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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Mathematics of Computation is 1.98.

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On measures of ill-conditioning for nonlinear equations
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by Werner C. Rheinboldt PDF
Math. Comp. 30 (1976), 104-111 Request permission

Abstract:

Let ${x^\ast }$ be a solution of the nonlinear equation $Fx = b$ on a normed linear space and ${y^\ast }$ that of a perturbed equation $Gx = c$. Estimates for the relativized error between ${x^\ast }$ and ${y^\ast }$ are derived which extend a known estimate for the corresponding matrix case. The condition number of F depends now also on the domain, and special considerations are needed to determine the existence of the solution of the perturbed equation. For differentiable F, when the domain shrinks to a point, the condition number of F is shown to reduce to that of the derivative at that point.
References
  • George J. Fix and Kate Larsen, On the convergence of SOR iterations for finite element approximations to elliptic boundary value problems, SIAM J. Numer. Anal. 8 (1971), 536–547. MR 293859, DOI 10.1137/0708051
  • J. M. Ortega and W. C. Rheinboldt, Iterative solution of nonlinear equations in several variables, Academic Press, New York-London, 1970. MR 0273810
  • R. S. Varga, Functional analysis and approximation theory in numerical analysis, Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 3, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1971. MR 0310504
  • J. H. Wilkinson, Rounding errors in algebraic processes, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1963. MR 0161456
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Additional Information
  • © Copyright 1976 American Mathematical Society
  • Journal: Math. Comp. 30 (1976), 104-111
  • MSC: Primary 65J05
  • DOI: https://doi.org/10.1090/S0025-5718-1976-0400702-1
  • MathSciNet review: 0400702