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Clenshaw algorithm

In the mathematical subfield of numerical analysis the Clenshaw algorithm (Invented by Charles William Clenshaw) 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

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

where Tn is the nth Chebyshev polynomial.

Clenshaw algorithm

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

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

we define

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

then

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