| <html> |
| <head> |
| <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> |
| <title>Polynomial and Rational Function Evaluation</title> |
| <link rel="stylesheet" href="../../../../../../../../doc/src/boostbook.css" type="text/css"> |
| <meta name="generator" content="DocBook XSL Stylesheets V1.74.0"> |
| <link rel="home" href="../../../index.html" title="Math Toolkit"> |
| <link rel="up" href="../internals1.html" title="Reused Utilities"> |
| <link rel="prev" href="cf.html" title="Continued Fraction Evaluation"> |
| <link rel="next" href="roots.html" title="Root Finding With Derivatives"> |
| </head> |
| <body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> |
| <table cellpadding="2" width="100%"><tr> |
| <td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../../../boost.png"></td> |
| <td align="center"><a href="../../../../../../../../index.html">Home</a></td> |
| <td align="center"><a href="../../../../../../../../libs/libraries.htm">Libraries</a></td> |
| <td align="center"><a href="http://www.boost.org/users/people.html">People</a></td> |
| <td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td> |
| <td align="center"><a href="../../../../../../../../more/index.htm">More</a></td> |
| </tr></table> |
| <hr> |
| <div class="spirit-nav"> |
| <a accesskey="p" href="cf.html"><img src="../../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../internals1.html"><img src="../../../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../../index.html"><img src="../../../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="roots.html"><img src="../../../../../../../../doc/src/images/next.png" alt="Next"></a> |
| </div> |
| <div class="section" lang="en"> |
| <div class="titlepage"><div><div><h4 class="title"> |
| <a name="math_toolkit.toolkit.internals1.rational"></a><a class="link" href="rational.html" title="Polynomial and Rational Function Evaluation"> Polynomial |
| and Rational Function Evaluation</a> |
| </h4></div></div></div> |
| <a name="math_toolkit.toolkit.internals1.rational.synopsis"></a><h5> |
| <a name="id1196860"></a> |
| <a class="link" href="rational.html#math_toolkit.toolkit.internals1.rational.synopsis">synopsis</a> |
| </h5> |
| <p> |
| |
| </p> |
| <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> |
| </pre> |
| <p> |
| </p> |
| <pre class="programlisting"><span class="comment">// Polynomials: |
| </span><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> |
| <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> |
| |
| <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> |
| <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> |
| |
| <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="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> |
| |
| <span class="comment">// Even polynomials: |
| </span><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> |
| <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> |
| |
| <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> |
| <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> |
| |
| <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="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> |
| |
| <span class="comment">// Odd polynomials |
| </span><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> |
| <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> |
| |
| <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> |
| <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> |
| |
| <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="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> |
| |
| <span class="comment">// Rational Functions: |
| </span><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> |
| <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> |
| |
| <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> |
| <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> |
| |
| <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> |
| <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> |
| </pre> |
| <a name="math_toolkit.toolkit.internals1.rational.description"></a><h5> |
| <a name="id1198990"></a> |
| <a class="link" href="rational.html#math_toolkit.toolkit.internals1.rational.description">Description</a> |
| </h5> |
| <p> |
| Each of the functions come in three variants: a pair of overloaded functions |
| where the order of the polynomial or rational function is evaluated at |
| compile time, and an overload that accepts a runtime variable for the size |
| of the coefficient array. Generally speaking, compile time evaluation of |
| the array size results in better type safety, is less prone to programmer |
| errors, and may result in better optimised code. The polynomial evaluation |
| functions in particular, are specialised for various array sizes, allowing |
| for loop unrolling, and one hopes, optimal inline expansion. |
| </p> |
| <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> |
| <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> |
| |
| <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> |
| <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> |
| |
| <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="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> |
| </pre> |
| <p> |
| Evaluates the <a href="http://en.wikipedia.org/wiki/Polynomial" target="_top">polynomial</a> |
| described by the coefficients stored in <span class="emphasis"><em>poly</em></span>. |
| </p> |
| <p> |
| If the size of the array is specified at runtime, then the polynomial most |
| have order <span class="emphasis"><em>count-1</em></span> with <span class="emphasis"><em>count</em></span> |
| coefficients. Otherwise it has order <span class="emphasis"><em>N-1</em></span> with <span class="emphasis"><em>N</em></span> |
| coefficients. |
| </p> |
| <p> |
| Coefficients should be stored such that the coefficients for the x<sup>i </sup> terms |
| are in poly[i]. |
| </p> |
| <p> |
| The types of the coefficients and of variable <span class="emphasis"><em>z</em></span> may |
| differ as long as <span class="emphasis"><em>*poly</em></span> is convertible to type <span class="emphasis"><em>U</em></span>. |
| This allows, for example, for the coefficient table to be a table of integers |
| if this is appropriate. |
| </p> |
| <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> |
| <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> |
| |
| <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> |
| <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> |
| |
| <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="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> |
| </pre> |
| <p> |
| As above, but evaluates an even polynomial: one where all the powers of |
| <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>. |
| </p> |
| <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> |
| <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> |
| |
| <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> |
| <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> |
| |
| <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="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> |
| </pre> |
| <p> |
| As above but evaluates a polynomial where all the powers are odd numbers. |
| 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> <span class="identifier">count</span><span class="special">-</span><span class="number">1</span><span class="special">)</span> |
| <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>. |
| </p> |
| <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> |
| <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> |
| |
| <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> |
| <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> |
| |
| <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> |
| <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> |
| </pre> |
| <p> |
| Evaluates the rational function (the ratio of two polynomials) described |
| by the coefficients stored in <span class="emphasis"><em>num</em></span> and <span class="emphasis"><em>demom</em></span>. |
| </p> |
| <p> |
| If the size of the array is specified at runtime then both polynomials |
| most have order <span class="emphasis"><em>count-1</em></span> with <span class="emphasis"><em>count</em></span> |
| coefficients. Otherwise both polynomials have order <span class="emphasis"><em>N-1</em></span> |
| with <span class="emphasis"><em>N</em></span> coefficients. |
| </p> |
| <p> |
| Array <span class="emphasis"><em>num</em></span> describes the numerator, and <span class="emphasis"><em>demon</em></span> |
| the denominator. |
| </p> |
| <p> |
| Coefficients should be stored such that the coefficients for the x<sup>i </sup> terms |
| are in num[i] and denom[i]. |
| </p> |
| <p> |
| The types of the coefficients and of variable <span class="emphasis"><em>v</em></span> may |
| differ as long as <span class="emphasis"><em>*num</em></span> and <span class="emphasis"><em>*denom</em></span> |
| are convertible to type <span class="emphasis"><em>V</em></span>. This allows, for example, |
| for one or both of the coefficient tables to be a table of integers if |
| this is appropriate. |
| </p> |
| <p> |
| These functions are designed to safely evaluate the result, even when the |
| value <span class="emphasis"><em>z</em></span> is very large. As such they do not take advantage |
| of compile time array sizes to make any optimisations. These functions |
| are best reserved for situations where <span class="emphasis"><em>z</em></span> may be large: |
| if you can be sure that numerical overflow will not occur then polynomial |
| evaluation with compile-time array sizes may offer slightly better performance. |
| </p> |
| <a name="math_toolkit.toolkit.internals1.rational.implementation"></a><h5> |
| <a name="id1200830"></a> |
| <a class="link" href="rational.html#math_toolkit.toolkit.internals1.rational.implementation">Implementation</a> |
| </h5> |
| <p> |
| Polynomials are evaluated by <a href="http://en.wikipedia.org/wiki/Horner_algorithm" target="_top">Horners |
| method</a>. If the array size is known at compile time then the functions |
| dispatch to size-specific implementations that unroll the evaluation loop. |
| </p> |
| <p> |
| Rational evaluation is by <a href="http://en.wikipedia.org/wiki/Horner_algorithm" target="_top">Horners |
| method</a>: with the two polynomials being evaluated in parallel to |
| make the most of the processors floating-point pipeline. If <span class="emphasis"><em>v</em></span> |
| is greater than one, then the polynomials are evaluated in reverse order |
| as polynomials in <span class="emphasis"><em>1/v</em></span>: this avoids unnecessary numerical |
| overflow when the coefficients are large. |
| </p> |
| <p> |
| Both the polynomial and rational function evaluation algorithms can be |
| tuned using various configuration macros to provide optimal performance |
| for a particular combination of compiler and platform. This includes support |
| for second-order Horner's methods. The various options are <a class="link" href="../../perf/tuning.html" title="Performance Tuning Macros">documented |
| here</a>. However, the performance benefits to be gained from these |
| are marginal on most current hardware, consequently it's best to run the |
| <a class="link" href="../../perf/perf_test_app.html" title="The Performance Test Application">performance test application</a> |
| before changing the default settings. |
| </p> |
| </div> |
| <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> |
| <td align="left"></td> |
| <td align="right"><div class="copyright-footer">Copyright © 2006 , 2007, 2008, 2009, 2010 John Maddock, Paul A. Bristow, |
| Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani and |
| Thijs van den Berg<p> |
| Distributed under the Boost Software License, Version 1.0. (See accompanying |
| 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>) |
| </p> |
| </div></td> |
| </tr></table> |
| <hr> |
| <div class="spirit-nav"> |
| <a accesskey="p" href="cf.html"><img src="../../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../internals1.html"><img src="../../../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../../index.html"><img src="../../../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="roots.html"><img src="../../../../../../../../doc/src/images/next.png" alt="Next"></a> |
| </div> |
| </body> |
| </html> |
