Editing Clenshaw algorithm (section)

===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
<math display="block">
  m(\theta_1)-m(\theta_2) = C_0(\theta_1-\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).
</math>
Clenshaw summation can be applied in this case<ref>{{ cite journal
| last = Karney
| first = C. F. F.
| year = 2024
| volume = 68
| number = 3-4
| pages = 99-120
| title = The area of rhumb polygons
| journal = Stud. Geophys. Geod.
| doi = 10.1007/s11200-024-0709-z
| doi-access = free
| postscript = Appendix B| arxiv = 2303.03219
}}</ref>
provided we simultaneously compute <math>m(\theta_1)+m(\theta_2)</math>
and perform a matrix summation,
<math display="block">
  \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),
</math>
where
<math display="block"> \begin{align}
  \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{align} </math>
The first element of <math>\mathsf M(\theta_1,\theta_2)</math> is the average
value of <math>m</math> and the second element is the average slope.
<math>\mathsf F_k(\theta_1,\theta_2)</math> satisfies the recurrence
relation
<math display="block">
  \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),
</math>
where
<math display="block">
   \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}
</math>
takes the place of <math>\alpha</math> in the recurrence relation, and <math>\beta=-1</math>.
The standard Clenshaw algorithm can now be applied to yield
<math display="block"> \begin{align}
  \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\ge k \ge 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{align}</math>
where <math>\mathsf B_k</math> are 2×2 matrices.  Finally we have
<math display="block">
  \frac{m(\theta_1) - m(\theta_2)}{\theta_1 - \theta_2} =
  \mathsf M_2(\theta_1, \theta_2).
</math>
This technique can be used in the [[limit (mathematics)|limit]] <math>\theta_2 = \theta_1 = \mu</math> and <math> \delta = 0 </math> to simultaneously compute  <math>m(\mu)</math> and the [[derivative]] <math>dm(\mu)/d\mu</math>, provided that, in evaluating <math>\mathsf F_1</math> and <math>\mathsf A</math>, we take <math>\lim_{\delta \to 0} (\sin k \delta)/\delta = k</math>.