Notes on linear factor polynomial deflation in polynomial bases

Abstract

Scalar polynomials as approximations to more general scalar functions lead to the study of scalar polynomials represented in a variety of classical systems of polynomials, including orthogonal systems and Lagrange polynomials, for example. This article, motivated in part by analogy with the existing methods for linear factor polynomial deflation in the monomial basis, finds forward and backward deflation formulae for several such representations. It also finds the sensitivity factor of the deflation process for each representation.

Keywords:

• polynomials
• polynomial bases
• forward/backward deflation

AMS Subject Classifications::

• 15A22
• 65H04
• 65H10

Acknowledgements

The author would like to thank anonymous reviewers for their constructive comments. The author is grateful to Dr Robert Corless for his generous support and perceptive comments.

Notes

Note

1. It is a rough definition, because we can always re-order the nodes and get different cases based on what ordering we use and which node has turned out to be z _n . We follow through assuming that z _n is to be eliminated.