. In this paper classical matrix perturbation theory is approached from a probabilistic point of view. The perturbed quantity is approximated by a first-order perturbation expansion, in which the perturbation is assumed to be random. This permits the computation of statistics estimating the variation in the perturbed quantity. Up to the higher-order terms that are ignored in the expansion, these statistics tend to be more realistic than perturbation bounds obtained in terms of norms. The technique is applied to a number of problems in matrix perturbation theory, including least squares and the eigenvalue problem. Key words. perturbation theory, random matrix, linear system, least squares, eigenvalue, eigenvector, invariant subspace, singular value AMS(MOS) subject classifications. 15A06, 15A12, 15A18, 15A52, 15A60 1. Introduction. Let A be a matrix and let F be a matrix valued function of A. Two principal problems of matrix perturbation theory are the following. Given a matrix E, pr...

It is known that finite precision versus exact computation is a crucial issue only when the computation takes place in the neighbourhood of a singularity. In such a situation, it is essential to know the distance to singularity. Much attention has been dedicated to the relationship between the distance to singularity ffi and the condition number K of the problem under study. The wellknown Turing theorem states that, for a linear system Ax = b, the distance to singularity, in a normwise measure, is the reciprocal of the normwise condition number kA \Gamma1 kkAk. In this paper, we examine the possibility of extending this theorem for nonlinear problems in the neighbourhood of algebraic singularities. After reviewing the literature on that topic (Demmel (1987, 1990), Shub and Smale (1992)), we propose and check on the computer a conjecture which makes more explicit Demmel's asymptotic bounds on the distance to singularity. 1 Survey of finite precision computation We consider the mathe...

alled Qualitative Computing. It consists of two main objectives : 1. for computations where the influence of finite precision arithmetic remains moderate, to control the global error on the computed result, 2. for computations dominated by round-off (such as chaotic computations), to use the errors to reveal mathematical properties which are out of reach of any finite precision computation. In such cases, computations are said to be "impossible" (because all significant digits are wrong") and the quantification --exact computing, for example-- becomes meaningless. However some information, of a more qualitative type, can be extracted from results which are "wrong" in a classical sense. 0 ZAMM . Z. angew. Math. Mech. 74 (1994) 2, pp. 105--113. y Laboratoire Central de Recherche, Thomson-CSF, 91404 Orsay Cedex, and University Paris IX Dauphine. Email : chatelin@thomson-lcr.fr z CERFACS, 42 av. Coriolis, 3105

The authors derive a formulation for the structured condition number and bounds for structured backward error for the linear system A*Ax = b when the square matrix A is subject to normwise perturbations. Perturbations on the data and the solution are measured in the Frobenius norm. Then numerical experiments that shows the relevance of this condition number in the prediction of the computing error in solving this system are presented. Finally the methodology PRECISE is proposed as an alternative way to estimate the condition number and predict the forward error in the absence of a mathematical formulation for these quantities.

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