Clenshaw algorithm

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(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

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