Iterative refinement implies numerical stability

Abstract

Abstract.Suppose that a method q~ computes an approximation of the exact solution of a linear system Ax=b with the relative error q, q < 1. We prove that if all computations are performed in floating point arithmetic fl and single precision, then q~ with iterative refinement is numerically stable and well-behaved whenever qHAll ItA-1LL is at most of order unity.

Newton's Method in Floating Point Arithmetic and Iterative Refinement of Generalized Eigenvalue Problems

“…Corollaries 3.1 and 3.2 are standard normwise results in the literature [17], [18], [20], [22], [27]. They show that we do not lose anything by using our general analysis.…”

“…For many problems the backward error is a scaled residual norm, in which case we can use our results to bound the backward error. The idea of using iterative refinement to obtain a small backward error with a potentially unstable solution method has been investigated for linear systems by several authors, including Jankowski and Woźniakowski [18], Skeel [22], and Higham [17], and more recently for the algebraic Riccati equation by Ghavimi and Laub [11]. The idea does not seem to have been applied previously to the generalized eigenvalue problem.…”

Stability Analysis of Algorithms for Solving Confluent Vandermonde-Like Systems

“…This does not appear to be due to errors in computing the single precision residual via Algorithm 2.3, but seems to indicate that in order to guarantee the success of iterative refinement in single precision a certain level of precision is required relative to the degree of instability (indeed this is implied by the results in [13]). …”

Stability analysis of a general toeplitz systems solver

“…Consider algorithms for solving a nonsingular, n × n linear system Ax = b, so m = n. There are many definitions of numerical stability in the literature, for example [5,6,9,12,16,22,36,48,57,61,70]. Definitions 1 and 2 below are taken from Bunch [17].…”

“…In practice this gives an accurate solution in a small number of iterations so long as the residual is computed accurately and the working precision is sufficient to ensure convergence. For details see [5,36,37,48,69,84].…”