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Mathematics of Computation

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Orthogonal polyanalytic polynomials and normal matrices

Author: Marko Huhtanen
Journal: Math. Comp. 72 (2003), 355-373
MSC (2000): Primary 42C05; Secondary 15A57
Published electronically: February 22, 2002
MathSciNet review: 1933825
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Abstract: The Hermitian Lanczos method for Hermitian matrices has a well-known connection with a 3-term recurrence for polynomials orthogonal on a discrete subset of $\mathbb{R}$. This connection disappears for normal matrices with the Arnoldi method. In this paper we consider an iterative method that is more faithful to the normality than the Arnoldi iteration. The approach is based on enlarging the set of polynomials to the set of polyanalytic polynomials. Denoting by $d$ the number of elements computed so far, the arising scheme yields a recurrence of length bounded by $\sqrt{8d}$ for polyanalytic polynomials orthogonal on a discrete subset of $\mathbb{C}$. Like this slowly growing length of the recurrence, the method preserves, at least partially, the properties of the Hermitian Lanczos method. We employ the algorithm in least squares approximation and bivariate Lagrange interpolation.

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Additional Information

Marko Huhtanen
Affiliation: SCCM program, Computer Science Department, Stanford University, Stanford, California 94305
Address at time of publication: Department of Mathematics, MIT, 77 Massachusetts Avenue, Cambridge, Massachusetts 01239

Keywords: Normal matrix, orthogonal polynanalytic polynomial, slowly growing length of the recurrence, least squares approximation, bivariate Lagrange interpolation
Received by editor(s): September 12, 2000
Received by editor(s) in revised form: March 1, 2001
Published electronically: February 22, 2002
Additional Notes: This work was supported by the Academy of Finland and the Alfred Kordelin Foundation
Article copyright: © Copyright 2002 American Mathematical Society