Abstract
In this paper, a parallel multisplitting iterative method with the self-adaptive weighting matrices is presented for the linear system of equations when the coefficient matrix is an H-matrix. The zero pattern in weighting matrices is determined in advance, while the non-zero entries of weighting matrices are determined by finding the optimal solution in a hyperplane of α points generated by the parallel multisplitting iterations. Especially, the nonnegative restriction of weighting matrices is released. The convergence theory is established for the parallel multisplitting method with self-adaptive weightings. Finally, a numerical example shows that the parallel multisplitting iterative method with the self-adaptive weighting matrices is effective.
Highlights
Introduction and preliminariesTo solve the large sparse linear system of equationsAx = b, A ∈ Rn×n nonsingular and b ∈ Rn, ( . )O’Leary and White [ ] first proposed parallel methods based on multisplittings of matrices in, where several basic convergence results may be found
We focus on the H-matrix, which originates from Ostrowski [ ]
2 Description of the method Here, we present a parallel multisplitting iterative method with the self-adaptive weighting matrices
Summary
Solve the following optimization problem: min Ax – b e(j i,k) ∈S(k ) α s.t. Ei(k) = I. We obtain an approximated solution, its complexity is about n flops. Let A = M – N be H-compatible splitting of the strictly diagonally dominant matrix A, . From the H-compatible splitting A = M – |N| and the strict diagonal dominance of A , it holds that n j=. Let A = M – N be H-compatible splitting of the H-matrix A. We use the standard Ritz-Galerkin finite element method by P conforming triangular element to approximate the following continuous solutions u = x · y · ( – x) · ( – y) in the domain = [ , ] × [ , ], the step-sizes along both x and y directions are the same, that is, h m
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