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  25 <div class="section math_toolkit_toolkit_internals1_rational">
  26 <div class="titlepage"><div><div><h4 class="title">
  27 <a name="math_toolkit.toolkit.internals1.rational"></a><a class="link" href="rational.html" title="Polynomial and Rational Function Evaluation">Polynomial
  28         and Rational Function Evaluation</a>
  29 </h4></div></div></div>
  30 <h5>
  31 <a name="math_toolkit.toolkit.internals1.rational.h0"></a>
  32           <span><a name="math_toolkit.toolkit.internals1.rational.synopsis"></a></span><a class="link" href="rational.html#math_toolkit.toolkit.internals1.rational.synopsis">synopsis</a>
  33         </h5>
  34 <p>
  35 </p>
  36 <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">tools</span><span class="special">/</span><span class="identifier">rational</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
  37 </pre>
  38 <p>
  39         </p>
  40 <pre class="programlisting"><span class="comment">// Polynomials:</span>
  41 <span class="keyword">template</span> <span class="special">&lt;</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">size_t</span> <span class="identifier">N</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">V</span><span class="special">&gt;</span>
  42 <span class="identifier">V</span> <span class="identifier">evaluate_polynomial</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">T</span><span class="special">(&amp;</span><span class="identifier">poly</span><span class="special">)[</span><span class="identifier">N</span><span class="special">],</span> <span class="keyword">const</span> <span class="identifier">V</span><span class="special">&amp;</span> <span class="identifier">val</span><span class="special">);</span>
  43
  44 <span class="keyword">template</span> <span class="special">&lt;</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">size_t</span> <span class="identifier">N</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">V</span><span class="special">&gt;</span>
  45 <span class="identifier">V</span> <span class="identifier">evaluate_polynomial</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">array</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">,</span><span class="identifier">N</span><span class="special">&gt;&amp;</span> <span class="identifier">poly</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">V</span><span class="special">&amp;</span> <span class="identifier">val</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> <span class="identifier">U</span><span class="special">&gt;</span>
  48 <span class="identifier">U</span> <span class="identifier">evaluate_polynomial</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">T</span><span class="special">*</span> <span class="identifier">poly</span><span class="special">,</span> <span class="identifier">U</span> <span class="identifier">z</span><span class="special">,</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">size_t</span> <span class="identifier">count</span><span class="special">);</span>
  49
  50 <span class="comment">// Even polynomials:</span>
  51 <span class="keyword">template</span> <span class="special">&lt;</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">size_t</span> <span class="identifier">N</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">V</span><span class="special">&gt;</span>
  52 <span class="identifier">V</span> <span class="identifier">evaluate_even_polynomial</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">T</span><span class="special">(&amp;</span><span class="identifier">poly</span><span class="special">)[</span><span class="identifier">N</span><span class="special">],</span> <span class="keyword">const</span> <span class="identifier">V</span><span class="special">&amp;</span> <span class="identifier">z</span><span class="special">);</span>
  53
  54 <span class="keyword">template</span> <span class="special">&lt;</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">size_t</span> <span class="identifier">N</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">V</span><span class="special">&gt;</span>
  55 <span class="identifier">V</span> <span class="identifier">evaluate_even_polynomial</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">array</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">,</span><span class="identifier">N</span><span class="special">&gt;&amp;</span> <span class="identifier">poly</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">V</span><span class="special">&amp;</span> <span class="identifier">z</span><span class="special">);</span>
  56
  57 <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> <span class="identifier">U</span><span class="special">&gt;</span>
  58 <span class="identifier">U</span> <span class="identifier">evaluate_even_polynomial</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">T</span><span class="special">*</span> <span class="identifier">poly</span><span class="special">,</span> <span class="identifier">U</span> <span class="identifier">z</span><span class="special">,</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">size_t</span> <span class="identifier">count</span><span class="special">);</span>
  59
  60 <span class="comment">// Odd polynomials   </span>
  61 <span class="keyword">template</span> <span class="special">&lt;</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">size_t</span> <span class="identifier">N</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">V</span><span class="special">&gt;</span>
  62 <span class="identifier">V</span> <span class="identifier">evaluate_odd_polynomial</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">T</span><span class="special">(&amp;</span><span class="identifier">a</span><span class="special">)[</span><span class="identifier">N</span><span class="special">],</span> <span class="keyword">const</span> <span class="identifier">V</span><span class="special">&amp;</span> <span class="identifier">z</span><span class="special">);</span>
  63
  64 <span class="keyword">template</span> <span class="special">&lt;</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">size_t</span> <span class="identifier">N</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">V</span><span class="special">&gt;</span>
  65 <span class="identifier">V</span> <span class="identifier">evaluate_odd_polynomial</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">array</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">,</span><span class="identifier">N</span><span class="special">&gt;&amp;</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">V</span><span class="special">&amp;</span> <span class="identifier">z</span><span class="special">);</span>
  66
  67 <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> <span class="identifier">U</span><span class="special">&gt;</span>
  68 <span class="identifier">U</span> <span class="identifier">evaluate_odd_polynomial</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">T</span><span class="special">*</span> <span class="identifier">poly</span><span class="special">,</span> <span class="identifier">U</span> <span class="identifier">z</span><span class="special">,</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">size_t</span> <span class="identifier">count</span><span class="special">);</span>
  69
  70 <span class="comment">// Rational Functions:</span>
  71 <span class="keyword">template</span> <span class="special">&lt;</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">size_t</span> <span class="identifier">N</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">V</span><span class="special">&gt;</span>
  72 <span class="identifier">V</span> <span class="identifier">evaluate_rational</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">T</span><span class="special">(&amp;</span><span class="identifier">a</span><span class="special">)[</span><span class="identifier">N</span><span class="special">],</span> <span class="keyword">const</span> <span class="identifier">T</span><span class="special">(&amp;</span><span class="identifier">b</span><span class="special">)[</span><span class="identifier">N</span><span class="special">],</span> <span class="keyword">const</span> <span class="identifier">V</span><span class="special">&amp;</span> <span class="identifier">z</span><span class="special">);</span>
  73
  74 <span class="keyword">template</span> <span class="special">&lt;</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">size_t</span> <span class="identifier">N</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">V</span><span class="special">&gt;</span>
  75 <span class="identifier">V</span> <span class="identifier">evaluate_rational</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">array</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">,</span><span class="identifier">N</span><span class="special">&gt;&amp;</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">array</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">,</span><span class="identifier">N</span><span class="special">&gt;&amp;</span> <span class="identifier">b</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">V</span><span class="special">&amp;</span> <span class="identifier">z</span><span class="special">);</span>
  76
  77 <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> <span class="identifier">U</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">V</span><span class="special">&gt;</span>
  78 <span class="identifier">V</span> <span class="identifier">evaluate_rational</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">T</span><span class="special">*</span> <span class="identifier">num</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">U</span><span class="special">*</span> <span class="identifier">denom</span><span class="special">,</span> <span class="identifier">V</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">count</span><span class="special">);</span>
  79 </pre>
  80 <h5>
  81 <a name="math_toolkit.toolkit.internals1.rational.h1"></a>
  82           <span><a name="math_toolkit.toolkit.internals1.rational.description"></a></span><a class="link" href="rational.html#math_toolkit.toolkit.internals1.rational.description">Description</a>
  83         </h5>
  84 <p>
  85           Each of the functions come in three variants: a pair of overloaded functions
  86           where the order of the polynomial or rational function is evaluated at
  87           compile time, and an overload that accepts a runtime variable for the size
  88           of the coefficient array. Generally speaking, compile time evaluation of
  89           the array size results in better type safety, is less prone to programmer
  90           errors, and may result in better optimised code. The polynomial evaluation
  91           functions in particular, are specialised for various array sizes, allowing
  92           for loop unrolling, and one hopes, optimal inline expansion.
  93         </p>
  94 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">size_t</span> <span class="identifier">N</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">V</span><span class="special">&gt;</span>
  95 <span class="identifier">V</span> <span class="identifier">evaluate_polynomial</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">T</span><span class="special">(&amp;</span><span class="identifier">poly</span><span class="special">)[</span><span class="identifier">N</span><span class="special">],</span> <span class="keyword">const</span> <span class="identifier">V</span><span class="special">&amp;</span> <span class="identifier">val</span><span class="special">);</span>
  96
  97 <span class="keyword">template</span> <span class="special">&lt;</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">size_t</span> <span class="identifier">N</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">V</span><span class="special">&gt;</span>
  98 <span class="identifier">V</span> <span class="identifier">evaluate_polynomial</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">array</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">,</span><span class="identifier">N</span><span class="special">&gt;&amp;</span> <span class="identifier">poly</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">V</span><span class="special">&amp;</span> <span class="identifier">val</span><span class="special">);</span>
  99
 100 <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> <span class="identifier">U</span><span class="special">&gt;</span>
 101 <span class="identifier">U</span> <span class="identifier">evaluate_polynomial</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">T</span><span class="special">*</span> <span class="identifier">poly</span><span class="special">,</span> <span class="identifier">U</span> <span class="identifier">z</span><span class="special">,</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">size_t</span> <span class="identifier">count</span><span class="special">);</span>
 102 </pre>
 103 <p>
 104           Evaluates the <a href="http://en.wikipedia.org/wiki/Polynomial" target="_top">polynomial</a>
 105           described by the coefficients stored in <span class="emphasis"><em>poly</em></span>.
 106         </p>
 107 <p>
 108           If the size of the array is specified at runtime, then the polynomial most
 109           have order <span class="emphasis"><em>count-1</em></span> with <span class="emphasis"><em>count</em></span>
 110           coefficients. Otherwise it has order <span class="emphasis"><em>N-1</em></span> with <span class="emphasis"><em>N</em></span>
 111           coefficients.
 112         </p>
 113 <p>
 114           Coefficients should be stored such that the coefficients for the x<sup>i </sup> terms
 115           are in poly[i].
 116         </p>
 117 <p>
 118           The types of the coefficients and of variable <span class="emphasis"><em>z</em></span> may
 119           differ as long as <span class="emphasis"><em>*poly</em></span> is convertible to type <span class="emphasis"><em>U</em></span>.
 120           This allows, for example, for the coefficient table to be a table of integers
 121           if this is appropriate.
 122         </p>
 123 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">size_t</span> <span class="identifier">N</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">V</span><span class="special">&gt;</span>
 124 <span class="identifier">V</span> <span class="identifier">evaluate_even_polynomial</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">T</span><span class="special">(&amp;</span><span class="identifier">poly</span><span class="special">)[</span><span class="identifier">N</span><span class="special">],</span> <span class="keyword">const</span> <span class="identifier">V</span><span class="special">&amp;</span> <span class="identifier">z</span><span class="special">);</span>
 125
 126 <span class="keyword">template</span> <span class="special">&lt;</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">size_t</span> <span class="identifier">N</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">V</span><span class="special">&gt;</span>
 127 <span class="identifier">V</span> <span class="identifier">evaluate_even_polynomial</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">array</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">,</span><span class="identifier">N</span><span class="special">&gt;&amp;</span> <span class="identifier">poly</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">V</span><span class="special">&amp;</span> <span class="identifier">z</span><span class="special">);</span>
 128
 129 <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> <span class="identifier">U</span><span class="special">&gt;</span>
 130 <span class="identifier">U</span> <span class="identifier">evaluate_even_polynomial</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">T</span><span class="special">*</span> <span class="identifier">poly</span><span class="special">,</span> <span class="identifier">U</span> <span class="identifier">z</span><span class="special">,</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">size_t</span> <span class="identifier">count</span><span class="special">);</span>
 131 </pre>
 132 <p>
 133           As above, but evaluates an even polynomial: one where all the powers of
 134           <span class="emphasis"><em>z</em></span> are even numbers. Equivalent to calling <code class="computeroutput"><span class="identifier">evaluate_polynomial</span><span class="special">(</span><span class="identifier">poly</span><span class="special">,</span> <span class="identifier">z</span><span class="special">*</span><span class="identifier">z</span><span class="special">,</span> <span class="identifier">count</span><span class="special">)</span></code>.
 135         </p>
 136 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">size_t</span> <span class="identifier">N</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">V</span><span class="special">&gt;</span>
 137 <span class="identifier">V</span> <span class="identifier">evaluate_odd_polynomial</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">T</span><span class="special">(&amp;</span><span class="identifier">a</span><span class="special">)[</span><span class="identifier">N</span><span class="special">],</span> <span class="keyword">const</span> <span class="identifier">V</span><span class="special">&amp;</span> <span class="identifier">z</span><span class="special">);</span>
 138
 139 <span class="keyword">template</span> <span class="special">&lt;</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">size_t</span> <span class="identifier">N</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">V</span><span class="special">&gt;</span>
 140 <span class="identifier">V</span> <span class="identifier">evaluate_odd_polynomial</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">array</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">,</span><span class="identifier">N</span><span class="special">&gt;&amp;</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">V</span><span class="special">&amp;</span> <span class="identifier">z</span><span class="special">);</span>
 141
 142 <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> <span class="identifier">U</span><span class="special">&gt;</span>
 143 <span class="identifier">U</span> <span class="identifier">evaluate_odd_polynomial</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">T</span><span class="special">*</span> <span class="identifier">poly</span><span class="special">,</span> <span class="identifier">U</span> <span class="identifier">z</span><span class="special">,</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">size_t</span> <span class="identifier">count</span><span class="special">);</span>
 144 </pre>
 145 <p>
 146           As above but evaluates a polynomial where all the powers are odd numbers.
 147           Equivalent to <code class="computeroutput"><span class="identifier">evaluate_polynomial</span><span class="special">(</span><span class="identifier">poly</span><span class="special">+</span><span class="number">1</span><span class="special">,</span>
 148           <span class="identifier">z</span><span class="special">*</span><span class="identifier">z</span><span class="special">,</span> <span class="identifier">count</span><span class="special">-</span><span class="number">1</span><span class="special">)</span>
 149           <span class="special">*</span> <span class="identifier">z</span>
 150           <span class="special">+</span> <span class="identifier">poly</span><span class="special">[</span><span class="number">0</span><span class="special">]</span></code>.
 151         </p>
 152 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">size_t</span> <span class="identifier">N</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">U</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">V</span><span class="special">&gt;</span>
 153 <span class="identifier">V</span> <span class="identifier">evaluate_rational</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">T</span><span class="special">(&amp;</span><span class="identifier">num</span><span class="special">)[</span><span class="identifier">N</span><span class="special">],</span> <span class="keyword">const</span> <span class="identifier">U</span><span class="special">(&amp;</span><span class="identifier">denom</span><span class="special">)[</span><span class="identifier">N</span><span class="special">],</span> <span class="keyword">const</span> <span class="identifier">V</span><span class="special">&amp;</span> <span class="identifier">z</span><span class="special">);</span>
 154
 155 <span class="keyword">template</span> <span class="special">&lt;</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">size_t</span> <span class="identifier">N</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">U</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">V</span><span class="special">&gt;</span>
 156 <span class="identifier">V</span> <span class="identifier">evaluate_rational</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">array</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">,</span><span class="identifier">N</span><span class="special">&gt;&amp;</span> <span class="identifier">num</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">array</span><span class="special">&lt;</span><span class="identifier">U</span><span class="special">,</span><span class="identifier">N</span><span class="special">&gt;&amp;</span> <span class="identifier">denom</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">V</span><span class="special">&amp;</span> <span class="identifier">z</span><span class="special">);</span>
 157
 158 <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> <span class="identifier">U</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">V</span><span class="special">&gt;</span>
 159 <span class="identifier">V</span> <span class="identifier">evaluate_rational</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">T</span><span class="special">*</span> <span class="identifier">num</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">U</span><span class="special">*</span> <span class="identifier">denom</span><span class="special">,</span> <span class="identifier">V</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">count</span><span class="special">);</span>
 160 </pre>
 161 <p>
 162           Evaluates the rational function (the ratio of two polynomials) described
 163           by the coefficients stored in <span class="emphasis"><em>num</em></span> and <span class="emphasis"><em>demom</em></span>.
 164         </p>
 165 <p>
 166           If the size of the array is specified at runtime then both polynomials
 167           most have order <span class="emphasis"><em>count-1</em></span> with <span class="emphasis"><em>count</em></span>
 168           coefficients. Otherwise both polynomials have order <span class="emphasis"><em>N-1</em></span>
 169           with <span class="emphasis"><em>N</em></span> coefficients.
 170         </p>
 171 <p>
 172           Array <span class="emphasis"><em>num</em></span> describes the numerator, and <span class="emphasis"><em>demon</em></span>
 173           the denominator.
 174         </p>
 175 <p>
 176           Coefficients should be stored such that the coefficients for the x<sup>i </sup> terms
 177           are in num[i] and denom[i].
 178         </p>
 179 <p>
 180           The types of the coefficients and of variable <span class="emphasis"><em>v</em></span> may
 181           differ as long as <span class="emphasis"><em>*num</em></span> and <span class="emphasis"><em>*denom</em></span>
 182           are convertible to type <span class="emphasis"><em>V</em></span>. This allows, for example,
 183           for one or both of the coefficient tables to be a table of integers if
 184           this is appropriate.
 185         </p>
 186 <p>
 187           These functions are designed to safely evaluate the result, even when the
 188           value <span class="emphasis"><em>z</em></span> is very large. As such they do not take advantage
 189           of compile time array sizes to make any optimisations. These functions
 190           are best reserved for situations where <span class="emphasis"><em>z</em></span> may be large:
 191           if you can be sure that numerical overflow will not occur then polynomial
 192           evaluation with compile-time array sizes may offer slightly better performance.
 193         </p>
 194 <h5>
 195 <a name="math_toolkit.toolkit.internals1.rational.h2"></a>
 196           <span><a name="math_toolkit.toolkit.internals1.rational.implementation"></a></span><a class="link" href="rational.html#math_toolkit.toolkit.internals1.rational.implementation">Implementation</a>
 197         </h5>
 198 <p>
 199           Polynomials are evaluated by <a href="http://en.wikipedia.org/wiki/Horner_algorithm" target="_top">Horners
 200           method</a>. If the array size is known at compile time then the functions
 201           dispatch to size-specific implementations that unroll the evaluation loop.
 202         </p>
 203 <p>
 204           Rational evaluation is by <a href="http://en.wikipedia.org/wiki/Horner_algorithm" target="_top">Horners
 205           method</a>: with the two polynomials being evaluated in parallel to
 206           make the most of the processors floating-point pipeline. If <span class="emphasis"><em>v</em></span>
 207           is greater than one, then the polynomials are evaluated in reverse order
 208           as polynomials in <span class="emphasis"><em>1/v</em></span>: this avoids unnecessary numerical
 209           overflow when the coefficients are large.
 210         </p>
 211 <p>
 212           Both the polynomial and rational function evaluation algorithms can be
 213           tuned using various configuration macros to provide optimal performance
 214           for a particular combination of compiler and platform. This includes support
 215           for second-order Horner's methods. The various options are <a class="link" href="../../perf/tuning.html" title="Performance Tuning Macros">documented
 216           here</a>. However, the performance benefits to be gained from these
 217           are marginal on most current hardware, consequently it's best to run the
 218           <a class="link" href="../../perf/perf_test_app.html" title="The Performance Test Application">performance test application</a>
 219           before changing the default settings.
 220         </p>
 221 </div>
 222 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 223 <td align="left"></td>
 224 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010 John Maddock, Paul A. Bristow, Hubert Holin, Xiaogang Zhang, Bruno
 225       Lalande, Johan R&#229;de, Gautam Sewani, Thijs van den Berg and Benjamin Sobotta<p>
 226         Distributed under the Boost Software License, Version 1.0. (See accompanying
 227         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>)
 228       </p>
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