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 \phi_k(x):

S(x) = \sum_{k=0}^n a_k \phi_k(x)

where \phi_k,\; k=0, 1, \ldots is a sequence of functions that satisfy the linear recurrence relation

\phi_{k+1}(x) = \alpha_k(x)\,\phi_k(x) + \beta_k(x)\,\phi_{k-1}(x),

where the coefficients \alpha_k(x) and \beta_k(x) are known in advance.

The algorithm is most useful when \phi_k(x) are complex functions to compute directly, but \alpha_k(x) and \beta_k(x) are particularly simple. In the most common applications, \alpha(x) does not depend on k, and \beta is a constant that depends on neither x nor k.

To perform the summation for given series of coefficients a_0, \ldots, a_n, compute the values b_k(x) by the "reverse" recurrence formula:


  \begin{align}
  b_{n+1}(x) &= b_{n+2}(x) = 0, \\
  b_k(x) &= a_k + \alpha_k(x)\,b_{k+1}(x) + \beta_{k+1}(x)\,b_{k+2}(x).
  \end{align}

Note that this computation makes no direct reference to the functions \phi_k(x). After computing b_2(x) and b_1(x), the desired sum can be expressed in terms of them and the simplest functions \phi_0(x) and \phi_1(x):

S(x) = \phi_0(x)\,a_0 + \phi_1(x)\,b_1(x) + \beta_1(x)\,\phi_0(x)\,b_2(x).

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

S(x) = \sum_{k=0}^n a_k x^k.

The functions are simply


  \begin{align}
  \phi_0(x) &= 1, \\
  \phi_k(x) &= x^k = x\phi_{k-1}(x)
  \end{align}

and are produced by the recurrence coefficients \alpha(x) = x and \beta = 0.

In this case, the recurrence formula to compute the sum is

b_k(x) = a_k + x b_{k+1}(x)

and, in this case, the sum is simply

S(x) = a_0 + x b_1(x) = b_0(x),

which is exactly the usual Horner's method.

Special case for Chebyshev series[edit]

Consider a truncated Chebyshev series

p_n(x) = a_0 + a_1T_1(x) + a_2T_2(x) + \cdots + a_nT_n(x).

The coefficients in the recursion relation for the Chebyshev polynomials are

\alpha(x) = 2x, \quad \beta = -1,

with the initial conditions

T_0(x) = 1, \quad T_1(x) = x.

Thus, the recurrence is

b_k(x) = a_k + 2xb_{k+1}(x) - b_{k+2}(x)

and the final sum is

p_n(x) = a_0 + xb_1(x) - b_2(x).

One way to evaluate this is to continue the recurrence one more step, and compute

b_0(x) = 2a_0 + 2xb_1(x) - b_2(x),

(note the doubled a0 coefficient) followed by

p_n(x) = \frac{1}{2}\left[b_0(x) - b_2(x)\right].

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

m(\theta) = C_0\,\theta + C_1\sin \theta + C_2\sin 2\theta + \cdots + C_n\sin n\theta.

Leaving off the initial C_0\,\theta term, the remainder is a summation of the appropriate form. There is no leading term because \phi_0(\theta) = \sin 0\theta = \sin 0 = 0.

The recurrence relation for \sin k\theta is

\sin (k+1)\theta = 2 \cos\theta \sin k\theta - \sin (k-1)\theta,

making the coefficients in the recursion relation

\alpha_k(\theta) = 2\cos\theta, \quad \beta_k = -1.

and the evaluation of the series is given by


  \begin{align}
  b_{n+1}(\theta) &= b_{n+2}(\theta) = 0, \\
  b_k(\theta) &= C_k + 2\cos \theta \,b_{k+1}(\theta) - b_{k+2}(\theta),\quad\mathrm{for\ } n\ge k \ge 1.
  \end{align}

The final step is made particularly simple because \phi_0(\theta) = \sin 0 = 0, so the end of the recurrence is simply b_1(\theta)\sin(\theta); the C_0\,\theta term is added separately:

m(\theta) = C_0\,\theta + b_1(\theta)\sin \theta.

Note that the algorithm requires only the evaluation of two trigonometric quantities \cos \theta and \sin \theta.

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


  m(\theta_1)-m(\theta_2) = \sum_{k=1}^n 2 C_k
  \sin\bigl({\textstyle\frac12}k(\theta_1-\theta_2)\bigr)
  \cos\bigl({\textstyle\frac12}k(\theta_1+\theta_2)\bigr).

Clenshaw summation can be applied in this case[5] provided we simultaneously compute m(\theta_1)+m(\theta_2) and perform a matrix summation,


  \mathsf M(\theta_1,\theta_2) = \begin{bmatrix}
  (m(\theta_1) + m(\theta_2)) / 2\\
  (m(\theta_1) - m(\theta_2)) / (\theta_1 - \theta_2)
  \end{bmatrix} =
  C_0 \begin{bmatrix}\mu\\1\end{bmatrix} +
  \sum_{k=1}^n C_k \mathsf F_k(\theta_1,\theta_2),

where


  \begin{align}
  \delta &= {\textstyle\frac12}(\theta_1-\theta_2), \\
  \mu &= {\textstyle\frac12}(\theta_1+\theta_2), \\
  \mathsf F_k(\theta_1,\theta_2) &=
    \begin{bmatrix}
       \cos k \delta \sin k \mu\\
    \displaystyle\frac{\sin k \delta}\delta \cos k \mu
    \end{bmatrix}.
  \end{align}

The first element of \mathsf M(\theta_1,\theta_2) is the average value of m and the second element is the average slope. \mathsf F_k(\theta_1,\theta_2) satisfies the recurrence relation


  \mathsf F_{k+1}(\theta_1,\theta_2) =
  \mathsf A(\theta_1,\theta_2) \mathsf F_k(\theta_1,\theta_2) -
  \mathsf F_{k-1}(\theta_1,\theta_2),

where


   \mathsf A(\theta_1,\theta_2) = 2\begin{bmatrix}
      \cos \delta \cos \mu & -\delta\sin \delta \sin \mu \\
      - \displaystyle\frac{\sin \delta}\delta \sin \mu &   \cos \delta \cos \mu
  \end{bmatrix}

takes the place of \alpha in the recurrence relation, and \beta=-1. The standard Clenshaw algorithm can now be applied to yield


  \begin{align}
  \mathsf B_{n+1} &= \mathsf B_{n+2} = \mathsf 0, \\
  \mathsf B_k &= C_k \mathsf I + \mathsf A \mathsf B_{k+1} -
  \mathsf B_{k+2}, \qquad\mathrm{for\ } n\ge k \ge 1,\\
  \mathsf M(\theta_1,\theta_2) &=
    C_0 \begin{bmatrix}\mu\\1\end{bmatrix} +
    \mathsf B_1 \mathsf F_1(\theta_1,\theta_2),
  \end{align}

where \mathsf B_k are 2×2 matrices. Finally we have


  \frac{m(\theta_1) - m(\theta_2)}{\theta_1 - \theta_2} =
  \mathsf M_2(\theta_1, \theta_2).

This technique can be used in the limit \theta_2 = \theta_1 = \mu and \delta = 0\, to compute the derivative dm(\mu)/d\mu, provided that, in evaluating \mathsf F_1 and \mathsf
A, we take \lim_{\delta\rightarrow0}(\sin \delta)/\delta =
1.

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.  edit Note that this paper is written in terms of the Shifted Chebyshev polynomials of the first kind T^*_n(x) = T_n(2x-1).
  2. ^ a b Tscherning, C. C.; Poder, K. (1982), "Some Geodetic applications of Clenshaw Summation", 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