Improved methods and starting values to solve the matrix equations iteratively

Authors:
Ivan G. Ivanov, Vejdi I. Hasanov and Frank Uhlig

Translated by:

Journal:
Math. Comp. **74** (2005), 263-278

MSC (2000):
Primary 65F10

Published electronically:
January 27, 2004

MathSciNet review:
2085410

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The two matrix iterations are known to converge linearly to a positive definite solution of the matrix equations , respectively, for known choices of and under certain restrictions on . The convergence for previously suggested starting matrices is generally very slow. This paper explores different initial choices of in both iterations that depend on the extreme singular values of 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 Newton's method in part.

**1.**W. N. Anderson, T. D. Morley and G. E. Trapp,*Positive Solution to*, Linear Algebra Appl.,**134**(1990), 53-62. MR**91c:47031****2.**B. L. Buzbee, G. H. Golub and C. W. Nielson,*On Direct Methods for Solving Poisson's Equations*, SIAM J. Numer. Anal.,**7**(1970), 627-656. MR**44:4920****3.**S. M. El-Sayed,*Theorems for the Existence and Computing of Positive Definite Solutions for Two Nonlinear Matrix Equation*, Proc. of 25 Spring Conference of the Union of Bulgarian Mathematicians, Kazanlak, 1996, pp.155-161, (in Bulgarian).**4.**S. M. El-Sayed and A. C. M. Ran,*On an Iteration Method for Solving a Class of Nonlinear Matrix Equations*, SIAM J. Matrix Anal. Appl.,**23**(2001), 632-645. MR**2002m:15023****5.**J. C. Engwerda, A. C. M. Ran and A. L. Rijkeboer,*Necessary and Sufficient Conditions for the Existence of a Positive Definite Solution of the Matrix Equation*, Linear Algebra Appl.,**186**(1993), 255-275. MR**94j:15012****6.**J. C. Engwerda,*On the Existence of a Positive Definite Solution of the Matrix Equation*, Linear Algebra Appl.,**194**(1993), 91-108. MR**94j:15013****7.**A. Ferrante and B. Levy,*Hermitian Solutions of the Equation*, Linear Algebra Appl.,**247**(1996) 359-373. MR**97m:93071****8.**C. H. Golub and C. F. Van Loan,*Matrix Computations*, Second Edition, John Hopkins University Press, Baltimore, 1989. MR**90d:65055****9.**C.-H. Guo and P. Lancaster,*Iterative Solution of Two Matrix Equations*, Math. Comp.,**68**(1999), 1589-1603. MR**99m:65061****10.**A. S. Householder,*The Theory of Matrices in Numerical Analysis*, Blaisdell, New York, 1964. MR**30:5475****11.**I. G. Ivanov and S. M. El-Sayed,*Properties of Positive Definite Solutions of the Equation*, Linear Algebra Appl.,**279**(1998), 303-316. MR**99c:15019****12.**J. M. Ortega and W. C. Rheinboldt,*Iterative Solution of Nonlinear Equations in Several Variables*, Academic Press, New York, 1970 (in Russian 1975). MR**42:8686****13.**O. Taussky,*Matrices with*, J. Algebra,**1**(1965), 5-10. MR**28:5069****14.**J. H. Wilkinson,*The Algebraic Eigenvalue Problem*, Oxford University Press, London, 1965. MR**32:1894****15.**X. Zhan,*Computing the Extremal Positive Definite Solutions of a Matrix Equation*, SIAM J. Sci. Computing,**17**(1996), 1167-1174. MR**97g:65074**

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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 36849–5310

Email:
uhligfd@auburn.edu

DOI:
https://doi.org/10.1090/S0025-5718-04-01636-9

Keywords:
Matrix equation,
positive definite solution,
iterative method,
Newton's method

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.

Article copyright:
© Copyright 2004
American Mathematical Society