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

Authors: Ivan G. Ivanov, Vejdi I. Hasanov and Frank Uhlig
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Journal: Math. Comp. 74 (2005), 263-278
MSC (2000): Primary 65F10
Published electronically: January 27, 2004
MathSciNet review: 2085410
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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 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

Vejdi I. Hasanov
Affiliation: Laboratory of Mathematical Modelling, Shumen University, Shumen 9712, Bulgaria

Frank Uhlig
Affiliation: Department of Mathematics, Auburn University, Auburn, Alabama 36849–5310

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