Continuous phase modulation

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.

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

Suppose that $\phi _{k},\;k=0,1,\ldots$ 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 $\alpha _{k}$ and $\beta _{k}$ are known in advance. Note that in the most common applications, $\alpha (x)$ does not depend on $k$, and $\beta$ is a constant that depends on neither $x$ nor $k$.

Our goal is to evaluate a weighted sum of these functions

$S(x)=\sum _{k=0}^{n}a_{k}\phi _{k}(x)$

Given the coefficients $a_{0},\ldots ,a_{n}$, compute the values $b_{k}(x)$ by the "reverse" recurrence formula:

${\begin{aligned}b_{n+1}(x)&=b_{n+2}(x)=0,\\[.5em]b_{k}(x)&=a_{k}+\alpha _{k}(x)\,b_{k+1}(x)+\beta _{k+1}(x)\,b_{k+2}(x).\end{aligned}}$

The linear combination of the $\phi _{k}$ satisfies:

$S(x)=\phi _{0}(x)a_{0}+\phi _{1}(x)b_{1}(x)+\beta _{1}(x)\phi _{0}(x)b_{2}(x).$

See Fox and Parker for more information and stability analyses.

Horner as a special case of Clenshaw

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

${\begin{aligned}\phi _{0}(x)&=1,\\\phi _{k}(x)&=x^{k}=x\phi _{k-1}(x)\end{aligned}}$

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

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

$b_{k}(x)=a_{k}+xb_{k+1}(x)$

and, in this case, the sum is simply

$S(x)=a_{0}+xb_{1}(x)=b_{0}(x)$,

which is exactly the usual Horner's method.

Special case for Chebyshev series

Consider a truncated Chebyshev series

$p_{n}(x)=a_{0}+a_{1}T_{1}(x)+a_{2}T_{2}(x)+\cdots +a_{n}T_{n}(x).$

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

$p_{n}(x)=b_{0}(x)-xb_{1}(x)-a_{0}={\frac {1}{2}}\left[b_{0}(x)-b_{2}(x)\right].$

Geodetic applications

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

$m(\theta )=C_{0}\,\theta +C_{1}\sin \theta +C_{2}\sin 2\theta +\cdots +C_{n}\sin n\theta .$

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

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

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

${\begin{aligned}b_{n+1}(\theta )&=b_{n+2}(\theta )=0,\\[.3em]b_{k}(\theta )&=C_{k}+2b_{k+1}(\theta )\cos \theta -b_{k+2}(\theta )\quad (n\geq k\geq 1).\end{aligned}}$

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 $\sin \theta$.

See also

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