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Inversion error, condition number, and approximate inverses of uncertain matrices. (English) Zbl 0997.65049
The author proposes an approach to rigorously measure, and reduce the effect, of possibly large, structured perturbation in the computation of an inverse matrix. He defines the structured maximal inversion error, that takes into account the structure and not-necessarily small perturbation size. For infinitesimal perturbation he gets a structure condition number. For a wide class of perturbation structures, he shows how to use the convex semidefinte programming to compute bounds on the structured maximal inversion error and structure condition number, and also to compute an approximate inverse. He points out that when the perturbation in unstructured and additive the classical condition number is recovered and the approximate inverse is an operator related to the total least squares.

MSC:
65F05 Direct numerical methods for linear systems and matrix inversion
65F35 Numerical computation of matrix norms, conditioning, scaling
90C22 Semidefinite programming
Software:
mctoolbox; Sp
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