \begin {center} Listing 3. Inclusions of least squares solution. \end {center} \vspace {10 pt} For each of these intervals, XL(J) differs from XU(J) by one unit in the last hexadecimal place. These interval answers verify that we have computed the answers with full double precision accuracy. This example illustrates the power of interval techniques for automated error analysis. An interval answer carries with it a guarantee of its uncertainty. By contrast, a point value carries no measure of its uncertainty. Even a sound error analysis can only estimate the error. Interval techniques should also be considered when input data or parameters in the problem are uncertain. For example, if we replace the point valued coefficients of $ A $ in Equation (\ref{Least_Squares}) by intervals whose endpoints differ in the 4th decimal place, we get intervals containing the solution. The width of the solution intervals is a measure of the uncertainty in the answer as a result of uncertainty in the data. This is a type of sensitivity analysis which can replace costly, repeated simulation runs. However, if the interval coefficients of $ A $ in our problem are only one double precision unit wide, ACRITH's linear equation solver finds that $ A $ contains a singular matrix. That is, the data differs from a singular problem by less than double precision accuracy. Since the data is certainly not known with double precision accuracy, the tight inclusions shown in Listing 3 are meaningless. In the terminology of \cite{Kuli88a}, we have solved the specified least squares problem with high accuracy, but the specified problem has no physical meaning. We would prefer to be assured that everything is fine, but if we are attempting to solve a meaningless problem, we welcome a warning of that fact. This example illustrates that \begin {enumerate} \item It is possible to solve poorly conditioned linear systems with high accuracy. \item It is possible to be guaranteed of the accuracy attained. \item ACRITH can tell us when our problems are too poorly conditioned relative to their input data to render any computed solution physically meaningless. \end {enumerate} \beginsection {5. An example - Nonlinear systems} Another example of the application of interval techniques is the Nordgren crack model \cite{Nord72a}. The underlying mathematical problem is a partial differential equation with a moving boundary. Let $ w(x,t) $ be the width of the crack at position $ x $ and time $ t $. Let $ L(t) $ be the length of the crack at time $ t $, and $ \tau (x) $ be the time required for the crack to reach length $ x $. Then $ w(x, t) $ satisfies the PDE: \begin {eqnarray} (w^4)_{xx} & = & w_t + \frac {1}{\sqrt {t - \tau (x)}}, \quad 0 \le x \le L(t), 0 \le t, \nonumber \\ -(w^4)_x(0, t) & = & 1 \quad \mbox {(flow from wellbore is constant)} \label {Nordgren} \\ w^4(x,0) & = & 0 \quad \mbox {(crack starts opening at $ t = 0 $)} \nonumber \\ w^4(L(t), t) & = & 0 \quad \mbox {(no flow at the crack tip)} \nonumber \\ -(w^4)_x (L(t), t) & = & \frac {dL}{dt} \cdot w(L(t), t) \nonumber \\ && \qquad \qquad \mbox {(fluid velocity equals tip propagation speed)} \nonumber \\ & \mbox {or} & \nonumber \\ -(w^4)_x (L(t), t) & = & 0 \quad \mbox {(no volume of fluid flow at the tip).} \nonumber \end {eqnarray} A standard algorithm uses a discretization in $ x $ and $ t $ to form finite difference approximations to the partial derivatives appearing in Equation (\ref{Nordgren}). At each time step $ t_j $, one must solve a nonlinear system of equations for $ w_{i,j} = w(x_i, t_j) $ using a Newton iteration. A conventional implementation of this algorithm gave results which agreed with asymptotic results and with results computed by Nordgren. However, after about 100 time steps, the solution failed when the program attempted to take the square root of a negative number. Consider the approximations made in modeling the crack: \begin {enumerate} \item The physical behavior of cracking rock is approximated by Equation (\ref{Nordgren}). \item Equation (\ref{Nordgren}) has exact solution $ w(x,t) $. \item Equation (\ref{Nordgren}) is approximated by finite difference equations. \item The finite difference equations have exact solution $ w_{i,j} $ which approximates $ w(x,t) $. \item At time step $ t_j $, $ w_{i,j} $ is approximated by a sequence of Newton iterates based on approximations to $ w_{i,j-1} $. \end {enumerate} No interval algorithms for Problem (\ref{Nordgren}) are known, but we can use interval techniques to explore the behavior of the point algorithm. We do not attempt to compute an inclusion for $ w(x, t) $. Instead, we take as our specified problem the finite difference equations which define $ w_{i,j} $. Suppose we assume that $ w_{i,j-1} $ at time step $ t_j $ are known exactly. Then the values $ w_{i,j} $ satisfy a system of $ j + 1 $ nonlinear equations. The interval Newton algorithm \cite{Moor79a} computes 1 ULP inclusions for $ w_{i,j} $. This gives inclusions for $ w_{i,j} $, not for $ w(x, t) $, and not for the width of the physical crack. Further, the algorithm of assumes that $ w_{i,j-1} $ are exact, which they are not. Hence, we have an inclusion of the answer which the point algorithm in Step 5 above should have computed. The interval program described in the preceding paragraph showed that as time passes, the Newton iteration becomes extremely sensitive to the initial guess for $ w_{i,j} $ used to start the iteration. The problem is not with the convergence of the Newton iteration, but that a very carefully constructed initial guess is required to evaluate the first Newton iteration. The interval algorithm gives us a better understanding of the behavior of the point algorithm, so we can improve its performance. Another approach is to view the entire set of $ w_{i,j} $ for $ 0 < j \le N $ as determined by a single system of $ (N + 1)(N + 2) / 2 $ nonlinear equations. Solving as a single system rather than solving one time step at a time allows us to compute inclusions for $ w_{i,j} $ without assuming that $ w_{i,j-1} $ are exact. Further, the widths at earlier times are automatically computed with sufficient accuracy to allow the widths at later times to be computed with 1 ULP inclusions. This is a slight improvement over the previous algorithm because it gives inclusions for all the $ w_{i,j} $ without assuming that some are known exactly. The result is {\em not\/} an inclusion for $ w(x,t) $ because we have taken the finite difference equations as our specified problem. To compute an inclusion for $ w(x,t) $ requires an interval algorithm, not just an interval version of a point algorithm. Interval algorithms for PDE's are under development. One could capture the truncation error in the finite difference approximations to the derivatives, but a better interval algorithm will probably result from an application of a maximum principle or from reformulating the PDE as an integral equation and applying a contractive mapping. The nonlinear equation solver in ACRITH was overly restrictive for this application, so we had to write our own. We estimate that it took approximately 60 -- 100 times as long to write our own interval Newton algorithm following \cite{Moor79a} than would be required to simply use an existing point nonlinear equation solver from the IMSL or NAG libraries. \beginsection {6. What are some limitations of interval techniques?} We do not pretend that interval techniques are the solution to all scientific computation problems. Among the difficulties involved in using interval techniques are: \begin {enumerate} \item What do we get an inclusion of? \item How do we develop interval algorithms? \item How do we develop interval models? \item Is it worth the cost? \end {enumerate} We address each of these in turn. \subsect {6.1. What do we get an inclusion of?} Interval problems should be interpreted in the sense of sets. For example, consider the interval valued system of linear equations \begin {equation} A \vector {x} = \vector {b}, \label {Interval_System} \end {equation} where the elements of the matrix $ A $ and of the vector $ \vector {b} $ are intervals. Let $ \tilde {A} $ be any point valued matrix such that $ \tilde {A}_{i,j} \in A_{i,j} $, and let $ \tilde {\vector {b}} $ be any point valued vector with $ \tilde {b}_i \in b_i $. Let $ \tilde {\vector {x}} $ be the solution to the point valued problem $ \tilde {A} \tilde {\vector {x}} = \tilde {\vector {b}} $. Then, the meaning of having an interval valued vector $ \vector {x} $ which solves Equation (\ref{Interval_System}) is that it must hold that $ \tilde {x}_i \in x_i $. That is, the solution to the interval problem contains the solution to every possible point problem. Having an interval answer does not guarantee that it includes anything of interest, as we saw in the discussion above of the Nordgren crack model. To achieve a meaningful inclusion requires careful mathematical reasoning at all stages of algorithm development and implementation. For example, suppose that we have a code which computes a numerical solution to a system of ordinary differential equations using a Runge-Kutta algorithm. Suppose that we are concerned about the accuracy of the computed solution, so we convert the point arithmetic in the code to interval arithmetic using calls to ACRITH routines. The resulting interval answers are {\em useless\/} for two reasons: \begin {enumerate} \item They enclose the Runge-Kutta approximation to the solution, not the solution itself. In order to enclose the solution, we must extend the algorithm to include an inclusion of the truncation error term. \item They are hopelessly wide. In order to achieve the desired {\em tight\/} inclusions, algorithms must be interval algorithms, not naive interval versions of point algorithms. \end {enumerate} Both of these reasons suggest that it is often necessary to develop algorithms which are designed from the beginning as interval algorithms. As a practical matter, it may not be necessary or cost-effective to develop an interval algorithm for an entire problem. Instead, interval techniques can often be profitably applied to small subproblems of a larger problem. For example, the original algorithm may require the solution of a linear system, a nonlinear system, or an optimization problem. If the accuracy or reliability of one of these subproblems is suspected, an interval solution of the subproblem is suggested. The result will not be an inclusion for the solution of the entire problem, but the result may provide increased confidence or insight. The example given above of the nonlinear system from a Nordgren crack model illustrates these points. There, we applied interval techniques the subproblem of solving the nonlinear system, either at one time step at a time, or for the entire time domain of interest. We were very careful to identify what we were computing an inclusion of. The use of interval techniques was valuable because we gained insights which allowed us to improve performance of our calculations. \subsect {6.2. How do we develop interval algorithms?} Good point algorithms rarely make good interval algorithms. Hence, when developing interval algorithms, it is necessary to discard much of the conventional wisdom of numerical analysis and perform a fresh analysis of each problem. In what follows, we give some useful hints. The development of good interval algorithms begins with a search of computer and written libraries for existing tools or algorithms which can be applied. The ACRITH library contains many useful problem solving routines. Interval algorithms often involve computing a point approximation, followed by computing either an interval inclusion near the approximation or an interval inclusion of the error of the approximation. When developing an algorithm using such a mix of point and interval valued objects, special care must be taken to keep track of which objects are points and which are intervals. It is usually helpful to view an interval as a primitive data object on which operations are performed using calls to ACRITH routines. In computer science jargon, intervals form an abstract data type. Thinking of the two endpoints as separate objects often leads to confusion. Fortran-SC \cite{Metz87a} supports interval as a primitive data type at the language level. Interval algorithms are developed to enclose mathematical results. For example, if $ a \in A = [1, 2] $, then $ a - a = 0 $, while $ A - A = [-1, 1] $. Similarly, if $ B = [-1, 2] $, then $ B * B = [-2, 4] $, while $ B^2 = [0, 4] $. Hence, equivalent mathematical expressions are not necessarily equivalent interval expressions. Fundamental tools for achieving a guaranteed inclusion use interval arithmetic and must capture of all sources of errors including modeling errors, approximation errors, truncation errors, discretization errors, and roundoff errors. Differentiation arithmetic \cite{Rall81a} is often helpful to capture derivatives evaluated at unknown points which appear in error terms for many algorithms. Differentiation arithmetic uses recurrence relations for the fast, accurate generation of the terms of the Taylor series. Achieving an inclusion is often a separate issue from achieving a {\em tight\/} inclusion. Mathematical ``tricks'' which are commonly used in good interval algorithms to achieve inclusions which are tight include \begin {itemize} \item interval arithmetic, \item formula + remainder, \item subinterval adaptation, \item monotonicity, \item use of point values, \item high order methods, \item differentiation arithmetic, \item intersection of multiple inclusions, and \item contractive iteration. \end {itemize} In \cite{Corl88a}, Corliss illustrates these techniques using several algorithms for computing bounds for the range of a function on an interval. \subsect {6.3. How do we develop interval models?} Pursuit of an interval strategy often affects the way we build the original mathematical models. In general, we want to solve a problem as close as possible to the original one. If we formulate problem $ P_1 $, do some approximation to get problem $ P_2 $, do some further approximation to get problem $ P_3 $, and finally solve several subproblems, we have many sources of error which must be bounded in order to get a guaranteed inclusion of the solution to the original problem. It is usually better to solve problem $ P_1 $, if we can. We cannot always solve $ P_1 $. Even when we can, we may not be able to compute an inclusion. As a compromise, it may be useful to find guaranteed inclusions of the solution to some subproblems or of some approximate solution to the original problem. In this case, be careful to understand clearly what the answer is an inclusion of. Pursuit of an interval strategy often encourages one to develop models in terms of lower and upper bounds. This is especially useful when details of the physical problem are uncertain but can be shown to have little affect on the outcome. \subsect {6.4. Is it worth the cost?} Interval algorithms incur costs for learning an unfamiliar approach, for programmer time, and for CPU time. Interval techniques are not usually as well known among scientists and engineers as point techniques, so there is a cost associated with learning new approaches and new tools. There are fewer ready-made interval programs in either computer software libraries or in the literature. Further, an interval algorithm is usually more difficult to write than a point algorithm for the same problem. For these reasons, the design and implementation of a complete integral algorithm may require 4 -- 8 times as much programmer time as the design and implementation of a point algorithm. If the choice is between designing and writing your own interval algorithm and using an existing high quality point routine, the cost may be a factor of hundreds. Hopefully, this will be less frequent as more high quality interval routines are developed. This cost should be balanced against the value of increased insight provided by verified bounds for the computed answers. An interval algorithm carries lower and upper bounds throughout its calculations. Hence, it nearly always requires twice as much CPU time as a corresponding point algorithm. In practice, a good interval algorithm typically takes about 3 -- 5 times as much CPU time as a good point algorithm. It is possible for an interval algorithm to be faster than a point algorithm, especially when a tolerance is requested, because the interval program can determine exactly when the requested tolerance is met. It is also possible for an interval algorithm to take much longer. The increased CPU cost should be balanced against the cost of repeated runs for simulation, with perturbed data, or in higher precisions. \beginsection {7. What should you DO?} We recommend: \begin {itemize} \item Determine whether the ACRITH accurate scalar product routines are faster or slower than the corresponding DO loop on your machine. If they are faster, use them whenever possible. \item Use ACRITH accurate scalar product routines for vector summations and scalar products. \item Include the ACRITH library among the libraries you customarily search when you are looking for existing software to help solve your problem. \item Consider older, simpler algorithms whose bad reputation is due to an accumulation of roundoff errors which can be minimized by an accurate scalar product. \item Determine whether the potential benefits of ACRITH's high accuracy tools justify your effort to learn how to use them by applying them first to small problems to gain experience in your own setting. \item Learn to recognize problems for which interval techniques should be considered because you would like to know the accuracy of the answer you have computed. \item Learn to recognize problems for which interval techniques should be considered because the input data or the parameters in the problem are uncertain. \item Consider interval techniques as an alternative to repeated simulation runs for sensitivity analysis. \item Apply interval techniques to small subproblems of larger, more complicated problems. \end {itemize} The following references are good places to start. R. E. Moore, {\sl Methods and Applications of Interval Analysis,\/} \cite{Moor79a} and G. Alefeld and J. Herzberger, {\sl Introduction to Interval Computations,\/} \cite{Alef83a} are good introductory surveys of interval methods. K. Nickel, ed., {\sl Interval Mathematics, 1975,\/} \cite{Nick75a}, {\sl Interval Mathematics, 1980,\/} \cite{Nick80a}, and {\sl Interval Mathematics, 1985,\/} \cite{Nick85a} are each collections of papers presented at a prestigious conference held in Freiburg. R. E. Moore, ed., {\sl Reliability in Computing: The Role of Interval Methods in Scientific Computing,\/} \cite{Moor88a} is a collection of 22 research articles in the field. The ACRITH User's Guide \cite{IBM86a} and the Pascal-SC User's Manual \cite{Kuli87a} are excellent descriptions of two commercially available products which make interval tools widely accessible. \beginsection {Acknowledgments} The author wishes to thank a client whose wish to remain anonymous prevents giving the appropriate credit for work done by several of its scientists. Professor Edgar Kaucher suggested the matrix reformulations in Section 2. The author also wishes to thank Professors L. B. Rall and Gary S. Krenz for many helpful discussions. \begin {thebibliography}{99} \bibitem {Alle74a} Allen, D. M. The relationship between variable selection and data augmentation and a method for prediction, {\sl Techonometrics,\/} {\bf 16} (1974), 125--127. \bibitem {Alef83a} Alefeld, G.; and Herzberger, J. {\sl Introduction to Interval Computations,\/} Academic Press, New York 1983. \bibitem {Corl88a} Corliss, George F. How can you apply interval techniques in an industrial setting? Technical Report No. 301, Department of Mathematics, Statistics and Computer Science, Marquette University, Milwaukee, WI 1988. \bibitem {Delv87a} Delves, L. M.; Ford, B.; and Hodgson, G. S. The MAP 750 Ada Library Project, {\sl NAG Newsletter,\/} {\bf 2} (1987), 29--38. \bibitem {Hest52a} Hestenes, M.; and Stiefel, E. Methods of conjugate gradients for solving linear systems, {\sl Nat. Bur. Standards J. Res.,\/} {\bf 49} (1952), 409--436. \bibitem {Hoag78a} Hoaglin, David C.; and Welsch, Roy E. The hat matrix in regression and ANOVA, {\sl The American Statistician,\/} {\bf 32} (1978) 1, 17--22. \bibitem {IBM86a} IBM. {\sl High-Accuracy Arithmetic Subroutine Library (ACRITH),\/} Program Description and User's Guide, SC 33--6164--2, 3rd Edition 1986. \bibitem {Kuli81a} Kulisch, Ulrich W.; and Miranker, Willard L. {\sl Computer Arithmetic in Theory and Practice,\/} Academic Press, New York 1981. \bibitem {Kuli83a} Kulisch, Ulrich W.; and Miranker, Willard L. {\sl A New Approach to Scientific Computation,\/} Academic Press, New York 1983. \bibitem {Kuli87a} Kulisch, Ulrich W. (ed.). {\sl Pascal-SC: A PASCAL Extension for Scientific Computation,\/} Information Manual and Floppy Disks, Teubner, Stuttgart, and Wiley, New York 1987. \bibitem {Kuli88a} Kulisch, Ulrich W.; and Stetter, Hans J. Automatic result verification. In: Kulisch, Ulrich W.; and Stetter, Hans J. (eds.): {\sl Scientific Computation with Automatic Result Verification,\/} Springer, New York 1988, pp. 1 -- 8. \bibitem {Metz87a} Metzger, M. FORTRAN-SC, A FORTRAN extension for engineering/scientific computing with access to ACRITH. In: Moore, Ramon E. (ed.): {\sl Reliability in Computing: The Role of Interval Methods in Scientific Computing,\/} Academic Press, New York 1988, pp. 63--80. \bibitem {Moor66a} Moore, Ramon E. {\sl Interval Analysis\/}, Prentice-Hall, Englewood Cliffs, NJ 1966. \bibitem {Moor79a} Moore, Ramon E. {\sl Methods and Applications of Interval Analysis,\/} SIAM, Philadelphia, PA 1979. \bibitem {Moor88a} Moore, Ramon E. (ed.). {\sl Reliability in Computing: The Role of Interval Methods in Scientific Computing,\/} Academic Press, New York 1988. \bibitem {Nick75a} Nickel, Karl L. E. (ed.). {\sl Interval Mathematics, 1975,\/} Springer, New York 1975. \bibitem {Nick80a} Nickel, Karl L. E. (ed.). {\sl Interval Mathematics, 1980,\/} Academic Press, New York 1980. \bibitem {Nick85a} Nickel, Karl L. E. (ed.). {\sl Interval Mathematics, 1985,\/} Springer, New York 1985. \bibitem {Nord72a} Nordgren, R. P. Propagation of vertical hydraulic fracture, {\sl Society of Petroleum Engineers Journal\/} {\bf 12} (1972) 4, 306--314. \bibitem {Rall81a} Rall, Louis B. {\sl Automatic Differentiation: Techniques and Applications,\/} Lecture Notes in Computer Science No. 120, Springer, Berlin 1981. \bibitem {Siem86a} Siemens. {\sl Arithmos, (BS 2000) Benutzerhandbuch,\/} U2900--J--Z87--1, Siemens, GmbH 1986. \end {thebibliography} \end {document} ----- End Included Message ----- From owner-reliable_computing Sat Dec 23 09:31:03 1995 Received: by interval.usl.edu id AA15007 (5.65c/IDA-1.4.4 for reliable_computing-outgoing); Sat, 23 Dec 1995 03:03:04 -0600 Received: from c220.unimo.it by interval.usl.edu with SMTP id AA15001 (5.65c/IDA-1.4.4 for ); Sat, 23 Dec 1995 03:02:34 -0600 Received: from dsi.unimo.it (iago.dsi.unimo.it) by c220.unimo.it with ESMTP (1.39.111.2/16.2) id AA285336294; Sat, 23 Dec 1995 09:58:14 -0100 Received: (from benedett@localhost) by dsi.unimo.it (8.6.11/8.6.9) id IAA00112; Sat, 23 Dec 1995 08:31:03 +0100 Date: Sat, 23 Dec 1995 08:31:03 +0100 From: Arrigo Benedetti Message-Id: <199512230731.IAA00112 [at] dsi [dot] unimo.it> To: georgec [at] boris [dot] mscs.mu.edu Cc: reliable_computing [at] interval [dot] usl.edu In-Reply-To: (georgec [at] boris [dot] mscs.mu.edu) Subject: Re: Applications of intervals (long) Sender: owner-reliable_computing Precedence: bulk Date: Fri, 22 Dec 95 15:33 CST From: georgec [at] boris [dot] mscs.mu.edu (Dr. George F. Corliss MU MSCS) X-Sun-Charset: US-ASCII Sender: owner-reliable_computing [at] interval [dot] usl.edu Precedence: bulk Raj, > I am looking for help in collecting following information to convince > a group of people in using Interval Arithmetic for verified computing > in problems where safety issues are important in the design of a > physical system. Here are some stories I know. Interval techniques are no "silver bullet", but they CAN provide insight. [omitted] 7. I headr a wonderful talk by Wolfram Klein at SCAN 95 in Wuppertal about using interval techniques and automatic differentiation for a circuit simulation problem. [omitted] Dear Prof. Corliss, I am very interested in this point, as one of my interests is in the application of interval methods to circuit design and simulation problems. Could you please give me contact informations for Wolfram Klein, so I can get in touch with him? Thanks in advance, -Arrigo Benedetti -- Arrigo Benedetti e-mail: benedett [at] dsi [dot] unimo.it University of Modena abenedetti [at] deis [dot] unibo.it address: Via Vivaldi, 70 41100 MODENA - ITALY phone: (home) + 39 59 224929 (office) +39 59 304057 (fax) +39 59 220727 http://deis12.cineca.it/~benedett/ <-- under construction ! From owner-reliable_computing Sat Dec 23 05:32:49 1995 Received: by interval.usl.edu id AA15335 (5.65c/IDA-1.4.4 for reliable_computing-outgoing); Sat, 23 Dec 1995 11:33:21 -0600 Received: by interval.usl.edu id AA15325 (5.65c/IDA-1.4.4 for reliable_computing [at] interval [dot] usl.edu); Sat, 23 Dec 1995 11:32:49 -0600 Date: Sat, 23 Dec 1995 11:32:49 -0600 From: "Kearfott R. Baker" Message-Id: <199512231732.AA15325 [at] interval [dot] usl.edu> To: georgec [at] boris [dot] mscs.mu.edu, benedett [at] dsi [dot] unimo.it Subject: Re: Applications of intervals (long) Cc: reliable_computing [at] interval [dot] usl.edu Sender: owner-reliable_computing Precedence: bulk > Dear Prof. Corliss, > > I am very interested in this point, as one of my interests is in the > application of interval methods to circuit design and simulation problems. > Could you please give me contact informations for Wolfram Klein, so I can > get in touch with him? > While we are on this topic, I might add that Kohshi Okumura (kohshi [at] kuee [dot] kyoto-u.ac.jp) has applied interval methods very effectively to circuit problems. --------------------------------------------------------------- R. Baker Kearfott, rbk [at] usl [dot] edu (318) 482-5346 (fax) (318) 482-5270 (work) (318) 981-9744 (home) URL: ftp://interval.usl.edu/pub/interval_math/www/kearfott.html Department of Mathematics, University of Southwestern Louisiana --------------------------------------------------------------- From owner-reliable_computing Sat Dec 23 18:28:16 1995 Received: by interval.usl.edu id AA15511 (5.65c/IDA-1.4.4 for reliable_computing-outgoing); Sat, 23 Dec 1995 11:59:58 -0600 Received: from c220.unimo.it by interval.usl.edu with SMTP id AA15501 (5.65c/IDA-1.4.4 for ); Sat, 23 Dec 1995 11:59:38 -0600 Received: from dsi.unimo.it (iago.dsi.unimo.it) by c220.unimo.it with ESMTP (1.39.111.2/16.2) id AA015148521; Sat, 23 Dec 1995 18:55:21 -0100 Received: (from benedett@localhost) by dsi.unimo.it (8.6.11/8.6.9) id RAA00117; Sat, 23 Dec 1995 17:28:16 +0100 Date: Sat, 23 Dec 1995 17:28:16 +0100 From: Arrigo Benedetti Message-Id: <199512231628.RAA00117 [at] dsi [dot] unimo.it> To: rbk5287 [at] interval [dot] usl.edu Cc: georgec [at] boris [dot] mscs.mu.edu, reliable_computing [at] interval [dot] usl.edu In-Reply-To: <199512231732.AA15325 [at] interval [dot] usl.edu> (rbk5287 [at] interval [dot] usl.edu) Subject: Re: Applications of intervals (long) Sender: owner-reliable_computing Precedence: bulk Date: Sat, 23 Dec 1995 11:32:49 -0600 From: "Kearfott R. Baker" Cc: reliable_computing [at] interval [dot] usl.edu > Dear Prof. Corliss, > > I am very interested in this point, as one of my interests is in the > application of interval methods to circuit design and simulation problems. > Could you please give me contact informations for Wolfram Klein, so I can > get in touch with him? > While we are on this topic, I might add that Kohshi Okumura (kohshi [at] kuee [dot] kyoto-u.ac.jp) has applied interval methods very effectively to circuit problems. Nice to hear from you again Prof. Baker Kearfott, and thank you very much for this information. By the way, recently Prof. Neumaier, when asked to comment a paper that we wrote recently, suggested to read your papers on existence conditions for undetermined systems as a way to overcome some of the problems that we have with an algorithm for tracing solution curves of nonlinear circuits. If you are interested, I can send you a postscript copy of our paper, that will be presented at ISCAS'96 (IEEE International Symposium on Circuits and Systems). Let me give you my best wishes for a merry Xmas and happy New Year, -Arrigo Benedetti -- Arrigo Benedetti e-mail: benedett [at] dsi [dot] unimo.it University of Modena abenedetti [at] deis [dot] unibo.it address: Via Vivaldi, 70 41100 MODENA - ITALY phone: (home) + 39 59 224929 (office) +39 59 304057 (fax) +39 59 220727 http://deis12.cineca.it/~benedett/ <-- under construction ! From owner-reliable_computing Tue Dec 26 03:35:31 1995 Received: by interval.usl.edu id AA17529 (5.65c/IDA-1.4.4 for reliable_computing-outgoing); Tue, 26 Dec 1995 13:50:28 -0600 Received: from mercury.Sun.COM by interval.usl.edu with SMTP id AA17523 (5.65c/IDA-1.4.4 for ); Tue, 26 Dec 1995 13:50:22 -0600 Received: from Eng.Sun.COM by mercury.Sun.COM (Sun.COM) id LAA25954; Tue, 26 Dec 1995 11:49:29 -0800 Received: from roundoff.Eng.Sun.COM by Eng.Sun.COM (5.x/SMI-5.3) id AA12951; Tue, 26 Dec 1995 11:49:26 -0800 Received: by roundoff.Eng.Sun.COM (5.x/SMI-SVR4) id AA09085; Tue, 26 Dec 1995 11:35:31 -0800 Date: Tue, 26 Dec 1995 11:35:31 -0800 From: Douglas.Priest [at] eng [dot] sun.com (Douglas Priest) Message-Id: <9512261935.AA09085 [at] roundoff [dot] Eng.Sun.COM> To: reliable_computing [at] interval [dot] usl.edu Subject: Handling exceptions in interval operations Sender: owner-reliable_computing Precedence: bulk I am looking for references which describe methods for dealing with floating point exceptions, in particular IEEE 754 invalid operation exceptions, in basic interval arithmetic operations. Any information readers of this mailing list can provide would be greatly appreciated. Thank you, Douglas M. Priest From owner-reliable_computing Fri Dec 29 13:23:02 1995 Received: by interval.usl.edu id AA04360 (5.65c/IDA-1.4.4 for reliable_computing-outgoing); Fri, 29 Dec 1995 21:23:09 -0600 Received: from cs.utep.edu by interval.usl.edu with SMTP id AA04354 (5.65c/IDA-1.4.4 for ); Fri, 29 Dec 1995 21:23:05 -0600 Received: from earth.cs.utep.edu by cs.utep.edu (4.1/SMI-4.1) id AA02324; Fri, 29 Dec 95 20:23:02 MST Date: Fri, 29 Dec 95 20:23:02 MST From: vladik [at] cs [dot] utep.edu (Vladik Kreinovich) Message-Id: <9512300323.AA02324 [at] cs [dot] utep.edu> To: reliable_computing [at] interval [dot] usl.edu Subject: Happy New Year Sender: owner-reliable_computing Precedence: bulk * * * . * * * . . . . . . _ _ * * . * * * . (*) (*) . . * . * . * . | |_| | __ _ _ __ _ __ * _ _ . * . * * | _ | / _' )( '_ \ ( '_ \ ( ) ( ) . . . * |=| |=|(=(_|=||=(_)=)|=(_)=)|=\_/=| * * * (_) (_) \__,_)| ,__/ | ,__/ \__ ( . * | | | | ____) | * . . * . * . (_) (_) (_____/ * _ _ . * . . . _ . _ * * . _ * * (*) (*) . * . . (*) (*) . * (*) | \| | ____ _ _ _ \ \_/ / ____ __ _ _ __ | | * * | \ | / __ \( ) ( ) ( ) . \ / / __ \ / _' )( '__)|_| . . |=|\==|(=====/|=\_/=\_/=| |=| (=====/(=(_|=||=| = (_) (_) \____) \___^___/ * (_) \____) \__,_)(_) (_) * . * * . * . * . . . * . . * * . * *