Abstract
Suppose that a method ϕ computes an approximation of the exact solution of a linear systemAx=b with the relative errorq,q<1. We prove that if all computations are performed in floating point arithmeticfl and single precision, then ϕ with iterative refinement is numerically stable and well-behaved wheneverq∥A∥ ∥A −1∥ is at most of order unity.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Å. Björck,Solving Linear Least Squares Problems by Gram-Schmidt Orthogonalization, BIT (1967), 1–21.
A. Kieybasinski,Numerical Analysis of the Gram-Schmidt Orthogonalization Algorithm, Mat. Stosowana 2 (1974), 15–35 (in Polish).
C. B. Moler,Iterative Refinement in Floating Point, Journal of the ACM, Vol. 14 (1967), 316–321.
P. Pohl,Iterative Improvement Without Double Precision in a Boundary Value Problem, BIT 14 (1974), 361–365.
J. H. Wilksinson,Rounding Errors in Algebraic Processes, Prentice-Hall, Englewood Cliffs, N.J., 1963.
J. H. Wilkinson,The Algebraic Eigenvalue Problem, Clarendon Press, Oxford, 1965.
H. Woźniakowski,Numerical Stability of the Chebyshev Method for the Solution of Large Linear Systems, Dept. of Computer Science Report, Carnegie-Mellon University, 1975.
H. Woźniakowski,Round-off Error Analysis of Iterations for Large Linear Systems, Dept. of Computer Science Report, Carnegie-Mellon University, 1977, to appear.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Jankowski, M., Woźniakowski, H. Iterative refinement implies numerical stability. BIT 17, 303–311 (1977). https://doi.org/10.1007/BF01932150
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01932150