Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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



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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • 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
  • 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, 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
  • J. H. Wilkinson, Rounding errors in algebraic processes, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1963. MR 0161456

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65J05

Retrieve articles in all journals with MSC: 65J05

Additional Information

Article copyright: © Copyright 1976 American Mathematical Society