Inversion of a triangular matrix can be accomplished in several ways. The standard methods are characterised by the loop ordering, whether matrix-vector multiplication, solution of a triangular system, or a rank-1 update is done inside the outer loop, and whether the method is blocked or unblocked. The numerical stability properties of these methods are investigated. It is shown that unblocked methods satisfy pleasing bounds on the left or right residual. However, for one of the block methods it is necessary to convert a matrix multiplication into the solution of a multiple right-hand side triangular system in order to have an acceptable residual bound. The inversion of a full matrix given a factorization PA=LU is also considered, including the special cases of symmetric inde nite and symmetric positive de nite matrices. Three popular methods are shown to possess satisfactory residual bounds, subject to a certain requirement on the implementation, and an attractive new method is described. This work was motivated by the question of what inversion methods should be used in LAPACK.
The problem of solving large full sets of linear equations on a computer with a paged virtual memory is considered and a block column algorithm proposed. Details of software design are considered and results of experimental runs on five different computer systems are reported.
If a reliable, high quality numerical algorithms library is to be developed then it is essential that we recognize the need for collaboration between different technical communities in the development of the library. This paper suggests an ultimate design for the library and describes the implications of that design for the people involved in the development of the library.
DESCRIPTIONBLCFAC is a routine for performing Oaussian elimination with partial pivoting on a real square matrix A, with the operations on blocks of consecutive columns grouped together to minimize the number of page swaps on a machine with a paged virtual store, as described in [1]. Given that A has been factorized by BLCFAC, the routine BLCSOL will solve either of the systems AXffiB or ATX= B with multiple right-hand sides. The operations on blocks of consecutive righthand sides are grouped together for the same reason. BLCFAC and BLCSOL are essentially the same as the routines F01BTF and F04AYF in the NAG FORTRAN Library.
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