Improved methods and starting values to solve the matrix equations $X\pm A^*X^{-1}A=I$ iteratively
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- by Ivan G. Ivanov, Vejdi I. Hasanov and Frank Uhlig;
- Math. Comp. 74 (2005), 263-278
- DOI: https://doi.org/10.1090/S0025-5718-04-01636-9
- Published electronically: January 27, 2004
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Abstract:
The two matrix iterations $X_{k+1}=I\mp A^*X_k^{-1}A$ are known to converge linearly to a positive definite solution of the matrix equations $X\pm A^*X^{-1}A=I$, respectively, for known choices of $X_0$ and under certain restrictions on $A$. The convergence for previously suggested starting matrices $X_0$ is generally very slow. This paper explores different initial choices of $X_0$ in both iterations that depend on the extreme singular values of $A$ and lead to much more rapid convergence. Further, the paper offers a new algorithm for solving the minus sign equation and explores mixed algorithms that use Newtonโs method in part.References
- W. N. Anderson Jr., T. D. Morley, and G. E. Trapp, Positive solutions to $X=A-BX^{-1}B^*$, Linear Algebra Appl. 134 (1990), 53โ62. MR 1060009, DOI 10.1016/0024-3795(90)90005-W
- B. L. Buzbee, G. H. Golub, and C. W. Nielson, On direct methods for solving Poissonโs equations, SIAM J. Numer. Anal. 7 (1970), 627โ656. MR 287717, DOI 10.1137/0707049
- S. M. El-Sayed, Theorems for the Existence and Computing of Positive Definite Solutions for Two Nonlinear Matrix Equation, Proc. of 25$^{th}$ Spring Conference of the Union of Bulgarian Mathematicians, Kazanlak, 1996, pp.155-161, (in Bulgarian).
- Salah M. El-Sayed and Andrรฉ C. M. Ran, On an iteration method for solving a class of nonlinear matrix equations, SIAM J. Matrix Anal. Appl. 23 (2001/02), no.ย 3, 632โ645. MR 1896810, DOI 10.1137/S0895479899345571
- Jacob C. Engwerda, Andrรฉ C. M. Ran, and Arie L. Rijkeboer, Necessary and sufficient conditions for the existence of a positive definite solution of the matrix equation $X+A^*X^{-1}A=Q$, Linear Algebra Appl. 186 (1993), 255โ275. MR 1217209, DOI 10.1016/0024-3795(93)90295-Y
- Jacob C. Engwerda, On the existence of a positive definite solution of the matrix equation $X+A^{\mathsf T}X^{-1}A=I$, Linear Algebra Appl. 194 (1993), 91โ108. MR 1243822, DOI 10.1016/0024-3795(93)90115-5
- Augusto Ferrante and Bernard C. Levy, Hermitian solutions of the equation $X=Q+NX^{-1}N^*$, Linear Algebra Appl. 247 (1996), 359โ373. MR 1412761, DOI 10.1016/0024-3795(95)00121-2
- Gene H. Golub and Charles F. Van Loan, Matrix computations, 2nd ed., Johns Hopkins Series in the Mathematical Sciences, vol. 3, Johns Hopkins University Press, Baltimore, MD, 1989. MR 1002570
- Chun-Hua Guo and Peter Lancaster, Iterative solution of two matrix equations, Math. Comp. 68 (1999), no.ย 228, 1589โ1603. MR 1651757, DOI 10.1090/S0025-5718-99-01122-9
- Alston S. Householder, The theory of matrices in numerical analysis, Blaisdell Publishing Co. [Ginn and Co.], New York-Toronto-London, 1964. MR 175290
- Ivan G. Ivanov and Salah M. El-sayed, Properties of positive definite solutions of the equation $X+A^*X^{-2}A=I$, Linear Algebra Appl. 279 (1998), no.ย 1-3, 303โ316. MR 1637909, DOI 10.1016/S0024-3795(98)00023-8
- J. M. Ortega and W. C. Rheinboldt, Iterative solution of nonlinear equations in several variables, Academic Press, New York-London, 1970. MR 273810
- Olga Taussky, Matrices $C$ with $C^{n}\rightarrow 0$, J. Algebra 1 (1964), 5โ10. MR 161865, DOI 10.1016/0021-8693(64)90003-1
- J. H. Wilkinson, The algebraic eigenvalue problem, Clarendon Press, Oxford, 1965. MR 184422
- Xingzhi Zhan, Computing the extremal positive definite solutions of a matrix equation, SIAM J. Sci. Comput. 17 (1996), no.ย 5, 1167โ1174. MR 1404867, DOI 10.1137/S1064827594277041
Bibliographic Information
- Ivan G. Ivanov
- Affiliation: Faculty of Economics and Business Administration, 125 Tzarigradsko chaussee, bl.3, Sofia University, Sofia 1113, Bulgaria
- Email: i_-ivanov@feb.uni-sofia.bg
- Vejdi I. Hasanov
- Affiliation: Laboratory of Mathematical Modelling, Shumen University, Shumen 9712, Bulgaria
- Email: v.hasanov@fmi.shu-bg.net
- Frank Uhlig
- Affiliation: Department of Mathematics, Auburn University, Auburn, Alabama 36849โ5310
- Email: uhligfd@auburn.edu
- Received by editor(s): May 29, 2001
- Received by editor(s) in revised form: May 7, 2003
- Published electronically: January 27, 2004
- Additional Notes: This work is partially supported by Shumen University under Grant #3/04.06.2001.
- © Copyright 2004 American Mathematical Society
- Journal: Math. Comp. 74 (2005), 263-278
- MSC (2000): Primary 65F10
- DOI: https://doi.org/10.1090/S0025-5718-04-01636-9
- MathSciNet review: 2085410