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24 | </div> | |
25 | <div class="section"> | |
26 | <div class="titlepage"><div><div><h3 class="title"> | |
27 | <a name="math_toolkit.number_series.bernoulli_numbers"></a><a class="link" href="bernoulli_numbers.html" title="Bernoulli Numbers">Bernoulli | |
28 | Numbers</a> | |
29 | </h3></div></div></div> | |
30 | <p> | |
31 | <a href="https://en.wikipedia.org/wiki/Bernoulli_number" target="_top">Bernoulli numbers</a> | |
32 | are a sequence of rational numbers useful for the Taylor series expansion, | |
33 | Euler-Maclaurin formula, and the Riemann zeta function. | |
34 | </p> | |
35 | <p> | |
36 | Bernoulli numbers are used in evaluation of some Boost.Math functions, including | |
37 | the <a class="link" href="../sf_gamma/tgamma.html" title="Gamma">tgamma</a>, <a class="link" href="../sf_gamma/lgamma.html" title="Log Gamma">lgamma</a> | |
38 | and polygamma functions. | |
39 | </p> | |
40 | <h5> | |
41 | <a name="math_toolkit.number_series.bernoulli_numbers.h0"></a> | |
42 | <span class="phrase"><a name="math_toolkit.number_series.bernoulli_numbers.single_bernoulli_number"></a></span><a class="link" href="bernoulli_numbers.html#math_toolkit.number_series.bernoulli_numbers.single_bernoulli_number">Single | |
43 | Bernoulli number</a> | |
44 | </h5> | |
45 | <h5> | |
46 | <a name="math_toolkit.number_series.bernoulli_numbers.h1"></a> | |
47 | <span class="phrase"><a name="math_toolkit.number_series.bernoulli_numbers.synopsis"></a></span><a class="link" href="bernoulli_numbers.html#math_toolkit.number_series.bernoulli_numbers.synopsis">Synopsis</a> | |
48 | </h5> | |
49 | <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">special_functions</span><span class="special">/</span><span class="identifier">bernoulli</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> | |
50 | </pre> | |
51 | <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> | |
52 | ||
53 | <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> | |
54 | <span class="identifier">T</span> <span class="identifier">bernoulli_b2n</span><span class="special">(</span><span class="keyword">const</span> <span class="keyword">int</span> <span class="identifier">n</span><span class="special">);</span> <span class="comment">// Single Bernoulli number (default policy).</span> | |
55 | ||
56 | <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">Policy</span><span class="special">></span> | |
57 | <span class="identifier">T</span> <span class="identifier">bernoulli_b2n</span><span class="special">(</span><span class="keyword">const</span> <span class="keyword">int</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Policy</span> <span class="special">&</span><span class="identifier">pol</span><span class="special">);</span> <span class="comment">// User policy for errors etc.</span> | |
58 | ||
59 | <span class="special">}}</span> <span class="comment">// namespaces</span> | |
60 | </pre> | |
61 | <h5> | |
62 | <a name="math_toolkit.number_series.bernoulli_numbers.h2"></a> | |
63 | <span class="phrase"><a name="math_toolkit.number_series.bernoulli_numbers.description"></a></span><a class="link" href="bernoulli_numbers.html#math_toolkit.number_series.bernoulli_numbers.description">Description</a> | |
64 | </h5> | |
65 | <p> | |
66 | Both return the (2 * n)<sup>th</sup> Bernoulli number B<sub>2n</sub>. | |
67 | </p> | |
68 | <p> | |
69 | Note that since all odd numbered Bernoulli numbers are zero (apart from B<sub>1</sub> which | |
70 | is -½) the interface will only return the even numbered Bernoulli numbers. | |
71 | </p> | |
72 | <p> | |
73 | This function uses fast table lookup for low-indexed Bernoulli numbers, while | |
74 | larger values are calculated as needed and then cached. The caching mechanism | |
75 | requires a certain amount of thread safety code, so <code class="computeroutput"><span class="identifier">unchecked_bernoulli_b2n</span></code> | |
76 | may provide a better interface for performance critical code. | |
77 | </p> | |
78 | <p> | |
79 | The final <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can | |
80 | be used to control the behaviour of the function: how it handles errors, | |
81 | what level of precision to use, etc. | |
82 | </p> | |
83 | <p> | |
84 | Refer to <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policies</a> for more details. | |
85 | </p> | |
86 | <h5> | |
87 | <a name="math_toolkit.number_series.bernoulli_numbers.h3"></a> | |
88 | <span class="phrase"><a name="math_toolkit.number_series.bernoulli_numbers.examples"></a></span><a class="link" href="bernoulli_numbers.html#math_toolkit.number_series.bernoulli_numbers.examples">Examples</a> | |
89 | </h5> | |
90 | <p> | |
91 | A simple example computes the value of B<sub>4</sub> where the return type is <code class="computeroutput"><span class="keyword">double</span></code>, note that the argument to bernoulli_b2n | |
92 | is <span class="emphasis"><em>2</em></span> not <span class="emphasis"><em>4</em></span> since it computes B<sub>2N</sub>. | |
93 | </p> | |
94 | <pre class="programlisting"><span class="keyword">try</span> | |
95 | <span class="special">{</span> <span class="comment">// It is always wise to use try'n'catch blocks around Boost.Math functions</span> | |
96 | <span class="comment">// so that any informative error messages can be displayed in the catch block.</span> | |
97 | <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> | |
98 | <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setprecision</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="keyword">double</span><span class="special">>::</span><span class="identifier">digits10</span><span class="special">)</span> | |
99 | <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">bernoulli_b2n</span><span class="special"><</span><span class="keyword">double</span><span class="special">>(</span><span class="number">2</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> | |
100 | </pre> | |
101 | <p> | |
102 | So B<sub>4</sub> == -1/30 == -0.0333333333333333 | |
103 | </p> | |
104 | <p> | |
105 | If we use Boost.Multiprecision and its 50 decimal digit floating-point type | |
106 | <code class="computeroutput"><span class="identifier">cpp_dec_float_50</span></code>, we can | |
107 | calculate the value of much larger numbers like B<sub>200</sub> | |
108 | and also obtain much | |
109 | higher precision. | |
110 | </p> | |
111 | <pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> | |
112 | <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setprecision</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">boost</span><span class="special">::</span><span class="identifier">multiprecision</span><span class="special">::</span><span class="identifier">cpp_dec_float_50</span><span class="special">>::</span><span class="identifier">digits10</span><span class="special">)</span> | |
113 | <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">bernoulli_b2n</span><span class="special"><</span><span class="identifier">boost</span><span class="special">::</span><span class="identifier">multiprecision</span><span class="special">::</span><span class="identifier">cpp_dec_float_50</span><span class="special">>(</span><span class="number">100</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> | |
114 | </pre> | |
115 | <pre class="programlisting"><span class="special">-</span><span class="number">3.6470772645191354362138308865549944904868234686191e+215</span> | |
116 | </pre> | |
117 | <h5> | |
118 | <a name="math_toolkit.number_series.bernoulli_numbers.h4"></a> | |
119 | <span class="phrase"><a name="math_toolkit.number_series.bernoulli_numbers.single_unchecked_bernoulli_numbe"></a></span><a class="link" href="bernoulli_numbers.html#math_toolkit.number_series.bernoulli_numbers.single_unchecked_bernoulli_numbe">Single | |
120 | (unchecked) Bernoulli number</a> | |
121 | </h5> | |
122 | <h5> | |
123 | <a name="math_toolkit.number_series.bernoulli_numbers.h5"></a> | |
124 | <span class="phrase"><a name="math_toolkit.number_series.bernoulli_numbers.synopsis0"></a></span><a class="link" href="bernoulli_numbers.html#math_toolkit.number_series.bernoulli_numbers.synopsis0">Synopsis</a> | |
125 | </h5> | |
126 | <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">special_functions</span><span class="special">/</span><span class="identifier">bernoulli</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> | |
127 | </pre> | |
128 | <pre class="programlisting"><span class="keyword">template</span> <span class="special"><></span> | |
129 | <span class="keyword">struct</span> <span class="identifier">max_bernoulli_b2n</span><span class="special"><</span><span class="identifier">T</span><span class="special">>;</span> | |
130 | ||
131 | <span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> | |
132 | <span class="keyword">inline</span> <span class="identifier">T</span> <span class="identifier">unchecked_bernoulli_b2n</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">);</span> | |
133 | </pre> | |
134 | <p> | |
135 | <code class="computeroutput"><span class="identifier">unchecked_bernoulli_b2n</span></code> provides | |
136 | access to Bernoulli numbers <span class="bold"><strong>without any checks for | |
137 | overflow or invalid parameters</strong></span>. It is implemented as a direct | |
138 | (and very fast) table lookup, and while not recommended for general use it | |
139 | can be useful inside inner loops where the ultimate performance is required, | |
140 | and error checking is moved outside the loop. | |
141 | </p> | |
142 | <p> | |
143 | The largest value you can pass to <code class="computeroutput"><span class="identifier">unchecked_bernoulli_b2n</span><span class="special"><></span></code> is <code class="computeroutput"><span class="identifier">max_bernoulli_b2n</span><span class="special"><>::</span><span class="identifier">value</span></code>: | |
144 | passing values greater than that will result in a buffer overrun error, so | |
145 | it's clearly important to place the error handling in your own code when | |
146 | using this direct interface. | |
147 | </p> | |
148 | <p> | |
149 | The value of <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">max_bernoulli_b2n</span><span class="special"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">value</span></code> varies by the type T, for types | |
150 | <code class="computeroutput"><span class="keyword">float</span></code>/<code class="computeroutput"><span class="keyword">double</span></code>/<code class="computeroutput"><span class="keyword">long</span> <span class="keyword">double</span></code> | |
151 | it's the largest value which doesn't overflow the target type: for example, | |
152 | <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">max_bernoulli_b2n</span><span class="special"><</span><span class="keyword">double</span><span class="special">>::</span><span class="identifier">value</span></code> is 129. However, for multiprecision | |
153 | types, it's the largest value for which the result can be represented as | |
154 | the ratio of two 64-bit integers, for example <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">max_bernoulli_b2n</span><span class="special"><</span><span class="identifier">boost</span><span class="special">::</span><span class="identifier">multiprecision</span><span class="special">::</span><span class="identifier">cpp_dec_float_50</span><span class="special">>::</span><span class="identifier">value</span></code> | |
155 | is just 17. Of course larger indexes can be passed to <code class="computeroutput"><span class="identifier">bernoulli_b2n</span><span class="special"><</span><span class="identifier">T</span><span class="special">>(</span><span class="identifier">n</span><span class="special">)</span></code>, but | |
156 | then you lose fast table lookup (i.e. values may need to be calculated). | |
157 | </p> | |
158 | <pre class="programlisting"><span class="comment">/*For example: | |
159 | */</span> | |
160 | <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"boost::math::max_bernoulli_b2n<float>::value = "</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">max_bernoulli_b2n</span><span class="special"><</span><span class="keyword">float</span><span class="special">>::</span><span class="identifier">value</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> | |
161 | <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Maximum Bernoulli number using float is "</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">bernoulli_b2n</span><span class="special"><</span><span class="keyword">float</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">max_bernoulli_b2n</span><span class="special"><</span><span class="keyword">float</span><span class="special">>::</span><span class="identifier">value</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> | |
162 | <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"boost::math::max_bernoulli_b2n<double>::value = "</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">max_bernoulli_b2n</span><span class="special"><</span><span class="keyword">double</span><span class="special">>::</span><span class="identifier">value</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> | |
163 | <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Maximum Bernoulli number using double is "</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">bernoulli_b2n</span><span class="special"><</span><span class="keyword">double</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">max_bernoulli_b2n</span><span class="special"><</span><span class="keyword">double</span><span class="special">>::</span><span class="identifier">value</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> | |
164 | </pre> | |
165 | <pre class="programlisting"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">max_bernoulli_b2n</span><span class="special"><</span><span class="keyword">float</span><span class="special">>::</span><span class="identifier">value</span> <span class="special">=</span> <span class="number">32</span> | |
166 | <span class="identifier">Maximum</span> <span class="identifier">Bernoulli</span> <span class="identifier">number</span> <span class="keyword">using</span> <span class="keyword">float</span> <span class="identifier">is</span> <span class="special">-</span><span class="number">2.0938e+038</span> | |
167 | <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">max_bernoulli_b2n</span><span class="special"><</span><span class="keyword">double</span><span class="special">>::</span><span class="identifier">value</span> <span class="special">=</span> <span class="number">129</span> | |
168 | <span class="identifier">Maximum</span> <span class="identifier">Bernoulli</span> <span class="identifier">number</span> <span class="keyword">using</span> <span class="keyword">double</span> <span class="identifier">is</span> <span class="number">1.33528e+306</span> | |
169 | </pre> | |
170 | <h5> | |
171 | <a name="math_toolkit.number_series.bernoulli_numbers.h6"></a> | |
172 | <span class="phrase"><a name="math_toolkit.number_series.bernoulli_numbers.multiple_bernoulli_numbers"></a></span><a class="link" href="bernoulli_numbers.html#math_toolkit.number_series.bernoulli_numbers.multiple_bernoulli_numbers">Multiple | |
173 | Bernoulli Numbers</a> | |
174 | </h5> | |
175 | <h5> | |
176 | <a name="math_toolkit.number_series.bernoulli_numbers.h7"></a> | |
177 | <span class="phrase"><a name="math_toolkit.number_series.bernoulli_numbers.synopsis1"></a></span><a class="link" href="bernoulli_numbers.html#math_toolkit.number_series.bernoulli_numbers.synopsis1">Synopsis</a> | |
178 | </h5> | |
179 | <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">special_functions</span><span class="special">/</span><span class="identifier">bernoulli</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> | |
180 | </pre> | |
181 | <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> | |
182 | ||
183 | <span class="comment">// Multiple Bernoulli numbers (default policy).</span> | |
184 | <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">OutputIterator</span><span class="special">></span> | |
185 | <span class="identifier">OutputIterator</span> <span class="identifier">bernoulli_b2n</span><span class="special">(</span> | |
186 | <span class="keyword">int</span> <span class="identifier">start_index</span><span class="special">,</span> | |
187 | <span class="keyword">unsigned</span> <span class="identifier">number_of_bernoullis_b2n</span><span class="special">,</span> | |
188 | <span class="identifier">OutputIterator</span> <span class="identifier">out_it</span><span class="special">);</span> | |
189 | ||
190 | <span class="comment">// Multiple Bernoulli numbers (user policy).</span> | |
191 | <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">OutputIterator</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Policy</span><span class="special">></span> | |
192 | <span class="identifier">OutputIterator</span> <span class="identifier">bernoulli_b2n</span><span class="special">(</span> | |
193 | <span class="keyword">int</span> <span class="identifier">start_index</span><span class="special">,</span> | |
194 | <span class="keyword">unsigned</span> <span class="identifier">number_of_bernoullis_b2n</span><span class="special">,</span> | |
195 | <span class="identifier">OutputIterator</span> <span class="identifier">out_it</span><span class="special">,</span> | |
196 | <span class="keyword">const</span> <span class="identifier">Policy</span><span class="special">&</span> <span class="identifier">pol</span><span class="special">);</span> | |
197 | <span class="special">}}</span> <span class="comment">// namespaces</span> | |
198 | </pre> | |
199 | <h5> | |
200 | <a name="math_toolkit.number_series.bernoulli_numbers.h8"></a> | |
201 | <span class="phrase"><a name="math_toolkit.number_series.bernoulli_numbers.description0"></a></span><a class="link" href="bernoulli_numbers.html#math_toolkit.number_series.bernoulli_numbers.description0">Description</a> | |
202 | </h5> | |
203 | <p> | |
204 | Two versions of the Bernoulli number function are provided to compute multiple | |
205 | Bernoulli numbers with one call (one with default policy and the other allowing | |
206 | a user-defined policy). | |
207 | </p> | |
208 | <p> | |
209 | These return a series of Bernoulli numbers: | |
210 | </p> | |
211 | <div class="blockquote"><blockquote class="blockquote"><p> | |
212 | B<sub>2*start_index</sub>,B<sub>2*(start_index+1)</sub>,...,B<sub>2*(start_index+number_of_bernoullis_b2n-1)</sub> | |
213 | </p></blockquote></div> | |
214 | <h5> | |
215 | <a name="math_toolkit.number_series.bernoulli_numbers.h9"></a> | |
216 | <span class="phrase"><a name="math_toolkit.number_series.bernoulli_numbers.examples0"></a></span><a class="link" href="bernoulli_numbers.html#math_toolkit.number_series.bernoulli_numbers.examples0">Examples</a> | |
217 | </h5> | |
218 | <p> | |
219 | We can compute and save all the float-precision Bernoulli numbers from one | |
220 | call. | |
221 | </p> | |
222 | <pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span><span class="keyword">float</span><span class="special">></span> <span class="identifier">bn</span><span class="special">;</span> <span class="comment">// Space for 32-bit `float` precision Bernoulli numbers.</span> | |
223 | ||
224 | <span class="comment">// Start with Bernoulli number 0.</span> | |
225 | <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">bernoulli_b2n</span><span class="special"><</span><span class="keyword">float</span><span class="special">>(</span><span class="number">0</span><span class="special">,</span> <span class="number">32</span><span class="special">,</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">back_inserter</span><span class="special">(</span><span class="identifier">bn</span><span class="special">));</span> <span class="comment">// Fill vector with even Bernoulli numbers.</span> | |
226 | ||
227 | <span class="keyword">for</span><span class="special">(</span><span class="identifier">size_t</span> <span class="identifier">i</span> <span class="special">=</span> <span class="number">0</span><span class="special">;</span> <span class="identifier">i</span> <span class="special"><</span> <span class="identifier">bn</span><span class="special">.</span><span class="identifier">size</span><span class="special">();</span> <span class="identifier">i</span><span class="special">++)</span> | |
228 | <span class="special">{</span> <span class="comment">// Show vector of even Bernoulli numbers, showing all significant decimal digits.</span> | |
229 | <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setprecision</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="keyword">float</span><span class="special">>::</span><span class="identifier">digits10</span><span class="special">)</span> | |
230 | <span class="special"><<</span> <span class="identifier">i</span><span class="special">*</span><span class="number">2</span> <span class="special"><<</span> <span class="char">' '</span> | |
231 | <span class="special"><<</span> <span class="identifier">bn</span><span class="special">[</span><span class="identifier">i</span><span class="special">]</span> | |
232 | <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> | |
233 | <span class="special">}</span> | |
234 | </pre> | |
235 | <pre class="programlisting"><span class="number">0</span> <span class="number">1</span> | |
236 | <span class="number">2</span> <span class="number">0.166667</span> | |
237 | <span class="number">4</span> <span class="special">-</span><span class="number">0.0333333</span> | |
238 | <span class="number">6</span> <span class="number">0.0238095</span> | |
239 | <span class="number">8</span> <span class="special">-</span><span class="number">0.0333333</span> | |
240 | <span class="number">10</span> <span class="number">0.0757576</span> | |
241 | <span class="number">12</span> <span class="special">-</span><span class="number">0.253114</span> | |
242 | <span class="number">14</span> <span class="number">1.16667</span> | |
243 | <span class="number">16</span> <span class="special">-</span><span class="number">7.09216</span> | |
244 | <span class="number">18</span> <span class="number">54.9712</span> | |
245 | <span class="number">20</span> <span class="special">-</span><span class="number">529.124</span> | |
246 | <span class="number">22</span> <span class="number">6192.12</span> | |
247 | <span class="number">24</span> <span class="special">-</span><span class="number">86580.3</span> | |
248 | <span class="number">26</span> <span class="number">1.42552e+006</span> | |
249 | <span class="number">28</span> <span class="special">-</span><span class="number">2.72982e+007</span> | |
250 | <span class="number">30</span> <span class="number">6.01581e+008</span> | |
251 | <span class="number">32</span> <span class="special">-</span><span class="number">1.51163e+010</span> | |
252 | <span class="number">34</span> <span class="number">4.29615e+011</span> | |
253 | <span class="number">36</span> <span class="special">-</span><span class="number">1.37117e+013</span> | |
254 | <span class="number">38</span> <span class="number">4.88332e+014</span> | |
255 | <span class="number">40</span> <span class="special">-</span><span class="number">1.92966e+016</span> | |
256 | <span class="number">42</span> <span class="number">8.41693e+017</span> | |
257 | <span class="number">44</span> <span class="special">-</span><span class="number">4.03381e+019</span> | |
258 | <span class="number">46</span> <span class="number">2.11507e+021</span> | |
259 | <span class="number">48</span> <span class="special">-</span><span class="number">1.20866e+023</span> | |
260 | <span class="number">50</span> <span class="number">7.50087e+024</span> | |
261 | <span class="number">52</span> <span class="special">-</span><span class="number">5.03878e+026</span> | |
262 | <span class="number">54</span> <span class="number">3.65288e+028</span> | |
263 | <span class="number">56</span> <span class="special">-</span><span class="number">2.84988e+030</span> | |
264 | <span class="number">58</span> <span class="number">2.38654e+032</span> | |
265 | <span class="number">60</span> <span class="special">-</span><span class="number">2.14e+034</span> | |
266 | <span class="number">62</span> <span class="number">2.0501e+036</span> | |
267 | </pre> | |
268 | <p> | |
269 | Of course, for any floating-point type, there is a maximum Bernoulli number | |
270 | that can be computed before it overflows the exponent. By default policy, | |
271 | if we try to compute too high a Bernoulli number, an exception will be thrown. | |
272 | </p> | |
273 | <pre class="programlisting"><span class="keyword">try</span> | |
274 | <span class="special">{</span> | |
275 | <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> | |
276 | <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setprecision</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="keyword">float</span><span class="special">>::</span><span class="identifier">digits10</span><span class="special">)</span> | |
277 | <span class="special"><<</span> <span class="string">"Bernoulli number "</span> <span class="special"><<</span> <span class="number">33</span> <span class="special">*</span> <span class="number">2</span> <span class="special"><<</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> | |
278 | ||
279 | <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</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">bernoulli_b2n</span><span class="special"><</span><span class="keyword">float</span><span class="special">>(</span><span class="number">33</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> | |
280 | <span class="special">}</span> | |
281 | <span class="keyword">catch</span> <span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">exception</span> <span class="identifier">ex</span><span class="special">)</span> | |
282 | <span class="special">{</span> | |
283 | <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Thrown Exception caught: "</span> <span class="special"><<</span> <span class="identifier">ex</span><span class="special">.</span><span class="identifier">what</span><span class="special">()</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> | |
284 | <span class="special">}</span> | |
285 | </pre> | |
286 | <p> | |
287 | and we will get a helpful error message (provided try'n'catch blocks are | |
288 | used). | |
289 | </p> | |
290 | <pre class="programlisting"><span class="identifier">Bernoulli</span> <span class="identifier">number</span> <span class="number">66</span> | |
291 | <span class="identifier">Thrown</span> <span class="identifier">Exception</span> <span class="identifier">caught</span><span class="special">:</span> <span class="identifier">Error</span> <span class="identifier">in</span> <span class="identifier">function</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">bernoulli_b2n</span><span class="special"><</span><span class="keyword">float</span><span class="special">>(</span><span class="identifier">n</span><span class="special">):</span> | |
292 | <span class="identifier">Overflow</span> <span class="identifier">evaluating</span> <span class="identifier">function</span> <span class="identifier">at</span> <span class="number">33</span> | |
293 | </pre> | |
294 | <p> | |
295 | The source of this example is at <a href="../../../../example/bernoulli_example.cpp" target="_top">bernoulli_example.cpp</a> | |
296 | </p> | |
297 | <h5> | |
298 | <a name="math_toolkit.number_series.bernoulli_numbers.h10"></a> | |
299 | <span class="phrase"><a name="math_toolkit.number_series.bernoulli_numbers.accuracy"></a></span><a class="link" href="bernoulli_numbers.html#math_toolkit.number_series.bernoulli_numbers.accuracy">Accuracy</a> | |
300 | </h5> | |
301 | <p> | |
302 | All the functions usually return values within one ULP (unit in the last | |
303 | place) for the floating-point type. | |
304 | </p> | |
305 | <h5> | |
306 | <a name="math_toolkit.number_series.bernoulli_numbers.h11"></a> | |
307 | <span class="phrase"><a name="math_toolkit.number_series.bernoulli_numbers.implementation"></a></span><a class="link" href="bernoulli_numbers.html#math_toolkit.number_series.bernoulli_numbers.implementation">Implementation</a> | |
308 | </h5> | |
309 | <p> | |
310 | The implementation details are in <a href="../../../../include/boost/math/special_functions/detail/bernoulli_details.hpp" target="_top">bernoulli_details.hpp</a> | |
311 | and <a href="../../../../include/boost/math/special_functions/detail/unchecked_bernoulli.hpp" target="_top">unchecked_bernoulli.hpp</a>. | |
312 | </p> | |
313 | <p> | |
314 | For <code class="computeroutput"><span class="identifier">i</span> <span class="special"><=</span> | |
315 | <span class="identifier">max_bernoulli_index</span><span class="special"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">value</span></code> this is implemented by simple table | |
316 | lookup from a statically initialized table; for larger values of <code class="computeroutput"><span class="identifier">i</span></code>, this is implemented by the Tangent Numbers | |
317 | algorithm as described in the paper: Fast Computation of Bernoulli, Tangent | |
318 | and Secant Numbers, Richard P. Brent and David Harvey, <a href="http://arxiv.org/pdf/1108.0286v3.pdf" target="_top">http://arxiv.org/pdf/1108.0286v3.pdf</a> | |
319 | (2011). | |
320 | </p> | |
321 | <p> | |
322 | <a href="http://mathworld.wolfram.com/TangentNumber.html" target="_top">Tangent (or | |
323 | Zag) numbers</a> (an even alternating permutation number) are defined | |
324 | and their generating function is also given therein. | |
325 | </p> | |
326 | <p> | |
327 | The relation of Tangent numbers with Bernoulli numbers <span class="emphasis"><em>B<sub>i</sub></em></span> | |
328 | is given by Brent and Harvey's equation 14: | |
329 | </p> | |
330 | <p> | |
331 |    <span class="inlinemediaobject"><img src="../../../equations/tangent_numbers.svg"></span> | |
332 | </p> | |
333 | <p> | |
334 | Their relation with Bernoulli numbers <span class="emphasis"><em>B<sub>i</sub></em></span> are defined | |
335 | by | |
336 | </p> | |
337 | <p> | |
338 | if i > 0 and i is even then    <span class="inlinemediaobject"><img src="../../../equations/bernoulli_numbers.svg"></span> <br> elseif | |
339 | i == 0 then <span class="emphasis"><em>B<sub>i</sub></em></span> = 1 <br> elseif i == 1 then <span class="emphasis"><em>B<sub>i</sub></em></span> | |
340 | = -1/2 <br> elseif i < 0 or i is odd then <span class="emphasis"><em>B<sub>i</sub></em></span> = | |
341 | 0 | |
342 | </p> | |
343 | <p> | |
344 | Note that computed values are stored in a fixed-size table, access is thread | |
345 | safe via atomic operations (i.e. lock free programming), this imparts a much | |
346 | lower overhead on access to cached values than might otherwise be expected | |
347 | - typically for multiprecision types the cost of thread synchronisation is | |
348 | negligible, while for built in types this code is not normally executed anyway. | |
349 | For very large arguments which cannot be reasonably computed or stored in | |
350 | our cache, an asymptotic expansion <a href="http://www.luschny.de/math/primes/bernincl.html" target="_top">due | |
351 | to Luschny</a> is used: | |
352 | </p> | |
353 | <p> | |
354 | <span class="inlinemediaobject"><img src="../../../equations/bernoulli_numbers2.svg"></span> | |
355 | </p> | |
356 | </div> | |
357 | <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> | |
358 | <td align="left"></td> | |
359 | <td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal, | |
360 | Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert | |
361 | Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani, | |
362 | Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p> | |
363 | Distributed under the Boost Software License, Version 1.0. (See accompanying | |
364 | 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>) | |
365 | </p> | |
366 | </div></td> | |
367 | </tr></table> | |
368 | <hr> | |
369 | <div class="spirit-nav"> | |
370 | <a accesskey="p" href="../number_series.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../number_series.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="tangent_numbers.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> | |
371 | </div> | |
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373 | </html> |