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Clenshaw algorithm

In the mathematical subfield of numerical analysis the Clenshaw algorithm is a recursive method to evaluate polynomials in Chebyshev form.

Polynomial in Chebyshev form

A polynomial of degree N in Chebyshev form is a polynomial p(x) of the form

$p(x) = \sum_{n=0}^{N} a_n T_n(x)$

where Tn is the nth Chebyshev polynomial.

Clenshaw algorithm

The Clenshaw algorithm can be used to evaluate a polynomial in the Chebyshev form. Given

$p(x) = \sum_{n=0}^{N} a_n T_n(x)$

we define

 $b_{N} \,\!$ $:= a_{N} \,$ $b_{N-1} \,\!$ $:= 2 x b_{N} + a_{N-1} \,$ $b_{N-n} \,\!$ $:= 2 x b_{N-n+1} + a_{N-n} + b_{N-n+2} \,,\; n=2,\ldots,N-1 \,$ $b_{0} \,\!$ $:= x b_{1} + a_{0} - b_{2} \,$

then

$p(x) = \sum_{n=0}^{N} a_n T_n(x) = b_{0}.$