Iterative solution of two matrix equations

Guo, Chun-Hua; Lancaster, Peter

doi:10.1090/S0025-5718-99-01122-9

Iterative solution of two matrix equations
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by Chun-Hua Guo and Peter Lancaster PDF

Math. Comp. 68 (1999), 1589-1603 Request permission

Abstract:

We study iterative methods for finding the maximal Hermitian positive definite solutions of the matrix equations $X+A^*X^{-1}A=Q$ and $X-A^*X^{-1}A=Q$, where $Q$ is Hermitian positive definite. General convergence results are given for the basic fixed point iteration for both equations. Newton’s method and inversion free variants of the basic fixed point iteration are discussed in some detail for the first equation. Numerical results are reported to illustrate the convergence behaviour of various algorithms.

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