Iterative methods for the extremal positive definite solution of the matrix equation \(X+A^{*}X^{-\alpha}A=Q\). (English) Zbl 1118.65029

Peng, Zhen-Yun; El-Sayed, Salah M.; Zhang, Xiang-Lin

J. Comput. Appl. Math. 200, No. 2, 520-527 (2007).

The authors propose two algorithms that avoid matrix inversion for every iteration, called inversion free variant of the basic point iteration. The first algorithm computes the maximal positive definite solution \(X_{+}\) of the nonlinear equation \(X+A^{\ast }X^{-\alpha }A=Q,\) where \(A\) is a nonsingular matrix, \(Q\) is a Hermitian positive definite matrix and \(\alpha \in (0,1].\) The second algorithm computes the minimal positive definite solution \(X_{-}\) of the same nonlinear equation with the case \(\alpha \in [ 1,\infty )\). Convergence theorems are provided. Some numerical examples are added to illustrate the convergence features.

Reviewer: Raffaella Pavani (Milano)

MSC:

65F30	Other matrix algorithms (MSC2010)
65F10	Iterative numerical methods for linear systems
15A24	Matrix equations and identities

Keywords:

positive definite matrix; matrix equation; iterative methods; maximal solution; minimal solution; algorithms; inversion free; convergence; numerical examples

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