A Preconditioning Technique for Computing Functions of Triangular Matrices
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We propose a simple preconditioning technique that, if incorporated into algorithms for computing functions of triangular matrices, can make them more efficient. Basically, such a technique consists in a similarity transformation that reduces the departure from normality of a triangular matrix, thus decreasing its norm and in general its function condition number. It can easily be extended to non triangular matrices, provided that it is combined with algorithms involving a prior Schur decomposition. Special attention is devoted to particular algorithms like the inverse scaling and squaring to the matrix logarithm and the scaling and squaring to the matrix exponential. The advantages of our proposal are supported by theoretical results and illustrated with numerical experiments.
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