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SIAM. J. Matrix Anal. & Appl. 31, pp. 1279-1302 (24 pages)
Verified Computation of Square Roots of a Matrix
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Add to FacebookWe 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 $n\times n$ 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.
© 2009 Society for Industrial and Applied Mathematics
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Received April 27, 2009
Accepted September 08, 2009
Published online November 04, 2009
Accepted September 08, 2009
Published online November 04, 2009
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