Clenshaw algorithm

In numerical analysis, the Clenshaw algorithm, also called Clenshaw summation, is a recursive method to evaluate a linear combination of Chebyshev polynomials.^[1]^[2] The method was published by Charles William Clenshaw in 1955. It is a generalization of Horner's method for evaluating a linear combination of monomials.

It generalizes to more than just Chebyshev polynomials; it applies to any class of functions that can be defined by a three-term recurrence relation.^[3]

^ Clenshaw, C. W. (July 1955). "A note on the summation of Chebyshev series". Mathematical Tables and Other Aids to Computation. 9 (51): 118. doi:10.1090/S0025-5718-1955-0071856-0. ISSN 0025-5718. Note that this paper is written in terms of the Shifted Chebyshev polynomials of the first kind $T_{n}^{*}(x)=T_{n}(2x-1)$ .
^ Cite error: The named reference Tscherning82 was invoked but never defined (see the help page).
^ Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007), "Section 5.4.2. Clenshaw's Recurrence Formula", Numerical Recipes: The Art of Scientific Computing (3rd ed.), New York: Cambridge University Press, ISBN 978-0-521-88068-8

[Clenshaw55-1] Clenshaw, C. W. (July 1955). "A note on the summation of Chebyshev series". Mathematical Tables and Other Aids to Computation. 9 (51): 118. doi:10.1090/S0025-5718-1955-0071856-0. ISSN 0025-5718. Note that this paper is written in terms of the Shifted Chebyshev polynomials of the first kind $T_{n}^{*}(x)=T_{n}(2x-1)$ .

[Tscherning82-2] Cite error: The named reference Tscherning82 was invoked but never defined (see the help page).

[3] Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007), "Section 5.4.2. Clenshaw's Recurrence Formula", Numerical Recipes: The Art of Scientific Computing (3rd ed.), New York: Cambridge University Press, ISBN 978-0-521-88068-8

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