Newton’s method for the matrix square root
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- by Nicholas J. Higham PDF
- Math. Comp. 46 (1986), 537-549 Request permission
Abstract:
One approach to computing a square root of a matrix A is to apply Newton’s method to the quadratic matrix equation $F(X) \equiv {X^2} - A = 0$. Two widely-quoted matrix square root iterations obtained by rewriting this Newton iteration are shown to have excellent mathematical convergence properties. However, by means of a perturbation analysis and supportive numerical examples, it is shown that these simplified iterations are numerically unstable. A further variant of Newton’s method for the matrix square root, recently proposed in the literature, is shown to be, for practical purposes, numerically stable.References
-
R. H. Bartels & G. W. Stewart, "Solution of the matrix equation $AX + XB = C$," Comm. ACM, v. 15, 1972, pp. 820-826.
- Ȧke Björck and Sven Hammarling, A Schur method for the square root of a matrix, Linear Algebra Appl. 52/53 (1983), 127–140. MR 709347, DOI 10.1016/0024-3795(83)80010-X
- Eugene D. Denman, Roots of real matrices, Linear Algebra Appl. 36 (1981), 133–139. MR 604336, DOI 10.1016/0024-3795(81)90226-3
- Eugene D. Denman and Alex N. Beavers Jr., The matrix sign function and computations in systems, Appl. Math. Comput. 2 (1976), no. 1, 63–94. MR 393075, DOI 10.1016/0096-3003(76)90020-5 F. R. Gantmacher, The Theory of Matrices, Vol. I, Chelsea, New York, 1959.
- G. H. Golub, S. Nash, and C. Van Loan, A Hessenberg-Schur method for the problem $AX+XB=C$, IEEE Trans. Automat. Control 24 (1979), no. 6, 909–913. MR 566448, DOI 10.1109/TAC.1979.1102170
- Gene H. Golub and Charles F. Van Loan, Matrix computations, Johns Hopkins Series in the Mathematical Sciences, vol. 3, Johns Hopkins University Press, Baltimore, MD, 1983. MR 733103
- Robert T. Gregory and David L. Karney, A collection of matrices for testing computational algorithms, Wiley-Interscience [A division of John Wiley & Sons, Inc.], New York-London-Sydney, 1969. MR 0253538
- Nicholas J. Higham, Computing real square roots of a real matrix, Linear Algebra Appl. 88/89 (1987), 405–430. MR 882456, DOI 10.1016/0024-3795(87)90118-2
- Nicholas J. Higham, Computing the polar decomposition—with applications, SIAM J. Sci. Statist. Comput. 7 (1986), no. 4, 1160–1174. MR 857788, DOI 10.1137/0907079 W. D. Hoskins & D. J. Walton, "A faster method of computing the square root of a matrix," IEEE Trans. Automat. Control, v. AC-23, 1978, pp. 494-495.
- W. D. Hoskins and D. J. Walton, A faster, more stable method for computing the $p$th roots of positive definite matrices, Linear Algebra Appl. 26 (1979), 139–163. MR 535683, DOI 10.1016/0024-3795(79)90176-9
- Pentti Laasonen, On the iterative solution of the matrix equation $AX^{2}-I=0$, Math. Tables Aids Comput. 12 (1958), 109–116. MR 99107, DOI 10.1090/S0025-5718-1958-0099107-4
- James M. Ortega, Numerical analysis. A second course, Computer Science and Applied Mathematics, Academic Press, New York-London, 1972. MR 0403154
- G. W. Stewart, Introduction to matrix computations, Computer Science and Applied Mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1973. MR 0458818
- Carl-Erik Fröberg, Introduction to numerical analysis, 2nd ed., Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969. MR 0245168
- Peter Henrici, Elements of numerical analysis, John Wiley & Sons, Inc., New York-London-Sydney, 1964. MR 0166900
Additional Information
- © Copyright 1986 American Mathematical Society
- Journal: Math. Comp. 46 (1986), 537-549
- MSC: Primary 65F30; Secondary 65H10
- DOI: https://doi.org/10.1090/S0025-5718-1986-0829624-5
- MathSciNet review: 829624