# Fast computation of the zeros of a trigonometric polynomial

I am wondering what is the fastest way to compute the zeroes of the trigonometric polynomial

$$T(x) = \sum_{i=1}^La_i\sin\big(2\pi(a_i x) - \phi_i \big), \\ a_i \in \mathbb{N}, \\ \phi_i \in \left[-\pi,\pi\right[, \\ x \in \left[-\frac{1}{2},+\frac{1}{2}\right[.$$

• Did you try converting the sum to a Chebyshev series and then using the colleague matrix approach (i.e. calculating the eigenvalues of a special companion matrix)? Jan 28 '20 at 9:47
• What programming language are you using? Jan 28 '20 at 16:59
• @GertVdE yes that seems to be way to go. I have collected some references in my reply below Jan 28 '20 at 19:09
• @nicoguaro mainly MATLAB and Julia. I know that Chebfun can compute the zeroes easily, however I'm more interested in the algorithm than in the software implementation right now Jan 28 '20 at 19:12
• In that case, have you checked "Approximation Theory and Approximation Practice" by Nick Trefethen? Jan 28 '20 at 19:23