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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}.

See also

Last updated: 10-18-2005 16:23:24
Science Fair Project Encyclopedia Contents Page
03-10-2013 05:06:04
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