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<a name="math_toolkit.polynomials"></a><a class="link" href="polynomials.html" title="Polynomials">Polynomials</a>
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<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">tools</span><span class="special">/</span><span class="identifier">polynomial</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
</pre>
<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>
<span class="keyword">namespace</span> <span class="identifier">tools</span> <span class="special">{</span>
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
<span class="keyword">class</span> <span class="identifier">polynomial</span>
<span class="special">{</span>
<span class="keyword">public</span><span class="special">:</span>
<span class="comment">// typedefs:</span>
<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"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">value_type</span> <span class="identifier">value_type</span><span class="special">;</span>
<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"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">size_type</span> <span class="identifier">size_type</span><span class="special">;</span>
<span class="comment">// construct:</span>
<span class="identifier">polynomial</span><span class="special">(){}</span>
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">U</span><span class="special">></span>
<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>
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">Iterator</span><span class="special">></span>
<span class="identifier">polynomial</span><span class="special">(</span><span class="identifier">Iterator</span> <span class="identifier">first</span><span class="special">,</span> <span class="identifier">Iterator</span> <span class="identifier">last</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">U</span><span class="special">></span>
<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">&</span> <span class="identifier">point</span><span class="special">,</span>
<span class="keyword">typename</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">enable_if</span><span class="special"><</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">is_convertible</span><span class="special"><</span><span class="identifier">U</span><span class="special">,</span> <span class="identifier">T</span><span class="special">></span> <span class="special">>::</span><span class="identifier">type</span><span class="special">*</span> <span class="special">=</span> <span class="keyword">nullptr</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">Range</span><span class="special">></span>
<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">&</span> <span class="identifier">r</span><span class="special">,</span>
<span class="keyword">typename</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">enable_if</span><span class="special"><</span><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">tools</span><span class="special">::</span><span class="identifier">detail</span><span class="special">::</span><span class="identifier">is_const_iterable</span><span class="special"><</span><span class="identifier">Range</span><span class="special">></span> <span class="special">>::</span><span class="identifier">type</span><span class="special">*</span> <span class="special">=</span> <span class="keyword">nullptr</span><span class="special">);</span> <span class="comment">// C++14</span>
<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"><</span><span class="identifier">T</span><span class="special">></span> <span class="identifier">l</span><span class="special">);</span> <span class="comment">// C++11</span>
<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"><</span><span class="identifier">T</span><span class="special">>&&</span> <span class="identifier">p</span><span class="special">);</span>
<span class="comment">// access:</span>
<span class="identifier">size_type</span> <span class="identifier">size</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
<span class="identifier">size_type</span> <span class="identifier">degree</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
<span class="identifier">value_type</span><span class="special">&</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="identifier">value_type</span><span class="special">&</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>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="keyword">const</span><span class="special">&</span> <span class="identifier">data</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span><span class="identifier">T</span><span class="special">>&</span> <span class="identifier">data</span><span class="special">();</span>
<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>
<span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="identifier">prime</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span>
<span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="identifier">integrate</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span>
<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>
<span class="comment">// modify:</span>
<span class="keyword">void</span> <span class="identifier">set_zero</span><span class="special">();</span>
<span class="comment">// operators:</span>
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">U</span><span class="special">></span>
<span class="identifier">polynomial</span><span class="special">&</span> <span class="keyword">operator</span> <span class="special">+=(</span><span class="keyword">const</span> <span class="identifier">U</span><span class="special">&</span> <span class="identifier">value</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">U</span><span class="special">></span>
<span class="identifier">polynomial</span><span class="special">&</span> <span class="keyword">operator</span> <span class="special">-=(</span><span class="keyword">const</span> <span class="identifier">U</span><span class="special">&</span> <span class="identifier">value</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">U</span><span class="special">></span>
<span class="identifier">polynomial</span><span class="special">&</span> <span class="keyword">operator</span> <span class="special">*=(</span><span class="keyword">const</span> <span class="identifier">U</span><span class="special">&</span> <span class="identifier">value</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">U</span><span class="special">></span>
<span class="identifier">polynomial</span><span class="special">&</span> <span class="keyword">operator</span> <span class="special">/=(</span><span class="keyword">const</span> <span class="identifier">U</span><span class="special">&</span> <span class="identifier">value</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">U</span><span class="special">></span>
<span class="identifier">polynomial</span><span class="special">&</span> <span class="keyword">operator</span> <span class="special">%=(</span><span class="keyword">const</span> <span class="identifier">U</span><span class="special">&</span> <span class="identifier">value</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">U</span><span class="special">></span>
<span class="identifier">polynomial</span><span class="special">&</span> <span class="keyword">operator</span> <span class="special">+=(</span><span class="keyword">const</span> <span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">U</span><span class="special">>&</span> <span class="identifier">value</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">U</span><span class="special">></span>
<span class="identifier">polynomial</span><span class="special">&</span> <span class="keyword">operator</span> <span class="special">-=(</span><span class="keyword">const</span> <span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">U</span><span class="special">>&</span> <span class="identifier">value</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">U</span><span class="special">></span>
<span class="identifier">polynomial</span><span class="special">&</span> <span class="keyword">operator</span> <span class="special">*=(</span><span class="keyword">const</span> <span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">U</span><span class="special">>&</span> <span class="identifier">value</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">U</span><span class="special">></span>
<span class="identifier">polynomial</span><span class="special">&</span> <span class="keyword">operator</span> <span class="special">/=(</span><span class="keyword">const</span> <span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">U</span><span class="special">>&</span> <span class="identifier">value</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">U</span><span class="special">></span>
<span class="identifier">polynomial</span><span class="special">&</span> <span class="keyword">operator</span> <span class="special">%=(</span><span class="keyword">const</span> <span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">U</span><span class="special">>&</span> <span class="identifier">value</span><span class="special">);</span>
<span class="special">};</span>
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
<span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">></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"><</span><span class="identifier">T</span><span class="special">>&</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">>&</span> <span class="identifier">b</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
<span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">></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"><</span><span class="identifier">T</span><span class="special">>&</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">>&</span> <span class="identifier">b</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
<span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">></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"><</span><span class="identifier">T</span><span class="special">>&</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">>&</span> <span class="identifier">b</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
<span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">></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"><</span><span class="identifier">T</span><span class="special">>&</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">>&</span> <span class="identifier">b</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
<span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">></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"><</span><span class="identifier">T</span><span class="special">>&</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">>&</span> <span class="identifier">b</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">U</span><span class="special">></span>
<span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">></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"><</span><span class="identifier">T</span><span class="special">>&</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">U</span><span class="special">&</span> <span class="identifier">b</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">U</span><span class="special">></span>
<span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">></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"><</span><span class="identifier">T</span><span class="special">>&</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">U</span><span class="special">&</span> <span class="identifier">b</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">U</span><span class="special">></span>
<span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">></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"><</span><span class="identifier">T</span><span class="special">>&</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">U</span><span class="special">&</span> <span class="identifier">b</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">U</span><span class="special">></span>
<span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">></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"><</span><span class="identifier">T</span><span class="special">>&</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">U</span><span class="special">&</span> <span class="identifier">b</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">U</span><span class="special">></span>
<span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">></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"><</span><span class="identifier">T</span><span class="special">>&</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">U</span><span class="special">&</span> <span class="identifier">b</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">U</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
<span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">></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">&</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">>&</span> <span class="identifier">b</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">U</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
<span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">></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">&</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">>&</span> <span class="identifier">b</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">U</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
<span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">></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">&</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">>&</span> <span class="identifier">b</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
<span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">></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"><</span><span class="identifier">T</span><span class="special">>&</span> <span class="identifier">a</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
<span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">></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">&</span> <span class="identifier">a</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
<span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="keyword">operator</span> <span class="special">>></span> <span class="special">(</span><span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="keyword">const</span> <span class="special">&</span><span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">U</span><span class="special">&</span> <span class="identifier">b</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
<span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">></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">&</span> <span class="identifier">a</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
<span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="keyword">operator</span> <span class="special"><<</span> <span class="special">(</span><span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="keyword">const</span> <span class="special">&</span><span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">U</span><span class="special">&</span> <span class="identifier">b</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
<span class="keyword">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"><</span><span class="identifier">T</span><span class="special">></span> <span class="special">&</span><span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="special">&</span><span class="identifier">b</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
<span class="keyword">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"><</span><span class="identifier">T</span><span class="special">></span> <span class="special">&</span><span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="special">&</span><span class="identifier">b</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
<span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="identifier">pow</span><span class="special">(</span><span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="identifier">base</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">exp</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special"><</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">></span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">basic_ostream</span><span class="special"><</span><span class="identifier">charT</span><span class="special">,</span> <span class="identifier">traits</span><span class="special">>&</span> <span class="keyword">operator</span> <span class="special"><<</span>
<span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">basic_ostream</span><span class="special"><</span><span class="identifier">charT</span><span class="special">,</span> <span class="identifier">traits</span><span class="special">>&</span> <span class="identifier">os</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">>&</span> <span class="identifier">poly</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">typename</span> <span class="identifier">T</span><span class="special">></span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span> <span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">>,</span> <span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="special">></span>
<span class="identifier">quotient_remainder</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">>&</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">>&</span> <span class="identifier">b</span><span class="special">);</span>
<span class="special">}</span> <span class="comment">// namespace tools</span>
<span class="special">}}</span> <span class="comment">// namespace boost { namespace math</span>
</pre>
<h5>
<a name="math_toolkit.polynomials.h1"></a>
<span class="phrase"><a name="math_toolkit.polynomials.description"></a></span><a class="link" href="polynomials.html#math_toolkit.polynomials.description">Description</a>
</h5>
<p>
This is a somewhat trivial class for polynomial manipulation.
</p>
<p>
See
</p>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
<li class="listitem">
<a href="https://en.wikipedia.org/wiki/Polynomial" target="_top">Polynomial</a>
and
</li>
<li class="listitem">
Donald E. Knuth, The Art of Computer Programming: Volume 2, Third edition,
(1998) Chapter 4.6.1, Algorithm D: Division of polynomials over a field.
</li>
</ul></div>
<p>
Implementation is currently of the "naive" variety, with 𝑶(N<sup>2</sup>) multiplication,
for example. This class should not be used in high-performance computing environments:
it is intended for the simple manipulation of small polynomials, typically
generated for special function approximation.
</p>
<p>
It does has division for polynomials over a <a href="https://en.wikipedia.org/wiki/Field_%28mathematics%29" target="_top">field</a>
(here floating point, complex, etc) and over a unique factorization domain
(integers). Division of polynomials over a field is compatible with <a href="https://en.wikipedia.org/wiki/Euclidean_algorithm" target="_top">Euclidean GCD</a>.
</p>
<p>
Division of polynomials over a UFD is compatible with the subresultant algorithm
for GCD (implemented as subresultant_gcd), but a serious word of warning is
required: the intermediate value swell of that algorithm will cause single-precision
integral types to overflow very easily. So although the algorithm will work
on single-precision integral types, an overload of the gcd function is only
provided for polynomials with multi-precision integral types, to prevent nasty
surprises. This is done somewhat crudely by disabling the overload for non-POD
integral types.
</p>
<p>
Advanced manipulations: the FFT, factorisation etc are not currently provided.
Submissions for these are of course welcome :-)
</p>
<h5>
<a name="math_toolkit.polynomials.h2"></a>
<span class="phrase"><a name="math_toolkit.polynomials.polynomial_examples"></a></span><a class="link" href="polynomials.html#math_toolkit.polynomials.polynomial_examples">Polynomial
Arithmetic Examples</a>
</h5>
<p>
First include the essential polynomial header (and others) to make the example:
</p>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">tools</span><span class="special">/</span><span class="identifier">polynomial</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
</pre>
<p>
and some using statements are convenient:
</p>
<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>
<span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">exception</span><span class="special">;</span>
<span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span><span class="special">;</span>
<span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">abs</span><span class="special">;</span>
<span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special">;</span>
<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="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>
<span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">lexical_cast</span><span class="special">;</span>
</pre>
<p>
Store the coefficients in a convenient way to access them, then create some
polynomials using construction from an iterator range, and finally output in
a 'pretty' formula format.
</p>
<div class="tip"><table border="0" summary="Tip">
<tr>
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Tip]" src="../../../../../doc/src/images/tip.png"></td>
<th align="left">Tip</th>
</tr>
<tr><td align="left" valign="top"><p>
Although we might conventionally write a polynomial from left to right in
descending order of degree, Boost.Math stores in <span class="bold"><strong>ascending
order of degree</strong></span>.
</p></td></tr>
</table></div>
<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>
<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>
</pre>
<pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">array</span><span class="special"><</span><span class="keyword">double</span><span class="special">,</span> <span class="number">4</span><span class="special">></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>
<span class="identifier">polynomial</span><span class="special"><</span><span class="keyword">double</span><span class="special">></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>
<span class="comment">// With C++11 and later, you can also use initializer_list construction.</span>
<span class="identifier">polynomial</span><span class="special"><</span><span class="keyword">double</span><span class="special">></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>
<span class="comment">// formula_format() converts from Boost storage to human notation.</span>
<span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"a = "</span> <span class="special"><<</span> <span class="identifier">formula_format</span><span class="special">(</span><span class="identifier">a</span><span class="special">)</span>
<span class="special"><<</span> <span class="string">"\nb = "</span> <span class="special"><<</span> <span class="identifier">formula_format</span><span class="special">(</span><span class="identifier">b</span><span class="special">)</span> <span class="special"><<</span> <span class="string">"\n\n"</span><span class="special">;</span>
</pre>
<p>
for output:
</p>
<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>
<span class="identifier">b</span> <span class="special">=</span> <span class="identifier">x</span> <span class="special">-</span> <span class="number">2</span>
</pre>
<pre class="programlisting"><span class="comment">// Now we can do arithmetic with the usual infix operators: + - * / and %.</span>
<span class="identifier">polynomial</span><span class="special"><</span><span class="keyword">double</span><span class="special">></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>
<span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"a + b = "</span> <span class="special"><<</span> <span class="identifier">formula_format</span><span class="special">(</span><span class="identifier">s</span><span class="special">)</span> <span class="special"><<</span> <span class="string">"\n"</span><span class="special">;</span>
<span class="identifier">polynomial</span><span class="special"><</span><span class="keyword">double</span><span class="special">></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>
<span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"a - b = "</span> <span class="special"><<</span> <span class="identifier">formula_format</span><span class="special">(</span><span class="identifier">d</span><span class="special">)</span> <span class="special"><<</span> <span class="string">"\n"</span><span class="special">;</span>
<span class="identifier">polynomial</span><span class="special"><</span><span class="keyword">double</span><span class="special">></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>
<span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"a * b = "</span> <span class="special"><<</span> <span class="identifier">formula_format</span><span class="special">(</span><span class="identifier">p</span><span class="special">)</span> <span class="special"><<</span> <span class="string">"\n"</span><span class="special">;</span>
<span class="identifier">polynomial</span><span class="special"><</span><span class="keyword">double</span><span class="special">></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>
<span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"a / b = "</span> <span class="special"><<</span> <span class="identifier">formula_format</span><span class="special">(</span><span class="identifier">q</span><span class="special">)</span> <span class="special"><<</span> <span class="string">"\n"</span><span class="special">;</span>
<span class="identifier">polynomial</span><span class="special"><</span><span class="keyword">double</span><span class="special">></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>
<span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"a % b = "</span> <span class="special"><<</span> <span class="identifier">formula_format</span><span class="special">(</span><span class="identifier">r</span><span class="special">)</span> <span class="special"><<</span> <span class="string">"\n"</span><span class="special">;</span>
</pre>
<p>
for output
</p>
<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>
<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>
<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>
<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>
<span class="identifier">a</span> <span class="special">%</span> <span class="identifier">b</span> <span class="special">=</span> <span class="number">6</span>
</pre>
<h6>
<a name="math_toolkit.polynomials.h3"></a>
<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,
Quotient and Remainder</a>
</h6>
<p>
Division is a special case where you can calculate two for the price of one.
</p>
<p>
Actually, quotient and remainder are always calculated together due to the
nature of the algorithm: the infix operators return one result and throw the
other away.
</p>
<p>
If you are doing a lot of division and want both the quotient and remainder,
then you don't want to do twice the work necessary.
</p>
<p>
In that case you can call the underlying function, <code class="literal">quotient_remainder</code>,
to get both results together as a pair.
</p>
<pre class="programlisting"> <span class="identifier">pair</span><span class="special"><</span> <span class="identifier">polynomial</span><span class="special"><</span><span class="keyword">double</span><span class="special">>,</span> <span class="identifier">polynomial</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="special">></span> <span class="identifier">result</span><span class="special">;</span>
<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>
<span class="comment">// Reassure ourselves that the result is the same.</span>
<span class="identifier">BOOST_MATH_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>
<span class="identifier">BOOST_MATH_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>
</pre>
<p>
The source code is at <a href="../../../example/polynomial_arithmetic.cpp" target="_top">polynomial_arithmetic.cpp</a>
</p>
</div>
<div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
Walker and Xiaogang Zhang<p>
Distributed under the Boost Software License, Version 1.0. (See accompanying
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
</p>
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