Spectral methods in MatLab
Computing Zeros on a Real Interval through Chebyshev Expansion and Polynomial Rootfinding
SIAM Journal on Numerical Analysis
An Extension of MATLAB to Continuous Functions and Operators
SIAM Journal on Scientific Computing
Numerical Polynomial Algebra
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Roots of Polynomials Expressed in Terms of Orthogonal Polynomials
SIAM Journal on Numerical Analysis
Computing real roots of a polynomial in Chebyshev series form through subdivision
Applied Numerical Mathematics
Journal of Computational and Applied Mathematics
New series for the cosine lemniscate function and the polynomialization of the lemniscate integral
Journal of Computational and Applied Mathematics
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In recent years, good algorithms have been developed for finding the zeros of trigonometric polynomials and of ordinary polynomials when written in the form of a truncated Chebyshev polynomial or Legendre polynomial series. In each case, the roots can be found from the eigenvalues of a generalized Frobenius companion matrix whose elements are trivial functions of the Fourier coefficients or Chebyshev coefficients. However, the QR method for computing the companion matrix eigenvalues has a cost that grows proportionally to N^3 where N is the polynomial degree. (By exploiting the special structure of the companion matrices, the cost can be reduced to O(N^2), but only for large N.) Here, we show that if the polynomial has definite parity, such as a trigonometric polynomial composed only of cosines or a polynomial that is a sum only of Chebyshev polynomials of odd degree, one can exploit these symmetries to halve the size of the problem. This reduces costs in the companion matrix method by a factor ranging between four and eight. For trigonometric polynomials, we give transformations that dramatically reduce costs even if the roots are found by an algorithm other than the companion matrix procedure. We further give reductions for trigonometric polynomials with double parity symmetries which save a factor of sixteen to a factor of sixty-four in the companion matrix algorithm. Special functions such as spherical harmonics, Mathieu functions, prolate spheroidal wavefunctions and Hough functions, all represented by truncated Fourier series with double parity, are a rich source of applications.