A Fourier Companion Matrix (Multiplication Matrix) with Real-Valued Elements: Finding the Roots of a Trigonometric Polynomial by Matrix Eigensolving
Year: 2013
Numerical Mathematics: Theory, Methods and Applications, Vol. 6 (2013), Iss. 4 : pp. 586–599
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 $M_{jk}$. This matrix $\vec{\vec{M}}$ is a multiplication matrix in the sense that, after first defining a vector $\vec{\phi}$ whose elements are the first $2N$ basis functions, $\vec{\vec{M}}\vec{\phi}$ = 2cos($t$)$\vec{\phi}$. This relationship is the eigenproblem; the zeros $t_{k}$ are the arccosine function of $\lambda_{k}/2$ where the $\lambda_{k}$ are the eigenvalues of $\vec{\vec {M}}$. 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.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/nmtma.2013.1220nm
Numerical Mathematics: Theory, Methods and Applications, Vol. 6 (2013), Iss. 4 : pp. 586–599
Published online: 2013-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 14
Keywords: Fourier series trigonometric polynomial root-finding secular companion matrix.