Clenshaw algorithm

In numerical analysis, the Clenshaw algorithm, also called Clenshaw summation, is a recursive method to evaluate a linear combination of Chebyshev polynomials. Note that this paper is written in terms of the ''Shifted'' Chebyshev polynomials of the first kind $T^*_n(x) = T_n(2x-1)$. 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 $\phi_k(x)$:
:$S(x) = \sum_^n a_k \phi_k(x)$

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

:$\phi_(x) = \alpha_k(x)\,\phi_k(x) + \beta_k(x)\,\phi_(x),$

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

The algorithm is most useful when $\phi_k(x)$ are functions that are complicated 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:

:$\begin b_(x) &= b_(x) = 0, \\ b_k(x) &= a_k + \alpha_k(x)\,b_(x) + \beta_(x)\,b_(x). \end$

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 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
:$S(x) = \sum_^n a_k x^k$.
The functions are simply
:$\begin \phi_0(x) &= 1, \\ \phi_k(x) &= x^k = x\phi_(x) \end$
and are produced by the recurrence coefficients $\alpha(x) = x$ and $\beta = 0$.

In this case, the recurrence formula to compute the sum is
:$b_k(x) = a_k + x b_(x)$
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

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
:$b_k(x) = a_k + 2xb_(x) - b_(x)$
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 ''a''₀ coefficient) followed by
:p_n(x) = \frac\left _0(x) - b_2(x)\right

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

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

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

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

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
:$m(\theta_1)-m(\theta_2) = C_0(\theta_1-\theta_2) + \sum_^n 2 C_k \sin\bigl(k(\theta_1-\theta_2)\bigr) \cos\bigl(k(\theta_1+\theta_2)\bigr).$
Clenshaw summation can be applied in this case

provided we simultaneously compute $m(\theta_1)+m(\theta_2)$
and perform a matrix summation,
:$\mathsf M(\theta_1,\theta_2) = \begin (m(\theta_1) + m(\theta_2)) / 2\\ (m(\theta_1) - m(\theta_2)) / (\theta_1 - \theta_2) \end = C_0 \begin\mu\\1\end + \sum_^n C_k \mathsf F_k(\theta_1,\theta_2),$
where
:$\begin \delta &= (\theta_1-\theta_2), \\ \mu &= (\theta_1+\theta_2), \\ \mathsf F_k(\theta_1,\theta_2) &= \begin \cos k \delta \sin k \mu\\ \displaystyle\frac\delta \cos k \mu \end. \end$
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
:$\mathsf F_(\theta_1,\theta_2) = \mathsf A(\theta_1,\theta_2) \mathsf F_k(\theta_1,\theta_2) - \mathsf F_(\theta_1,\theta_2),$
where
:$\mathsf A(\theta_1,\theta_2) = 2\begin \cos \delta \cos \mu & -\delta\sin \delta \sin \mu \\ - \displaystyle\frac\delta \sin \mu & \cos \delta \cos \mu \end$
takes the place of $\alpha$ in the recurrence relation, and $\beta=-1$.
The standard Clenshaw algorithm can now be applied to yield
:$\begin \mathsf B_ &= \mathsf B_ = \mathsf 0, \\ \mathsf B_k &= C_k \mathsf I + \mathsf A \mathsf B_ - \mathsf B_, \qquad\mathrm n\ge k \ge 1,\\ \mathsf M(\theta_1,\theta_2) &= C_0 \begin\mu\\1\end + \mathsf B_1 \mathsf F_1(\theta_1,\theta_2), \end$
where $\mathsf B_k$ are 2×2 matrices. Finally
we have
:$\frac = \mathsf M_2(\theta_1, \theta_2).$
This technique can be used in the limit $\theta_2 = \theta_1 = \mu$
and $\delta = 0\,$ to simultaneously compute $m(\mu)$ and the derivative
$dm(\mu)/d\mu$, provided that, in evaluating $\mathsf F_1$ and $\mathsf A$,
we take $\lim_(\sin k \delta)/\delta = k$.

See also

* Horner scheme to evaluate polynomials in monomial form
* De Casteljau's algorithm to evaluate polynomials in Bézier form

References

{{DEFAULTSORT:Clenshaw Algorithm
Numerical analysis