In numerical analysis, the Clenshaw algorithm, also called Clenshaw summation, is a recursive method to evaluate a linear combination of Chebyshev polynomials. The method was published by Charles William Clenshaw in 1955. It is a generalization of Horner's method for evaluating a linear combination of monomials.

It generalizes to more than just Chebyshev polynomials; it applies to any class of functions that can be defined by a three-term recurrence relation.

Clenshaw algorithm

are known in advance.

.

See Fox and Parker for more information and stability analyses.

Examples

Horner as a special case of Clenshaw

.

which is exactly the usual Horner's method.

Special case for Chebyshev series

Consider a truncated Chebyshev series

The coefficients in the recursion relation for the Chebyshev polynomials

Meridian arc length on the ellipsoid

Clenshaw summation is extensively used in geodetic applications. A simple application is summing the trigonometric series to compute the meridian arc

.

The \sin k\theta

.

Difference in meridian arc lengths

Clenshaw summation can be applied in this case This technique can be used in the limit and the derivative.

See also

• Horner scheme to evaluate polynomials in monomial form

• De Casteljau's algorithm to evaluate polynomials in Bézier form

References