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<title>Clenshaw algorithm</title>

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Clenshaw algorithm

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr">In <a href="Numerical_analysis" title="Numerical analysis">numerical analysis</a>, the Clenshaw algorithm, also called Clenshaw summation, is a <a href="Recursion" title="Recursion">recursive</a> method to evaluate a <a href="Linear_combination" title="Linear combination">linear combination</a> of <a href="Chebyshev_polynomials" title="Chebyshev polynomials">Chebyshev polynomials</a>.<a href="#cite_note-Clenshaw55-1">[1]</a><a href="#cite_note-Tscherning82-2">[2]</a> The method was published by <a href="Charles_William_Clenshaw" title="Charles William Clenshaw">Charles William Clenshaw</a> in 1955. It is a generalization of <a href="Horner's_method" title="Horner's method">Horner's method</a> for evaluating a linear combination of <a href="Monomial" title="Monomial">monomials</a>.

It generalizes to more than just Chebyshev polynomials; it applies to any class of functions that can be defined by a three-term <a href="Recurrence_relation" title="Recurrence relation">recurrence relation</a>.<a href="#cite_note-3">[3]</a>

<div class="mw-heading mw-heading2"><h2 id="Clenshaw_algorithm">Clenshaw algorithm</h2></div>

In full generality, the Clenshaw algorithm computes the weighted sum of a finite series of functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{k}(x)}">

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<annotation encoding="application/x-tex">{\displaystyle \phi _{k}(x)}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle S(x)=\sum _{k=0}^{n}a_{k}\phi _{k}(x)}</annotation>

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where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{k},\;k=0,1,\ldots }">

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<annotation encoding="application/x-tex">{\displaystyle \phi _{k},\;k=0,1,\ldots }</annotation>

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</math><img src="./650360a5f576470d98a0d5d8f60f946447789286.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.579ex; height:2.509ex;" alt="{\displaystyle \phi _{k},\;k=0,1,\ldots }" loading="lazy"> is a sequence of functions that satisfy the linear recurrence relation

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<annotation encoding="application/x-tex">{\displaystyle \phi _{k+1}(x)=\alpha _{k}(x)\,\phi _{k}(x)+\beta _{k}(x)\,\phi _{k-1}(x),}</annotation>

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where the coefficients <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{k}(x)}">

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<annotation encoding="application/x-tex">{\displaystyle \alpha _{k}(x)}</annotation>

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</math><img src="./ed623fbfa54ca215e7adba3823ec241fb9d847fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.715ex; height:2.843ex;" alt="{\displaystyle \alpha _{k}(x)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta _{k}(x)}">

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<annotation encoding="application/x-tex">{\displaystyle \beta _{k}(x)}</annotation>

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The algorithm is most useful when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{k}(x)}">

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<annotation encoding="application/x-tex">{\displaystyle \phi _{k}(x)}</annotation>

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</math><img src="./1326e4bfae37f629e340a85daa93c15f71780a44.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.613ex; height:2.843ex;" alt="{\displaystyle \phi _{k}(x)}" loading="lazy"> are functions that are complicated to compute directly, but <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{k}(x)}">

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<annotation encoding="application/x-tex">{\displaystyle \alpha _{k}(x)}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \beta _{k}(x)}</annotation>

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</math><img src="./25aa81e95befdd716fc246ec3130b44bc05becea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.543ex; height:2.843ex;" alt="{\displaystyle \beta _{k}(x)}" loading="lazy"> are particularly simple. In the most common applications, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha (x)}">

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<annotation encoding="application/x-tex">{\displaystyle \alpha (x)}</annotation>

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</math><img src="./43941ef40b4eb751a8daea118d0012ea998f5610.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.627ex; height:2.843ex;" alt="{\displaystyle \alpha (x)}" loading="lazy"> does not depend on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

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</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }">

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<annotation encoding="application/x-tex">{\displaystyle \beta }</annotation>

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</math><img src="./7ed48a5e36207156fb792fa79d29925d2f7901e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }" loading="lazy"> is a constant that depends on neither <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

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</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> nor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

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To perform the summation for given series of coefficients <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{0},\ldots ,a_{n}}">

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<annotation encoding="application/x-tex">{\displaystyle a_{0},\ldots ,a_{n}}</annotation>

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</math><img src="./403d004cd76aacd38b6432fdc449ddbc226075f7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.911ex; height:2.009ex;" alt="{\displaystyle a_{0},\ldots ,a_{n}}" loading="lazy">, compute the values <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{k}(x)}">

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<annotation encoding="application/x-tex">{\displaystyle b_{k}(x)}</annotation>

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</math><img src="./46806196e2a5d6a17790af9ba50d2d5d0195361f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.225ex; height:2.843ex;" alt="{\displaystyle b_{k}(x)}" loading="lazy"> by the "reverse" recurrence formula:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}b_{n+1}(x)&=b_{n+2}(x)=0,\\b_{k}(x)&=a_{k}+\alpha _{k}(x)\,b_{k+1}(x)+\beta _{k+1}(x)\,b_{k+2}(x).\end{aligned}}}</annotation>

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Note that this computation makes no direct reference to the functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{k}(x)}">

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<annotation encoding="application/x-tex">{\displaystyle \phi _{k}(x)}</annotation>

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</math><img src="./1326e4bfae37f629e340a85daa93c15f71780a44.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.613ex; height:2.843ex;" alt="{\displaystyle \phi _{k}(x)}" loading="lazy">. After computing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{2}(x)}">

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<annotation encoding="application/x-tex">{\displaystyle b_{2}(x)}</annotation>

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</math><img src="./467e00b6750755523a8158df518553b78306b662.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.191ex; height:2.843ex;" alt="{\displaystyle b_{2}(x)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{1}(x)}">

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<annotation encoding="application/x-tex">{\displaystyle b_{1}(x)}</annotation>

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</math><img src="./d4001397252899f67e06a5eb52ab6c60cc64ab28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.191ex; height:2.843ex;" alt="{\displaystyle b_{1}(x)}" loading="lazy">,

the desired sum can be expressed in terms of them and the simplest functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{0}(x)}">

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<annotation encoding="application/x-tex">{\displaystyle \phi _{0}(x)}</annotation>

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</math><img src="./0fc4c33e56244177ed16ac29df7022f7f782eddf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.579ex; height:2.843ex;" alt="{\displaystyle \phi _{0}(x)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{1}(x)}">

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<annotation encoding="application/x-tex">{\displaystyle \phi _{1}(x)}</annotation>

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</math><img src="./54815ecb905d342819c87c8a05a4440ec722bbf2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.579ex; height:2.843ex;" alt="{\displaystyle \phi _{1}(x)}" loading="lazy">:

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<annotation encoding="application/x-tex">{\displaystyle S(x)=\phi _{0}(x)\,a_{0}+\phi _{1}(x)\,b_{1}(x)+\beta _{1}(x)\,\phi _{0}(x)\,b_{2}(x).}</annotation>

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See Fox and Parker<a href="#cite_note-FoxParker-4">[4]</a> for more information and stability analyses.

<div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Horner_as_a_special_case_of_Clenshaw">Horner as a special case of Clenshaw</h3></div>

A particularly simple case occurs when evaluating a polynomial of the form

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<annotation encoding="application/x-tex">{\displaystyle S(x)=\sum _{k=0}^{n}a_{k}x^{k}.}</annotation>

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The functions are simply

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\phi _{0}(x)&=1,\\\phi _{k}(x)&=x^{k}=x\phi _{k-1}(x)\end{aligned}}}</annotation>

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and are produced by the recurrence coefficients <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha (x)=x}">

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<annotation encoding="application/x-tex">{\displaystyle \alpha (x)=x}</annotation>

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</math><img src="./938b2abc2d8c615e798ab802bd6bf7e060566818.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.055ex; height:2.843ex;" alt="{\displaystyle \alpha (x)=x}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta =0}">

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<annotation encoding="application/x-tex">{\displaystyle \beta =0}</annotation>

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</math><img src="./60b5e78663eba7ba08e0dd4915251e6261f4f35c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.593ex; height:2.509ex;" alt="{\displaystyle \beta =0}" loading="lazy">.

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

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<annotation encoding="application/x-tex">{\displaystyle b_{k}(x)=a_{k}+xb_{k+1}(x)}</annotation>

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and, in this case, the sum is simply

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<annotation encoding="application/x-tex">{\displaystyle S(x)=a_{0}+xb_{1}(x)=b_{0}(x),}</annotation>

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which is exactly the usual <a href="Horner's_method" title="Horner's method">Horner's method</a>.

<div class="mw-heading mw-heading3"><h3 id="Special_case_for_Chebyshev_series">Special case for Chebyshev series</h3></div>

Consider a truncated <a href="Chebyshev_series" class="mw-redirect" title="Chebyshev series">Chebyshev series</a>

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<annotation encoding="application/x-tex">{\displaystyle p_{n}(x)=a_{0}+a_{1}T_{1}(x)+a_{2}T_{2}(x)+\cdots +a_{n}T_{n}(x).}</annotation>

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The coefficients in the recursion relation for the <a href="Chebyshev_polynomials" title="Chebyshev polynomials">Chebyshev polynomials</a> are

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<annotation encoding="application/x-tex">{\displaystyle \alpha (x)=2x,\quad \beta =-1,}</annotation>

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with the initial conditions

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<annotation encoding="application/x-tex">{\displaystyle T_{0}(x)=1,\quad T_{1}(x)=x.}</annotation>

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Thus, the recurrence is

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<annotation encoding="application/x-tex">{\displaystyle b_{k}(x)=a_{k}+2xb_{k+1}(x)-b_{k+2}(x)}</annotation>

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and the final results are

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<annotation encoding="application/x-tex">{\displaystyle b_{0}(x)=a_{0}+2xb_{1}(x)-b_{2}(x),}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle p_{n}(x)={\tfrac {1}{2}}\left[a_{0}+b_{0}(x)-b_{2}(x)\right].}</annotation>

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An equivalent expression for the sum is given by

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<annotation encoding="application/x-tex">{\displaystyle p_{n}(x)=a_{0}+xb_{1}(x)-b_{2}(x).}</annotation>

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<div class="mw-heading mw-heading3"><h3 id="Meridian_arc_length_on_the_ellipsoid">Meridian arc length on the ellipsoid</h3></div>

Clenshaw summation is extensively used in <a href="Geodesy" title="Geodesy">geodetic</a> applications.<a href="#cite_note-Tscherning82-2">[2]</a> A simple application is summing the <a href="Trigonometric_series" title="Trigonometric series">trigonometric series</a> to compute the <a href="Meridian_arc" title="Meridian arc">meridian arc</a> distance on the surface of an ellipsoid. These have the form

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<annotation encoding="application/x-tex">{\displaystyle m(\theta )=C_{0}\,\theta +C_{1}\sin \theta +C_{2}\sin 2\theta +\cdots +C_{n}\sin n\theta .}</annotation>

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Leaving off the initial <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{0}\,\theta }">

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<annotation encoding="application/x-tex">{\displaystyle C_{0}\,\theta }</annotation>

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</math><img src="./d714e6d60e715bccc9d696a482173244abadc427.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.194ex; height:2.509ex;" alt="{\displaystyle C_{0}\,\theta }" loading="lazy"> term, the remainder is a summation of the appropriate form. There is no leading term because <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{0}(\theta )=\sin 0\theta =\sin 0=0}">

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<annotation encoding="application/x-tex">{\displaystyle \phi _{0}(\theta )=\sin 0\theta =\sin 0=0}</annotation>

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</math><img src="./4a9f5ac05e214b79db51b229242d6501c4a56f8a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.698ex; height:2.843ex;" alt="{\displaystyle \phi _{0}(\theta )=\sin 0\theta =\sin 0=0}" loading="lazy">.

The <a href="List_of_trigonometric_identities#Chebyshev_method" title="List of trigonometric identities">recurrence relation for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin k\theta }">

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<annotation encoding="application/x-tex">{\displaystyle \sin k\theta }</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \sin(k+1)\theta =2\cos \theta \sin k\theta -\sin(k-1)\theta ,}</annotation>

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making the coefficients in the recursion relation

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<annotation encoding="application/x-tex">{\displaystyle \alpha _{k}(\theta )=2\cos \theta ,\quad \beta _{k}=-1.}</annotation>

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and the evaluation of the series is given by

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<annotation encoding="application/x-tex">{\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}}}</annotation>

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The final step is made particularly simple because <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{0}(\theta )=\sin 0=0}">

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<annotation encoding="application/x-tex">{\displaystyle \phi _{0}(\theta )=\sin 0=0}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle m(\theta )=C_{0}\,\theta +b_{1}(\theta )\sin \theta .}</annotation>

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Note that the algorithm requires only the evaluation of two trigonometric quantities <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos \theta }">

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<annotation encoding="application/x-tex">{\displaystyle \cos \theta }</annotation>

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</math><img src="./efa733f6703578b0c3af870a3170b4ab0dd99c00.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.333ex; height:2.176ex;" alt="{\displaystyle \sin \theta }" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="Difference_in_meridian_arc_lengths">Difference in meridian arc lengths</h3></div>

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

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<annotation encoding="application/x-tex">{\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 )}.}</annotation>

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Clenshaw summation can be applied in this case<a href="#cite_note-5">[5]</a>

provided we simultaneously compute <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m(\theta _{1})+m(\theta _{2})}">

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<annotation encoding="application/x-tex">{\displaystyle m(\theta _{1})+m(\theta _{2})}</annotation>

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and perform a matrix summation,

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<annotation encoding="application/x-tex">{\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}),}</annotation>

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where

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<annotation encoding="application/x-tex">{\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}}}</annotation>

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The first element of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {M}}(\theta _{1},\theta _{2})}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathsf {M}}(\theta _{1},\theta _{2})}</annotation>

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</math><img src="./9fd222e99910539025ea8ad267dab15e6fcbc991.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.166ex; height:2.843ex;" alt="{\displaystyle {\mathsf {M}}(\theta _{1},\theta _{2})}" loading="lazy"> is the average

value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

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<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

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</math><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"> and the second element is the average slope.

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<annotation encoding="application/x-tex">{\displaystyle {\mathsf {F}}_{k}(\theta _{1},\theta _{2})}</annotation>

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</math><img src="./47f251c2e3d8e4c170f6ae56bffe2b7bf8bc99ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.544ex; height:2.843ex;" alt="{\displaystyle {\mathsf {F}}_{k}(\theta _{1},\theta _{2})}" loading="lazy"> satisfies the recurrence

relation

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<annotation encoding="application/x-tex">{\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}),}</annotation>

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where

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<annotation encoding="application/x-tex">{\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}}}</annotation>

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takes the place of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

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<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>

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</math><img src="./b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"> in the recurrence relation, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta =-1}">

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<annotation encoding="application/x-tex">{\displaystyle \beta =-1}</annotation>

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</math><img src="./eebc0b3c576d5448d3bdf0d2c360514ecfa7d57c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.401ex; height:2.509ex;" alt="{\displaystyle \beta =-1}" loading="lazy">.

The standard Clenshaw algorithm can now be applied to yield

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<annotation encoding="application/x-tex">{\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}}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle {\mathsf {B}}_{k}}</annotation>

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</math><img src="./1b4a9b2c592d6e6f97530e008405c9b6ff31e124.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.639ex; height:2.509ex;" alt="{\displaystyle {\mathsf {B}}_{k}}" loading="lazy"> are 2×2 matrices. Finally we have

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<annotation encoding="application/x-tex">{\displaystyle {\frac {m(\theta _{1})-m(\theta _{2})}{\theta _{1}-\theta _{2}}}={\mathsf {M}}_{2}(\theta _{1},\theta _{2}).}</annotation>

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This technique can be used in the <a href="Limit_(mathematics)" title="Limit (mathematics)">limit</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta _{2}=\theta _{1}=\mu }">

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<annotation encoding="application/x-tex">{\displaystyle \theta _{2}=\theta _{1}=\mu }</annotation>

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</math><img src="./51dff71a4eb599abff4480075120fa2bd2be8904.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.888ex; height:2.676ex;" alt="{\displaystyle \theta _{2}=\theta _{1}=\mu }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta =0}">

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<annotation encoding="application/x-tex">{\displaystyle \delta =0}</annotation>

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</math><img src="./8bbc4a4d0a8fcabdf770691af61b994e64b81468.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.31ex; height:2.343ex;" alt="{\displaystyle \delta =0}" loading="lazy"> to simultaneously compute <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m(\mu )}">

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<annotation encoding="application/x-tex">{\displaystyle m(\mu )}</annotation>

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</math><img src="./78c3c34de5fd0d66f9939fc984422b97bc0329f2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.251ex; height:2.843ex;" alt="{\displaystyle m(\mu )}" loading="lazy"> and the <a href="Derivative" title="Derivative">derivative</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dm(\mu )/d\mu }">

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<annotation encoding="application/x-tex">{\displaystyle dm(\mu )/d\mu }</annotation>

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</math><img src="./20e006f4ff92e8e3d27c14ba1aa8392ef777c070.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.247ex; height:2.843ex;" alt="{\displaystyle dm(\mu )/d\mu }" loading="lazy">, provided that, in evaluating <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {F}}_{1}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathsf {F}}_{1}}</annotation>

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</math><img src="./ba85ee3158798cbe72ab0fe65031e8e4b455d899.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.377ex; height:2.509ex;" alt="{\displaystyle {\mathsf {F}}_{1}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {A}}}">

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle {\mathsf {A}}}</annotation>

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</math><img src="./102043e55e1bdbf81aad8c9c1419ba91d44d6755.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle {\mathsf {A}}}" loading="lazy">, we take <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{\delta \to 0}(\sin k\delta )/\delta =k}">

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<annotation encoding="application/x-tex">{\displaystyle \lim _{\delta \to 0}(\sin k\delta )/\delta =k}</annotation>

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</math><img src="./f2bd4fcdaa82184d398babb6e32b04eb7442e2ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:17.062ex; height:4.176ex;" alt="{\displaystyle \lim _{\delta \to 0}(\sin k\delta )/\delta =k}" loading="lazy">.

<ul><li><a href="Horner_scheme" class="mw-redirect" title="Horner scheme">Horner scheme</a> to evaluate polynomials in <a href="Monomial_form" class="mw-redirect" title="Monomial form">monomial form</a></li>

<li><a href="De_Casteljau's_algorithm" title="De Casteljau's algorithm">De Casteljau's algorithm</a> to evaluate polynomials in <a href="B%C3%A9zier_form" class="mw-redirect" title="Bézier form">Bézier form</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFClenshaw1955" class="citation journal cs1">Clenshaw, C. W. (July 1955). <a rel="nofollow" class="external text" href="https://www.ams.org/journals/mcom/1955-09-051/S0025-5718-1955-0071856-0/">"A note on the summation of Chebyshev series"</a>. Mathematical Tables and Other Aids to Computation. 9 (51): 118. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0025-5718-1955-0071856-0">10.1090/S0025-5718-1955-0071856-0</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0025-5718">0025-5718</a>.</cite> Note that this paper is written in terms of the Shifted Chebyshev polynomials of the first kind <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{n}^{*}(x)=T_{n}(2x-1)}">

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<annotation encoding="application/x-tex">{\displaystyle T_{n}^{*}(x)=T_{n}(2x-1)}</annotation>

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<li id="cite_note-Tscherning82-2">^ <a href="#cite_ref-Tscherning82_2-0">a</a> <a href="#cite_ref-Tscherning82_2-1">b</a> <cite id="CITEREFTscherningPoder1982" class="citation cs2">Tscherning, C. C.; Poder, K. (1982), <a rel="nofollow" class="external text" href="https://web.archive.org/web/20070612091533/http://cct.gfy.ku.dk/publ_cct/cct80.pdf">"Some Geodetic applications of Clenshaw Summation"</a> (PDF), Bolletino di Geodesia e Scienze Affini, 41 (4): 349–375, archived from <a rel="nofollow" class="external text" href="http://cct.gfy.ku.dk/publ_cct/cct80.pdf">the original</a> (PDF) on 2007-06-12, retrieved 2012-08-02</cite>

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<li id="cite_note-3"><a href="#cite_ref-3">^</a> <cite id="CITEREFPressTeukolskyVetterlingFlannery2007" class="citation cs2">Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007), <a rel="nofollow" class="external text" href="http://apps.nrbook.com/empanel/index.html?pg=222">"Section 5.4.2. Clenshaw's Recurrence Formula"</a>, Numerical Recipes: The Art of Scientific Computing (3rd ed.), New York: Cambridge University Press, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-88068-8</bdi></cite>

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<li id="cite_note-FoxParker-4"><a href="#cite_ref-FoxParker_4-0">^</a> <cite id="CITEREFFoxParker1968" class="citation cs2">Fox, Leslie; Parker, Ian B. (1968), Chebyshev Polynomials in Numerical Analysis, Oxford University Press, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-19-859614-6</bdi></cite>

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<li id="cite_note-5"><a href="#cite_ref-5">^</a> <cite id="CITEREFKarney2024" class="citation journal cs1">Karney, C. F. F. (2024). <a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs11200-024-0709-z">"The area of rhumb polygons"</a>. Stud. Geophys. Geod. 68 (3–4): 99–120. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/2303.03219">2303.03219</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs11200-024-0709-z">10.1007/s11200-024-0709-z</a>Appendix B</cite><code class="cs1-code">{{cite journal}}</code>: CS1 maint: postscript (link)

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