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 In full generality, the Clenshaw algorithm computes the weighted sum of a finite series of functions : where is a sequence of functions that satisfy the linear recurrence relation where the coefficients and are known in advance. The algorithm is most useful when are functions that are complicated to compute directly, but and are particularly simple. In the most common applications, does not depend on , and is a constant that depends on neither nor . To perform the summation for given series of coefficients , compute the values by the "reverse" recurrence formula: Note that this computation makes no direct reference to the functions . After computing and , the desired sum can be expressed in terms of them and the simplest functions and : See Fox and Parker for more information and stability analyses. Examples Horner as a special case of Clenshaw A particularly simple case occurs when evaluating a polynomial of the form . The functions are simply and are produced by the recurrence coefficients and . In this case, the recurrence formula to compute the sum is and, in this case, the sum is simply , 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 are with the initial conditions Thus, the recurrence is and the final sum is One way to evaluate this is to continue the recurrence one more step, and compute (note the doubled a0 coefficient) followed by 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 distance on the surface of an ellipsoid. These have the form Leaving off the initial term, the remainder is a summation of the appropriate form. There is no leading term because . The recurrence relation for is , making the coefficients in the recursion relation and the evaluation