Clenshaw algorithm

Horner as a special case of Clenshaw

A particularly simple case occurs when evaluating a polynomial of the form $${\displaystyle S(x)=\sum _{k=0}^{n}a_{k}x^{k}.}$$ The functions are simply $${\displaystyle {\begin{aligned}\phi _{0}(x)&=1,\\\phi _{k}(x)&=x^{k}=x\phi _{k-1}(x)\end{aligned}}}$$ and are produced by the recurrence coefficients ${\displaystyle \alpha (x)=x}$ and ${\displaystyle \beta =0}$.

In this case, the recurrence formula to compute the sum is $${\displaystyle b_{k}(x)=a_{k}+xb_{k+1}(x)}$$ and, in this case, the sum is simply $${\displaystyle S(x)=a_{0}+xb_{1}(x)=b_{0}(x),}$$ which is exactly the usual Horner's method.

Special case for Chebyshev series

Consider a truncated Chebyshev series $${\displaystyle p_{n}(x)=a_{0}+a_{1}T_{1}(x)+a_{2}T_{2}(x)+\cdots +a_{n}T_{n}(x).}$$

The coefficients in the recursion relation for the Chebyshev polynomials are $${\displaystyle \alpha (x)=2x,\quad \beta =-1,}$$ with the initial conditions $${\displaystyle T_{0}(x)=1,\quad T_{1}(x)=x.}$$

Thus, the recurrence is $${\displaystyle b_{k}(x)=a_{k}+2xb_{k+1}(x)-b_{k+2}(x)}$$ and the final results are $${\displaystyle b_{0}(x)=a_{0}+2xb_{1}(x)-b_{2}(x),}$$ $${\displaystyle p_{n}(x)={\tfrac {1}{2}}\left[a_{0}+b_{0}(x)-b_{2}(x)\right].}$$

An equivalent expression for the sum is given by $${\displaystyle p_{n}(x)=a_{0}+xb_{1}(x)-b_{2}(x).}$$

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

Leaving off the initial ${\displaystyle C_{0}\,\theta }$ term, the remainder is a summation of the appropriate form. There is no leading term because ${\displaystyle \phi _{0}(\theta )=\sin 0\theta =\sin 0=0}$.

The recurrence relation for ${\displaystyle \sin k\theta }$ is $${\displaystyle \sin(k+1)\theta =2\cos \theta \sin k\theta -\sin(k-1)\theta ,}$$ making the coefficients in the recursion relation $${\displaystyle \alpha _{k}(\theta )=2\cos \theta ,\quad \beta _{k}=-1.}$$ and the evaluation of the series is given by $${\displaystyle {\begin{aligned}b_{n+1}(\theta )&=b_{n+2}(\theta )=0,\\b_{k}(\theta )&=C_{k}+2\cos \theta \,b_{k+1}(\theta )-b_{k+2}(\theta ),\quad \mathrm {for\ } n\geq k\geq 1.\end{aligned}}}$$ The final step is made particularly simple because ${\displaystyle \phi _{0}(\theta )=\sin 0=0}$, so the end of the recurrence is simply ${\displaystyle b_{1}(\theta )\sin(\theta )}$; the ${\displaystyle C_{0}\,\theta }$ term is added separately: $${\displaystyle m(\theta )=C_{0}\,\theta +b_{1}(\theta )\sin \theta .}$$

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

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 $${\displaystyle m(\theta _{1})-m(\theta _{2})=C_{0}(\theta _{1}-\theta _{2})+\sum _{k=1}^{n}2C_{k}\sin {\bigl (}{\textstyle {\frac {1}{2}}}k(\theta _{1}-\theta _{2}){\bigr )}\cos {\bigl (}{\textstyle {\frac {1}{2}}}k(\theta _{1}+\theta _{2}){\bigr )}.}$$ Clenshaw summation can be applied in this case provided we simultaneously compute ${\displaystyle m(\theta _{1})+m(\theta _{2})}$ and perform a matrix summation, $${\displaystyle {\mathsf {M}}(\theta _{1},\theta _{2})={\begin{bmatrix}(m(\theta _{1})+m(\theta _{2}))/2\\(m(\theta _{1})-m(\theta _{2}))/(\theta _{1}-\theta _{2})\end{bmatrix}}=C_{0}{\begin{bmatrix}\mu \\1\end{bmatrix}}+\sum _{k=1}^{n}C_{k}{\mathsf {F}}_{k}(\theta _{1},\theta _{2}),}$$ where $${\displaystyle {\begin{aligned}\delta &={\tfrac {1}{2}}(\theta _{1}-\theta _{2}),\\[1ex]\mu &={\tfrac {1}{2}}(\theta _{1}+\theta _{2}),\\[1ex]{\mathsf {F}}_{k}(\theta _{1},\theta _{2})&={\begin{bmatrix}\cos k\delta \sin k\mu \\{\dfrac {\sin k\delta }{\delta }}\cos k\mu \end{bmatrix}}.\end{aligned}}}$$ The first element of ${\displaystyle {\mathsf {M}}(\theta _{1},\theta _{2})}$ is the average value of ${\displaystyle m}$ and the second element is the average slope. ${\displaystyle {\mathsf {F}}_{k}(\theta _{1},\theta _{2})}$ satisfies the recurrence relation $${\displaystyle {\mathsf {F}}_{k+1}(\theta _{1},\theta _{2})={\mathsf {A}}(\theta _{1},\theta _{2}){\mathsf {F}}_{k}(\theta _{1},\theta _{2})-{\mathsf {F}}_{k-1}(\theta _{1},\theta _{2}),}$$ where $${\displaystyle {\mathsf {A}}(\theta _{1},\theta _{2})=2{\begin{bmatrix}\cos \delta \cos \mu &-\delta \sin \delta \sin \mu \\-\displaystyle {\frac {\sin \delta }{\delta }}\sin \mu &\cos \delta \cos \mu \end{bmatrix}}}$$ takes the place of ${\displaystyle \alpha }$ in the recurrence relation, and ${\displaystyle \beta =-1}$. The standard Clenshaw algorithm can now be applied to yield $${\displaystyle {\begin{aligned}{\mathsf {B}}_{n+1}&={\mathsf {B}}_{n+2}={\mathsf {0}},\\[1ex]{\mathsf {B}}_{k}&=C_{k}{\mathsf {I}}+{\mathsf {A}}{\mathsf {B}}_{k+1}-{\mathsf {B}}_{k+2},\qquad \mathrm {for\ } n\geq k\geq 1,\\[1ex]{\mathsf {M}}(\theta _{1},\theta _{2})&=C_{0}{\begin{bmatrix}\mu \\1\end{bmatrix}}+{\mathsf {B}}_{1}{\mathsf {F}}_{1}(\theta _{1},\theta _{2}),\end{aligned}}}$$ where ${\displaystyle {\mathsf {B}}_{k}}$ are 2×2 matrices. Finally we have $${\displaystyle {\frac {m(\theta _{1})-m(\theta _{2})}{\theta _{1}-\theta _{2}}}={\mathsf {M}}_{2}(\theta _{1},\theta _{2}).}$$ This technique can be used in the limit ${\displaystyle \theta _{2}=\theta _{1}=\mu }$ and ${\displaystyle \delta =0}$ to simultaneously compute ${\displaystyle m(\mu )}$ and the derivative ${\displaystyle dm(\mu )/d\mu }$, provided that, in evaluating ${\displaystyle {\mathsf {F}}_{1}}$ and ${\displaystyle {\mathsf {A}}}$, we take ${\displaystyle \lim _{\delta \to 0}(\sin k\delta )/\delta =k}$.