1<html> 2<head> 3<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> 4<title>Chebyshev Polynomials</title> 5<link rel="stylesheet" href="../../math.css" type="text/css"> 6<meta name="generator" content="DocBook XSL Stylesheets V1.79.1"> 7<link rel="home" href="../../index.html" title="Math Toolkit 2.12.0"> 8<link rel="up" href="../sf_poly.html" title="Polynomials"> 9<link rel="prev" href="hermite.html" title="Hermite Polynomials"> 10<link rel="next" href="sph_harm.html" title="Spherical Harmonics"> 11</head> 12<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> 13<table cellpadding="2" width="100%"><tr> 14<td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../boost.png"></td> 15<td align="center"><a href="../../../../../../index.html">Home</a></td> 16<td align="center"><a href="../../../../../../libs/libraries.htm">Libraries</a></td> 17<td align="center"><a href="http://www.boost.org/users/people.html">People</a></td> 18<td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td> 19<td align="center"><a href="../../../../../../more/index.htm">More</a></td> 20</tr></table> 21<hr> 22<div class="spirit-nav"> 23<a accesskey="p" href="hermite.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../sf_poly.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="sph_harm.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> 24</div> 25<div class="section"> 26<div class="titlepage"><div><div><h3 class="title"> 27<a name="math_toolkit.sf_poly.chebyshev"></a><a class="link" href="chebyshev.html" title="Chebyshev Polynomials">Chebyshev Polynomials</a> 28</h3></div></div></div> 29<h5> 30<a name="math_toolkit.sf_poly.chebyshev.h0"></a> 31 <span class="phrase"><a name="math_toolkit.sf_poly.chebyshev.synopsis"></a></span><a class="link" href="chebyshev.html#math_toolkit.sf_poly.chebyshev.synopsis">Synopsis</a> 32 </h5> 33<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">special_functions</span><span class="special">/</span><span class="identifier">chebyshev</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> 34</pre> 35<pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span> 36 37<span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">Real1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Real2</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Real3</span><span class="special">></span> 38<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">chebyshev_next</span><span class="special">(</span><span class="identifier">Real1</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">Real2</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">Tn</span><span class="special">,</span> <span class="identifier">Real3</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">Tn_1</span><span class="special">);</span> 39 40<span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">Real</span><span class="special">></span> 41<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">chebyshev_t</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="identifier">Real</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">x</span><span class="special">);</span> 42 43<span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">Real</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> 44<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">chebyshev_t</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="identifier">Real</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> 45 46<span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">Real</span><span class="special">></span> 47<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">chebyshev_u</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="identifier">Real</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">x</span><span class="special">);</span> 48 49<span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">Real</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> 50<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">chebyshev_u</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="identifier">Real</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> 51 52<span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">Real</span><span class="special">></span> 53<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">chebyshev_t_prime</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="identifier">Real</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">x</span><span class="special">);</span> 54 55<span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">Real1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Real2</span><span class="special">></span> 56<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">chebyshev_clenshaw_recurrence</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">Real</span><span class="special">*</span> <span class="keyword">const</span> <span class="identifier">c</span><span class="special">,</span> <span class="identifier">size_t</span> <span class="identifier">length</span><span class="special">,</span> <span class="identifier">Real2</span> <span class="identifier">x</span><span class="special">);</span> 57 58<span class="special">}}</span> <span class="comment">// namespaces</span> 59</pre> 60<p> 61 <span class="emphasis"><em>"Real analysts cannot do without Fourier, complex analysts 62 cannot do without Laurent, and numerical analysts cannot do without Chebyshev"</em></span> 63 --Lloyd N. Trefethen 64 </p> 65<p> 66 The Chebyshev polynomials of the first kind are defined by the recurrence 67 <span class="emphasis"><em>T</em></span><sub>n+1</sub>(<span class="emphasis"><em>x</em></span>) := <span class="emphasis"><em>2xT</em></span><sub>n</sub>(<span class="emphasis"><em>x</em></span>) 68 - <span class="emphasis"><em>T</em></span><sub>n-1</sub>(<span class="emphasis"><em>x</em></span>), <span class="emphasis"><em>n > 0</em></span>, 69 where <span class="emphasis"><em>T</em></span><sub>0</sub>(<span class="emphasis"><em>x</em></span>) := 1 and <span class="emphasis"><em>T</em></span><sub>1</sub>(<span class="emphasis"><em>x</em></span>) 70 := <span class="emphasis"><em>x</em></span>. These can be calculated in Boost using the following 71 simple code 72 </p> 73<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">x</span> <span class="special">=</span> <span class="number">0.5</span><span class="special">;</span> 74<span class="keyword">double</span> <span class="identifier">T12</span> <span class="special">=</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">chebyshev_t</span><span class="special">(</span><span class="number">12</span><span class="special">,</span> <span class="identifier">x</span><span class="special">);</span> 75</pre> 76<p> 77 Calculation of derivatives is also straightforward: 78 </p> 79<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">T12_prime</span> <span class="special">=</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">chebyshev_t_prime</span><span class="special">(</span><span class="number">12</span><span class="special">,</span> <span class="identifier">x</span><span class="special">);</span> 80</pre> 81<p> 82 The complexity of evaluation of the <span class="emphasis"><em>n</em></span>-th Chebyshev polynomial 83 by these functions is linear. So they are unsuitable for use in calculation 84 of (say) a Chebyshev series, as a sum of linear scaling functions scales 85 quadratically. Though there are very sophisticated algorithms for the evaluation 86 of Chebyshev series, a linear time algorithm is presented below: 87 </p> 88<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">x</span> <span class="special">=</span> <span class="number">0.5</span><span class="special">;</span> 89<span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="identifier">c</span><span class="special">{</span><span class="number">14.2</span><span class="special">,</span> <span class="special">-</span><span class="number">13.7</span><span class="special">,</span> <span class="number">82.3</span><span class="special">,</span> <span class="number">96</span><span class="special">};</span> 90<span class="keyword">double</span> <span class="identifier">T0</span> <span class="special">=</span> <span class="number">1</span><span class="special">;</span> 91<span class="keyword">double</span> <span class="identifier">T1</span> <span class="special">=</span> <span class="identifier">x</span><span class="special">;</span> 92<span class="keyword">double</span> <span class="identifier">f</span> <span class="special">=</span> <span class="identifier">c</span><span class="special">[</span><span class="number">0</span><span class="special">]*</span><span class="identifier">T0</span><span class="special">/</span><span class="number">2</span><span class="special">;</span> 93<span class="keyword">unsigned</span> <span class="identifier">l</span> <span class="special">=</span> <span class="number">1</span><span class="special">;</span> 94<span class="keyword">while</span><span class="special">(</span><span class="identifier">l</span> <span class="special"><</span> <span class="identifier">c</span><span class="special">.</span><span class="identifier">size</span><span class="special">())</span> 95<span class="special">{</span> 96 <span class="identifier">f</span> <span class="special">+=</span> <span class="identifier">c</span><span class="special">[</span><span class="identifier">l</span><span class="special">]*</span><span class="identifier">T1</span><span class="special">;</span> 97 <span class="identifier">std</span><span class="special">::</span><span class="identifier">swap</span><span class="special">(</span><span class="identifier">T0</span><span class="special">,</span> <span class="identifier">T1</span><span class="special">);</span> 98 <span class="identifier">T1</span> <span class="special">=</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">chebyshev_next</span><span class="special">(</span><span class="identifier">x</span><span class="special">,</span> <span class="identifier">T0</span><span class="special">,</span> <span class="identifier">T1</span><span class="special">);</span> 99 <span class="special">++</span><span class="identifier">l</span><span class="special">;</span> 100<span class="special">}</span> 101</pre> 102<p> 103 This uses the <code class="computeroutput"><span class="identifier">chebyshev_next</span></code> 104 function to evaluate each term of the Chebyshev series in constant time. 105 However, this naive algorithm has a catastrophic loss of precision as <span class="emphasis"><em>x</em></span> 106 approaches 1. A method to mitigate this way given by <a href="http://www.ams.org/journals/mcom/1955-09-051/S0025-5718-1955-0071856-0/S0025-5718-1955-0071856-0.pdf" target="_top">Clenshaw</a>, 107 and is implemented in boost as 108 </p> 109<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">x</span> <span class="special">=</span> <span class="number">0.5</span><span class="special">;</span> 110<span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="identifier">c</span><span class="special">{</span><span class="number">14.2</span><span class="special">,</span> <span class="special">-</span><span class="number">13.7</span><span class="special">,</span> <span class="number">82.3</span><span class="special">,</span> <span class="number">96</span><span class="special">};</span> 111<span class="keyword">double</span> <span class="identifier">f</span> <span class="special">=</span> <span class="identifier">chebyshev_clenshaw_recurrence</span><span class="special">(</span><span class="identifier">c</span><span class="special">.</span><span class="identifier">data</span><span class="special">(),</span> <span class="identifier">c</span><span class="special">.</span><span class="identifier">size</span><span class="special">(),</span> <span class="identifier">Real</span> <span class="identifier">x</span><span class="special">);</span> 112</pre> 113<p> 114 N.B.: There is factor of <span class="emphasis"><em>2</em></span> difference in our definition 115 of the first coefficient in the Chebyshev series from Clenshaw's original 116 work. This is because two traditions exist in notation for the Chebyshev 117 series expansion, 118 </p> 119<div class="blockquote"><blockquote class="blockquote"><p> 120 <span class="emphasis"><em>f</em></span>(<span class="emphasis"><em>x</em></span>) ≈ ∑<sub>n=0</sub><sup>N-1</sup> <span class="emphasis"><em>a</em></span><sub>n</sub><span class="emphasis"><em>T</em></span><sub>n</sub>(<span class="emphasis"><em>x</em></span>) 121 </p></blockquote></div> 122<p> 123 and 124 </p> 125<div class="blockquote"><blockquote class="blockquote"><p> 126 <span class="emphasis"><em>f</em></span>(<span class="emphasis"><em>x</em></span>) ≈ <span class="emphasis"><em>c</em></span><sub>0</sub>/2 127 + ∑<sub>n=1</sub><sup>N-1</sup> <span class="emphasis"><em>c</em></span><sub>n</sub><span class="emphasis"><em>T</em></span><sub>n</sub>(<span class="emphasis"><em>x</em></span>) 128 </p></blockquote></div> 129<p> 130 <span class="emphasis"><em><span class="bold"><strong>boost math always uses the second convention, 131 with the factor of 1/2 on the first coefficient.</strong></span></em></span> 132 </p> 133<p> 134 Chebyshev polynomials of the second kind can be evaluated via <code class="computeroutput"><span class="identifier">chebyshev_u</span></code>: 135 </p> 136<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">x</span> <span class="special">=</span> <span class="special">-</span><span class="number">0.23</span><span class="special">;</span> 137<span class="keyword">double</span> <span class="identifier">U1</span> <span class="special">=</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">chebyshev_u</span><span class="special">(</span><span class="number">1</span><span class="special">,</span> <span class="identifier">x</span><span class="special">);</span> 138</pre> 139<p> 140 The evaluation of Chebyshev polynomials by a three-term recurrence is known 141 to be <a href="https://link.springer.com/article/10.1007/s11075-014-9925-x" target="_top">mixed 142 forward-backward stable</a> for <span class="emphasis"><em>x</em></span> ∊ [-1, 143 1]. However, the author does not know of a similar result for <span class="emphasis"><em>x</em></span> 144 outside [-1, 1]. For this reason, evaluation of Chebyshev polynomials outside 145 of [-1, 1] is strongly discouraged. That said, small rounding errors in the 146 course of a computation will often lead to this situation, and termination 147 of the computation due to these small problems is very discouraging. For 148 this reason, <code class="computeroutput"><span class="identifier">chebyshev_t</span></code> 149 and <code class="computeroutput"><span class="identifier">chebyshev_u</span></code> have code 150 paths for <span class="emphasis"><em>x > 1</em></span> and <span class="emphasis"><em>x < -1</em></span> 151 which do not use three-term recurrences. These code paths are <span class="emphasis"><em>much 152 slower</em></span>, and should be avoided if at all possible. 153 </p> 154<p> 155 Evaluation of a Chebyshev series is relatively simple. The real challenge 156 is <span class="emphasis"><em>generation</em></span> of the Chebyshev series. For this purpose, 157 boost provides a <span class="emphasis"><em>Chebyshev transform</em></span>, a projection operator 158 which projects a function onto a finite-dimensional span of Chebyshev polynomials. 159 But before we discuss the API, let's analyze why we might want to project 160 a function onto a span of Chebyshev polynomials. 161 </p> 162<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> 163<li class="listitem"> 164 We want a numerically stable way to evaluate the function's derivative. 165 </li> 166<li class="listitem"> 167 Our function is expensive to evaluate, and we wish to find a less expensive 168 way to estimate its value. An example are the standard library transcendental 169 functions: These functions are guaranteed to evaluate to within 1 ulp 170 of the exact value, but often this accuracy is not needed. A projection 171 onto the Chebyshev polynomials with a low accuracy requirement can vastly 172 accelerate the computation of these functions. 173 </li> 174<li class="listitem"> 175 We wish to numerically integrate the function. 176 </li> 177</ul></div> 178<p> 179 The API is given below. 180 </p> 181<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">special_functions</span><span class="special">/</span><span class="identifier">chebyshev_transform</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> 182</pre> 183<pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span> 184 185<span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">Real</span><span class="special">></span> 186<span class="keyword">class</span> <span class="identifier">chebyshev_transform</span> 187<span class="special">{</span> 188<span class="keyword">public</span><span class="special">:</span> 189 <span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">></span> 190 <span class="identifier">chebyshev_transform</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">F</span><span class="special">&</span> <span class="identifier">f</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">tol</span><span class="special">=</span><span class="number">500</span><span class="special">*</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">Real</span><span class="special">>::</span><span class="identifier">epsilon</span><span class="special">());</span> 191 192 <span class="identifier">Real</span> <span class="keyword">operator</span><span class="special">()(</span><span class="identifier">Real</span> <span class="identifier">x</span><span class="special">)</span> <span class="keyword">const</span> 193 194 <span class="identifier">Real</span> <span class="identifier">integrate</span><span class="special">()</span> <span class="keyword">const</span> 195 196 <span class="keyword">const</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span><span class="identifier">Real</span><span class="special">>&</span> <span class="identifier">coefficients</span><span class="special">()</span> <span class="keyword">const</span> 197 198 <span class="identifier">Real</span> <span class="identifier">prime</span><span class="special">(</span><span class="identifier">Real</span> <span class="identifier">x</span><span class="special">)</span> <span class="keyword">const</span> 199<span class="special">};</span> 200 201<span class="special">}}//</span> <span class="identifier">end</span> <span class="identifier">namespaces</span> 202</pre> 203<p> 204 The Chebyshev transform takes a function <span class="emphasis"><em>f</em></span> and returns 205 a <span class="emphasis"><em>near-minimax</em></span> approximation to <span class="emphasis"><em>f</em></span> 206 in terms of Chebyshev polynomials. By <span class="emphasis"><em>near-minimax</em></span>, 207 we mean that the resulting Chebyshev polynomial is "very close" 208 the polynomial <span class="emphasis"><em>p</em></span><sub>n</sub> which minimizes the uniform norm of 209 <span class="emphasis"><em>f</em></span> - <span class="emphasis"><em>p</em></span><sub>n</sub>. The notion of "very 210 close" can be made rigorous; see Trefethen's "Approximation Theory 211 and Approximation Practice" for details. 212 </p> 213<p> 214 The Chebyshev transform works by creating a vector of values by evaluating 215 the input function at the Chebyshev points, and then performing a discrete 216 cosine transform on the resulting vector. In order to do this efficiently, 217 we have used <a href="http://www.fftw.org/" target="_top">FFTW3</a>. So to compile, 218 you must have <code class="computeroutput"><span class="identifier">FFTW3</span></code> installed, 219 and link with <code class="computeroutput"><span class="special">-</span><span class="identifier">lfftw3</span></code> 220 for double precision, <code class="computeroutput"><span class="special">-</span><span class="identifier">lfftw3f</span></code> 221 for float precision, <code class="computeroutput"><span class="special">-</span><span class="identifier">lfftw3l</span></code> 222 for long double precision, and -lfftwq for quad (<code class="computeroutput"><span class="identifier">__float128</span></code>) 223 precision. After the coefficients of the Chebyshev series are known, the 224 routine goes back through them and filters out all the coefficients whose 225 absolute ratio to the largest coefficient are less than the tolerance requested 226 in the constructor. 227 </p> 228</div> 229<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> 230<td align="left"></td> 231<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar 232 Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, 233 Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan 234 Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, 235 Daryle Walker and Xiaogang Zhang<p> 236 Distributed under the Boost Software License, Version 1.0. (See accompanying 237 file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>) 238 </p> 239</div></td> 240</tr></table> 241<hr> 242<div class="spirit-nav"> 243<a accesskey="p" href="hermite.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../sf_poly.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="sph_harm.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> 244</div> 245</body> 246</html> 247
