Skip to Main Content

Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Rigorous sensitivity analysis for systems of linear and nonlinear equations
HTML articles powered by AMS MathViewer

by Siegfried M. Rump PDF
Math. Comp. 54 (1990), 721-736 Request permission

Abstract:

Methods are presented for performing a rigorous sensitivity analysis of numerical problems with independent, noncorrelated data for general systems of linear and nonlinear equations. The methods may serve for the following two purposes. First, to bound the dependency of the solution on changes in the input data. In contrast to condition numbers, a componentwise sensitivity analysis of the solution vector is performed. Second, to estimate the true solution set for problems whose input data are subject to tolerances. The methods presented are very effective and have the additional property that, owing to an automatic error control mechanism, every computed result is guaranteed to be correct. Examples are given for linear systems, demonstrating that the computed bounds are in general very sharp. Interesting comparisons to traditional condition numbers are given.
References
    ACRITH, High-Accuracy Arithmetic Subroutine Library, Program Description and User’s Guide, IBM Publications, Document Number SC 33-6164-3, 1986.
  • Götz Alefeld and Jürgen Herzberger, Introduction to interval computations, Computer Science and Applied Mathematics, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983. Translated from the German by Jon Rokne. MR 733988
  • H. Bauch, K.-U. Jahn, D. Oelschlägel, H. Süsse, and V. Wiebigke, Intervallmathematik, Mathematisch-Naturwissenschaftliche Bibliothek [Mathematical-Scientific Library], vol. 72, BSB B. G. Teubner Verlagsgesellschaft, Leipzig, 1987 (German). Theorie und Anwendungen. [Theory and applications]. MR 927085
  • Walter Baur and Volker Strassen, The complexity of partial derivatives, Theoret. Comput. Sci. 22 (1983), no. 3, 317–330. MR 693063, DOI 10.1016/0304-3975(83)90110-X
  • IEEE 754 Standard for Floating-Point Arithmetic, 1986. W. Kahan and E. LeBlanc, Anomalies in the IBM ACRITH package, Proc. 7th Sympos. on Computer Arithmetic (Kai Hwang, ed.), Urbana, Illinois, 1985.
  • R. Krawczyk, Newton-Algorithmen zur Bestimmung von Nullstellen mit Fehlerschranken, Computing (Arch. Elektron. Rechnen) 4 (1969), 187–201 (German, with English summary). MR 255046, DOI 10.1007/bf02234767
  • R. Krawczyk and A. Neumaier, Interval slopes for rational functions and associated centered forms, SIAM J. Numer. Anal. 22 (1985), no. 3, 604–616. MR 787580, DOI 10.1137/0722037
  • Ulrich Kulisch, Grundlagen des numerischen Rechnens, Mathematische Begründung der Rechnerarithmetik. Reihe Informatik, Band 19, Bibliographisches Institut, Mannheim-Vienna-Zurich, 1976 (German). MR 0426403
  • Ulrich W. Kulisch and Willard L. Miranker, Computer arithmetic in theory and practice, Computer Science and Applied Mathematics, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. MR 606741
  • —, (eds.), A new approach to scientific computation, Academic Press, New York, 1981.
  • Ramon E. Moore, Interval analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1966. MR 0231516
  • R. E. Moore, A test for existence of solutions to nonlinear systems, SIAM J. Numer. Anal. 14 (1977), no. 4, 611–615. MR 657002, DOI 10.1137/0714040
  • —, Methods and applications of interval analysis, SIAM, Philadelphia, 1979.
  • A. Neumaier, Overestimation in linear interval equations, SIAM J. Numer. Anal. 24 (1987), no. 1, 207–214. MR 874746, DOI 10.1137/0724017
  • Arnold Neumaier, Rigorous sensitivity analysis for parameter-dependent systems of equations, J. Math. Anal. Appl. 144 (1989), no. 1, 16–25 (English, with German summary). MR 1022559, DOI 10.1016/0022-247X(89)90357-0
  • L. B. Rall, Automatic differentiation: Techniques and applications. Lecture Notes in Computer Science, no. 120, Springer-Verlag, Berlin, Heidelberg, and New York, 1981. S. M. Rump, Solving algebraic problems with high accuracy, in A New Approach to Scientific Computation (U. Kulisch and W. L. Miranker, eds.), Academic Press, New York, 1981, pp. 51-120.
  • S. M. Rump, Solving nonlinear systems with least significant bit accuracy, Computing 29 (1982), no. 3, 183–200 (English, with German summary). MR 680469, DOI 10.1007/BF02241697
  • Siegfried M. Rump, New results on verified inclusions, Accurate scientific computations (Bad Neuenahr, 1985) Lecture Notes in Comput. Sci., vol. 235, Springer, Berlin, 1986, pp. 31–69. MR 868284, DOI 10.1007/3-540-16798-6_{4}
  • B. Speelpennig, Compiling fast partial derivatives of functions given by algorithms, Ph. D. thesis, University of Illinois, Urbana, Illinois, 1980. P. Wongwises, Experimentelle Untersuchungen zur numerischen Auflösung von linearen Gleichungs-systemen mit Fehlererfassung, Interner Bericht 75/1, Institut für Praktische Mathematik, Universität Karlsruhe, 1975.
Similar Articles
Additional Information
  • © Copyright 1990 American Mathematical Society
  • Journal: Math. Comp. 54 (1990), 721-736
  • MSC: Primary 65G10; Secondary 65G05, 65H10
  • DOI: https://doi.org/10.1090/S0025-5718-1990-1011445-5
  • MathSciNet review: 1011445