Proceedings of Symposia in Applied Mathematics

Volume 22

1978

THREE RESEARCH PROBLEMS IN NUMERICAL LINEAR ALGEBRA*

Cleve Moler

ABSTRACT. This article is intended to introduce its readers to a small

sample of current research work in numerical linear algebra by describing three

unsolved research problems. All three problems have arisen in the development

and application of LINPACK and EISPACK, two collections of Fortran subroutines

for matrix computation. Briefly, the problems involve

• Estimating the "nearness to singularity" of a matrix.

• Convergence of a method for the nonsymmetric eigenvalue problem.

• Use of matrix factorizations to approximate matrices.

1. THE LINPACK CONDITION ESTIMATOR

LINPACK is a collection of Fortran subroutines for solving various types

of simultaneous linear equations and for analyzing certain types of matrices

which is currently under development at Argonne National Laboratory and three

universities. The simplest problem addressed by LINPACK is the solution of

the equation

Ax = b

where A is a given n by n matrix of real or complex numbers and b is a

given n vector. The order n is limited by the amount of memory available

on any particular computer; today this usually means n is at most a few

hundred.

A mathematician facing this apparently simple problem would first be con-

cerned with existence and uniqueness of the solution x and hence be concerned

about whether or not A is singular. In numerical work with inexact data and

imprecise arithmetic, it is usually inappropriate to ask whether or not a

matrix is singular; the distinction between singular and nonsingular becomes a

bit fuzzy. It is more appropriate to formulate a quantitative notion of

"nearness to singularity" and develop METHODS for computing it.

*

Derived from a lecture given at the American Mathematical Society short course

in Numerical Analysis, Atlanta, January 1978. Supported in part by NSF Grant

NCS76-03052.

Copyright © 1978, American Mathematical Society

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http://dx.doi.org/10.1090/psapm/022/533048