On measures of ill-conditioning for nonlinear equations

Author:
Werner C. Rheinboldt

Journal:
Math. Comp. **30** (1976), 104-111

MSC:
Primary 65J05

MathSciNet review:
0400702

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Abstract: Let be a solution of the nonlinear equation on a normed linear space and that of a perturbed equation . Estimates for the relativized error between and 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.

**[1]**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**0293859****[2]**J. M. Ortega and W. C. Rheinboldt,*Iterative solution of nonlinear equations in several variables*, Academic Press, New York-London, 1970. MR**0273810****[3]**R. S. Varga,*Functional analysis and approximation theory in numerical analysis*, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1971. Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 3. MR**0310504****[4]**J. H. Wilkinson,*Rounding errors in algebraic processes*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1963. MR**0161456**

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DOI:
https://doi.org/10.1090/S0025-5718-1976-0400702-1

Article copyright:
© Copyright 1976
American Mathematical Society