A fast algorithm for the evaluation of Legendre expansions
SIAM Journal on Scientific and Statistical Computing
Remark on algorithms to find roots of polynomials
SIAM Journal on Scientific Computing
Spectral methods in MatLab
Numerical Methods for Scientists and Engineers
Numerical Methods for Scientists and Engineers
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
Roots of Polynomials Expressed in Terms of Orthogonal Polynomials
SIAM Journal on Numerical Analysis
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
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An arbitrary polynomial of degree N, fN (x), can always be represented as a truncated Chebyshev polynomial series ("Chebyshev form"). This representation is much better conditioned than the usual "power form" of a polynomial. We describe two families of algorithms for finding the real roots of fN in Chebyshev form. We briefly review existing companion matrix methods--robust, but relatively expensive. We then describe a broad family of new rootfinders employing subdivision. These new methods partition the canonical interval, x ∈ [-1, 1], into Ns subintervals and approximate fN by a low degree Chebyshev interpolant on each subdomain. We derive a rigorous error bound that allows tight control of the error in these local approximations. Because the cost of companion matrix methods grows as the cube of the degree, it is much less expensive for N 50 to calculate the roots of many low degree polynomials, one polynomial on each subdivision, than to directly compute the roots of a single polynomial of high degree.