# The Scaling and Squaring Method for the Matrix Exponential Revisited

@article{Higham2009TheSA,
title={The Scaling and Squaring Method for the Matrix Exponential Revisited},
author={Nicholas John Higham},
journal={Siam Review},
year={2009},
volume={51},
pages={747-764}
}
• N. Higham
• Published 1 November 2009
• Mathematics
• Siam Review
The scaling and squaring method is the most widely used method for computing the matrix exponential, not least because it is the method implemented in the MATLAB function expm. The method scales the matrix by a power of 2 to reduce the norm to order 1, computes a Pade approximant to the matrix exponential, and then repeatedly squares to undo the effect of the scaling. We give a new backward error analysis of the method (in exact arithmetic) that employs sharp bounds for the truncation errors… Expand
219 Citations

#### Figures and Tables from this paper

An Arbitrary Precision Scaling and Squaring Algorithm for the Matrix Exponential
• Mathematics, Computer Science
• SIAM J. Matrix Anal. Appl.
• 2019
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 Pade approximation below the unitroundoff. Expand
Scaled and Squared Subdiagonal Padé Approximation for the Matrix Exponential
• Computer Science, Mathematics
• SIAM J. Matrix Anal. Appl.
• 2016
An efficient variant that uses a much smaller squaring factor when $\|A\| \gg 1$ and a subdiagonal Pade approximant of low degree is introduced, thereby significantly reducing the overall cost and avoiding the potential instability caused by overscaling. Expand
A New Scaling and Squaring Algorithm for the Matrix Exponential
• Mathematics, Computer Science
• SIAM J. Matrix Anal. Appl.
• 2009
The numerical experiments show that the new algorithm generally provides accuracy at least as good as the existing algorithm of Higham at no higher cost, while for matrices that are triangular or cause overscaling it usually yields significant improvements in accuracy, cost, or both. Expand
Efficient orthogonal matrix polynomial based method for computing matrix exponential
• Mathematics, Computer Science
• Appl. Math. Comput.
• 2011
This paper presents an efficient method for computing matrix exponentials based on Hermite matrix polynomial expansions and shows that the application of floating point arithmetic bounds to the intermediate results may reduce considerably computational costs. Expand
Scaling and modified squaring method for the matrix exponential
• Computer Science, Mathematics
• JSIAM Lett.
• 2016
This work proposes a modified squaring process for the scaling and squaring methods of the Padé approximant and the matrix exponential, and shows an accuracy improvement about 1/100 in the best case. Expand
Polynomial Form of the Matrix Exponential
• Mathematics
• 2016
An algorithm for numerically computing the exponential of a matrix is presented. We have derived a polynomial expansion of $e^x$ by computing it as an initial value problem using a symbolicExpand
Aggressively Truncated Taylor Series Method for Accurate Computation of Exponentials of Essentially Nonnegative Matrices
• Mathematics, Computer Science
• SIAM J. Matrix Anal. Appl.
• 2014
An a priori componentwise relative error bound of truncation is established, from which one can choose the degree of Taylor series expansion and the scale factor so that the exponential is computed with desired componentwiserelative accuracy. Expand
Computing the Fréchet Derivative of the Matrix Exponential, with an Application to Condition Number Estimation
• Computer Science, Mathematics
• SIAM J. Matrix Anal. Appl.
• 2008
It is shown that the implementation of the scaling and squaring method can be extended to compute both $e^A$ and the Frechet derivative at $A$ in the direction of E, denoted by $L(A,E)$, at a cost about three times that for computing $e^{A$ alone. Expand
The Krylov Subspace Methods for the Computation of Matrix Exponentials
OF DISSERTATION The Krylov Subspace Methods for the Computation of Matrix Exponentials The problem of computing the matrix exponential e arises in many theoretical and practical problems. ManyExpand
New Scaling-Squaring Taylor Algorithms for Computing the Matrix Exponential
• Mathematics, Computer Science
• SIAM J. Sci. Comput.
• 2015
Two modifications are presented that provably reduce the number of matrix multiplications needed to satisfy the required accuracy bounds, and a detailed comparison of the several algorithmic variants is provided. Expand

#### References

SHOWING 1-10 OF 23 REFERENCES
A Schur--Fréchet Algorithm for Computing the Logarithm and Exponential of a Matrix
• Mathematics
• 1998
The Schur--Frechet method of evaluating matrix functions consists of putting the matrix in upper triangular form, computing the scalar function values along the main diagonal, and then using theExpand
A Schur-Parlett Algorithm for Computing Matrix Functions
• Mathematics, Computer Science
• SIAM J. Matrix Anal. Appl.
• 2003
An algorithm for computing matrix functions is presented. It employs a Schur decomposition with reordering and blocking followed by the block form of a recurrence of Parlett, with functions of theExpand
Condition Estimates for Matrix Functions
• Mathematics
• 1989
A sensitivity theory based on Frechet derivatives is presented that has both theoretical and computational advantages. Theoretical results such as a generalization of Van Loan’s work on the matrixExpand
Numerical Computation of the Matrix Exponential with Accuracy Estimate
This paper presents and analyzes an algorithm for computing the exponential of an arbitrary $n \times n$ matrix. Diagonal Pade table approximations are used in conjunction with several techniques forExpand
Expokit: a software package for computing matrix exponentials
Expokit provides a set of routines aimed at computing matrix exponentials. More precisely, it computes either a small matrix exponential in full, the action of a large sparse matrix exponential on anExpand
Orthogonal Eigenvectors and Relative Gaps
• Computer Science, Mathematics
• SIAM J. Matrix Anal. Appl.
• 2003
This paper presents and analyzes a new algorithm for computing eigenvectors of symmetric tridiagonal matrices factored as LDLt, with D diagonal and L unit bidiagonal. If an eigenpair is well behavedExpand
A numerical study of large sparse matrix exponentials arising in Markov chains 1 1 This work has ben
• Mathematics
• 1999
Krylov subspace techniques have been shown to yield robust methods for the numerical computation of large sparse matrix exponentials and especially the transient solutions of Markov Chains. TheExpand
Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later
• Mathematics, Computer Science
• SIAM Rev.
• 2003
Methods involv- ing approximation theory, dierential equations, the matrix eigenvalues, and the matrix characteristic polynomial have been proposed, indicating that some of the methods are preferable to others, but that none are completely satisfactory. Expand
Matrix decompositions of two-dimensional nuclear magnetic resonance spectra.
• Medicine
• Proceedings of the National Academy of Sciences of the United States of America
• 1994
A physically significant decomposition of two-dimensional NMR spectra into a product of matrices is considered, which permits these spectra to be efficiently simulated and permits the corresponding inverse problems to be given an elegant mathematical formulation to which standard numerical methods are applicable. Expand
Numerical analysis: an introduction
This is a text in numerical analysis which is taken to mean the branch of mathematics that develops and analyzes computational methods dealing with problems arising in classical analysis,Expand