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  27 <a name="math_toolkit.polynomials"></a><a class="link" href="polynomials.html" title="Polynomials">Polynomials</a>
  28 </h2></div></div></div>
  29 <h5>
  30 <a name="math_toolkit.polynomials.h0"></a>
  31       <span class="phrase"><a name="math_toolkit.polynomials.synopsis"></a></span><a class="link" href="polynomials.html#math_toolkit.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="keyword">typename</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">enable_if</span><span class="special">&lt;</span><span class="identifier">boost</span><span class="special">::</span><span class="identifier">is_convertible</span><span class="special">&lt;</span><span class="identifier">U</span><span class="special">,</span> <span class="identifier">T</span><span class="special">&gt;</span> <span class="special">&gt;::</span><span class="identifier">type</span><span class="special">*</span> <span class="special">=</span> <span class="number">0</span><span class="special">);</span>
  55    <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Range</span><span class="special">&gt;</span>
  56    <span class="keyword">explicit</span> <span class="identifier">polynomial</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">Range</span><span class="special">&amp;</span> <span class="identifier">r</span><span class="special">,</span>
  57          <span class="keyword">typename</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">enable_if</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">detail</span><span class="special">::</span><span class="identifier">is_const_iterable</span><span class="special">&lt;</span><span class="identifier">Range</span><span class="special">&gt;</span> <span class="special">&gt;::</span><span class="identifier">type</span><span class="special">*</span> <span class="special">=</span> <span class="number">0</span><span class="special">);</span> <span class="comment">// C++14</span>
  58    <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>
  59
  60    <span class="identifier">polynomial</span><span class="special">(</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;&amp;&amp;</span> <span class="identifier">p</span><span class="special">);</span>
  61
  62    <span class="comment">// access:</span>
  63    <span class="identifier">size_type</span> <span class="identifier">size</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
  64    <span class="identifier">size_type</span> <span class="identifier">degree</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
  65    <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>
  66    <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>
  67    <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>
  68    <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>
  69    <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>
  70
  71    <span class="identifier">polynomial</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="identifier">prime</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span>
  72
  73    <span class="identifier">polynomial</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="identifier">integrate</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span>
  74
  75    <span class="identifier">T</span> <span class="keyword">operator</span><span class="special">()(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">)</span> <span class="keyword">const</span><span class="special">;</span>
  76
  77
  78
  79    <span class="comment">// modify:</span>
  80    <span class="keyword">void</span> <span class="identifier">set_zero</span><span class="special">();</span>
  81
  82    <span class="comment">// operators:</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">U</span><span class="special">&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">U</span><span class="special">&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">U</span><span class="special">&amp;</span> <span class="identifier">value</span><span class="special">);</span>
  89    <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>
  90    <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>
  91    <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>
  92    <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>
  93    <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>
  94    <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>
  95    <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>
  96    <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>
  97    <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>
  98    <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>
  99    <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>
 100    <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>
 101    <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>
 102    <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>
 103 <span class="special">};</span>
 104
 105 <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>
 106 <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>
 107 <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>
 108 <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>
 109 <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>
 110 <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>
 111 <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>
 112 <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>
 113 <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>
 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">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>
 115
 116 <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>
 117 <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>
 118 <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>
 119 <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>
 120 <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>
 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> <span class="keyword">const</span> <span class="identifier">U</span><span class="special">&amp;</span> <span class="identifier">b</span><span class="special">);</span>
 122 <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>
 123 <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>
 124 <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>
 125 <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>
 126
 127 <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>
 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">+</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>
 129 <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>
 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">-</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>
 131 <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>
 132 <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>
 133
 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="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>
 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="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>
 139 <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>
 140 <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>
 141 <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>
 142 <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>
 143 <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>
 144 <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>
 145
 146 <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>
 147 <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>
 148 <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>
 149 <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>
 150
 151 <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>
 152 <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>
 153
 154 <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>
 155 <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>
 156    <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>
 157
 158 <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>
 159 <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>
 160 <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>
 161
 162 <span class="special">}</span> <span class="comment">//    namespace tools</span>
 163 <span class="special">}}</span> <span class="comment">//    namespace boost { namespace math</span>
 164 </pre>
 165 <h5>
 166 <a name="math_toolkit.polynomials.h1"></a>
 167       <span class="phrase"><a name="math_toolkit.polynomials.description"></a></span><a class="link" href="polynomials.html#math_toolkit.polynomials.description">Description</a>
 168     </h5>
 169 <p>
 170       This is a somewhat trivial class for polynomial manipulation.
 171     </p>
 172 <p>
 173       See
 174     </p>
 175 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
 176 <li class="listitem">
 177           <a href="https://en.wikipedia.org/wiki/Polynomial" target="_top">Polynomial</a>
 178           and
 179         </li>
 180 <li class="listitem">
 181           Donald E. Knuth, The Art of Computer Programming: Volume 2, Third edition,
 182           (1998) Chapter 4.6.1, Algorithm D: Division of polynomials over a field.
 183         </li>
 184 </ul></div>
 185 <p>
 186       Implementation is currently of the "naive" variety, with &#119926;(N<sup>2</sup>) multiplication,
 187       for example. This class should not be used in high-performance computing environments:
 188       it is intended for the simple manipulation of small polynomials, typically
 189       generated for special function approximation.
 190     </p>
 191 <p>
 192       It does has division for polynomials over a <a href="https://en.wikipedia.org/wiki/Field_%28mathematics%29" target="_top">field</a>
 193       (here floating point, complex, etc) and over a unique factorization domain
 194       (integers). Division of polynomials over a field is compatible with <a href="https://en.wikipedia.org/wiki/Euclidean_algorithm" target="_top">Euclidean GCD</a>.
 195     </p>
 196 <p>
 197       Division of polynomials over a UFD is compatible with the subresultant algorithm
 198       for GCD (implemented as subresultant_gcd), but a serious word of warning is
 199       required: the intermediate value swell of that algorithm will cause single-precision
 200       integral types to overflow very easily. So although the algorithm will work
 201       on single-precision integral types, an overload of the gcd function is only
 202       provided for polynomials with multi-precision integral types, to prevent nasty
 203       surprises. This is done somewhat crudely by disabling the overload for non-POD
 204       integral types.
 205     </p>
 206 <p>
 207       Advanced manipulations: the FFT, factorisation etc are not currently provided.
 208       Submissions for these are of course welcome :-)
 209     </p>
 210 <h5>
 211 <a name="math_toolkit.polynomials.h2"></a>
 212       <span class="phrase"><a name="math_toolkit.polynomials.polynomial_examples"></a></span><a class="link" href="polynomials.html#math_toolkit.polynomials.polynomial_examples">Polynomial
 213       Arithmetic Examples</a>
 214     </h5>
 215 <p>
 216       First include the essential polynomial header (and others) to make the example:
 217     </p>
 218 <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>
 219 </pre>
 220 <p>
 221       and some using statements are convenient:
 222     </p>
 223 <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>
 224 <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">exception</span><span class="special">;</span>
 225 <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span><span class="special">;</span>
 226 <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">abs</span><span class="special">;</span>
 227 <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special">;</span>
 228
 229 <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>
 230 <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>
 231 <span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">lexical_cast</span><span class="special">;</span>
 232 </pre>
 233 <p>
 234       Store the coefficients in a convenient way to access them, then create some
 235       polynomials using construction from an iterator range, and finally output in
 236       a 'pretty' formula format.
 237     </p>
 238 <div class="tip"><table border="0" summary="Tip">
 239 <tr>
 240 <td rowspan="2" align="center" valign="top" width="25"><img alt="[Tip]" src="../../../../../doc/src/images/tip.png"></td>
 241 <th align="left">Tip</th>
 242 </tr>
 243 <tr><td align="left" valign="top"><p>
 244         Although we might conventionally write a polynomial from left to right in
 245         descending order of degree, Boost.Math stores in <span class="bold"><strong>ascending
 246         order of degree</strong></span>.
 247       </p></td></tr>
 248 </table></div>
 249 <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>
 250 <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>
 251 </pre>
 252 <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>
 253 <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>
 254
 255 <span class="comment">// With C++11 and later, you can also use initializer_list construction.</span>
 256 <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>
 257
 258 <span class="comment">// formula_format() converts from Boost storage to human notation.</span>
 259 <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>
 260 <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>
 261 </pre>
 262 <p>
 263       for output:
 264     </p>
 265 <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>
 266 <span class="identifier">b</span> <span class="special">=</span> <span class="identifier">x</span> <span class="special">-</span> <span class="number">2</span>
 267 </pre>
 268 <pre class="programlisting"><span class="comment">// Now we can do arithmetic with the usual infix operators: + - * / and %.</span>
 269 <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>
 270 <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>
 271 <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>
 272 <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>
 273 <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>
 274 <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>
 275 <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>
 276 <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>
 277 <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>
 278 <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>
 279 </pre>
 280 <p>
 281       for output
 282     </p>
 283 <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>
 284 <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>
 285 <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>
 286 <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>
 287 <span class="identifier">a</span> <span class="special">%</span> <span class="identifier">b</span> <span class="special">=</span> <span class="number">6</span>
 288 </pre>
 289 <h6>
 290 <a name="math_toolkit.polynomials.h3"></a>
 291       <span class="phrase"><a name="math_toolkit.polynomials.division_quotient_and_remainder"></a></span><a class="link" href="polynomials.html#math_toolkit.polynomials.division_quotient_and_remainder">Division,
 292       Quotient and Remainder</a>
 293     </h6>
 294 <p>
 295       Division is a special case where you can calculate two for the price of one.
 296     </p>
 297 <p>
 298       Actually, quotient and remainder are always calculated together due to the
 299       nature of the algorithm: the infix operators return one result and throw the
 300       other away.
 301     </p>
 302 <p>
 303       If you are doing a lot of division and want both the quotient and remainder,
 304       then you don't want to do twice the work necessary.
 305     </p>
 306 <p>
 307       In that case you can call the underlying function, <code class="literal">quotient_remainder</code>,
 308       to get both results together as a pair.
 309     </p>
 310 <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>
 311   <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>
 312 <span class="comment">// Reassure ourselves that the result is the same.</span>
 313   <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>
 314   <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>
 315 </pre>
 316 <p>
 317       The source code is at <a href="../../../example/polynomial_arithmetic.cpp" target="_top">polynomial_arithmetic.cpp</a>
 318     </p>
 319 </div>
 320 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 321 <td align="left"></td>
 322 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
 323       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
 324       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
 325       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
 326       Daryle Walker and Xiaogang Zhang<p>
 327         Distributed under the Boost Software License, Version 1.0. (See accompanying
 328         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>)
 329       </p>
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