Stable iterations for the matrix square root
 Nicholas J. Higham
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Any matrix with no nonpositive real eigenvalues has a unique square root for which every eigenvalue lies in the open right halfplane. A link between the matrix sign function and this square root is exploited to derive both old and new iterations for the square root from iterations for the sign function. One new iteration is a quadratically convergent Schulz iteration based entirely on matrix multiplication; it converges only locally, but can be used to compute the square root of any nonsingular Mmatrix. A new Padé iteration well suited to parallel implementation is also derived and its properties explained. Iterative methods for the matrix square root are notorious for suffering from numerical instability. It is shown that apparently innocuous algorithmic modifications to the Padé iteration can lead to instability, and a perturbation analysis is given to provide some explanation. Numerical experiments are included and advice is offered on the choice of iterative method for computing the matrix square root.
 Title
 Stable iterations for the matrix square root
 Journal

Numerical Algorithms
Volume 15, Issue 2 , pp 227242
 Cover Date
 199709
 DOI
 10.1023/A:1019150005407
 Print ISSN
 10171398
 Online ISSN
 15729265
 Publisher
 Kluwer Academic Publishers
 Additional Links
 Topics
 Keywords

 matrix square root
 matrix logarithm
 matrix sign function
 Mmatrix
 symmetric positive definite matrix
 Padé approximation
 numerical stability
 Newton's method
 Schulz method
 65F30
 Industry Sectors
 Authors

 Nicholas J. Higham ^{(1)}
 Author Affiliations

 1. Department of Mathematics, University of Manchester, Manchester, M13 9PL, England