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Direct methods for solving large sparse systems of equations based on the two by two block decomposition of the matrix. (Russian. English summary) Zbl 1004.65036
Summary: Several algorithms for reordering a sparse symmetric positive definite matrix to a $$2\times 2$$ block form are considered; a task of finding a permutation such that the filling is at minimum in the block $$(1,1)$$ and is located mainly in the blocks $$(2,1)$$, $$(2,2)$$ is posed. In this respect two algorithms from the widely known sparse matrix package SPARSPAK are analyzed: QMD – a quotient minimum degree algorithm and ND – the nested dissection algorithm; a new one is proposed which is called $$\text{BND}+\text{QMD}$$ – balanced ND with internal (influencing block $$(1,1)$$) QMD-ordering. The results of numerical experiments for a set of grid problems containing 10000-25000 unknown values are presented. These results show that the usage of the implicit solution scheme may provide up to 25-30% reduction of primary storage without visible increasing the number of operations required to solve the triangular system.
MSC:
 65F05 Direct numerical methods for linear systems and matrix inversion 65F50 Computational methods for sparse matrices
SPARSPAK
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