Computing Zeros on a Real Interval through Chebyshev Expansion and Polynomial Rootfinding
SIAM Journal on Numerical Analysis
Numerical Polynomial Algebra
Analysis of Some Padé--Chebyshev Approximants
SIAM Journal on Numerical Analysis
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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The Kepler equation for the parameters of an elliptical orbit, E-ε sin(E) = M, is reduced from a transcendental to a polynomial equation by expanding the sine as a series of Chebyshev polynomials. The single real root is found by applying standard polynomial rootfinders and accepting only the polynomial root that lies on the interval predicted by rigorous theoretical bounds. A complete Matlab implementation is given in full because it requires just seven lines. For a polynomial of degree fifteen, the maximum absolute error over the whole range ε ∈ [0, 1] and all M is only 4 × 10-10. Other transcendental equations can similarly be reduced to polynomial equations through Chebyshev expansions.