*I. S. Duff, A. M. Erisman, and J. K. Reid*

- Published in print:
- 2017
- Published Online:
- April 2017
- ISBN:
- 9780198508380
- eISBN:
- 9780191746420
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198508380.003.0003
- Subject:
- Mathematics, Numerical Analysis

We review the fundamental operations in the direct solution of linear equations without concern for rounding error caused by computer arithmetic. We consider the relationship between Gaussian ...
More

We review the fundamental operations in the direct solution of linear equations without concern for rounding error caused by computer arithmetic. We consider the relationship between Gaussian elimination and LU factorization. We compare the use of different computational sequences including left-looking and right-looking. We show that advantage can be taken of symmetry and we consider the use of blocking. Our choice of material is based on what will be useful in the sparse case.Less

We review the fundamental operations in the direct solution of linear equations without concern for rounding error caused by computer arithmetic. We consider the relationship between Gaussian elimination and LU factorization. We compare the use of different computational sequences including left-looking and right-looking. We show that advantage can be taken of symmetry and we consider the use of blocking. Our choice of material is based on what will be useful in the sparse case.

*I. S. Duff, A. M. Erisman, and J. K. Reid*

- Published in print:
- 2017
- Published Online:
- April 2017
- ISBN:
- 9780198508380
- eISBN:
- 9780191746420
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198508380.003.0007
- Subject:
- Mathematics, Numerical Analysis

We consider local strategies for pivot selection, that is, where decisions are made at each stage of the factorization without regard to how they might affect later stages. They include minimum ...
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We consider local strategies for pivot selection, that is, where decisions are made at each stage of the factorization without regard to how they might affect later stages. They include minimum degree for symmetric matrices, Markowitz’ strategy for unsymmetric matrices, minimum fill-in, and simpler strategies. We discuss the inclusion of numerical pivoting. We consider taking advantage of sparsity in the right-hand side and of wanting only part of the solution.Less

We consider local strategies for pivot selection, that is, where decisions are made at each stage of the factorization without regard to how they might affect later stages. They include minimum degree for symmetric matrices, Markowitz’ strategy for unsymmetric matrices, minimum fill-in, and simpler strategies. We discuss the inclusion of numerical pivoting. We consider taking advantage of sparsity in the right-hand side and of wanting only part of the solution.

*I. S. Duff, A. M. Erisman, and J. K. Reid*

- Published in print:
- 2017
- Published Online:
- April 2017
- ISBN:
- 9780198508380
- eISBN:
- 9780191746420
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198508380.003.0014
- Subject:
- Mathematics, Numerical Analysis

We examine the SOLVE phase in the direct solution of sparse systems. Here we assume that the factors have been computed and we study the efficient use of these to determine the solution through ...
More

We examine the SOLVE phase in the direct solution of sparse systems. Here we assume that the factors have been computed and we study the efficient use of these to determine the solution through forward and back-substitution. We use the trees described in the previous chapters to study the efficient solution of sparse right-hand sides including the computation of null-space bases. We consider the parallelization of the SOLVE phase.Less

We examine the SOLVE phase in the direct solution of sparse systems. Here we assume that the factors have been computed and we study the efficient use of these to determine the solution through forward and back-substitution. We use the trees described in the previous chapters to study the efficient solution of sparse right-hand sides including the computation of null-space bases. We consider the parallelization of the SOLVE phase.