Editing Clenshaw algorithm (section)

===Meridian arc length on the ellipsoid===
Clenshaw summation is extensively used in [[geodesy|geodetic]] applications.<ref name="Tscherning82">{{Citation
| last1=Tscherning
| first1=C. C.
| last2=Poder
| first2=K.
| year=1982
| title=Some Geodetic applications of Clenshaw Summation
| journal=Bolletino di Geodesia e Scienze Affini
| volume=41
| number=4
| pages=349–375
| url=http://cct.gfy.ku.dk/publ_cct/cct80.pdf
| access-date=2012-08-02
| archive-url=https://web.archive.org/web/20070612091533/http://cct.gfy.ku.dk/publ_cct/cct80.pdf
| archive-date=2007-06-12
| url-status=dead
}}</ref>  A simple application is summing the trigonometric series to compute the [[meridian arc]] distance on the surface of an ellipsoid.  These have the form
<math display="block">m(\theta) = C_0\,\theta + C_1\sin \theta + C_2\sin 2\theta + \cdots + C_n\sin n\theta.</math>

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

The [[List of trigonometric identities#Chebyshev method|recurrence relation for <math>\sin k\theta</math>]] is
<math display="block">\sin (k+1)\theta = 2 \cos\theta \sin k\theta - \sin (k-1)\theta,</math>
making the coefficients in the recursion relation
<math display="block">\alpha_k(\theta) = 2\cos\theta, \quad \beta_k = -1.</math>
and the evaluation of the series is given by
<math display="block">\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}</math>
The final step is made particularly simple because <math>\phi_0(\theta) = \sin 0 = 0</math>, so the end of the recurrence is simply <math>b_1(\theta)\sin(\theta)</math>; the <math>C_0\,\theta</math> term is added separately:
<math display="block">m(\theta) = C_0\,\theta + b_1(\theta)\sin \theta.</math>

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