http://interval.louisiana.edu/Moores_early_papers/bibliography.html

# Publications Related to Early Interval Work of R. E. Moore

## 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. IEEE-Illinois 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 LMSC4-22-66-1, 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 61-130. 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 non-password 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 103-140. 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 non-password 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. Prentice-Hall, Englewood Cliffs N. J., 1966.
[9]
R. E. Moore. The dawning. Reliable Computing, 5:423-424, 1999.
[10]
R. E. Moore, J. A. Davison, H. R. Jaschke, and S. Shayer. DIFEQ integration routine - user's manual. Technical Report LMSC6-90-64-6, 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, ARO-D Report 67-3, 1967.
[15]
S. Shayer. Interval arithmetic with some applications for digital computers. Technical Report Report LMSD5-13-65-12, 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.