Numerical Algorithms

, Volume 15, Issue 2, pp 227–242

Stable iterations for the matrix square root

  • Nicholas J. Higham

DOI: 10.1023/A:1019150005407

Cite this article as:
Higham, N.J. Numerical Algorithms (1997) 15: 227. doi:10.1023/A:1019150005407


Any matrix with no nonpositive real eigenvalues has a unique square root for which every eigenvalue lies in the open right half-plane. 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 M-matrix. 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.

matrix square root matrix logarithm matrix sign function M-matrix symmetric positive definite matrix Padé approximation numerical stability Newton's method Schulz method 65F30 

Copyright information

© Kluwer Academic Publishers 1997

Authors and Affiliations

  • Nicholas J. Higham
    • 1
  1. 1.Department of MathematicsUniversity of ManchesterManchesterEngland

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