Journal
SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS
Volume 31, Issue 3, Pages 1279-1302Publisher
SIAM PUBLICATIONS
DOI: 10.1137/090757058
Keywords
matrix square root; Brouwer's fixed point theorem; Krawczyk's method; Kronecker structures; interval analysis; circular arithmetic
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We present methods to compute verified square roots of a square matrix A. Given an approximation X to the square root, obtained by a classical floating point algorithm, we use interval arithmetic to find an interval matrix which is guaranteed to contain the error of X. Our approach is based on the Krawczyk method, which we modify in two different ways in such a manner that the computational complexity for an n x n matrix is reduced to n(3). The methods are based on the spectral decomposition or, in the case that the eigenvector matrix is ill conditioned, on a similarity transformation to block diagonal form. Numerical experiments prove that our methods are computationally efficient and that they yield narrow enclosures provided X is a good approximation. This is particularly true for symmetric matrices, since their eigenvector matrix is perfectly conditioned.
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