Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J. Albrecht, Bemerkungen zu Iterationsverfahren zur Berechnung von A1/2 und A-1, Z. Angew. Math. Mech. 57 (1977) T262–T263.
] G. Alefeld and N. Schneider, On square roots of M-matrices, Linear Algebra Appl. 42 (1982) 119–132.
A. Berman and R.J. Plemmons, Nonnegative Matrices in the Mathematical Sciences (Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 1994; first published 1979 by Academic Press).
Å. Björck and S. Hammarling, A Schur method for the square root of a matrix, Linear Algebra Appl. 52/53 (1983) 127–140.
G.J. Butler, C.R. Johnson and H. Wolkowicz, Nonnegative solutions of a quadratic matrix equation arising from comparison theorems in ordinary differential equations, SIAM J. Alg. Disc. Meth. 6(1) (1985) 47–53.
R. Byers, Solving the algebraic Riccati equation with the matrix sign function, Linear Algebra Appl. 85 (1987) 267–279.
G.W. Cross and P. Lancaster, Square roots of complex matrices, Linear and Multilinear Algebra 1 (1974) 289–293.
E.D. Denman and A.N. Beavers, Jr., The matrix sign function and computations in systems, Appl. Math. Comput. 2 (1976) 63–94.
L. Dieci, B. Morini and A. Papini, Computational techniques for real logarithms of matrices, SIAM J. Matrix Anal. Appl. 17(3) (1996) 570–593.
G.H. Golub and C.F. Van Loan, Matrix Computations (Johns Hopkins Univ. Press, 3rd ed., Baltimore, MD, USA, 1996).
L.A. Hageman and D.M. Young, Applied Iterative Methods (Academic Press, New York, 1981).
N.J. Higham, Computing the polar decomposition – with applications, SIAM J. Sci. Statist. Comput. 7(4) (1986) 1160–1174.
N.J. Higham, Newton's method for the matrix square root, Math. Comp. 46(174) (1986) 537–549.
N.J. Higham, Computing real square roots of a real matrix, Linear Algebra Appl. 88/89 (1987) 405–430.
N.J. Higham, The Test Matrix Toolbox for Matlab (version 3.0), Numerical Analysis Report No. 276, Manchester Centre for Computational Mathematics, Manchester, England, September (1995) 70 pp.
N.J. Higham, The matrix sign decomposition and its relation to the polar decomposition, Linear Algebra Appl. 212/213 (1994) 3–20.
N.J. Higham, Accuracy and Stability of Numerical Algorithms (Society for Industrial and Applied Mathematics, Philadelphia, PA, 1996).
N.J. Higham and P. Papadimitriou, A parallel algorithm for computing the polar decomposition, Parallel Comput. 20(8) (1994) 1161–1173.
N.J. Higham and R.S. Schreiber, Fast polar decomposition of an arbitrary matrix, SIAM J. Sci. Statist. Comput. 11(4) (1990) 648–655.
R.A. Horn and C.R. Johnson, Matrix Analysis (Cambridge University Press, 1985).
R.A. Horn and C.R. Johnson, Topics in Matrix Analysis (Cambridge University Press, 1991).
W.D. Hoskins and D.J. Walton, A faster method of computing the square root of a matrix, IEEE Trans. Automat. Control. AC-23(3) (1978) 494–495.
T.J.R. Hughes, I. Levit and J. Winget, Element-by-element implicit algorithms for heat conduction, J. Eng. Mech. 109(2) (1983) 576–585.
C. Kenney and A.J. Laub, Condition estimates for matrix functions, SIAM J. Matrix Anal. Appl. 10(2) (1989) 191–209.
C. Kenney and A.J. Laub, Padé error estimates for the logarithm of a matrix, Internat J. Control 50(3) (1989) 707–730.
C. Kenney and A.J. Laub, Rational iterative methods for the matrix sign function, SIAM J. Matrix Anal. Appl. 12(2) (1991) 273–291.
C. Kenney and A.J. Laub, On scaling Newton's method for polar decomposition and the matrix sign function, SIAM J. Matrix Anal. Appl. 13(3) (1992) 688–706.
C.S. Kenney and A.J. Laub, A hyperbolic tangent identity and the geometry of Padé sign function iterations, Numerical Algorithms 7 (1994) 111–128.
C.S. Kenney and A.J. Laub, The matrix sign function, IEEE Trans. Automat. Control 40(8) (1995) 1330–1348.
P. Laasonen, On the iterative solution of the matrix equation AX2-I = 0, M.T.A.C. 12 (1958) 109–116.
P. Pandey, C. Kenney and A.J. Laub, A parallel algorithm for the matrix sign function, Internat. J. High Speed Comput. 2(2) (1990) 181–191.
B.N. Parlett, The Symmetric Eigenvalue Problem (Prentice-Hall, Englewood Cliffs, NJ, 1980).
P. Pulay, An iterative method for the determination of the square root of a positive definite matrix, Z. Angew. Math. Mech. 46 (1966) 151.
J.D. Roberts, Linear model reduction and solution of the algebraic Riccati equation by use of the sign function, Internat. J. Control 32(4) (1980) 677–687. First issued as report CUED/B-Control/TR13, Department of Engineering, University of Cambridge (1971).
B.A. Schmitt, An algebraic approximation for the matrix exponential in singularly perturbed boundary value problems, SIAM J. Numer. Anal. 27(1) (1990) 51–66.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Higham, N.J. Stable iterations for the matrix square root. Numerical Algorithms 15, 227–242 (1997). https://doi.org/10.1023/A:1019150005407
Issue Date:
DOI: https://doi.org/10.1023/A:1019150005407