ceph/src/boost/libs/math/doc/html/math_toolkit/roots/polynomials.html

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  27 <a name="math_toolkit.roots.polynomials"></a><a class="link" href="polynomials.html" title="Polynomials">Polynomials</a>
  28 </h3></div></div></div>
  29 <h5>
  30 <a name="math_toolkit.roots.polynomials.h0"></a>
  31         <span class="phrase"><a name="math_toolkit.roots.polynomials.synopsis"></a></span><a class="link" href="polynomials.html#math_toolkit.roots.polynomials.synopsis">Synopsis</a>
  32       </h5>
  33 <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">polynomial</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
  34 </pre>
  35 <pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span> <span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span> <span class="special">{</span>
  36 <span class="keyword">namespace</span> <span class="identifier">tools</span> <span class="special">{</span>
  37
  38 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
  39 <span class="keyword">class</span> <span class="identifier">polynomial</span>
  40 <span class="special">{</span>
  41 <span class="keyword">public</span><span class="special">:</span>
  42    <span class="comment">// typedefs:</span>
  43    <span class="keyword">typedef</span> <span class="keyword">typename</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;::</span><span class="identifier">value_type</span> <span class="identifier">value_type</span><span class="special">;</span>
  44    <span class="keyword">typedef</span> <span class="keyword">typename</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;::</span><span class="identifier">size_type</span>  <span class="identifier">size_type</span><span class="special">;</span>
  45
  46    <span class="comment">// construct:</span>
  47    <span class="identifier">polynomial</span><span class="special">(){}</span>
  48    <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">U</span><span class="special">&gt;</span>
  49    <span class="identifier">polynomial</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">U</span><span class="special">*</span> <span class="identifier">data</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">order</span><span class="special">);</span>
  50    <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">I</span><span class="special">&gt;</span>
  51    <span class="identifier">polynomial</span><span class="special">(</span><span class="identifier">I</span> <span class="identifier">first</span><span class="special">,</span> <span class="identifier">I</span> <span class="identifier">last</span><span class="special">);</span>
  52    <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">U</span><span class="special">&gt;</span>
  53    <span class="keyword">explicit</span> <span class="identifier">polynomial</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">U</span><span class="special">&amp;</span> <span class="identifier">point</span><span class="special">);</span>
  54    <span class="identifier">polynomial</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">initializer_list</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="identifier">l</span><span class="special">);</span> <span class="comment">// C++11</span>
  55
  56    <span class="comment">// access:</span>
  57    <span class="identifier">size_type</span> <span class="identifier">size</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
  58    <span class="identifier">size_type</span> <span class="identifier">degree</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
  59    <span class="identifier">value_type</span><span class="special">&amp;</span> <span class="keyword">operator</span><span class="special">[](</span><span class="identifier">size_type</span> <span class="identifier">i</span><span class="special">);</span>
  60    <span class="keyword">const</span> <span class="identifier">value_type</span><span class="special">&amp;</span> <span class="keyword">operator</span><span class="special">[](</span><span class="identifier">size_type</span> <span class="identifier">i</span><span class="special">)</span><span class="keyword">const</span><span class="special">;</span>
  61    <span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="keyword">const</span><span class="special">&amp;</span> <span class="identifier">data</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span>
  62    <span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;&amp;</span> <span class="identifier">data</span><span class="special">();</span>
  63    <span class="keyword">explicit</span> <span class="keyword">operator</span> <span class="keyword">bool</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span>
  64
  65    <span class="comment">// modify:</span>
  66    <span class="keyword">void</span> <span class="identifier">set_zero</span><span class="special">();</span>
  67
  68    <span class="comment">// operators:</span>
  69    <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">U</span><span class="special">&gt;</span>
  70    <span class="identifier">polynomial</span><span class="special">&amp;</span> <span class="keyword">operator</span> <span class="special">+=(</span><span class="keyword">const</span> <span class="identifier">U</span><span class="special">&amp;</span> <span class="identifier">value</span><span class="special">);</span>
  71    <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">U</span><span class="special">&gt;</span>
  72    <span class="identifier">polynomial</span><span class="special">&amp;</span> <span class="keyword">operator</span> <span class="special">-=(</span><span class="keyword">const</span> <span class="identifier">U</span><span class="special">&amp;</span> <span class="identifier">value</span><span class="special">);</span>
  73    <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">U</span><span class="special">&gt;</span>
  74    <span class="identifier">polynomial</span><span class="special">&amp;</span> <span class="keyword">operator</span> <span class="special">*=(</span><span class="keyword">const</span> <span class="identifier">U</span><span class="special">&amp;</span> <span class="identifier">value</span><span class="special">);</span>
  75    <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">U</span><span class="special">&gt;</span>
  76    <span class="identifier">polynomial</span><span class="special">&amp;</span> <span class="keyword">operator</span> <span class="special">/=(</span><span class="keyword">const</span> <span class="identifier">U</span><span class="special">&amp;</span> <span class="identifier">value</span><span class="special">);</span>
  77    <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">U</span><span class="special">&gt;</span>
  78    <span class="identifier">polynomial</span><span class="special">&amp;</span> <span class="keyword">operator</span> <span class="special">%=(</span><span class="keyword">const</span> <span class="identifier">U</span><span class="special">&amp;</span> <span class="identifier">value</span><span class="special">);</span>
  79    <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">U</span><span class="special">&gt;</span>
  80    <span class="identifier">polynomial</span><span class="special">&amp;</span> <span class="keyword">operator</span> <span class="special">+=(</span><span class="keyword">const</span> <span class="identifier">polynomial</span><span class="special">&lt;</span><span class="identifier">U</span><span class="special">&gt;&amp;</span> <span class="identifier">value</span><span class="special">);</span>
  81    <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">U</span><span class="special">&gt;</span>
  82    <span class="identifier">polynomial</span><span class="special">&amp;</span> <span class="keyword">operator</span> <span class="special">-=(</span><span class="keyword">const</span> <span class="identifier">polynomial</span><span class="special">&lt;</span><span class="identifier">U</span><span class="special">&gt;&amp;</span> <span class="identifier">value</span><span class="special">);</span>
  83    <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">U</span><span class="special">&gt;</span>
  84    <span class="identifier">polynomial</span><span class="special">&amp;</span> <span class="keyword">operator</span> <span class="special">*=(</span><span class="keyword">const</span> <span class="identifier">polynomial</span><span class="special">&lt;</span><span class="identifier">U</span><span class="special">&gt;&amp;</span> <span class="identifier">value</span><span class="special">);</span>
  85    <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">U</span><span class="special">&gt;</span>
  86    <span class="identifier">polynomial</span><span class="special">&amp;</span> <span class="keyword">operator</span> <span class="special">/=(</span><span class="keyword">const</span> <span class="identifier">polynomial</span><span class="special">&lt;</span><span class="identifier">U</span><span class="special">&gt;&amp;</span> <span class="identifier">value</span><span class="special">);</span>
  87    <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">U</span><span class="special">&gt;</span>
  88    <span class="identifier">polynomial</span><span class="special">&amp;</span> <span class="keyword">operator</span> <span class="special">%=(</span><span class="keyword">const</span> <span class="identifier">polynomial</span><span class="special">&lt;</span><span class="identifier">U</span><span class="special">&gt;&amp;</span> <span class="identifier">value</span><span class="special">);</span>
  89 <span class="special">};</span>
  90
  91 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
  92 <span class="identifier">polynomial</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="keyword">operator</span> <span class="special">+</span> <span class="special">(</span><span class="keyword">const</span> <span class="identifier">polynomial</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;&amp;</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">polynomial</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;&amp;</span> <span class="identifier">b</span><span class="special">);</span>
  93 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
  94 <span class="identifier">polynomial</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="keyword">operator</span> <span class="special">-</span> <span class="special">(</span><span class="keyword">const</span> <span class="identifier">polynomial</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;&amp;</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">polynomial</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;&amp;</span> <span class="identifier">b</span><span class="special">);</span>
  95 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
  96 <span class="identifier">polynomial</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="keyword">operator</span> <span class="special">*</span> <span class="special">(</span><span class="keyword">const</span> <span class="identifier">polynomial</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;&amp;</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">polynomial</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;&amp;</span> <span class="identifier">b</span><span class="special">);</span>
  97 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
  98 <span class="identifier">polynomial</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="keyword">operator</span> <span class="special">/</span> <span class="special">(</span><span class="keyword">const</span> <span class="identifier">polynomial</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;&amp;</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">polynomial</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;&amp;</span> <span class="identifier">b</span><span class="special">);</span>
  99 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
 100 <span class="identifier">polynomial</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="keyword">operator</span> <span class="special">%</span> <span class="special">(</span><span class="keyword">const</span> <span class="identifier">polynomial</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;&amp;</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">polynomial</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;&amp;</span> <span class="identifier">b</span><span class="special">);</span>
 101
 102 <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>
 103 <span class="identifier">polynomial</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="keyword">operator</span> <span class="special">+</span> <span class="special">(</span><span class="keyword">const</span> <span class="identifier">polynomial</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;&amp;</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">U</span><span class="special">&amp;</span> <span class="identifier">b</span><span class="special">);</span>
 104 <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>
 105 <span class="identifier">polynomial</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="keyword">operator</span> <span class="special">-</span> <span class="special">(</span><span class="keyword">const</span> <span class="identifier">polynomial</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;&amp;</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">U</span><span class="special">&amp;</span> <span class="identifier">b</span><span class="special">);</span>
 106 <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>
 107 <span class="identifier">polynomial</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="keyword">operator</span> <span class="special">*</span> <span class="special">(</span><span class="keyword">const</span> <span class="identifier">polynomial</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;&amp;</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">U</span><span class="special">&amp;</span> <span class="identifier">b</span><span class="special">);</span>
 108 <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>
 109 <span class="identifier">polynomial</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="keyword">operator</span> <span class="special">/</span> <span class="special">(</span><span class="keyword">const</span> <span class="identifier">polynomial</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;&amp;</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">U</span><span class="special">&amp;</span> <span class="identifier">b</span><span class="special">);</span>
 110 <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>
 111 <span class="identifier">polynomial</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="keyword">operator</span> <span class="special">%</span> <span class="special">(</span><span class="keyword">const</span> <span class="identifier">polynomial</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;&amp;</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">U</span><span class="special">&amp;</span> <span class="identifier">b</span><span class="special">);</span>
 112
 113 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">U</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
 114 <span class="identifier">polynomial</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="keyword">operator</span> <span class="special">+</span> <span class="special">(</span><span class="keyword">const</span> <span class="identifier">U</span><span class="special">&amp;</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">polynomial</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;&amp;</span> <span class="identifier">b</span><span class="special">);</span>
 115 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">U</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
 116 <span class="identifier">polynomial</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="keyword">operator</span> <span class="special">-</span> <span class="special">(</span><span class="keyword">const</span> <span class="identifier">U</span><span class="special">&amp;</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">polynomial</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;&amp;</span> <span class="identifier">b</span><span class="special">);</span>
 117 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">U</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
 118 <span class="identifier">polynomial</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="keyword">operator</span> <span class="special">*</span> <span class="special">(</span><span class="keyword">const</span> <span class="identifier">U</span><span class="special">&amp;</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">polynomial</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;&amp;</span> <span class="identifier">b</span><span class="special">);</span>
 119
 120 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
 121 <span class="identifier">polynomial</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="keyword">operator</span> <span class="special">-</span> <span class="special">(</span><span class="keyword">const</span> <span class="identifier">polynomial</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;&amp;</span> <span class="identifier">a</span><span class="special">);</span>
 122
 123 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
 124 <span class="identifier">polynomial</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="keyword">operator</span> <span class="special">&gt;&gt;=</span> <span class="special">(</span><span class="keyword">const</span> <span class="identifier">U</span><span class="special">&amp;</span> <span class="identifier">a</span><span class="special">);</span>
 125 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
 126 <span class="identifier">polynomial</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="keyword">operator</span> <span class="special">&gt;&gt;</span> <span class="special">(</span><span class="identifier">polynomial</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="keyword">const</span> <span class="special">&amp;</span><span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">U</span><span class="special">&amp;</span> <span class="identifier">b</span><span class="special">);</span>
 127 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
 128 <span class="identifier">polynomial</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="keyword">operator</span> <span class="special">&lt;&lt;=</span> <span class="special">(</span><span class="keyword">const</span> <span class="identifier">U</span><span class="special">&amp;</span> <span class="identifier">a</span><span class="special">);</span>
 129 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
 130 <span class="identifier">polynomial</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="keyword">operator</span> <span class="special">&lt;&lt;</span> <span class="special">(</span><span class="identifier">polynomial</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="keyword">const</span> <span class="special">&amp;</span><span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">U</span><span class="special">&amp;</span> <span class="identifier">b</span><span class="special">);</span>
 131
 132 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
 133 <span class="keyword">bool</span> <span class="keyword">operator</span> <span class="special">==</span> <span class="special">(</span><span class="keyword">const</span> <span class="identifier">polynomial</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="special">&amp;</span><span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">polynomial</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="special">&amp;</span><span class="identifier">b</span><span class="special">);</span>
 134 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
 135 <span class="keyword">bool</span> <span class="keyword">operator</span> <span class="special">!=</span> <span class="special">(</span><span class="keyword">const</span> <span class="identifier">polynomial</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="special">&amp;</span><span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">polynomial</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="special">&amp;</span><span class="identifier">b</span><span class="special">);</span>
 136
 137 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
 138 <span class="identifier">polynomial</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="identifier">pow</span><span class="special">(</span><span class="identifier">polynomial</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="identifier">base</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">exp</span><span class="special">);</span>
 139
 140 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">charT</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">traits</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
 141 <span class="identifier">std</span><span class="special">::</span><span class="identifier">basic_ostream</span><span class="special">&lt;</span><span class="identifier">charT</span><span class="special">,</span> <span class="identifier">traits</span><span class="special">&gt;&amp;</span> <span class="keyword">operator</span> <span class="special">&lt;&lt;</span>
 142    <span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">basic_ostream</span><span class="special">&lt;</span><span class="identifier">charT</span><span class="special">,</span> <span class="identifier">traits</span><span class="special">&gt;&amp;</span> <span class="identifier">os</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">polynomial</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;&amp;</span> <span class="identifier">poly</span><span class="special">);</span>
 143
 144 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">typename</span> <span class="identifier">T</span><span class="special">&gt;</span>
 145 <span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special">&lt;</span> <span class="identifier">polynomial</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;,</span> <span class="identifier">polynomial</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="special">&gt;</span>
 146 <span class="identifier">quotient_remainder</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">polynomial</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;&amp;</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">polynomial</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;&amp;</span> <span class="identifier">b</span><span class="special">);</span>
 147
 148 <span class="special">}</span> <span class="comment">//    namespace tools</span>
 149 <span class="special">}}</span> <span class="comment">//    namespace boost { namespace math</span>
 150 </pre>
 151 <h5>
 152 <a name="math_toolkit.roots.polynomials.h1"></a>
 153         <span class="phrase"><a name="math_toolkit.roots.polynomials.description"></a></span><a class="link" href="polynomials.html#math_toolkit.roots.polynomials.description">Description</a>
 154       </h5>
 155 <p>
 156         This is a somewhat trivial class for polynomial manipulation.
 157       </p>
 158 <p>
 159         See
 160       </p>
 161 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
 162 <li class="listitem">
 163             <a href="https://en.wikipedia.org/wiki/Polynomial" target="_top">Polynomial</a>
 164             and
 165           </li>
 166 <li class="listitem">
 167             Donald E. Knuth, The Art of Computer Programming: Volume 2, Third edition,
 168             (1998) Chapter 4.6.1, Algorithm D: Division of polynomials over a field.
 169           </li>
 170 </ul></div>
 171 <p>
 172         Implementation is currently of the "naive" variety, with &#119926;(N<sup>2</sup>) multiplication,
 173         for example. This class should not be used in high-performance computing
 174         environments: it is intended for the simple manipulation of small polynomials,
 175         typically generated for special function approximation.
 176       </p>
 177 <p>
 178         It does has division for polynomials over a <a href="https://en.wikipedia.org/wiki/Field_%28mathematics%29" target="_top">field</a>
 179         (here floating point, complex, etc) and over a unique factorization domain
 180         (integers). Division of polynomials over a field is compatible with <a href="https://en.wikipedia.org/wiki/Euclidean_algorithm" target="_top">Euclidean GCD</a>.
 181       </p>
 182 <p>
 183         Advanced manipulations: the FFT, factorisation etc are not currently provided.
 184         Submissions for these are of course welcome :-)
 185       </p>
 186 <h5>
 187 <a name="math_toolkit.roots.polynomials.h2"></a>
 188         <span class="phrase"><a name="math_toolkit.roots.polynomials.polynomial_examples"></a></span><a class="link" href="polynomials.html#math_toolkit.roots.polynomials.polynomial_examples">Polynomial
 189         Arithmetic Examples</a>
 190       </h5>
 191 <p>
 192         First include the essential polynomial header (and others) to make the example:
 193       </p>
 194 <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">polynomial</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
 195 </pre>
 196 <p>
 197         and some using statements are convenient:
 198       </p>
 199 <pre class="programlisting"><span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">string</span><span class="special">;</span>
 200 <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">exception</span><span class="special">;</span>
 201 <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span><span class="special">;</span>
 202 <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">abs</span><span class="special">;</span>
 203 <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special">;</span>
 204
 205 <span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">;</span>
 206 <span class="keyword">using</span> <span class="keyword">namespace</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="comment">// for polynomial</span>
 207 <span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">lexical_cast</span><span class="special">;</span>
 208 </pre>
 209 <p>
 210         Store the coefficients in a convenient way to access them, then create some
 211         polynomials using construction from an iterator range, and finally output
 212         in a 'pretty' formula format.
 213       </p>
 214 <div class="tip"><table border="0" summary="Tip">
 215 <tr>
 216 <td rowspan="2" align="center" valign="top" width="25"><img alt="[Tip]" src="../../../../../../doc/src/images/tip.png"></td>
 217 <th align="left">Tip</th>
 218 </tr>
 219 <tr><td align="left" valign="top"><p>
 220           Although we might conventionally write a polynomial from left to right
 221           in descending order of degree, Boost.Math stores in <span class="bold"><strong>ascending
 222           order of degree</strong></span>.
 223         </p></td></tr>
 224 </table></div>
 225 <pre class="programlisting"><span class="identifier">Read</span><span class="special">/</span><span class="identifier">write</span> <span class="keyword">for</span> <span class="identifier">humans</span><span class="special">:</span>    <span class="number">3</span><span class="identifier">x</span><span class="special">^</span><span class="number">3</span> <span class="special">-</span> <span class="number">4</span><span class="identifier">x</span><span class="special">^</span><span class="number">2</span> <span class="special">-</span> <span class="number">6</span><span class="identifier">x</span> <span class="special">+</span> <span class="number">10</span>
 226 <span class="identifier">Boost</span> <span class="identifier">polynomial</span> <span class="identifier">storage</span><span class="special">:</span> <span class="special">[</span> <span class="number">10</span><span class="special">,</span> <span class="special">-</span><span class="number">6</span><span class="special">,</span> <span class="special">-</span><span class="number">4</span><span class="special">,</span> <span class="number">3</span> <span class="special">]</span>
 227 </pre>
 228 <pre class="programlisting"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">array</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">,</span> <span class="number">4</span><span class="special">&gt;</span> <span class="keyword">const</span> <span class="identifier">d3a</span> <span class="special">=</span> <span class="special">{{</span><span class="number">10</span><span class="special">,</span> <span class="special">-</span><span class="number">6</span><span class="special">,</span> <span class="special">-</span><span class="number">4</span><span class="special">,</span> <span class="number">3</span><span class="special">}};</span>
 229 <span class="identifier">polynomial</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span> <span class="keyword">const</span> <span class="identifier">a</span><span class="special">(</span><span class="identifier">d3a</span><span class="special">.</span><span class="identifier">begin</span><span class="special">(),</span> <span class="identifier">d3a</span><span class="special">.</span><span class="identifier">end</span><span class="special">());</span>
 230
 231 <span class="comment">// With C++11 and later, you can also use initializer_list construction.</span>
 232 <span class="identifier">polynomial</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span> <span class="keyword">const</span> <span class="identifier">b</span><span class="special">{{-</span><span class="number">2.0</span><span class="special">,</span> <span class="number">1.0</span><span class="special">}};</span>
 233
 234 <span class="comment">// formula_format() converts from Boost storage to human notation.</span>
 235 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"a = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">formula_format</span><span class="special">(</span><span class="identifier">a</span><span class="special">)</span>
 236 <span class="special">&lt;&lt;</span> <span class="string">"\nb = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">formula_format</span><span class="special">(</span><span class="identifier">b</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="string">"\n\n"</span><span class="special">;</span>
 237 </pre>
 238 <p>
 239         for output:
 240       </p>
 241 <pre class="programlisting"><span class="identifier">a</span> <span class="special">=</span> <span class="number">3</span><span class="identifier">x</span><span class="special">^</span><span class="number">3</span> <span class="special">-</span> <span class="number">4</span><span class="identifier">x</span><span class="special">^</span><span class="number">2</span> <span class="special">-</span> <span class="number">6</span><span class="identifier">x</span> <span class="special">+</span> <span class="number">10</span>
 242 <span class="identifier">b</span> <span class="special">=</span> <span class="identifier">x</span> <span class="special">-</span> <span class="number">2</span>
 243 </pre>
 244 <pre class="programlisting"><span class="comment">// Now we can do arithmetic with the usual infix operators: + - * / and %.</span>
 245 <span class="identifier">polynomial</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span> <span class="identifier">s</span> <span class="special">=</span> <span class="identifier">a</span> <span class="special">+</span> <span class="identifier">b</span><span class="special">;</span>
 246 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"a + b = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">formula_format</span><span class="special">(</span><span class="identifier">s</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="string">"\n"</span><span class="special">;</span>
 247 <span class="identifier">polynomial</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span> <span class="identifier">d</span> <span class="special">=</span> <span class="identifier">a</span> <span class="special">-</span> <span class="identifier">b</span><span class="special">;</span>
 248 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"a - b = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">formula_format</span><span class="special">(</span><span class="identifier">d</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="string">"\n"</span><span class="special">;</span>
 249 <span class="identifier">polynomial</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span> <span class="identifier">p</span> <span class="special">=</span> <span class="identifier">a</span> <span class="special">*</span> <span class="identifier">b</span><span class="special">;</span>
 250 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"a * b = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">formula_format</span><span class="special">(</span><span class="identifier">p</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="string">"\n"</span><span class="special">;</span>
 251 <span class="identifier">polynomial</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span> <span class="identifier">q</span> <span class="special">=</span> <span class="identifier">a</span> <span class="special">/</span> <span class="identifier">b</span><span class="special">;</span>
 252 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"a / b = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">formula_format</span><span class="special">(</span><span class="identifier">q</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="string">"\n"</span><span class="special">;</span>
 253 <span class="identifier">polynomial</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span> <span class="identifier">r</span> <span class="special">=</span> <span class="identifier">a</span> <span class="special">%</span> <span class="identifier">b</span><span class="special">;</span>
 254 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"a % b = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">formula_format</span><span class="special">(</span><span class="identifier">r</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="string">"\n"</span><span class="special">;</span>
 255 </pre>
 256 <p>
 257         for output
 258       </p>
 259 <pre class="programlisting"><span class="identifier">a</span> <span class="special">+</span> <span class="identifier">b</span> <span class="special">=</span> <span class="number">3</span><span class="identifier">x</span><span class="special">^</span><span class="number">3</span> <span class="special">-</span> <span class="number">4</span><span class="identifier">x</span><span class="special">^</span><span class="number">2</span> <span class="special">-</span> <span class="number">5</span><span class="identifier">x</span> <span class="special">+</span> <span class="number">8</span>
 260 <span class="identifier">a</span> <span class="special">-</span> <span class="identifier">b</span> <span class="special">=</span> <span class="number">3</span><span class="identifier">x</span><span class="special">^</span><span class="number">3</span> <span class="special">-</span> <span class="number">4</span><span class="identifier">x</span><span class="special">^</span><span class="number">2</span> <span class="special">-</span> <span class="number">7</span><span class="identifier">x</span> <span class="special">+</span> <span class="number">12</span>
 261 <span class="identifier">a</span> <span class="special">*</span> <span class="identifier">b</span> <span class="special">=</span> <span class="number">3</span><span class="identifier">x</span><span class="special">^</span><span class="number">4</span> <span class="special">-</span> <span class="number">10</span><span class="identifier">x</span><span class="special">^</span><span class="number">3</span> <span class="special">+</span> <span class="number">2</span><span class="identifier">x</span><span class="special">^</span><span class="number">2</span> <span class="special">+</span> <span class="number">22</span><span class="identifier">x</span> <span class="special">-</span> <span class="number">20</span>
 262 <span class="identifier">a</span> <span class="special">/</span> <span class="identifier">b</span> <span class="special">=</span> <span class="number">3</span><span class="identifier">x</span><span class="special">^</span><span class="number">2</span> <span class="special">+</span> <span class="number">2</span><span class="identifier">x</span> <span class="special">-</span> <span class="number">2</span>
 263 <span class="identifier">a</span> <span class="special">%</span> <span class="identifier">b</span> <span class="special">=</span> <span class="number">6</span>
 264 </pre>
 265 <h6>
 266 <a name="math_toolkit.roots.polynomials.h3"></a>
 267         <span class="phrase"><a name="math_toolkit.roots.polynomials.division_quotient_and_remainder"></a></span><a class="link" href="polynomials.html#math_toolkit.roots.polynomials.division_quotient_and_remainder">Division,
 268         Quotient and Remainder</a>
 269       </h6>
 270 <p>
 271         Division is a special case where you can calculate two for the price of one.
 272       </p>
 273 <p>
 274         Actually, quotient and remainder are always calculated together due to the
 275         nature of the algorithm: the infix operators return one result and throw
 276         the other away.
 277       </p>
 278 <p>
 279         If you are doing a lot of division and want both the quotient and remainder,
 280         then you don't want to do twice the work necessary.
 281       </p>
 282 <p>
 283         In that case you can call the underlying function, <code class="literal">quotient_remainder</code>,
 284         to get both results together as a pair.
 285       </p>
 286 <pre class="programlisting">  <span class="identifier">pair</span><span class="special">&lt;</span> <span class="identifier">polynomial</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;,</span> <span class="identifier">polynomial</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span> <span class="special">&gt;</span> <span class="identifier">result</span><span class="special">;</span>
 287   <span class="identifier">result</span> <span class="special">=</span> <span class="identifier">quotient_remainder</span><span class="special">(</span><span class="identifier">a</span><span class="special">,</span> <span class="identifier">b</span><span class="special">);</span>
 288 <span class="comment">// Reassure ourselves that the result is the same.</span>
 289   <span class="identifier">BOOST_ASSERT</span><span class="special">(</span><span class="identifier">result</span><span class="special">.</span><span class="identifier">first</span> <span class="special">==</span> <span class="identifier">q</span><span class="special">);</span>
 290   <span class="identifier">BOOST_ASSERT</span><span class="special">(</span><span class="identifier">result</span><span class="special">.</span><span class="identifier">second</span> <span class="special">==</span> <span class="identifier">r</span><span class="special">);</span>
 291 </pre>
 292 <p>
 293         The source code is at <a href="../../../../example/polynomial_arithmetic.cpp" target="_top">polynomial_arithmetic.cpp</a>
 294       </p>
 295 </div>
 296 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 297 <td align="left"></td>
 298 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
 299       Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
 300       Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam Sewani,
 301       Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
 302         Distributed under the Boost Software License, Version 1.0. (See accompanying
 303         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>)
 304       </p>
 305 </div></td>
 306 </tr></table>
 307 <hr>
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