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Design features of a frontal code for solving sparse unsymmetric linear systems out-of-core. (English) Zbl 0541.65017

Consider the LU-decomposition of a matrix A, \(A=PL\cdot UQ\), with permutation matrices P,Q and lower, upper triangular matrices L,U, resp. Only PL,UQ factors are used for the solution. The frontal method is described by reference to its application in the solution of a finite element problem with \(A=\sum_{\ell}B^{(\ell)}\), each matrix \(B^{(\ell)}\) corresponding to the finite element \(\ell\), and with a permuted 2\(\times 2\) blocks partitioned frontal matrix. Pivots can be chosen from the left upper fully summed block. The extension includes: Each \(B^{(\ell)}\) corresponds to a single row of the assembled matrix. It enables to solve systems arising from finite difference discretizations. The equation entry frontal matrix is a rectangular one with pivots chosen from the left-hand side block which has fully summed rows and columns. A treshold pivoting criterion \(| a_{\ell k}| =u\cdot \max | a_{ik}|\), where \(u\in(0,1>\) is a parameter, is used as a test for a pivot \(a_{\ell k}\) in both cases. Various software implementation aspects of the unsymmetrical frontal code MA32 written in Fortran are discussed. The performance of the MA32 code is illustrated on a variety of problems solved on the IBM 3033 and using vectorization on the CRAY-1.
Reviewer: L.Bakule

MSC:

65F05 Direct numerical methods for linear systems and matrix inversion
65N22 Numerical solution of discretized equations for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65F50 Computational methods for sparse matrices
35J25 Boundary value problems for second-order elliptic equations

Software:

MA32
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