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Generalized stationary iterative method for solving linear systems. (English) Zbl 0913.65025

A classical iterative method to solve a system of linear equations \(Ax=b\) consists in the splitting of the matrix \(A\), i.e. in its representation as \(A=M-N\), where \(M\) is a nonsingular matrix, and in the calculation of successive approximations of the solution from the sequence of equations \(Mx^{(k+1)}=Nx^{(k)} +b\), for \(k=0,1,2,\ldots\), where \(x^{(0)}\) is an initial approximation. Iterative methods of this form are called stationary iterative methods. The authors consider the case, when the matrix \(A\) is an \(M\)-matrix, i.e. its entries \(a_{ij} \leq 0\) for \(i\neq j\) and \(A^{-1}\geq 0\) (\(A\) is called monotone).
For some special splittings, expressed as combinations of a diagonal matrix \(D\), upper triangular one \(U\) and lower triangular one \(L\), we obtain such well-known methods as Jacobi method (\(M=D\), \(N=L+U\)), Gauss-Seidel method (\(M=D-L\), \(N=U\)) or the successive overrelaxation (SOR) method (\(M=(\frac{1}{\omega})D-L\), \(N=((\frac{1}{\omega})-1)D+U\); \(1<\omega <2\)). The authors formulate a general paradigm for a class of such methods, named generalized stationary iterative (GSI) method, where the matrix \(M\) has the form \(M=\alpha D - \beta L\); \(\alpha\) and \(\beta\) denote some real parameters. Thus \(N=(\alpha-1)D+(1-\beta)L +U\). For \(\alpha=1\), \(\beta=0\) we obtain the Jacobi method, for \(\alpha=1\) and \(\beta=1\) – the Gauss-Seidel method, and for \(\alpha=\frac{1}{\omega}\), \(\beta=1\) – the SOR method.
Numerical experiments for some examples of the source problem suggest that there exist some pairs \((\alpha,\beta)\) for which the GSI method converges faster than the SOR method (such pairs of parameters were found by the authors as heuristic values for test examples). The authors suggest that the GSI method may provide a good idea for developing other types of stationary iterative methods.
Reviewer: S.Zabek (Lublin)

MSC:

65F10 Iterative numerical methods for linear systems

Software:

KELLEY
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