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25 | <div class="section"> | |
26 | <div class="titlepage"><div><div><h3 class="title"> | |
27 | <a name="math_toolkit.roots.polynomials"></a><a class="link" href="polynomials.html" title="Polynomials">Polynomials</a> | |
28 | </h3></div></div></div> | |
29 | <h5> | |
30 | <a name="math_toolkit.roots.polynomials.h0"></a> | |
31 | <span class="phrase"><a name="math_toolkit.roots.polynomials.synopsis"></a></span><a class="link" href="polynomials.html#math_toolkit.roots.polynomials.synopsis">Synopsis</a> | |
32 | </h5> | |
33 | <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</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> | |
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"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></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"><</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> | |
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"><</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> | |
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"><</span><span class="keyword">class</span> <span class="identifier">U</span><span class="special">></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"><</span><span class="keyword">class</span> <span class="identifier">I</span><span class="special">></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"><</span><span class="keyword">class</span> <span class="identifier">U</span><span class="special">></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">&</span> <span class="identifier">point</span><span class="special">);</span> | |
54 | <span class="identifier">polynomial</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">initializer_list</span><span class="special"><</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> | |
55 | ||
56 | <span class="comment">// access:</span> | |
57 | <span class="identifier">size_type</span> <span class="identifier">size</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> | |
58 | <span class="identifier">size_type</span> <span class="identifier">degree</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> | |
59 | <span class="identifier">value_type</span><span class="special">&</span> <span class="keyword">operator</span><span class="special">[](</span><span class="identifier">size_type</span> <span class="identifier">i</span><span class="special">);</span> | |
60 | <span class="keyword">const</span> <span class="identifier">value_type</span><span class="special">&</span> <span class="keyword">operator</span><span class="special">[](</span><span class="identifier">size_type</span> <span class="identifier">i</span><span class="special">)</span><span class="keyword">const</span><span class="special">;</span> | |
61 | <span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</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> | |
62 | <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> | |
63 | <span class="keyword">explicit</span> <span class="keyword">operator</span> <span class="keyword">bool</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span> | |
64 | ||
65 | <span class="comment">// modify:</span> | |
66 | <span class="keyword">void</span> <span class="identifier">set_zero</span><span class="special">();</span> | |
67 | ||
68 | <span class="comment">// operators:</span> | |
69 | <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">U</span><span class="special">></span> | |
70 | <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> | |
71 | <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">U</span><span class="special">></span> | |
72 | <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> | |
73 | <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">U</span><span class="special">></span> | |
74 | <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> | |
75 | <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">U</span><span class="special">></span> | |
76 | <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> | |
77 | <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">U</span><span class="special">></span> | |
78 | <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> | |
79 | <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">U</span><span class="special">></span> | |
80 | <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> | |
81 | <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">U</span><span class="special">></span> | |
82 | <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> | |
83 | <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">U</span><span class="special">></span> | |
84 | <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> | |
85 | <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">U</span><span class="special">></span> | |
86 | <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> | |
87 | <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">U</span><span class="special">></span> | |
88 | <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> | |
89 | <span class="special">};</span> | |
90 | ||
91 | <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> | |
92 | <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> | |
93 | <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> | |
94 | <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> | |
95 | <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> | |
96 | <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> | |
97 | <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> | |
98 | <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> | |
99 | <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> | |
100 | <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> | |
101 | ||
102 | <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> | |
103 | <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> | |
104 | <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> | |
105 | <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> | |
106 | <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> | |
107 | <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> | |
108 | <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> | |
109 | <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> | |
110 | <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> | |
111 | <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> | |
112 | ||
113 | <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> | |
114 | <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> | |
115 | <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> | |
116 | <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> | |
117 | <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> | |
118 | <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> | |
119 | ||
120 | <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> | |
121 | <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> | |
122 | ||
123 | <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> | |
124 | <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> | |
125 | <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> | |
126 | <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> | |
127 | <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> | |
128 | <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> | |
129 | <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> | |
130 | <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> | |
131 | ||
132 | <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> | |
133 | <span class="keyword">bool</span> <span class="keyword">operator</span> <span class="special">==</span> <span class="special">(</span><span class="keyword">const</span> <span class="identifier">polynomial</span><span class="special"><</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> | |
134 | <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> | |
135 | <span class="keyword">bool</span> <span class="keyword">operator</span> <span class="special">!=</span> <span class="special">(</span><span class="keyword">const</span> <span class="identifier">polynomial</span><span class="special"><</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> | |
136 | ||
137 | <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> | |
138 | <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> | |
139 | ||
140 | <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> | |
141 | <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> | |
142 | <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> | |
143 | ||
144 | <span class="keyword">template</span> <span class="special"><</span><span class="keyword">typename</span> <span class="identifier">T</span><span class="special">></span> | |
145 | <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> | |
146 | <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> | |
147 | ||
148 | <span class="special">}</span> <span class="comment">// namespace tools</span> | |
149 | <span class="special">}}</span> <span class="comment">// namespace boost { namespace math</span> | |
150 | </pre> | |
151 | <h5> | |
152 | <a name="math_toolkit.roots.polynomials.h1"></a> | |
153 | <span class="phrase"><a name="math_toolkit.roots.polynomials.description"></a></span><a class="link" href="polynomials.html#math_toolkit.roots.polynomials.description">Description</a> | |
154 | </h5> | |
155 | <p> | |
156 | This is a somewhat trivial class for polynomial manipulation. | |
157 | </p> | |
158 | <p> | |
159 | See | |
160 | </p> | |
161 | <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> | |
162 | <li class="listitem"> | |
163 | <a href="https://en.wikipedia.org/wiki/Polynomial" target="_top">Polynomial</a> | |
164 | and | |
165 | </li> | |
166 | <li class="listitem"> | |
167 | Donald E. Knuth, The Art of Computer Programming: Volume 2, Third edition, | |
168 | (1998) Chapter 4.6.1, Algorithm D: Division of polynomials over a field. | |
169 | </li> | |
170 | </ul></div> | |
171 | <p> | |
172 | Implementation is currently of the "naive" variety, with 𝑶(N<sup>2</sup>) multiplication, | |
173 | for example. This class should not be used in high-performance computing | |
174 | environments: it is intended for the simple manipulation of small polynomials, | |
175 | typically generated for special function approximation. | |
176 | </p> | |
177 | <p> | |
178 | It does has division for polynomials over a <a href="https://en.wikipedia.org/wiki/Field_%28mathematics%29" target="_top">field</a> | |
179 | (here floating point, complex, etc) and over a unique factorization domain | |
180 | (integers). Division of polynomials over a field is compatible with <a href="https://en.wikipedia.org/wiki/Euclidean_algorithm" target="_top">Euclidean GCD</a>. | |
181 | </p> | |
182 | <p> | |
183 | Advanced manipulations: the FFT, factorisation etc are not currently provided. | |
184 | Submissions for these are of course welcome :-) | |
185 | </p> | |
186 | <h5> | |
187 | <a name="math_toolkit.roots.polynomials.h2"></a> | |
188 | <span class="phrase"><a name="math_toolkit.roots.polynomials.polynomial_examples"></a></span><a class="link" href="polynomials.html#math_toolkit.roots.polynomials.polynomial_examples">Polynomial | |
189 | Arithmetic Examples</a> | |
190 | </h5> | |
191 | <p> | |
192 | First include the essential polynomial header (and others) to make the example: | |
193 | </p> | |
194 | <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</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> | |
195 | </pre> | |
196 | <p> | |
197 | and some using statements are convenient: | |
198 | </p> | |
199 | <pre class="programlisting"><span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">string</span><span class="special">;</span> | |
200 | <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">exception</span><span class="special">;</span> | |
201 | <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span><span class="special">;</span> | |
202 | <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">abs</span><span class="special">;</span> | |
203 | <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special">;</span> | |
204 | ||
205 | <span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">;</span> | |
206 | <span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">tools</span><span class="special">;</span> <span class="comment">// for polynomial</span> | |
207 | <span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">lexical_cast</span><span class="special">;</span> | |
208 | </pre> | |
209 | <p> | |
210 | Store the coefficients in a convenient way to access them, then create some | |
211 | polynomials using construction from an iterator range, and finally output | |
212 | in a 'pretty' formula format. | |
213 | </p> | |
214 | <div class="tip"><table border="0" summary="Tip"> | |
215 | <tr> | |
216 | <td rowspan="2" align="center" valign="top" width="25"><img alt="[Tip]" src="../../../../../../doc/src/images/tip.png"></td> | |
217 | <th align="left">Tip</th> | |
218 | </tr> | |
219 | <tr><td align="left" valign="top"><p> | |
220 | Although we might conventionally write a polynomial from left to right | |
221 | in descending order of degree, Boost.Math stores in <span class="bold"><strong>ascending | |
222 | order of degree</strong></span>. | |
223 | </p></td></tr> | |
224 | </table></div> | |
225 | <pre class="programlisting"><span class="identifier">Read</span><span class="special">/</span><span class="identifier">write</span> <span class="keyword">for</span> <span class="identifier">humans</span><span class="special">:</span> <span class="number">3</span><span class="identifier">x</span><span class="special">^</span><span class="number">3</span> <span class="special">-</span> <span class="number">4</span><span class="identifier">x</span><span class="special">^</span><span class="number">2</span> <span class="special">-</span> <span class="number">6</span><span class="identifier">x</span> <span class="special">+</span> <span class="number">10</span> | |
226 | <span class="identifier">Boost</span> <span class="identifier">polynomial</span> <span class="identifier">storage</span><span class="special">:</span> <span class="special">[</span> <span class="number">10</span><span class="special">,</span> <span class="special">-</span><span class="number">6</span><span class="special">,</span> <span class="special">-</span><span class="number">4</span><span class="special">,</span> <span class="number">3</span> <span class="special">]</span> | |
227 | </pre> | |
228 | <pre class="programlisting"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">array</span><span class="special"><</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> | |
229 | <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> | |
230 | ||
231 | <span class="comment">// With C++11 and later, you can also use initializer_list construction.</span> | |
232 | <span class="identifier">polynomial</span><span class="special"><</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> | |
233 | ||
234 | <span class="comment">// formula_format() converts from Boost storage to human notation.</span> | |
235 | <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> | |
236 | <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> | |
237 | </pre> | |
238 | <p> | |
239 | for output: | |
240 | </p> | |
241 | <pre class="programlisting"><span class="identifier">a</span> <span class="special">=</span> <span class="number">3</span><span class="identifier">x</span><span class="special">^</span><span class="number">3</span> <span class="special">-</span> <span class="number">4</span><span class="identifier">x</span><span class="special">^</span><span class="number">2</span> <span class="special">-</span> <span class="number">6</span><span class="identifier">x</span> <span class="special">+</span> <span class="number">10</span> | |
242 | <span class="identifier">b</span> <span class="special">=</span> <span class="identifier">x</span> <span class="special">-</span> <span class="number">2</span> | |
243 | </pre> | |
244 | <pre class="programlisting"><span class="comment">// Now we can do arithmetic with the usual infix operators: + - * / and %.</span> | |
245 | <span class="identifier">polynomial</span><span class="special"><</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> | |
246 | <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> | |
247 | <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> | |
248 | <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> | |
249 | <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> | |
250 | <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> | |
251 | <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> | |
252 | <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> | |
253 | <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> | |
254 | <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> | |
255 | </pre> | |
256 | <p> | |
257 | for output | |
258 | </p> | |
259 | <pre class="programlisting"><span class="identifier">a</span> <span class="special">+</span> <span class="identifier">b</span> <span class="special">=</span> <span class="number">3</span><span class="identifier">x</span><span class="special">^</span><span class="number">3</span> <span class="special">-</span> <span class="number">4</span><span class="identifier">x</span><span class="special">^</span><span class="number">2</span> <span class="special">-</span> <span class="number">5</span><span class="identifier">x</span> <span class="special">+</span> <span class="number">8</span> | |
260 | <span class="identifier">a</span> <span class="special">-</span> <span class="identifier">b</span> <span class="special">=</span> <span class="number">3</span><span class="identifier">x</span><span class="special">^</span><span class="number">3</span> <span class="special">-</span> <span class="number">4</span><span class="identifier">x</span><span class="special">^</span><span class="number">2</span> <span class="special">-</span> <span class="number">7</span><span class="identifier">x</span> <span class="special">+</span> <span class="number">12</span> | |
261 | <span class="identifier">a</span> <span class="special">*</span> <span class="identifier">b</span> <span class="special">=</span> <span class="number">3</span><span class="identifier">x</span><span class="special">^</span><span class="number">4</span> <span class="special">-</span> <span class="number">10</span><span class="identifier">x</span><span class="special">^</span><span class="number">3</span> <span class="special">+</span> <span class="number">2</span><span class="identifier">x</span><span class="special">^</span><span class="number">2</span> <span class="special">+</span> <span class="number">22</span><span class="identifier">x</span> <span class="special">-</span> <span class="number">20</span> | |
262 | <span class="identifier">a</span> <span class="special">/</span> <span class="identifier">b</span> <span class="special">=</span> <span class="number">3</span><span class="identifier">x</span><span class="special">^</span><span class="number">2</span> <span class="special">+</span> <span class="number">2</span><span class="identifier">x</span> <span class="special">-</span> <span class="number">2</span> | |
263 | <span class="identifier">a</span> <span class="special">%</span> <span class="identifier">b</span> <span class="special">=</span> <span class="number">6</span> | |
264 | </pre> | |
265 | <h6> | |
266 | <a name="math_toolkit.roots.polynomials.h3"></a> | |
267 | <span class="phrase"><a name="math_toolkit.roots.polynomials.division_quotient_and_remainder"></a></span><a class="link" href="polynomials.html#math_toolkit.roots.polynomials.division_quotient_and_remainder">Division, | |
268 | Quotient and Remainder</a> | |
269 | </h6> | |
270 | <p> | |
271 | Division is a special case where you can calculate two for the price of one. | |
272 | </p> | |
273 | <p> | |
274 | Actually, quotient and remainder are always calculated together due to the | |
275 | nature of the algorithm: the infix operators return one result and throw | |
276 | the other away. | |
277 | </p> | |
278 | <p> | |
279 | If you are doing a lot of division and want both the quotient and remainder, | |
280 | then you don't want to do twice the work necessary. | |
281 | </p> | |
282 | <p> | |
283 | In that case you can call the underlying function, <code class="literal">quotient_remainder</code>, | |
284 | to get both results together as a pair. | |
285 | </p> | |
286 | <pre class="programlisting"> <span class="identifier">pair</span><span class="special"><</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> | |
287 | <span class="identifier">result</span> <span class="special">=</span> <span class="identifier">quotient_remainder</span><span class="special">(</span><span class="identifier">a</span><span class="special">,</span> <span class="identifier">b</span><span class="special">);</span> | |
288 | <span class="comment">// Reassure ourselves that the result is the same.</span> | |
289 | <span class="identifier">BOOST_ASSERT</span><span class="special">(</span><span class="identifier">result</span><span class="special">.</span><span class="identifier">first</span> <span class="special">==</span> <span class="identifier">q</span><span class="special">);</span> | |
290 | <span class="identifier">BOOST_ASSERT</span><span class="special">(</span><span class="identifier">result</span><span class="special">.</span><span class="identifier">second</span> <span class="special">==</span> <span class="identifier">r</span><span class="special">);</span> | |
291 | </pre> | |
292 | <p> | |
293 | The source code is at <a href="../../../../example/polynomial_arithmetic.cpp" target="_top">polynomial_arithmetic.cpp</a> | |
294 | </p> | |
295 | </div> | |
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297 | <td align="left"></td> | |
298 | <td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal, | |
299 | Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert | |
300 | Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani, | |
301 | Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p> | |
302 | Distributed under the Boost Software License, Version 1.0. (See accompanying | |
303 | file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>) | |
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