ceph/src/boost/libs/math/doc/html/math_toolkit/sf_poly/legendre.html

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  24 </div>
  25 <div class="section">
  26 <div class="titlepage"><div><div><h3 class="title">
  27 <a name="math_toolkit.sf_poly.legendre"></a><a class="link" href="legendre.html" title="Legendre (and Associated) Polynomials">Legendre (and Associated)
  28       Polynomials</a>
  29 </h3></div></div></div>
  30 <h5>
  31 <a name="math_toolkit.sf_poly.legendre.h0"></a>
  32         <span class="phrase"><a name="math_toolkit.sf_poly.legendre.synopsis"></a></span><a class="link" href="legendre.html#math_toolkit.sf_poly.legendre.synopsis">Synopsis</a>
  33       </h5>
  34 <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</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">legendre</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
  35 </pre>
  36 <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>
  37
  38 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
  39 <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">legendre_p</span><span class="special">(</span><span class="keyword">int</span> <span class="identifier">n</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">x</span><span class="special">);</span>
  40
  41 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
  42 <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">legendre_p</span><span class="special">(</span><span class="keyword">int</span> <span class="identifier">n</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
  43
  44 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
  45 <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">legendre_p</span><span class="special">(</span><span class="keyword">int</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">m</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">x</span><span class="special">);</span>
  46
  47 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
  48 <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">legendre_p</span><span class="special">(</span><span class="keyword">int</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">m</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
  49
  50 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
  51 <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">legendre_q</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">x</span><span class="special">);</span>
  52
  53 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
  54 <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">legendre_q</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
  55
  56 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T3</span><span class="special">&gt;</span>
  57 <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">legendre_next</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">l</span><span class="special">,</span> <span class="identifier">T1</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">Pl</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">Plm1</span><span class="special">);</span>
  58
  59 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T3</span><span class="special">&gt;</span>
  60 <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">legendre_next</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">l</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">m</span><span class="special">,</span> <span class="identifier">T1</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">Pl</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">Plm1</span><span class="special">);</span>
  61
  62
  63 <span class="special">}}</span> <span class="comment">// namespaces</span>
  64 </pre>
  65 <p>
  66         The return type of these functions is computed using the <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>result
  67         type calculation rules</em></span></a>: note than when there is a single
  68         template argument the result is the same type as that argument or <code class="computeroutput"><span class="keyword">double</span></code> if the template argument is an integer
  69         type.
  70       </p>
  71 <p>
  72         The final <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
  73         be used to control the behaviour of the function: how it handles errors,
  74         what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">policy
  75         documentation for more details</a>.
  76       </p>
  77 <h5>
  78 <a name="math_toolkit.sf_poly.legendre.h1"></a>
  79         <span class="phrase"><a name="math_toolkit.sf_poly.legendre.description"></a></span><a class="link" href="legendre.html#math_toolkit.sf_poly.legendre.description">Description</a>
  80       </h5>
  81 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
  82 <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">legendre_p</span><span class="special">(</span><span class="keyword">int</span> <span class="identifier">l</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">x</span><span class="special">);</span>
  83
  84 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
  85 <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">legendre_p</span><span class="special">(</span><span class="keyword">int</span> <span class="identifier">l</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
  86 </pre>
  87 <p>
  88         Returns the Legendre Polynomial of the first kind:
  89       </p>
  90 <p>
  91         <span class="inlinemediaobject"><img src="../../../equations/legendre_0.svg"></span>
  92       </p>
  93 <p>
  94         Requires -1 &lt;= x &lt;= 1, otherwise returns the result of <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>.
  95       </p>
  96 <p>
  97         Negative orders are handled via the reflection formula:
  98       </p>
  99 <p>
 100         P<sub>-l-1</sub>(x) = P<sub>l</sub>(x)
 101       </p>
 102 <p>
 103         The following graph illustrates the behaviour of the first few Legendre Polynomials:
 104       </p>
 105 <p>
 106         <span class="inlinemediaobject"><img src="../../../graphs/legendre_p.svg" align="middle"></span>
 107       </p>
 108 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
 109 <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">legendre_p</span><span class="special">(</span><span class="keyword">int</span> <span class="identifier">l</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">m</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">x</span><span class="special">);</span>
 110
 111 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
 112 <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">legendre_p</span><span class="special">(</span><span class="keyword">int</span> <span class="identifier">l</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">m</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
 113 </pre>
 114 <p>
 115         Returns the associated Legendre polynomial of the first kind:
 116       </p>
 117 <p>
 118         <span class="inlinemediaobject"><img src="../../../equations/legendre_1.svg"></span>
 119       </p>
 120 <p>
 121         Requires -1 &lt;= x &lt;= 1, otherwise returns the result of <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>.
 122       </p>
 123 <p>
 124         Negative values of <span class="emphasis"><em>l</em></span> and <span class="emphasis"><em>m</em></span> are
 125         handled via the identity relations:
 126       </p>
 127 <p>
 128         <span class="inlinemediaobject"><img src="../../../equations/legendre_3.svg"></span>
 129       </p>
 130 <div class="caution"><table border="0" summary="Caution">
 131 <tr>
 132 <td rowspan="2" align="center" valign="top" width="25"><img alt="[Caution]" src="../../../../../../doc/src/images/caution.png"></td>
 133 <th align="left">Caution</th>
 134 </tr>
 135 <tr><td align="left" valign="top">
 136 <p>
 137           The definition of the associated Legendre polynomial used here includes
 138           a leading Condon-Shortley phase term of (-1)<sup>m</sup>. This matches the definition
 139           given by Abramowitz and Stegun (8.6.6) and that used by <a href="http://mathworld.wolfram.com/LegendrePolynomial.html" target="_top">Mathworld</a>
 140           and <a href="http://documents.wolfram.com/mathematica/functions/LegendreP" target="_top">Mathematica's
 141           LegendreP function</a>. However, uses in the literature do not always
 142           include this phase term, and strangely the specification for the associated
 143           Legendre function in the C++ TR1 (assoc_legendre) also omits it, in spite
 144           of stating that it uses Abramowitz and Stegun as the final arbiter on these
 145           matters.
 146         </p>
 147 <p>
 148           See:
 149         </p>
 150 <p>
 151           <a href="http://mathworld.wolfram.com/LegendrePolynomial.html" target="_top">Weisstein,
 152           Eric W. "Legendre Polynomial." From MathWorld--A Wolfram Web
 153           Resource</a>.
 154         </p>
 155 <p>
 156           Abramowitz, M. and Stegun, I. A. (Eds.). "Legendre Functions"
 157           and "Orthogonal Polynomials." Ch. 22 in Chs. 8 and 22 in Handbook
 158           of Mathematical Functions with Formulas, Graphs, and Mathematical Tables,
 159           9th printing. New York: Dover, pp. 331-339 and 771-802, 1972.
 160         </p>
 161 </td></tr>
 162 </table></div>
 163 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
 164 <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">legendre_q</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">x</span><span class="special">);</span>
 165
 166 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
 167 <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">legendre_q</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
 168 </pre>
 169 <p>
 170         Returns the value of the Legendre polynomial that is the second solution
 171         to the Legendre differential equation, for example:
 172       </p>
 173 <p>
 174         <span class="inlinemediaobject"><img src="../../../equations/legendre_2.svg"></span>
 175       </p>
 176 <p>
 177         Requires -1 &lt;= x &lt;= 1, otherwise <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
 178         is called.
 179       </p>
 180 <p>
 181         The following graph illustrates the first few Legendre functions of the second
 182         kind:
 183       </p>
 184 <p>
 185         <span class="inlinemediaobject"><img src="../../../graphs/legendre_q.svg" align="middle"></span>
 186       </p>
 187 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T3</span><span class="special">&gt;</span>
 188 <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">legendre_next</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">l</span><span class="special">,</span> <span class="identifier">T1</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">Pl</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">Plm1</span><span class="special">);</span>
 189 </pre>
 190 <p>
 191         Implements the three term recurrence relation for the Legendre polynomials,
 192         this function can be used to create a sequence of values evaluated at the
 193         same <span class="emphasis"><em>x</em></span>, and for rising <span class="emphasis"><em>l</em></span>. This
 194         recurrence relation holds for Legendre Polynomials of both the first and
 195         second kinds.
 196       </p>
 197 <p>
 198         <span class="inlinemediaobject"><img src="../../../equations/legendre_4.svg"></span>
 199       </p>
 200 <p>
 201         For example we could produce a vector of the first 10 polynomial values using:
 202       </p>
 203 <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>  <span class="comment">// Abscissa value</span>
 204 <span class="identifier">vector</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span> <span class="identifier">v</span><span class="special">;</span>
 205 <span class="identifier">v</span><span class="special">.</span><span class="identifier">push_back</span><span class="special">(</span><span class="identifier">legendre_p</span><span class="special">(</span><span class="number">0</span><span class="special">,</span> <span class="identifier">x</span><span class="special">));</span>
 206 <span class="identifier">v</span><span class="special">.</span><span class="identifier">push_back</span><span class="special">(</span><span class="identifier">legendre_p</span><span class="special">(</span><span class="number">1</span><span class="special">,</span> <span class="identifier">x</span><span class="special">));</span>
 207 <span class="keyword">for</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">l</span> <span class="special">=</span> <span class="number">1</span><span class="special">;</span> <span class="identifier">l</span> <span class="special">&lt;</span> <span class="number">10</span><span class="special">;</span> <span class="special">++</span><span class="identifier">l</span><span class="special">)</span>
 208    <span class="identifier">v</span><span class="special">.</span><span class="identifier">push_back</span><span class="special">(</span><span class="identifier">legendre_next</span><span class="special">(</span><span class="identifier">l</span><span class="special">,</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">v</span><span class="special">[</span><span class="identifier">l</span><span class="special">],</span> <span class="identifier">v</span><span class="special">[</span><span class="identifier">l</span><span class="special">-</span><span class="number">1</span><span class="special">]));</span>
 209 <span class="comment">// Double check values:</span>
 210 <span class="keyword">for</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">l</span> <span class="special">=</span> <span class="number">1</span><span class="special">;</span> <span class="identifier">l</span> <span class="special">&lt;</span> <span class="number">10</span><span class="special">;</span> <span class="special">++</span><span class="identifier">l</span><span class="special">)</span>
 211    <span class="identifier">assert</span><span class="special">(</span><span class="identifier">v</span><span class="special">[</span><span class="identifier">l</span><span class="special">]</span> <span class="special">==</span> <span class="identifier">legendre_p</span><span class="special">(</span><span class="identifier">l</span><span class="special">,</span> <span class="identifier">x</span><span class="special">));</span>
 212 </pre>
 213 <p>
 214         Formally the arguments are:
 215       </p>
 216 <div class="variablelist">
 217 <p class="title"><b></b></p>
 218 <dl class="variablelist">
 219 <dt><span class="term">l</span></dt>
 220 <dd><p>
 221               The degree of the last polynomial calculated.
 222             </p></dd>
 223 <dt><span class="term">x</span></dt>
 224 <dd><p>
 225               The abscissa value
 226             </p></dd>
 227 <dt><span class="term">Pl</span></dt>
 228 <dd><p>
 229               The value of the polynomial evaluated at degree <span class="emphasis"><em>l</em></span>.
 230             </p></dd>
 231 <dt><span class="term">Plm1</span></dt>
 232 <dd><p>
 233               The value of the polynomial evaluated at degree <span class="emphasis"><em>l-1</em></span>.
 234             </p></dd>
 235 </dl>
 236 </div>
 237 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T3</span><span class="special">&gt;</span>
 238 <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">legendre_next</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">l</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">m</span><span class="special">,</span> <span class="identifier">T1</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">Pl</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">Plm1</span><span class="special">);</span>
 239 </pre>
 240 <p>
 241         Implements the three term recurrence relation for the Associated Legendre
 242         polynomials, this function can be used to create a sequence of values evaluated
 243         at the same <span class="emphasis"><em>x</em></span>, and for rising <span class="emphasis"><em>l</em></span>.
 244       </p>
 245 <p>
 246         <span class="inlinemediaobject"><img src="../../../equations/legendre_5.svg"></span>
 247       </p>
 248 <p>
 249         For example we could produce a vector of the first m+10 polynomial values
 250         using:
 251       </p>
 252 <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>  <span class="comment">// Abscissa value</span>
 253 <span class="keyword">int</span> <span class="identifier">m</span> <span class="special">=</span> <span class="number">10</span><span class="special">;</span>      <span class="comment">// order</span>
 254 <span class="identifier">vector</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span> <span class="identifier">v</span><span class="special">;</span>
 255 <span class="identifier">v</span><span class="special">.</span><span class="identifier">push_back</span><span class="special">(</span><span class="identifier">legendre_p</span><span class="special">(</span><span class="identifier">m</span><span class="special">,</span> <span class="identifier">m</span><span class="special">,</span> <span class="identifier">x</span><span class="special">));</span>
 256 <span class="identifier">v</span><span class="special">.</span><span class="identifier">push_back</span><span class="special">(</span><span class="identifier">legendre_p</span><span class="special">(</span><span class="number">1</span> <span class="special">+</span> <span class="identifier">m</span><span class="special">,</span> <span class="identifier">m</span><span class="special">,</span> <span class="identifier">x</span><span class="special">));</span>
 257 <span class="keyword">for</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">l</span> <span class="special">=</span> <span class="number">1</span><span class="special">;</span> <span class="identifier">l</span> <span class="special">&lt;</span> <span class="number">10</span><span class="special">;</span> <span class="special">++</span><span class="identifier">l</span><span class="special">)</span>
 258    <span class="identifier">v</span><span class="special">.</span><span class="identifier">push_back</span><span class="special">(</span><span class="identifier">legendre_next</span><span class="special">(</span><span class="identifier">l</span> <span class="special">+</span> <span class="number">10</span><span class="special">,</span> <span class="identifier">m</span><span class="special">,</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">v</span><span class="special">[</span><span class="identifier">l</span><span class="special">],</span> <span class="identifier">v</span><span class="special">[</span><span class="identifier">l</span><span class="special">-</span><span class="number">1</span><span class="special">]));</span>
 259 <span class="comment">// Double check values:</span>
 260 <span class="keyword">for</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">l</span> <span class="special">=</span> <span class="number">1</span><span class="special">;</span> <span class="identifier">l</span> <span class="special">&lt;</span> <span class="number">10</span><span class="special">;</span> <span class="special">++</span><span class="identifier">l</span><span class="special">)</span>
 261    <span class="identifier">assert</span><span class="special">(</span><span class="identifier">v</span><span class="special">[</span><span class="identifier">l</span><span class="special">]</span> <span class="special">==</span> <span class="identifier">legendre_p</span><span class="special">(</span><span class="number">10</span> <span class="special">+</span> <span class="identifier">l</span><span class="special">,</span> <span class="identifier">m</span><span class="special">,</span> <span class="identifier">x</span><span class="special">));</span>
 262 </pre>
 263 <p>
 264         Formally the arguments are:
 265       </p>
 266 <div class="variablelist">
 267 <p class="title"><b></b></p>
 268 <dl class="variablelist">
 269 <dt><span class="term">l</span></dt>
 270 <dd><p>
 271               The degree of the last polynomial calculated.
 272             </p></dd>
 273 <dt><span class="term">m</span></dt>
 274 <dd><p>
 275               The order of the Associated Polynomial.
 276             </p></dd>
 277 <dt><span class="term">x</span></dt>
 278 <dd><p>
 279               The abscissa value
 280             </p></dd>
 281 <dt><span class="term">Pl</span></dt>
 282 <dd><p>
 283               The value of the polynomial evaluated at degree <span class="emphasis"><em>l</em></span>.
 284             </p></dd>
 285 <dt><span class="term">Plm1</span></dt>
 286 <dd><p>
 287               The value of the polynomial evaluated at degree <span class="emphasis"><em>l-1</em></span>.
 288             </p></dd>
 289 </dl>
 290 </div>
 291 <h5>
 292 <a name="math_toolkit.sf_poly.legendre.h2"></a>
 293         <span class="phrase"><a name="math_toolkit.sf_poly.legendre.accuracy"></a></span><a class="link" href="legendre.html#math_toolkit.sf_poly.legendre.accuracy">Accuracy</a>
 294       </h5>
 295 <p>
 296         The following table shows peak errors (in units of epsilon) for various domains
 297         of input arguments. Note that only results for the widest floating point
 298         type on the system are given as narrower types have <a class="link" href="../relative_error.html#math_toolkit.relative_error.zero_error">effectively
 299         zero error</a>.
 300       </p>
 301 <div class="table">
 302 <a name="math_toolkit.sf_poly.legendre.table_legendre_p"></a><p class="title"><b>Table&#160;6.32.&#160;Error rates for legendre_p</b></p>
 303 <div class="table-contents"><table class="table" summary="Error rates for legendre_p">
 304 <colgroup>
 305 <col>
 306 <col>
 307 <col>
 308 <col>
 309 <col>
 310 </colgroup>
 311 <thead><tr>
 312 <th>
 313               </th>
 314 <th>
 315                 <p>
 316                   Microsoft Visual C++ version 12.0<br> Win32<br> double
 317                 </p>
 318               </th>
 319 <th>
 320                 <p>
 321                   GNU C++ version 5.1.0<br> linux<br> double
 322                 </p>
 323               </th>
 324 <th>
 325                 <p>
 326                   GNU C++ version 5.1.0<br> linux<br> long double
 327                 </p>
 328               </th>
 329 <th>
 330                 <p>
 331                   Sun compiler version 0x5130<br> Sun Solaris<br> long double
 332                 </p>
 333               </th>
 334 </tr></thead>
 335 <tbody>
 336 <tr>
 337 <td>
 338                 <p>
 339                   Legendre Polynomials: Small Values
 340                 </p>
 341               </td>
 342 <td>
 343                 <p>
 344                   <span class="blue">Max = 211&#949; (Mean = 20.4&#949;)</span>
 345                 </p>
 346               </td>
 347 <td>
 348                 <p>
 349                   <span class="blue">Max = 0.732&#949; (Mean = 0.0619&#949;)</span><br>
 350                   <br> (<span class="emphasis"><em>GSL 1.16:</em></span> Max = 211&#949; (Mean = 20.4&#949;))
 351                 </p>
 352               </td>
 353 <td>
 354                 <p>
 355                   <span class="blue">Max = 69.2&#949; (Mean = 9.58&#949;)</span><br> <br>
 356                   (<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> Max = 124&#949; (Mean = 13.2&#949;))
 357                 </p>
 358               </td>
 359 <td>
 360                 <p>
 361                   <span class="blue">Max = 69.2&#949; (Mean = 9.58&#949;)</span>
 362                 </p>
 363               </td>
 364 </tr>
 365 <tr>
 366 <td>
 367                 <p>
 368                   Legendre Polynomials: Large Values
 369                 </p>
 370               </td>
 371 <td>
 372                 <p>
 373                   <span class="blue">Max = 300&#949; (Mean = 33.2&#949;)</span>
 374                 </p>
 375               </td>
 376 <td>
 377                 <p>
 378                   <span class="blue">Max = 0.632&#949; (Mean = 0.0693&#949;)</span><br>
 379                   <br> (<span class="emphasis"><em>GSL 1.16:</em></span> Max = 300&#949; (Mean = 33.2&#949;))
 380                 </p>
 381               </td>
 382 <td>
 383                 <p>
 384                   <span class="blue">Max = 699&#949; (Mean = 59.6&#949;)</span><br> <br>
 385                   (<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> Max = 343&#949; (Mean = 32.1&#949;))
 386                 </p>
 387               </td>
 388 <td>
 389                 <p>
 390                   <span class="blue">Max = 699&#949; (Mean = 59.6&#949;)</span>
 391                 </p>
 392               </td>
 393 </tr>
 394 </tbody>
 395 </table></div>
 396 </div>
 397 <br class="table-break"><div class="table">
 398 <a name="math_toolkit.sf_poly.legendre.table_legendre_q"></a><p class="title"><b>Table&#160;6.33.&#160;Error rates for legendre_q</b></p>
 399 <div class="table-contents"><table class="table" summary="Error rates for legendre_q">
 400 <colgroup>
 401 <col>
 402 <col>
 403 <col>
 404 <col>
 405 <col>
 406 </colgroup>
 407 <thead><tr>
 408 <th>
 409               </th>
 410 <th>
 411                 <p>
 412                   Microsoft Visual C++ version 12.0<br> Win32<br> double
 413                 </p>
 414               </th>
 415 <th>
 416                 <p>
 417                   GNU C++ version 5.1.0<br> linux<br> double
 418                 </p>
 419               </th>
 420 <th>
 421                 <p>
 422                   GNU C++ version 5.1.0<br> linux<br> long double
 423                 </p>
 424               </th>
 425 <th>
 426                 <p>
 427                   Sun compiler version 0x5130<br> Sun Solaris<br> long double
 428                 </p>
 429               </th>
 430 </tr></thead>
 431 <tbody>
 432 <tr>
 433 <td>
 434                 <p>
 435                   Legendre Polynomials: Small Values
 436                 </p>
 437               </td>
 438 <td>
 439                 <p>
 440                   <span class="blue">Max = 46.4&#949; (Mean = 7.32&#949;)</span>
 441                 </p>
 442               </td>
 443 <td>
 444                 <p>
 445                   <span class="blue">Max = 0.612&#949; (Mean = 0.0517&#949;)</span><br>
 446                   <br> (<span class="emphasis"><em>GSL 1.16:</em></span> Max = 46.4&#949; (Mean = 7.46&#949;))
 447                 </p>
 448               </td>
 449 <td>
 450                 <p>
 451                   <span class="blue">Max = 50.9&#949; (Mean = 9&#949;)</span>
 452                 </p>
 453               </td>
 454 <td>
 455                 <p>
 456                   <span class="blue">Max = 50.9&#949; (Mean = 8.98&#949;)</span>
 457                 </p>
 458               </td>
 459 </tr>
 460 <tr>
 461 <td>
 462                 <p>
 463                   Legendre Polynomials: Large Values
 464                 </p>
 465               </td>
 466 <td>
 467                 <p>
 468                   <span class="blue">Max = 4.6e+003&#949; (Mean = 366&#949;)</span>
 469                 </p>
 470               </td>
 471 <td>
 472                 <p>
 473                   <span class="blue">Max = 2.49&#949; (Mean = 0.202&#949;)</span><br> <br>
 474                   (<span class="emphasis"><em>GSL 1.16:</em></span> Max = 4.6e+03&#949; (Mean = 366&#949;))
 475                 </p>
 476               </td>
 477 <td>
 478                 <p>
 479                   <span class="blue">Max = 5.98e+03&#949; (Mean = 478&#949;)</span>
 480                 </p>
 481               </td>
 482 <td>
 483                 <p>
 484                   <span class="blue">Max = 5.98e+03&#949; (Mean = 478&#949;)</span>
 485                 </p>
 486               </td>
 487 </tr>
 488 </tbody>
 489 </table></div>
 490 </div>
 491 <br class="table-break"><div class="table">
 492 <a name="math_toolkit.sf_poly.legendre.table_legendre_p_associated_"></a><p class="title"><b>Table&#160;6.34.&#160;Error rates for legendre_p (associated)</b></p>
 493 <div class="table-contents"><table class="table" summary="Error rates for legendre_p (associated)">
 494 <colgroup>
 495 <col>
 496 <col>
 497 <col>
 498 <col>
 499 <col>
 500 </colgroup>
 501 <thead><tr>
 502 <th>
 503               </th>
 504 <th>
 505                 <p>
 506                   Microsoft Visual C++ version 12.0<br> Win32<br> double
 507                 </p>
 508               </th>
 509 <th>
 510                 <p>
 511                   GNU C++ version 5.1.0<br> linux<br> double
 512                 </p>
 513               </th>
 514 <th>
 515                 <p>
 516                   GNU C++ version 5.1.0<br> linux<br> long double
 517                 </p>
 518               </th>
 519 <th>
 520                 <p>
 521                   Sun compiler version 0x5130<br> Sun Solaris<br> long double
 522                 </p>
 523               </th>
 524 </tr></thead>
 525 <tbody><tr>
 526 <td>
 527                 <p>
 528                   Associated Legendre Polynomials: Small Values
 529                 </p>
 530               </td>
 531 <td>
 532                 <p>
 533                   <span class="blue">Max = 121&#949; (Mean = 7.14&#949;)</span>
 534                 </p>
 535               </td>
 536 <td>
 537                 <p>
 538                   <span class="blue">Max = 0.999&#949; (Mean = 0.05&#949;)</span><br> <br>
 539                   (<span class="emphasis"><em>GSL 1.16:</em></span> Max = 121&#949; (Mean = 6.75&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_legendre_p_associated__GSL_1_16_Associated_Legendre_Polynomials_Small_Values">And
 540                   other failures.</a>)
 541                 </p>
 542               </td>
 543 <td>
 544                 <p>
 545                   <span class="blue">Max = 175&#949; (Mean = 9.88&#949;)</span><br> <br>
 546                   (<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> Max = 175&#949; (Mean = 9.36&#949;)
 547                   <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_long_double_legendre_p_associated___tr1_cmath__Associated_Legendre_Polynomials_Small_Values">And
 548                   other failures.</a>)
 549                 </p>
 550               </td>
 551 <td>
 552                 <p>
 553                   <span class="blue">Max = 77.7&#949; (Mean = 5.59&#949;)</span>
 554                 </p>
 555               </td>
 556 </tr></tbody>
 557 </table></div>
 558 </div>
 559 <br class="table-break"><p>
 560         Note that the worst errors occur when the order increases, values greater
 561         than ~120 are very unlikely to produce sensible results, especially in the
 562         associated polynomial case when the degree is also large. Further the relative
 563         errors are likely to grow arbitrarily large when the function is very close
 564         to a root.
 565       </p>
 566 <h5>
 567 <a name="math_toolkit.sf_poly.legendre.h3"></a>
 568         <span class="phrase"><a name="math_toolkit.sf_poly.legendre.testing"></a></span><a class="link" href="legendre.html#math_toolkit.sf_poly.legendre.testing">Testing</a>
 569       </h5>
 570 <p>
 571         A mixture of spot tests of values calculated using functions.wolfram.com,
 572         and randomly generated test data are used: the test data was computed using
 573         <a href="http://shoup.net/ntl/doc/RR.txt" target="_top">NTL::RR</a> at 1000-bit
 574         precision.
 575       </p>
 576 <h5>
 577 <a name="math_toolkit.sf_poly.legendre.h4"></a>
 578         <span class="phrase"><a name="math_toolkit.sf_poly.legendre.implementation"></a></span><a class="link" href="legendre.html#math_toolkit.sf_poly.legendre.implementation">Implementation</a>
 579       </h5>
 580 <p>
 581         These functions are implemented using the stable three term recurrence relations.
 582         These relations guarantee low absolute error but cannot guarantee low relative
 583         error near one of the roots of the polynomials.
 584       </p>
 585 </div>
 586 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 587 <td align="left"></td>
 588 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
 589       Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
 590       Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam Sewani,
 591       Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
 592         Distributed under the Boost Software License, Version 1.0. (See accompanying
 593         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>)
 594       </p>
 595 </div></td>
 596 </tr></table>
 597 <hr>
 598 <div class="spirit-nav">
 599 <a accesskey="p" href="../sf_poly.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="laguerre.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
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