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The condition of gram matrices and related problems

Published online by Cambridge University Press:  14 November 2011

J. M. Taylor
J. M. Taylor
Affiliation:
Mathematics Division, University of Sussex
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Synopsis

It has been known for some time that certain least-squares problems are “ill-conditioned”, and that it is therefore difficult to compute an accurate solution. The degree of ill-conditioning depends on the basis chosen for the subspace in which it is desired to find an approximation. This paper characterizes the degree of ill-conditioning, for a general inner-product space, in terms of the basis.

The results are applied to least-squares polynomial approximation. It is shown, for example, that the powers {1, z, z2,…} are a universally bad choice of basis. In this case, the condition numbers of the associated matrices of the normal equations grow at least as fast as 4n, where n is the degree of the approximating polynomial.

Analogous results are given for the problem of finite interpolation, which is closely related to the least-squares problem.

Applications of the results are given to two algorithms—the Method of Moments for solving linear equations and Krylov's Method for computing the characteristic polynomial of a matrix.

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 80 , Issue 1-2 , 1978 , pp. 45 - 56
Copyright
Copyright © Royal Society of Edinburgh 1978

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