A. M. Erisman

On computing certain elements of the inverse of a sparse matrix

A recursive algorithm for computing the inverse of a matrix from the LU factors based on relationships in Takahashi, et al., is examined. The formulas for the algorithm are given; the dependency relationships are derived; the computational costs are developed; and some general comments on application and stability are made.

On George’s Nested Dissection Method

We make a precise recommendation for the choice of dissection sets for George’s [4] nested dissection on a square

Sparsity structure and Gaussian elimination

n \times n

Monitoring the stability of the triangular factorization of a sparse matrix

grid with n not a power of 2 and make some experimental comparisons with the direct use of the minimal degree algorithm of Markowitz [5]. Our results additionally indicate that the performance of the minimal degree algorithm can be significantly influenced by the way in which ties are broken. We also discuss the dissection of irregularly shaped regions and the extension of George’s ordering to three-dimensional regions.

Exploiting problem characteristics in the sparse matrix approach to frequency domain analysis

In this paper we collate and discuss some results on the sparsity structure of a matrix. If a matrix is irreducible, Gaussian elimination yields an LU factorization in which L has at least one entry beneath the diagonal in every column except the last and U has at least one entry to the right of the diagonal in every row except the last. If this factorization is used to solve the equation Ax=b, the intermediate vector has an entry in its last component and the solution x is full. Furthermore, the inverse of A is full.Where the matrix is reducible, these results are applicable to the diagonal blocks of its block triangular form.

Hybrid sparse-matrix methods

We propose a method for bounding the numerical instability in Gaussian elimination which may result from the use of interchanges calculated from the sparsity pattern alone or calculated for another matrix having the same sparsity pattern. The computational cost of obtaining the bound is small compared with that of the elimination. Also we propose the use of a variant of iterative refinement to estimate the accuracy of the solution obtained.

Direct Methods for Sparse Matrices

Most recent papers on sparse matrix methods in circuit analysis are written to describe large problem capability, but usually the illustrations given are for relatively small problems. In this paper a computer program for frequency domain analysis of large RLCM networks is described. It is regularly used on 3000-4000 node networks with many large sets of mutually coupled inductors. Timing and storage figures are given for a 3300 node problem. In the first part, the generation of the admittance description for networks with mutual inductors is described. Then the sparse matrix techniques are discussed. Some are standard, but the unique use of storage utilized in the construction of the LU factorization makes it possible to solve very large problems in a reasonable amount of core storage and time.

Analysis of descriptor systems using numerical algorithms

In computer-aided design for large systems, the efficient solution of large sparse systems of linear equations is important. We critically examine two parts of the solution of sparse systems of linear equations: ordering and the factorization of the ordered matrix. After examining several candidates from a theoretical point of view, as well as with examples, we conclude that the Markowitz ordering criteria, with extension for the variability-type problem, is the most practical ordering algorithm. A combination of solution processes provides a very efficient hybrid algorithm for factoring the ordered sparse matrix. The hybrid is outlined so that it may be tailored to minimum storage utilization, minimum central processing unit (CPU) time, or may be built entirely in a high-level language to reduce programming time. A number of examples are included.