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

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

Published electronically:
January 27, 2004

MathSciNet review:
2085410

Full-text PDF

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.

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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