# 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} }

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

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#### 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 the… Expand

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 the… Expand

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 matrix… Expand

Numerical Computation of the Matrix Exponential with Accuracy Estimate

- Mathematics
- 1977

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 for… Expand

Expokit: a software package for computing matrix exponentials

- Mathematics, Computer Science
- TOMS
- 1998

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 an… Expand

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 behaved… Expand

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. The… Expand

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

- Mathematics
- 1997

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