Numer. Math. Theor. Meth. Appl., 6 (2013), pp. 586-599.
Published online: 2013-06
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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.
}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.2013.1220nm}, url = {http://global-sci.org/intro/article_detail/nmtma/5920.html} }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.