Direct Methods for Sparse Matrices

Abstract

Direct Methods for Sparse Matrices, second edition, is a complete rewrite of the first edition published 30 years ago. Much has changed since that time. Problems have grown greatly in size and complexity; nearly all our examples were of order less than 5,000 in the first edition, and are often more than a million in the second edition. Computer architectures are now much more complex, requiring new ways of adapting algorithms to parallel environments with memory hierarchies. Because the area is such an important one to all of computational science and engineering, a huge amount of research has been done since the first edition, some of it by the authors. This new research is integrated into the text with a clear explanation of the underlying mathematics and algorithms. New research that is described includes new techniques for scaling and error control, new orderings, new combinatorial techniques for partitioning both symmetric and unsymmetric problems, and a detailed description of the multifrontal approach to solving systems that was pioneered by the research of the authors and colleagues. This includes a discussion of techniques for exploiting parallel architectures and new work for indefinite and unsymmetric systems.

CurveLP-A MATLAB implementation of an infeasible interior-point algorithm for linear programming

“…The dependent rows are among the rows left in the process. Then, Gaussian elimination method can be applied with pivot selection using Markowitz criterion [31,32]. Some implementation details include (a) break ties by choosing element with the largest magnitude, and (b) use threshold pivoting.…”

Symbolic elimination in dynamic optimization based on block-triangular ordering

“…The details of this are discussed in Section 4. An important part of this is the use of tearing [8,25,27,38]. Tearing is a method for solving sparse systems of equations that lack significant structure, such as being triangular or block diagonal, by ordering the variables and equations to get a partitioning of the system…”

“…Since pivots are selected based on numerical values to prevent numerical instability (caused by error growth due to numerical round-off), a priori selection based entirely on structural, as opposed to numerical, information can lead to numerical instability, as mentioned in Section 3.4.2 and discussed by Duff et al [25].…”

Parallelized Preconditioned Model Building Algorithm for Matrix Factorization

“…For matrix factorization, the pattern and the numerical values of a sparse matrix is stored in both the compressed row storage (CRS) or compressed column storage (CCS) formats. These are well known storage formats for sparse matrices (see, e.g., Section 2.7 of Duff et al [3]). Consider an m × n sparse matrix A with τ nonzeros.…”