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A Fourier Companion Matrix (Multiplication Matrix) with Real-Valued Elements: Finding the Roots of a Trigonometric Polynomial by Matrix Eigensolving

Published online by Cambridge University Press:  28 May 2015

John P. Boyd
John P. Boyd*
Affiliation:
Department of Atmospheric, Oceanic & Space Science, University of Michigan, 2455 Hayward Avenue, Ann Arbor MI 48109, USA
*
*Corresponding author.Email address:jpboyd@umich.edu
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Abstract

We show that the zeros of a trigonometric polynomial of degree N with the usual (2N + 1) terms can be calculated by computing the eigenvalues of a matrix of dimension 2N with real-valued elements Mjk. This matrix is a multiplication matrix in the sense that, after first defining a vector whose elements are the first 2N basis functions, . This relationship is the eigenproblem; the zeros tk are the arccosine function of λk/2 where the λk are the eigenvalues of . We dub this the “Fourier Division Companion Matrix”, or FDCM for short, because it is derived using trigonometric polynomial division. We show through examples that the algorithm computes both real and complex-valued roots, even double roots, to near machine precision accuracy.

Keywords

65H9965T4042A85Fourier seriestrigonometric polynomialrootfindingsecularcompanion matrix
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 6 , Issue 4 , November 2013 , pp. 586 - 599
Copyright
Copyright © Global Science Press Limited 2013

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