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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 T_n 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}.

See also

Horner scheme to evaluate polynomials in monomial form
De Casteljau's algorithm to evaluate polynomials in Bzier form

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