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

作者：Boyd John P^*

来源：Numerical Mathematics: Theory, Methods and Applications , 2013, 6(4): 586-599.

DOI：10.4208/nmtma.2013.1220nm

• Fourier series
• trigonometric polynomial
• rootfinding
• secular
• companion matrix

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 2 N with real-valued elements M-jk. This matrix (M) over right arrow is a multiplication matrix in the sense that, after first defining a vector (phi) over right arrow whose elements are the first 2N basis functions, (M) over right arrow (phi) over right arrow = 2cos(t)(phi) over right arrow. 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 (M) over right arrow. 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.

• 出版日期2013-11