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On the optimal stability of the Bernstein basis

Authors: R. T. Farouki and T. N. T. Goodman
Journal: Math. Comp. 65 (1996), 1553-1566
MSC (1991): Primary 65G99; Secondary 65D17
MathSciNet review: 1351201
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Abstract: We show that the Bernstein polynomial basis on a given interval is ``optimally stable,'' in the sense that no other nonnegative basis yields systematically smaller condition numbers for the values or roots of arbitrary polynomials on that interval. This result follows from a partial ordering of the set of all nonnegative bases that is induced by nonnegative basis transformations. We further show, by means of some low--degree examples, that the Bernstein form is not uniquely optimal in this respect. However, it is the only optimally stable basis whose elements have no roots on the interior of the chosen interval. These ideas are illustrated by comparing the stability properties of the power, Bernstein, and generalized Ball bases.

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Additional Information

R. T. Farouki
Affiliation: Department of Mechanical Engineering & Applied Mechanics, University of Michigan, Ann Arbor, Michigan 48109

T. N. T. Goodman
Affiliation: Department of Mathematics and Computer Science, University of Dundee, Dundee DD1 4HN, Scotland

Received by editor(s): March 2, 1995
Received by editor(s) in revised form: August 28, 1995
Article copyright: © Copyright 1996 American Mathematical Society

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