Computing the zeros of a Fourier series or a Chebyshev series or general orthogonal polynomial series with parity symmetries

DOI10.1016/j.camwa.2007.01.015zbMath1132.65039OpenAlexW2085902949MaRDI QIDQ2460590

John P. Boyd

Publication date: 12 November 2007

Published in: Computers \& Mathematics with Applications (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1016/j.camwa.2007.01.015

zbMATH Keywords

Chebyshev polynomialspseudospectral methodorthogonal polynomialsnumerical examplestrigonometric polynomialsFourier seriesprolate spheroidal wave functionsrootfindingMathieu functionsFrobenius companion matrixHough functions

Mathematics Subject Classification ID

Trigonometric approximation (42A10) Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) (33C45) Numerical computation of solutions to single equations (65H05) General harmonic expansions, frames (42C15) Numerical methods for trigonometric approximation and interpolation (65T40) Fourier coefficients, Fourier series of functions with special properties, special Fourier series (42A16) Numerical approximation and evaluation of special functions (33F05)

Related Items (5)

A comparison of companion matrix methods to find roots of a trigonometric polynomial ⋮ Using parity to accelerate Hermite function computations: zeros of truncated Hermite series, Gaussian quadrature and Clenshaw summation ⋮ New series for the cosine lemniscate function and the polynomialization of the lemniscate integral ⋮ Numerical, perturbative and Chebyshev inversion of the incomplete elliptic integral of the second kind ⋮ Using Parity to Accelerate the Computation of the Zeros of Truncated Legendre and Gegenbauer Polynomial Series and Gaussian Quadrature

Uses Software

• Matlab
• Chebfun

Cites Work

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• Computing the zeros, maxima and inflection points of Chebyshev, Legendre and Fourier series: solving transcendental equations by spectral interpolation and polynomial rootfinding
• A companion matrix analogue for orthogonal polynomials
• Fast and stable QR eigenvalue algorithms for generalized companion matrices and secular equations
• A Chebyshev polynomial interval-searching method (``Lanczos economization) for solving a nonlinear equation with application to the nonlinear eigenvalue problem
• Numerical experiments on the accuracy of the Chebyshev-Frobenius companion matrix method for finding the zeros of a truncated series of Chebyshev polynomials
• Computing real roots of a polynomial in Chebyshev series form through subdivision with linear testing and cubic solves
• Computing real roots of a polynomial in Chebyshev series form through subdivision
• Spectral Methods in MATLAB
• An Extension of MATLAB to Continuous Functions and Operators
• Computing Zeros on a Real Interval through Chebyshev Expansion and Polynomial Rootfinding
• Numerical Polynomial Algebra
• Roots of Polynomials Expressed in Terms of Orthogonal Polynomials

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