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  26 <div class="titlepage"><div><div><h2 class="title" style="clear: both">
  27 <a name="math_toolkit.rational"></a><a class="link" href="rational.html" title="Polynomial and Rational Function Evaluation">Polynomial and Rational Function
  28     Evaluation</a>
  29 </h2></div></div></div>
  30 <h5>
  31 <a name="math_toolkit.rational.h0"></a>
  32       <span class="phrase"><a name="math_toolkit.rational.synopsis"></a></span><a class="link" href="rational.html#math_toolkit.rational.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">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>
  35 </pre>
  36 <pre class="programlisting"><span class="comment">// Polynomials:</span>
  37 <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>
  38 <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>
  39
  40 <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>
  41 <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>
  42
  43 <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>
  44 <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>
  45
  46 <span class="comment">// Even polynomials:</span>
  47 <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>
  48 <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>
  49
  50 <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>
  51 <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>
  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> <span class="identifier">U</span><span class="special">&gt;</span>
  54 <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>
  55
  56 <span class="comment">// Odd polynomials</span>
  57 <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>
  58 <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>
  59
  60 <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>
  61 <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>
  62
  63 <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>
  64 <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>
  65
  66 <span class="comment">// Rational Functions:</span>
  67 <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>
  68 <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>
  69
  70 <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>
  71 <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>
  72
  73 <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>
  74 <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>
  75 </pre>
  76 <h5>
  77 <a name="math_toolkit.rational.h1"></a>
  78       <span class="phrase"><a name="math_toolkit.rational.description"></a></span><a class="link" href="rational.html#math_toolkit.rational.description">Description</a>
  79     </h5>
  80 <p>
  81       Each of the functions come in three variants: a pair of overloaded functions
  82       where the order of the polynomial or rational function is evaluated at compile
  83       time, and an overload that accepts a runtime variable for the size of the coefficient
  84       array. Generally speaking, compile time evaluation of the array size results
  85       in better type safety, is less prone to programmer errors, and may result in
  86       better optimised code. The polynomial evaluation functions in particular, are
  87       specialised for various array sizes, allowing for loop unrolling, and one hopes,
  88       optimal inline expansion.
  89     </p>
  90 <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>
  91 <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>
  92
  93 <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>
  94 <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>
  95
  96 <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>
  97 <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>
  98 </pre>
  99 <p>
 100       Evaluates the <a href="http://en.wikipedia.org/wiki/Polynomial" target="_top">polynomial</a>
 101       described by the coefficients stored in <span class="emphasis"><em>poly</em></span>.
 102     </p>
 103 <p>
 104       If the size of the array is specified at runtime, then the polynomial most
 105       have order <span class="emphasis"><em>count-1</em></span> with <span class="emphasis"><em>count</em></span> coefficients.
 106       Otherwise it has order <span class="emphasis"><em>N-1</em></span> with <span class="emphasis"><em>N</em></span>
 107       coefficients.
 108     </p>
 109 <p>
 110       Coefficients should be stored such that the coefficients for the x<sup>i</sup> terms are
 111       in poly[i].
 112     </p>
 113 <p>
 114       The types of the coefficients and of variable <span class="emphasis"><em>z</em></span> may differ
 115       as long as <span class="emphasis"><em>*poly</em></span> is convertible to type <span class="emphasis"><em>U</em></span>.
 116       This allows, for example, for the coefficient table to be a table of integers
 117       if this is appropriate.
 118     </p>
 119 <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>
 120 <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>
 121
 122 <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>
 123 <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>
 124
 125 <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>
 126 <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>
 127 </pre>
 128 <p>
 129       As above, but evaluates an even polynomial: one where all the powers of <span class="emphasis"><em>z</em></span>
 130       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>
 131       <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>.
 132     </p>
 133 <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>
 134 <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>
 135
 136 <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">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>
 138
 139 <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>
 140 <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>
 141 </pre>
 142 <p>
 143       As above but evaluates a polynomial where all the powers are odd numbers. Equivalent
 144       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> <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>
 145       <span class="special">*</span> <span class="identifier">z</span> <span class="special">+</span> <span class="identifier">poly</span><span class="special">[</span><span class="number">0</span><span class="special">]</span></code>.
 146     </p>
 147 <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>
 148 <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>
 149
 150 <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>
 151 <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>
 152
 153 <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>
 154 <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>
 155 </pre>
 156 <p>
 157       Evaluates the rational function (the ratio of two polynomials) described by
 158       the coefficients stored in <span class="emphasis"><em>num</em></span> and <span class="emphasis"><em>demom</em></span>.
 159     </p>
 160 <p>
 161       If the size of the array is specified at runtime then both polynomials most
 162       have order <span class="emphasis"><em>count-1</em></span> with <span class="emphasis"><em>count</em></span> coefficients.
 163       Otherwise both polynomials have order <span class="emphasis"><em>N-1</em></span> with <span class="emphasis"><em>N</em></span>
 164       coefficients.
 165     </p>
 166 <p>
 167       Array <span class="emphasis"><em>num</em></span> describes the numerator, and <span class="emphasis"><em>demon</em></span>
 168       the denominator.
 169     </p>
 170 <p>
 171       Coefficients should be stored such that the coefficients for the x<sup>i </sup> terms are
 172       in num[i] and denom[i].
 173     </p>
 174 <p>
 175       The types of the coefficients and of variable <span class="emphasis"><em>v</em></span> may differ
 176       as long as <span class="emphasis"><em>*num</em></span> and <span class="emphasis"><em>*denom</em></span> are convertible
 177       to type <span class="emphasis"><em>V</em></span>. This allows, for example, for one or both of
 178       the coefficient tables to be a table of integers if this is appropriate.
 179     </p>
 180 <p>
 181       These functions are designed to safely evaluate the result, even when the value
 182       <span class="emphasis"><em>z</em></span> is very large. As such they do not take advantage of
 183       compile time array sizes to make any optimisations. These functions are best
 184       reserved for situations where <span class="emphasis"><em>z</em></span> may be large: if you can
 185       be sure that numerical overflow will not occur then polynomial evaluation with
 186       compile-time array sizes may offer slightly better performance.
 187     </p>
 188 <h5>
 189 <a name="math_toolkit.rational.h2"></a>
 190       <span class="phrase"><a name="math_toolkit.rational.implementation"></a></span><a class="link" href="rational.html#math_toolkit.rational.implementation">Implementation</a>
 191     </h5>
 192 <p>
 193       Polynomials are evaluated by <a href="http://en.wikipedia.org/wiki/Horner_algorithm" target="_top">Horners
 194       method</a>. If the array size is known at compile time then the functions
 195       dispatch to size-specific implementations that unroll the evaluation loop.
 196     </p>
 197 <p>
 198       Rational evaluation is by <a href="http://en.wikipedia.org/wiki/Horner_algorithm" target="_top">Horners
 199       method</a>: with the two polynomials being evaluated in parallel to make
 200       the most of the processors floating-point pipeline. If <span class="emphasis"><em>v</em></span>
 201       is greater than one, then the polynomials are evaluated in reverse order as
 202       polynomials in <span class="emphasis"><em>1/v</em></span>: this avoids unnecessary numerical
 203       overflow when the coefficients are large.
 204     </p>
 205 <p>
 206       Both the polynomial and rational function evaluation algorithms can be tuned
 207       using various configuration macros to provide optimal performance for a particular
 208       combination of compiler and platform. This includes support for second-order
 209       Horner's methods. The various options are <a class="link" href="tuning.html" title="Performance Tuning Macros">documented
 210       here</a>. However, the performance benefits to be gained from these are
 211       marginal on most current hardware, consequently it's best to run the <a class="link" href="perf_test_app.html" title="The Performance Test Applications">performance test application</a> before
 212       changing the default settings.
 213     </p>
 214 </div>
 215 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 216 <td align="left"></td>
 217 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
 218       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
 219       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
 220       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
 221       Daryle Walker and Xiaogang Zhang<p>
 222         Distributed under the Boost Software License, Version 1.0. (See accompanying
 223         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>)
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