@Article{NMTMA-6-4,
author = {},
title = {A Fourier Companion Matrix (Multiplication Matrix) with Real-Valued Elements: Finding the Roots of a Trigonometric Polynomial by Matrix Eigensolving},
journal = {Numerical Mathematics: Theory, Methods and Applications},
year = {2013},
volume = {6},
number = {4},
pages = {586--599},
abstract = {<p style="text-align: justify;">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 &quot;Fourier Division Companion Matrix&#39;&#39;, 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.</p>},
issn = {2079-7338},
doi = {https://doi.org/10.4208/nmtma.2013.1220nm},
url = {https://global-sci.com/article/90648/a-fourier-companion-matrix-multiplication-matrix-with-real-valued-elements-finding-the-roots-of-a-trigonometric-polynomial-by-matrix-eigensolving}
}