Clenshaw algorithm

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In numerical analysis, the Clenshaw algorithm,[1] also called Clenshaw summation,[2] 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.

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.[3]

Clenshaw algorithm[edit]

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[4] for more information and stability analyses.

Examples[edit]

Horner as a special case of Clenshaw[edit]

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[edit]

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[edit]

Clenshaw summation is extensively used in geodetic applications.[2] 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[edit]

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[5] 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 .

See also[edit]

References[edit]

  1. ^ 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 .
  2. ^ a b Tscherning, C. C.; Poder, K. (1982), "Some Geodetic applications of Clenshaw Summation" (PDF), Bolletino di Geodesia e Scienze Affini, 41 (4): 349–375 
  3. ^ 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 
  4. ^ Fox, Leslie; Parker, Ian B. (1968), Chebyshev Polynomials in Numerical Analysis, Oxford University Press, ISBN 0-19-859614-6 
  5. ^ Karney, C. F. F. (2014), Clenshaw evaluation of differenced sums