## Asymmetric Hermitian and skew-Hermitian splitting methods for positive definite linear systems.(English)Zbl 1127.65019

Stationary iterative methods for a linear system of equations $$Ax = b$$ can be constructed by an additive splitting of the matrix $$A$$. Z.-Z. Bai, G. H. Golub and M. K. Ng [SIAM J. Matrix Anal. Appl. 24, No. 3, 603–626 (2003; Zbl 1036.65032)] considered the splitting $$A = H + S$$ with $$H = (A+A^*)/2$$ and $$S = (A-A^*)/2$$, leading to the so called HSS iteration $(\alpha I + H)x^{(k+1/2)}= (\alpha I - S) x^{(k)} + b, \quad (\alpha I + S) x^{(k+1)} = (\alpha I - H) x^{(k+1/2)} + b,$ where $$\alpha$$ is a fixed positive parameter. This paper proposes to allow for a different parameter in the second part of this iteration: $(\alpha I + H)x^{(k+1/2)} = (\alpha I - S) x^{(k)} + b, \quad (\beta I + S) x^{(k+1)} = (\beta I - H) x^{(k+1/2)} + b,$ where $$\alpha$$ is nonnegative and $$\beta$$ is positive. Bounds on $$\beta$$ are given for which this modified iteration converges to $$x$$. A bound on the spectral radius of the iteration matrix is given, depending on $$\alpha,\beta$$ and the eigenvalues of $$H,M$$. Optimal values of the parameters for a simplified bound are computed. It turns out that, for an arbitrary $$\alpha$$, the choice $$\beta = \alpha$$ can be far from optimal. On the other hand, numerical experiments for the finite difference discretization of a $$3D$$ convection-diffusion equation suggest that little improvement over the HSS iteration is made when $$\alpha$$ is chosen optimally. Finally, existing results on the influence of the inexact solution of the subsystems (e.g., by Krylov subspace methods) on the convergence are extended.

### MSC:

 65F10 Iterative numerical methods for linear systems 65N06 Finite difference methods for boundary value problems involving PDEs 35J25 Boundary value problems for second-order elliptic equations

Zbl 1036.65032
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### References:

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