The Padé method for computing the matrix exponential

Arioli, Mario; Codenotti, Bruno; Fassino, Claudia

doi:10.1016/0024-3795(94)00190-1

Cited by 35 publications

(26 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The error in the squaring phase is innocuous because the normality of A ensures that the hump effect does not occur, hence by the same analysis as in [5],…”

Section: 2mentioning

confidence: 99%

“…We now describe our algorithm sexpm. The standard scaling and squaring method is known to be forward stable for certain types of matrices, including normal matrices [5], and we aim to design an algorithm that retains this property. Recall that forward stability means that the forward error is about the same order as that of a backward stable algorithm [21, Sec.…”

Section: Initial Shiftingmentioning

confidence: 99%

“…Higham [23] established backward stability of scaling and squaring in exact arithmetic, though backward stability in finite precision arithmetic remains an open problem. Regarding forward stability, several results exist [5,9,10]. Among these, of particular relevance is that the standard scaling and squaring method is forward stable for normal matrices, as shown by Arioli, Codenotti, and Fassino [5]; see also [22, p. 248].…”

Section: Algorithm 31 Sexpm : Compute Ementioning

confidence: 99%

“…As in the analysis of [5] we denote by f l(x) a computed approximation to x, so the output of sexpm is f l((r k,m (A/2 s ))…”

Section: Algorithm 31 Sexpm : Compute Ementioning

confidence: 99%

See 3 more Smart Citations

Scaled and Squared Subdiagonal Padé Approximation for the Matrix Exponential

Güttel

Nakatsukasa

2016

SIAM J. Matrix Anal. & Appl.

View full text Add to dashboard Cite

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.Key words. matrix exponential, scaling and squaring, subdiagonal Padé approximation, stable matrix, conditioning, matrix function times a vector AMS subject classifications. 15A23, 65F30, 65G501. Introduction. For a given matrix A ∈ C n×n , the matrix exponential e A is among the most important matrix functions, one reason being its close connection to ordinary differential equations; see, e.g., [42, Chapter 8]. We explore the scaling and squaring method [2,23] for its computation, which is the most commonly used in practice, and implemented for example in Matlab's expm. It is based on a type (2 s m, 2 s m) rational approximation to the exponential in a region of the complex plane containing the eigenvalues Λ(A), where s and m are suitably chosen integer parameters. The scaling and squaring method computes e A by first choosing an integer s such that A/2 s has norm of order 1, then taking a rational type (m, m) Padé approximation r(z) to e z , where m is chosen such that e

show abstract

“…The error in the squaring phase is innocuous because the normality of A ensures that the hump effect does not occur, hence by the same analysis as in [5],…”

Section: 2mentioning

confidence: 99%

Section: Initial Shiftingmentioning

confidence: 99%

Section: Algorithm 31 Sexpm : Compute Ementioning

confidence: 99%

“…As in the analysis of [5] we denote by f l(x) a computed approximation to x, so the output of sexpm is f l((r k,m (A/2 s ))…”

Section: Algorithm 31 Sexpm : Compute Ementioning

confidence: 99%

See 2 more Smart Citations

Scaled and Squared Subdiagonal Padé Approximation for the Matrix Exponential

Güttel

Nakatsukasa

2016

SIAM J. Matrix Anal. & Appl.

View full text Add to dashboard Cite

show abstract

“…If A is normal, then the scaling and squaring method is guaranteed to be forward stable; hence, the square phase is innocuous and the error in the computed exponential is consistent with the conditioning of the problem. Another case where the scaling and squaring method is forward stable corresponds to matrices with nonnegative nondiagonal entries as shown in [12].…”

Section: Taylor Algorithmmentioning

confidence: 99%

Accurate and efficient matrix exponential computation

Sastre

Ibáñez

Ruiz

et al. 2013

International Journal of Computer Mathematics

View full text Add to dashboard Cite

This work gives a new formula for the forward relative error of matrix exponential Taylor approximation and proposes new bounds for it depending on the matrix size and the Taylor approximation order, providing a new efficient scaling and squaring Taylor algorithm for the matrix exponential. A Matlab version of the new algorithm is provided and compared with Padé state-of-the-art algorithms obtaining higher accuracy in the majority of tests at similar or even lower cost.

show abstract

Explicit Finite Element Methods for Large Deformation Problems in Solid Mechanics

Benson

2004

Encyclopedia of Computational Mechanics

View full text Add to dashboard Cite

Explicit finite methods are often the best choice for solving large deformation, high strain rate problems in solids mechanics. They are simple to implement, robust, efficient, and scale well on massively parallel computers. Their commercial applications include simulations of automobile crashes to the more prosaic drop testing of cell phones. This chapter presents an introduction of explicit finite element methods with an emphasis on the topics that distinguish the explicit formulation from the more traditional implicit one, including stable time step size estimation, energy conservation, shock viscosities, zero energy modes, and stress rotations.

show abstract

The Padé method for computing the matrix exponential

Cited by 35 publications

References 8 publications

Scaled and Squared Subdiagonal Padé Approximation for the Matrix Exponential

Scaled and Squared Subdiagonal Padé Approximation for the Matrix Exponential

Accurate and efficient matrix exponential computation

Explicit Finite Element Methods for Large Deformation Problems in Solid Mechanics

Contact Info

Product

Resources

About