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24 | </div> | |
25 | <div class="section"> | |
26 | <div class="titlepage"><div><div><h3 class="title"> | |
27 | <a name="math_toolkit.roots.rational"></a><a class="link" href="rational.html" title="Polynomial and Rational Function Evaluation">Polynomial and Rational | |
28 | Function Evaluation</a> | |
29 | </h3></div></div></div> | |
30 | <h5> | |
31 | <a name="math_toolkit.roots.rational.h0"></a> | |
32 | <span class="phrase"><a name="math_toolkit.roots.rational.synopsis"></a></span><a class="link" href="rational.html#math_toolkit.roots.rational.synopsis">synopsis</a> | |
33 | </h5> | |
34 | <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">tools</span><span class="special">/</span><span class="identifier">rational</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> | |
35 | </pre> | |
36 | <pre class="programlisting"><span class="comment">// Polynomials:</span> | |
37 | <span class="keyword">template</span> <span class="special"><</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">></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">(&</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">&</span> <span class="identifier">val</span><span class="special">);</span> | |
39 | ||
40 | <span class="keyword">template</span> <span class="special"><</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">></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"><</span><span class="identifier">T</span><span class="special">,</span><span class="identifier">N</span><span class="special">>&</span> <span class="identifier">poly</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">V</span><span class="special">&</span> <span class="identifier">val</span><span class="special">);</span> | |
42 | ||
43 | <span class="keyword">template</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> | |
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"><</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">></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">(&</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">&</span> <span class="identifier">z</span><span class="special">);</span> | |
49 | ||
50 | <span class="keyword">template</span> <span class="special"><</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">></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"><</span><span class="identifier">T</span><span class="special">,</span><span class="identifier">N</span><span class="special">>&</span> <span class="identifier">poly</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">V</span><span class="special">&</span> <span class="identifier">z</span><span class="special">);</span> | |
52 | ||
53 | <span class="keyword">template</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> | |
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"><</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">></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">(&</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">&</span> <span class="identifier">z</span><span class="special">);</span> | |
59 | ||
60 | <span class="keyword">template</span> <span class="special"><</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">></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"><</span><span class="identifier">T</span><span class="special">,</span><span class="identifier">N</span><span class="special">>&</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">V</span><span class="special">&</span> <span class="identifier">z</span><span class="special">);</span> | |
62 | ||
63 | <span class="keyword">template</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> | |
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"><</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">></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">(&</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">(&</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">&</span> <span class="identifier">z</span><span class="special">);</span> | |
69 | ||
70 | <span class="keyword">template</span> <span class="special"><</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">></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"><</span><span class="identifier">T</span><span class="special">,</span><span class="identifier">N</span><span class="special">>&</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"><</span><span class="identifier">T</span><span class="special">,</span><span class="identifier">N</span><span class="special">>&</span> <span class="identifier">b</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">V</span><span class="special">&</span> <span class="identifier">z</span><span class="special">);</span> | |
72 | ||
73 | <span class="keyword">template</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">></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.roots.rational.h1"></a> | |
78 | <span class="phrase"><a name="math_toolkit.roots.rational.description"></a></span><a class="link" href="rational.html#math_toolkit.roots.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 | |
84 | coefficient array. Generally speaking, compile time evaluation of the array | |
85 | size results in better type safety, is less prone to programmer errors, and | |
86 | may result in better optimised code. The polynomial evaluation functions | |
87 | in particular, are specialised for various array sizes, allowing for loop | |
88 | unrolling, and one hopes, optimal inline expansion. | |
89 | </p> | |
90 | <pre class="programlisting"><span class="keyword">template</span> <span class="special"><</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">></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">(&</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">&</span> <span class="identifier">val</span><span class="special">);</span> | |
92 | ||
93 | <span class="keyword">template</span> <span class="special"><</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">></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"><</span><span class="identifier">T</span><span class="special">,</span><span class="identifier">N</span><span class="special">>&</span> <span class="identifier">poly</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">V</span><span class="special">&</span> <span class="identifier">val</span><span class="special">);</span> | |
95 | ||
96 | <span class="keyword">template</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> | |
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 | |
111 | are in poly[i]. | |
112 | </p> | |
113 | <p> | |
114 | The types of the coefficients and of variable <span class="emphasis"><em>z</em></span> may | |
115 | differ 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"><</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">></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">(&</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">&</span> <span class="identifier">z</span><span class="special">);</span> | |
121 | ||
122 | <span class="keyword">template</span> <span class="special"><</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">></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"><</span><span class="identifier">T</span><span class="special">,</span><span class="identifier">N</span><span class="special">>&</span> <span class="identifier">poly</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">V</span><span class="special">&</span> <span class="identifier">z</span><span class="special">);</span> | |
124 | ||
125 | <span class="keyword">template</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> | |
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 | |
130 | <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>. | |
131 | </p> | |
132 | <pre class="programlisting"><span class="keyword">template</span> <span class="special"><</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">></span> | |
133 | <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">(&</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">&</span> <span class="identifier">z</span><span class="special">);</span> | |
134 | ||
135 | <span class="keyword">template</span> <span class="special"><</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">></span> | |
136 | <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"><</span><span class="identifier">T</span><span class="special">,</span><span class="identifier">N</span><span class="special">>&</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">V</span><span class="special">&</span> <span class="identifier">z</span><span class="special">);</span> | |
137 | ||
138 | <span class="keyword">template</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> | |
139 | <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> | |
140 | </pre> | |
141 | <p> | |
142 | As above but evaluates a polynomial where all the powers are odd numbers. | |
143 | 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> <span class="identifier">z</span><span class="special">*</span><span class="identifier">z</span><span class="special">,</span> | |
144 | <span class="identifier">count</span><span class="special">-</span><span class="number">1</span><span class="special">)</span> <span class="special">*</span> | |
145 | <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"><</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">></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">(&</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">(&</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">&</span> <span class="identifier">z</span><span class="special">);</span> | |
149 | ||
150 | <span class="keyword">template</span> <span class="special"><</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">></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"><</span><span class="identifier">T</span><span class="special">,</span><span class="identifier">N</span><span class="special">>&</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"><</span><span class="identifier">U</span><span class="special">,</span><span class="identifier">N</span><span class="special">>&</span> <span class="identifier">denom</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">V</span><span class="special">&</span> <span class="identifier">z</span><span class="special">);</span> | |
152 | ||
153 | <span class="keyword">template</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">></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 | |
158 | by 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 | |
172 | are 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 | |
176 | differ as long as <span class="emphasis"><em>*num</em></span> and <span class="emphasis"><em>*denom</em></span> | |
177 | are convertible to type <span class="emphasis"><em>V</em></span>. This allows, for example, | |
178 | for one or both of the coefficient tables to be a table of integers if this | |
179 | is appropriate. | |
180 | </p> | |
181 | <p> | |
182 | These functions are designed to safely evaluate the result, even when the | |
183 | value <span class="emphasis"><em>z</em></span> is very large. As such they do not take advantage | |
184 | of compile time array sizes to make any optimisations. These functions are | |
185 | best reserved for situations where <span class="emphasis"><em>z</em></span> may be large: if | |
186 | you can be sure that numerical overflow will not occur then polynomial evaluation | |
187 | with compile-time array sizes may offer slightly better performance. | |
188 | </p> | |
189 | <h5> | |
190 | <a name="math_toolkit.roots.rational.h2"></a> | |
191 | <span class="phrase"><a name="math_toolkit.roots.rational.implementation"></a></span><a class="link" href="rational.html#math_toolkit.roots.rational.implementation">Implementation</a> | |
192 | </h5> | |
193 | <p> | |
194 | Polynomials are evaluated by <a href="http://en.wikipedia.org/wiki/Horner_algorithm" target="_top">Horners | |
195 | method</a>. If the array size is known at compile time then the functions | |
196 | dispatch to size-specific implementations that unroll the evaluation loop. | |
197 | </p> | |
198 | <p> | |
199 | Rational evaluation is by <a href="http://en.wikipedia.org/wiki/Horner_algorithm" target="_top">Horners | |
200 | method</a>: with the two polynomials being evaluated in parallel to make | |
201 | the most of the processors floating-point pipeline. If <span class="emphasis"><em>v</em></span> | |
202 | is greater than one, then the polynomials are evaluated in reverse order | |
203 | as polynomials in <span class="emphasis"><em>1/v</em></span>: this avoids unnecessary numerical | |
204 | overflow when the coefficients are large. | |
205 | </p> | |
206 | <p> | |
207 | Both the polynomial and rational function evaluation algorithms can be tuned | |
208 | using various configuration macros to provide optimal performance for a particular | |
209 | combination of compiler and platform. This includes support for second-order | |
210 | Horner's methods. The various options are <a class="link" href="../tuning.html" title="Performance Tuning Macros">documented | |
211 | here</a>. However, the performance benefits to be gained from these are | |
212 | 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> | |
213 | before changing the default settings. | |
214 | </p> | |
215 | </div> | |
216 | <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> | |
217 | <td align="left"></td> | |
218 | <td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal, | |
219 | Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert | |
220 | Holin, Bruno Lalande, John Maddock, Johan Råde, Gautam Sewani, Benjamin Sobotta, | |
221 | Thijs van den Berg, 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>) | |
224 | </p> | |
225 | </div></td> | |
226 | </tr></table> | |
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