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A Fourier Companion Matrix (Multiplication Matrix) with Real-Valued Elements: Finding the Roots of a Trigonometric Polynomial by Matrix Eigensolving
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- Journal:
- Numerical Mathematics: Theory, Methods and Applications / Volume 6 / Issue 4 / November 2013
- Published online by Cambridge University Press:
- 28 May 2015, pp. 586-599
- Print publication:
- November 2013
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- Article
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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 Mjk. This matrix
is a multiplication matrix in the sense that, after first defining a vector 
whose elements are the first 2N basis functions, 
. This relationship is the eigenproblem; the zeros tk are the arccosine function of λk/2 where the λk are the eigenvalues of . 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.
is a multiplication matrix in the sense that, after first defining a vector 
whose elements are the first 2
. This relationship is the eigenproblem; the zeros 