Computing real roots of a polynomial in Chebyshev series form through subdivision
Applied Numerical Mathematics
Variable-stepsize Chebyshev-type methods for the integration of second-order I.V.P.'s
Journal of Computational and Applied Mathematics
Journal of Computational and Applied Mathematics
Computers & Mathematics with Applications
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
Sturm root counting using chebyshev expansion
ACM Communications in Computer Algebra
A Sinc Function Analogue of Chebfun
SIAM Journal on Scientific Computing
Analysis of cutoff wavelength of elliptical waveguide by regularized meshless method
International Journal of Numerical Modelling: Electronic Networks, Devices and Fields
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Robust polynomial rootfinders can be exploited to compute the roots on a real interval of a nonpolynomial function f(x) by the following: (i) expand f as a Chebyshev polynomial series, (ii) convert to a polynomial in ordinary, series-of-powers form, and (iii) apply the polynomial rootfinder. (Complex-valued roots and real roots outside the target interval are discarded.) The expansion is most efficiently done by adaptive Chebyshev interpolation with N equal to a power of two, where $N$ is the degree of the truncated Chebyshev series. All previous evaluations of f can then be reused when N is increased; adaption stops when N is sufficiently large so that further increases produce no significant change in the interpolant. We describe two conversion strategies. The "convert-to-powers" method uses multiplication by mildly ill-conditioned matrices to create a polynomial of degree N. The "degree-doubling" strategy defines a polynomial of larger degree 2N but is very well-conditioned. The "convert-to-powers" method, although faster, restricts N to moderate values; this can always be accomplished by subdividing the target interval. Both these strategies allow simultaneous approximation of many roots on an interval, whether simple or multiple, for nonpolynomial f(x).