Computing real roots of a polynomial in Chebyshev series form through subdivision

• Published in: Applied numerical mathematics Vol. 56; no. 8; pp. 1077 - 1091
• Main Author: Boyd, John P.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01.08.2006; Elsevier
• Subjects: Algebra; Chebyshev polynomials; Exact sciences and technology; Field theory and polynomials; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Numerical linear algebra; Rootfinding; Sciences and techniques of general use; Rootfinding; Chebyshev polynomials; Numerical analysis; Error estimation; Partition method; Applied mathematics; Chebyshev polynomial; Approximation error; Matrix method; Computing; Algorithm
• ISSN: 0168-9274; 1873-5460
• DOI: 10.1016/j.apnum.2005.09.007

An arbitrary polynomial of degree N, f N ( 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 f N 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 N s subintervals and approximate f N 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.