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Concept# Clenshaw algorithm

Summary

In numerical analysis, the Clenshaw algorithm, also called Clenshaw summation, is a recursive method to evaluate a linear combination of Chebyshev polynomials. 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.
Clenshaw algorithm
In full generality, the Clenshaw algorithm computes the weighted sum of a finite series of functions \phi_k(x):
:S(x) = \sum_{k=0}^n a_k \phi_k(x)
where \phi_k,; k=0, 1, \ldots is a sequence of functions that satisfy the linear recurrence relation
:\phi_{k+1}(x) = \alpha_k(x),\phi_k(x) + \beta_k(x),\phi_{k-1}(x),
where the coefficients \alpha_k(x) and \beta_k(x) are known in advance.
The al

Official source

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