Computing the zeros, maxima and inflection points of Chebyshev, Legendre and Fourier series: solving transcendental equations by spectral interpolation and polynomial rootfinding

Recently, both companion-matrix methods and subdivision algorithms have been developed for finding the zeros of a truncated spectral series. Since the Chebyshev or Legendre coefficients of derivatives of a function f(x) can be computed by trivial recurrences from those of the function itself, it follows that finding the maxima, minima and inflection points of a truncated Chebyshev or Fourier series f (N) (x) is also a problem of finding the zeros of a polynomial when written in truncated Chebyshev series form, or computing the roots of a trigonometric polynomial. Widely scattered results are reviewed and a few previously unpublished ideas sprinkled in. There are now robust zerofinders for all species of spectral series. A transcendental function f(x) can be approximated arbitrarily well on a real interval by a truncated Chebyshev series f (N) (x) of sufficiently high degree N. It follows that through Chebyshev interpolation and Chebyshev rootfinders, it is now possible to easily find all the real roots on an interval for any smooth transcendental function.