A new inversion free iteration for solving the equation \(X + A^{\star} X^{-1} A = Q\). (English) Zbl 1072.65060

El-Sayed, Salah M.; Al-Dbiban, Asmaa M.

J. Comput. Appl. Math. 181, No. 1, 148-156 (2005).

A new iterative method for solving the matrix equation \(X + A^\star X^{-1} A = I\), where \(I\) denotes the identity matrix, is proposed: \[ X_0 = Y_0 = I, \;Y_{n+1} = (I-X_n) Y_n + I_n, \;X_{n+1} = I-A^\star Y_{n+1} A. \] It is shown that \(X_n\) converges to the maximal positive definite solution. Based on numerical experiments with two \(3\times 3\) and \(4\times 4\) examples, the authors conclude that the new method is more accurate and requires less floating point operations than some existing methods.

Reviewer: Daniel Kressner (Zagreb)

MSC:

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

Keywords:

matrix equation; fixed point iteration; convergence; iterative method; maximal positive definite solution; numerical experiments

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