An Arbitrary Precision Scaling and Squaring Algorithm for the Matrix Exponential

Fasi, Massimiliano and Higham, Nicholas J. (2018) An Arbitrary Precision Scaling and Squaring Algorithm for the Matrix Exponential. [MIMS Preprint] (In Press)

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Abstract

The most popular algorithms for computing the matrix exponential are those based on the scaling and squaring technique. For optimal efficiency these are usually tuned to a particular precision of floating-point arithmetic. We design a new scaling and squaring algorithm that takes the unit roundoff of the arithmetic as input and chooses the algorithmic parameters in order to keep the forward error in the underlying Padé approximation below the unit roundoff. To do so, we derive an explicit expression for all the coefficients in an error expansion for Padé approximants to the exponential and use it to obtain a new bound for the truncation error. We also derive a new technique for selecting the internal parameters used by the algorithm, which at each step decides whether to scale or to increase the degree of the approximant. The algorithm can employ diagonal Padé approximants or Taylor approximants and can be used with a Schur decomposition or in transformation-free form. Our numerical experiments show that the new algorithm performs in a forward stable way for a wide range of precisions and that the most accurate of our implementations, the Taylor-based transformation-free variant, is superior to existing alternatives.

Item Type: MIMS Preprint
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: Mr Massimiliano Fasi
Date Deposited: 19 Jun 2019 14:28
Last Modified: 19 Jun 2019 14:28
URI: https://eprints.maths.manchester.ac.uk/id/eprint/2721

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