## Improved methods and starting values to solve the matrix equations $X\pm A^*X^{-1}A=I$ iteratively

HTML articles powered by AMS MathViewer

- by Ivan G. Ivanov, Vejdi I. Hasanov and Frank Uhlig PDF
- Math. Comp.
**74**(2005), 263-278 Request permission

## 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**0175290** - 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**0273810** - 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**0184422** - 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

## Additional 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