TY - JOUR T1 - A Fourier Companion Matrix (Multiplication Matrix) with Real-Valued Elements: Finding the Roots of a Trigonometric Polynomial by Matrix Eigensolving AU - John P. Boyd JO - Numerical Mathematics: Theory, Methods and Applications VL - 4 SP - 586 EP - 599 PY - 2013 DA - 2013/06 SN - 6 DO - http://doi.org/10.4208/nmtma.2013.1220nm UR - https://global-sci.org/intro/article_detail/nmtma/5920.html KW - Fourier series, trigonometric polynomial, root-finding, secular, companion matrix. AB -

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.