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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.

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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References

1Todd, J. On condition numbers. Programmation en Mathématiques Numériques. Actes Colloq. Internat. C.N.R.S. 165, Besancon (1966), 141–159. Paris: Editions C.N.R.S., 1968.Google Scholar
2Bellman, R.Introduction to matrix analysis (New York: McGraw-Hill, 1960).Google Scholar
3Szegö, G.Orthogonal polynomials (Providence, R.I.: Amer. Math. Soc, 1939).Google Scholar
4Knopp, K.Theory and application of infinite series (Glasgow: Blackie, 1949).Google Scholar
5Todd, J.The condition of the finite segments of the Hilbert matrix. Nat. Bureau Standards Appl. Math. Ser. 39 (1954), 109–116.Google Scholar
6Hille, E.Analytic function theory II (Massachussets: Ginn, 1959).Google Scholar
7Vorobyev, U. V.Moments method in applied mathematics. (Delhi: Hindustani Publishing, 1962).Google Scholar
8Taylor, A. E.Introduction to functional analysis (New York: Wiley, 1958).Google Scholar
9Wilkinson, J. H.The algebraic eigenvalue problem (Oxford: Clarendon Press, 1965).Google Scholar