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Polynomial evaluation and associated polynomials. (English) Zbl 0909.65007

Author’s abstract: We show a connection between the Clenshaw algorithm for evaluating a polynomial \(q_n\), expanded in terms of a system of orthogonal polynomials, and special linear combinations of associated polynomials. These results enable us to get the derivatives of \(q_{n}(z)\) analogously to the Horner algorithm for evaluating polynomials in monomial representations. Furthermore we show how a polynomial \(\widehat{q}_{n}(z)\) given in monomial representation can be evaluated for \(z \in C\) using the Clenshaw algorithm without complex arithmetic. From this we get a connection between zeros of polynomials expanded in terms of Chebychev polynomials and the corresponding polynomials in monomial representation with the same coefficients.

MSC:

65D20 Computation of special functions and constants, construction of tables
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
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