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A projection method for solving nonsymmetric linear systems on multiprocessors. (English) Zbl 0668.65028

For the solution of nonsingular linear \(n\times n\) systems, the method of S. Kaczmarz [Bull. Int. Acad. Polon. Sci. A 1937, 355-357 (1937; Zbl 0017.31703)] is considered. The authors organize this method in a block SSOR manner, show the optimal iteration parameter to be 1, and accelerate it by an outer conjugate gradient iteration. This leads to a method in which in every step a linear least squares problem has to be solved. If the latter can be broken up into independent least squares problems, a multiprocessor algorithm arises. It is shown how to do this for block tridiagonal systems. On a CRAY X-MP with one CPU and for discretized 2D elliptic equations, the method has been compared with those from the Yale PCGPAK and found to be more reliable. On a 4-processor CRAY X-MP, a speedup rate of 3.6 has been obtained.
Reviewer: G.Stoyan

MSC:

65F10 Iterative numerical methods for linear systems
65N22 Numerical solution of discretized equations for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations

Citations:

Zbl 0017.31703

Software:

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