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On inexact Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems. (English) Zbl 1135.65016

The authors discuss Hermitian/skew-Hermitian splitting methods for solving systems of linear equations \(Ax=b\), where \(A\) is a non-Hermitian and positive definite matrix. In each iteration step one has to solve a system of equations with the matrix \(\alpha I + H\), \(H = (A + A^\ast)/2\) and the matrix \(\alpha I + S\), \(S = (A - A^\ast)/2\). The systems with the matrix \(\alpha I + H\) are solved by means of the conjugate gradient method and the other systems by means of the Lanczos method or the conjugate gradient method applied to the normal equations. This approach leads to inexact Hermitian/skew-Hermitian splitting methods.
In the paper convergence properties of these methods are studied. It is shown that the contraction factor and the asymptotic convergence rates depend dominantly on the spectrum of the Hermitian part. Furthermore, the computational efficiency of the presented methods is analysed. Optimal choices of the number of inner iteration steps are discussed. The methods are compared by numerical examples.

MSC:

65F10 Iterative numerical methods for linear systems
65Y20 Complexity and performance of numerical algorithms
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