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
Computing real roots of a polynomial in Chebyshev series form through subdivision
Applied Numerical Mathematics
Computing real roots of a polynomial in Chebyshev series form through subdivision
Applied Numerical Mathematics
Computers & Mathematics with Applications
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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For a function f(x) that is smooth on the interval x@?[a,b] but otherwise arbitrary, the real-valued roots on the interval can always be found by the following two-part procedure. First, expand f(x) as a Chebyshev polynomial series on the interval and truncate for sufficiently large N. Second, find the zeros of the truncated Chebyshev series. The roots of an arbitrary polynomial of degree N, when written in the form of a truncated Chebyshev series, are the eigenvalues of an NxN matrix whose elements are simple, explicit functions of the coefficients of the Chebyshev series. This matrix is a generalization of the Frobenius companion matrix. We show by experimenting with random polynomials, Wilkinson's notoriously ill-conditioned polynomial, and polynomials with high-order roots that the Chebyshev companion matrix method is remarkably accurate for finding zeros on the target interval, yielding roots close to full machine precision. We also show that it is easy and cheap to apply Newton's iteration directly to the Chebyshev series so as to refine the roots to full machine precision, using the companion matrix eigenvalues as the starting point. Lastly, we derive a couple of theorems. The first shows that simple roots are stable under small perturbations of magnitude @e to a Chebyshev coefficient: the shift in the root x"* is bounded by @e/df/dx(x"*)+O(@e^2) for sufficiently small @e. Second, we show that polynomials with definite parity (only even or only odd powers of x) can be solved by a companion matrix whose size is one less than the number of nonzero coefficients, a vast cost-saving.