Computing the real roots of a Fourier series-plus-linear-polynomial: A Chebyshev companion matrix approach

作者：Boyd John P^*; Sadiq Burhan A

来源：Applied Mathematics and Computation, 2012, 219(3): 819-826.

DOI：10.1016/j.amc.2012.06.045

• Fourier series
• Trigonometric polynomial
• Rootfinding
• Secular
• Companion matrix
• Chebyshev polynomials

Fourier series often need to be generalized by appending a linear polynomial to the usual series of sines and cosines. The integral of a trigonometric polynomial is one example; another is a time series of climate data where the periodic oscillations of the diurnal and annual cycles are accompanied by a non-periodic trend (global warming). Stock market averages fluctuate about a generally upward trend. Such non-periodic variations with time are commonly called "secular trends". We borrow "secular" to label a truncated Fourier series plus a linear trend as a "linear", secular trigonometric polynomial. Standard Fourier root-finding methods are wrecked by the extra, nonperiodic term. Therefore, we introduce a new algorithm for computing the zeros of a Fourier polynomial-with-secular-trend. First, we expand the linear secular trigonometric polynomial f(N)(t) as a truncated Chebyshev series. Because of the special structure, it is easy to calculate a problem-dependent truncation M such that the error of the truncated Chebyshev series is guaranteed to be less than a user-specified tolerance. We then find the roots of the truncated Chebyshev series as the eigen-values of the Chebyshev companion matrix. This computes all roots, but we explain why the method is not reliable for complex-valued roots unless these are close to the real axis. No a priori information is required of the user except the coefficients of the linear secular trigonometric polynomial. Numerical examples show that 13 decimal place accuracy for real roots is typical.

• 出版日期2012-10-15