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
1
http://dx.doi.org/10.1090/psapm/022/533048
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