Scaled and squared subdiagonal Padé approximation for the matrix exponential

Güttel, Stefan and Yuji, Nakatsukasa (2015) Scaled and squared subdiagonal Padé approximation for the matrix exponential. [MIMS Preprint]

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Abstract

The scaling and squaring method is the most widely used algorithm for computing the exponential of a square matrix A. We introduce an efficient variant that uses a much smaller squaring factor when ||A||>>1 and a subdiagonal Padé approximant of low degree, thereby significantly reducing the overall cost and avoiding the potential instability caused by overscaling, while giving forward error of the same magnitude as the standard algorithm. The new algorithm performs well if a rough estimate of the rightmost eigenvalue of A is available and the rightmost eigenvalues do not have widely varying imaginary parts, and it achieves significant speedup over the conventional algorithm especially when A is of large norm. Our algorithm uses the partial fraction form to evaluate the Padé approximant, which makes it suitable for parallelization and directly applicable to computing the action of the matrix exponential exp(A)b, where b is a vector or a tall skinny matrix. For this problem the significantly smaller squaring factor has an even pronounced benefit for efficiency when evaluating the action of the Padé approximant.

Item Type:	MIMS Preprint
Uncontrolled Keywords:	matrix exponential, scaling and squaring, subdiagonal Padé approximation, stable matrix, conditioning, matrix function times a vector
Subjects:	MSC 2010, the AMS's Mathematics Subject Classification > 15 Linear and multilinear algebra; matrix theory MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis
Depositing User:	Stefan Güttel
Date Deposited:	26 Jun 2015
Last Modified:	08 Nov 2017 18:18
URI:	https://eprints.maths.manchester.ac.uk/id/eprint/2322

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