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.
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.
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 followed by
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 of the series is given by The final step is made particularly simple because, so the end of the recurrence is simply ; the term is added separately: Note that the algorithm requires only the evaluation of two trigonometric quantities and.
Difference in meridian arc lengths
Sometimes it necessary to compute the difference of two meridian arcs in a way that maintains high relative accuracy. This is accomplished by using trigonometric identities to write Clenshaw summation can be applied in this case provided we simultaneously compute and perform a matrix summation, where The first element of is the average value of and the second element is the average slope. satisfies the recurrence relation where takes the place of in the recurrence relation, and. The standard Clenshaw algorithm can now be applied to yield where are 2×2 matrices. Finally we have This technique can be used in the limit and to simultaneously compute and the derivative , provided that, in evaluating and, we take.
