http://interval.louisiana.edu/Moores_early_papers/bibliography.html
E. R. Hansen
Publications Related to Early Interval Work of R. E. Moore
August 13, 2001
Abstract
Interval analysis is often said to have begun with Moore's book [8].
The references below contain some of his earlier work. The references also
contain (or indicate) earlier works that appeared because of Moore's personal
influence. Moore's early papers are made available on this site to document
his early contributions and influence on the development of interval analysis.
1 The Record
In [9], Moore states that he conceived of interval arithmetic
and some of its ramifications in the spring of 1958. By January of 1959,
he had published [4] a report on how interval arithmetic
could be implemented on a computer. A 1959 report [12]
showed that interval computations could bound the range of rational functions
and integrals of rational functions. Theoretical and practical interval
arithmetic were differentiated. Reference [6] discusses
interval valued functions, interval contractions, a metric topology for
interval numbers, interval integrals, and contains an extensive discussion
of Moore's use of interval analysis to bound the solution of ordinary differential
equations. Further work on integrals appeared in [11].
Under Moore's direction and influence, general purpose interval arithmetic
became available for use on early computers (see [1]);
and a program for bounding the solution of ordinary differential equations
was produced (see [3], [10], [13],
[14]). The paper [7] was an early effort
to reduce the wrapping effect which was recognized and named by Moore.
Moore applied interval arithmetic to computational linear algebra problems
and observed the poor performance. A lecture of Moore's which noted
this fact stimulated Hansen to introduce the idea of preconditioning which
made it possible to obtain acceptable bounds for linear algebraic problems.
See [3].
Moore applied interval arithmetic to computational linear algebra problems
and observed the poor performance. A lecture of Moore's which noted this
fact stimulated Hansen to introduce the idea of preconditioning which made
it possible to obtain acceptable bounds for linear algebraic problems.
See [3].
Working under Moore's direction, other authors wrote reports on other
developments of interval analysis. See [13] for work
on bounding the remainder in Taylor expansions and other topics. See [2]
for a discussion of complex interval arithmetic. See also [14]
and [15].
Moore's early work at the Lockheed Research Labs. came to the attention
of George Forsythe (who became chairman of the world's first Computer Science
department). He invited Moore to do a Ph.D. dissertation on interval analysis
at Stanford University. Reference [5] was the resulting
dissertation.
Other early references related to interval analysis can be found in
Moore's book [8].
References

[1]

R. E. Boche. An
operational interval arithmetic. Chicago Ill., 1963. IEEEIllinois
Inst. of Tech.Northwestern Univ., Univ. of Illinois. Abstract of a paper
given at National Electronics Conference.

[2]

R. E. Boche. Complex
interval arithmetic with some applications. Technical Report Report
LMSC422661, Lockheed Missiles and Space Co., 1965.

[3]

M. E. Miller. Interval
arithmetic programs and references. Lockheed Missiles and Space Div.
Memo, 1965.

[4]

R. E. Moore. Automatic
error analysis in digital computation. Technical Report Space Div.
Report LMSD84821, Lockheed Missiles and Space Co., 1959.

[5]

R. E. Moore. Interval
Arithmetic and Automatic Error Analysis in Digital Computing. Ph.d.
Dissertation, Department of Mathematics, Stanford University, Stanford,
California, Nov. 1962. Published as Applied Mathematics and Statistics
Laboratories Technical Report No. 25.

[6]

R. E. Moore. The
automatic analysis and control of error in digital computing based on the
use of interval numbers. In L. B. Rall, editor, Error in Digital
Computation, Vol. I, chapter 2, pages 61130. John Wiley and Sons,
Inc., New York, 1965. John Wiley and Sons, Inc. has given permission for
this paper to be posted indefinitely on the nonpassword protected website
http://interval.louisiana.edu/Moores_early_papers/Moore_in_Rall_V1.pdf
for use by the interval research community.

[7]

R. E. Moore. Automatic
local coordinate transformations to reduce the growth of error bounds in
interval computation of solutions of ordinary differential equations.
volume II of Error in Digital Computation, pages 103140. John Wiley
and Sons, Inc., New York, New York, 1965. John Wiley and Sons, Inc. has
given permission for this paper to be posted indefinitely on the nonpassword
protected website http://interval.louisiana.edu/Moores_early_papers/Moore_in_Rall_V2.pdf
for use by the interval research community.

[8]

R. E. Moore. Interval Analysis. PrenticeHall, Englewood
Cliffs N. J., 1966.

[9]

R. E. Moore. The dawning. Reliable Computing, 5:423424,
1999.

[10]

R. E. Moore, J. A. Davison, H. R. Jaschke, and S. Shayer.
DIFEQ
integration routine  user's manual. Technical Report LMSC690646,
Lockheed Missiles and Space Co., 1964.

[11]

R. E. Moore, W. Strother, and C. T. Yang. Interval
integrals. Technical Report Space Div. Report LMSD703073, Lockheed
Missiles and Space Co., 1960.

[12]

R. E. Moore and C. T. Yang. Interval
analysis I. Technical Report Space Div. Report LMSD285875, Lockheed
Missiles and Space Co., 1959.

[13]

A. Reiter. Interval
arithmetic package (INTERVAL). Technical Report Report, Univ. of Wisconsin
Mathematics Research Center, 1965.

[14]

A. Reiter. Programming
interval arithmetic and applications. In Proceedings of the 1967
Army Numerical Analysis Conference, AROD Report 673, 1967.

[15]

S. Shayer. Interval
arithmetic with some applications for digital computers. Technical
Report Report LMSD5136512, Lockheed Missiles and Space Co., 1965.
Acknowledgement 1 Thanks to John Wiley & Sons, Inc. for permission
to post a copy of [6], and to Sun Microsystems, Inc. for
document scanning and funding the preparation of this note.
© 2001 Sun Microsystems, Inc. All rights reserved. Sun, Sun Microsystems,
and the Sun Logo are trademarks or registered trademarks of Sun Microsystems,
Inc. in the United States and other countries.
Addendum
(R. B. Kearfott, January 5, 2007)
File translated from T_{E}X by T_{T}H,
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On 28 Sep 2001, 16:38.