In numerical analysis, the Clenshaw algorithm is a recursive method to evaluate a linear combination of Chebyshev polynomials. It is a generalization of Horner's method for evaluating a linear combination of monomials.
Suppose that is a sequence of functions that satisfy the linear recurrence relation
where the coefficients and are known in advance. Note that in the most common applications, does not depend on , and is a constant that depends on neither nor .
Our goal is to evaluate a weighted sum of these functions
Given the coefficients , compute the values by the "reverse" recurrence formula:
The linear combination of the satisfies:
See Fox and Parker for more information and stability analyses.
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
Clenshaw's algorithm is extensively used in geodetic applications where it is usually referred to as Clenshaw summation. A simple application is summing the trigonometric series to compute the meridian arc. 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 .
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,
, and . The first element of is the average value of and the second element is the average slope. satisfies the recurrence relation
The standard Clenshaw algorithm can now be applied to yield
where are matrices. Finally we have
In the limit , this relation correctly evaluates the derivative , providing that, in evaluating and , we take .
- Horner scheme to evaluate polynomials in monomial form
- De Casteljau's algorithm to evaluate polynomials in Bézier form
- Clenshaw, C. W. (July 1955). "A note on the summation of Chebyshev series". Mathematical Tables and other Aids to Computation 9 (51): 118–110. doi:10.1090/S0025-5718-1955-0071856-0. ISSN 0025-5718. Note that this paper is written in terms of the Shifted Chebyshev polynomials of the first kind .
- Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007), "Section 5.4.2. Clenshaw's Recurrence Formula", Numerical Recipes: The Art of Scientific Computing (3rd ed.), New York: Cambridge University Press, ISBN 978-0-521-88068-8
- L. Fox; I. B. Parker (1968), Chebyshev Polynomials in Numerical Analysis, Oxford University Press, ISBN 0-19-859614-6
- Tscherning, C. C.; Poder, K. (1982), "Some Geodetic applications of Clenshaw Summation", Bolletino di Geodesia e Scienze Affini 41 (4): 349–375
- Karney, C. F. F. (2014), Clenshaw evaluation of differenced sums