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<div class="titlepage"><div><div><h4 class="title">
<a name="math_toolkit.special.zetas.zeta"></a><a class="link" href="zeta.html" title="Riemann Zeta Function"> Riemann Zeta Function</a>
</h4></div></div></div>
<a name="math_toolkit.special.zetas.zeta.synopsis"></a><h5>
<a name="id1144944"></a>
<a class="link" href="zeta.html#math_toolkit.special.zetas.zeta.synopsis">Synopsis</a>
</h5>
<p>
</p>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">special_functions</span><span class="special">/</span><span class="identifier">zeta</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
</pre>
<p>
</p>
<pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span>
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
<a class="link" href="../../main_overview/result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">zeta</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">></span>
<a class="link" href="../../main_overview/result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">zeta</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">&);</span>
<span class="special">}}</span> <span class="comment">// namespaces
</span></pre>
<p>
The return type of these functions is computed using the <a class="link" href="../../main_overview/result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>result
type calculation rules</em></span></a>: the return type is <code class="computeroutput"><span class="keyword">double</span></code> if T is an integer type, and T otherwise.
</p>
<p>
</p>
<p>
The final <a class="link" href="../../policy.html" title="Policies">Policy</a> argument
is optional and can be used to control the behaviour of the function:
how it handles errors, what level of precision to use etc. Refer to the
<a class="link" href="../../policy.html" title="Policies">policy documentation for more details</a>.
</p>
<p>
</p>
<a name="math_toolkit.special.zetas.zeta.description"></a><h5>
<a name="id1145205"></a>
<a class="link" href="zeta.html#math_toolkit.special.zetas.zeta.description">Description</a>
</h5>
<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
<a class="link" href="../../main_overview/result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">zeta</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">></span>
<a class="link" href="../../main_overview/result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">zeta</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">&);</span>
</pre>
<p>
Returns the <a href="http://mathworld.wolfram.com/RiemannZetaFunction.html" target="_top">zeta
function</a> of z:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../../equations/zeta1.png"></span>
</p>
<p>
<span class="inlinemediaobject"><img src="../../../../graphs/zeta1.png" align="middle"></span>
</p>
<p>
<span class="inlinemediaobject"><img src="../../../../graphs/zeta2.png" align="middle"></span>
</p>
<a name="math_toolkit.special.zetas.zeta.accuracy"></a><h5>
<a name="id1145429"></a>
<a class="link" href="zeta.html#math_toolkit.special.zetas.zeta.accuracy">Accuracy</a>
</h5>
<p>
The following table shows the peak errors (in units of epsilon) found on
various platforms with various floating point types, along with comparisons
to the <a href="http://www.gnu.org/software/gsl/" target="_top">GSL-1.9</a> and
<a href="http://www.netlib.org/cephes/" target="_top">Cephes</a> libraries. Unless
otherwise specified any floating point type that is narrower than the one
shown will have <a class="link" href="../../backgrounders/relative_error.html#zero_error">effectively zero error</a>.
</p>
<div class="table">
<a name="id1145457"></a><p class="title"><b>Table 44. Errors In the Function zeta(z)</b></p>
<div class="table-contents"><table class="table" summary="Errors In the Function zeta(z)">
<colgroup>
<col>
<col>
<col>
<col>
</colgroup>
<thead><tr>
<th>
<p>
Significand Size
</p>
</th>
<th>
<p>
Platform and Compiler
</p>
</th>
<th>
<p>
z > 0
</p>
</th>
<th>
<p>
z < 0
</p>
</th>
</tr></thead>
<tbody>
<tr>
<td>
<p>
53
</p>
</td>
<td>
<p>
Win32, Visual C++ 8
</p>
</td>
<td>
<p>
Peak=0.99 Mean=0.1
</p>
<p>
GSL Peak=8.7 Mean=1.0
</p>
<p>
<a href="http://www.netlib.org/cephes/" target="_top">Cephes</a> Peak=2.1
Mean=1.1
</p>
</td>
<td>
<p>
Peak=7.1 Mean=3.0
</p>
<p>
GSL Peak=137 Mean=14
</p>
<p>
<a href="http://www.netlib.org/cephes/" target="_top">Cephes</a> Peak=5084
Mean=470
</p>
</td>
</tr>
<tr>
<td>
<p>
64
</p>
</td>
<td>
<p>
RedHat Linux IA_EM64, gcc-4.1
</p>
</td>
<td>
<p>
Peak=0.99 Mean=0.5
</p>
</td>
<td>
<p>
Peak=570 Mean=60
</p>
</td>
</tr>
<tr>
<td>
<p>
64
</p>
</td>
<td>
<p>
Redhat Linux IA64, gcc-4.1
</p>
</td>
<td>
<p>
Peak=0.99 Mean=0.5
</p>
</td>
<td>
<p>
Peak=559 Mean=56
</p>
</td>
</tr>
<tr>
<td>
<p>
113
</p>
</td>
<td>
<p>
HPUX IA64, aCC A.06.06
</p>
</td>
<td>
<p>
Peak=1.0 Mean=0.4
</p>
</td>
<td>
<p>
Peak=1018 Mean=79
</p>
</td>
</tr>
</tbody>
</table></div>
</div>
<br class="table-break"><a name="math_toolkit.special.zetas.zeta.testing"></a><h5>
<a name="id1145686"></a>
<a class="link" href="zeta.html#math_toolkit.special.zetas.zeta.testing">Testing</a>
</h5>
<p>
The tests for these functions come in two parts: basic sanity checks use
spot values calculated using <a href="http://functions.wolfram.com/webMathematica/FunctionEvaluation.jsp?name=Zeta" target="_top">Mathworld's
online evaluator</a>, while accuracy checks use high-precision test
values calculated at 1000-bit precision with <a href="http://shoup.net/ntl/doc/RR.txt" target="_top">NTL::RR</a>
and this implementation. Note that the generic and type-specific versions
of these functions use differing implementations internally, so this gives
us reasonably independent test data. Using our test data to test other
"known good" implementations also provides an additional sanity
check.
</p>
<a name="math_toolkit.special.zetas.zeta.implementation"></a><h5>
<a name="id1145712"></a>
<a class="link" href="zeta.html#math_toolkit.special.zetas.zeta.implementation">Implementation</a>
</h5>
<p>
All versions of these functions first use the usual reflection formulas
to make their arguments positive:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../../equations/zeta3.png"></span>
</p>
<p>
The generic versions of these functions are implemented using the series:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../../equations/zeta1.png"></span>
</p>
<p>
for large z, and using the globally convergent series:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../../equations/zeta2.png"></span>
</p>
<p>
In all other cases. The crossover point for these is chosen so that the
first series is used only if it will converge reasonably quickly, the problem
with this series is that convergence become slower the more terms you take,
so we really do have to be certain of convergence before using this series,
even though the alternative is often quite slow.
</p>
<p>
When the significand (mantissa) size is recognised (currently for 53, 64
and 113-bit reals, plus single-precision 24-bit handled via promotion to
double) then a series of rational approximations <a class="link" href="../../backgrounders/implementation.html#math_toolkit.backgrounders.implementation.rational_approximations_used">devised
by JM</a> are used.
</p>
<p>
For 0 < z < 1 the approximating form is:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../../equations/zeta4.png"></span>
</p>
<p>
For a rational approximation R(1-z) and a constant C.
</p>
<p>
For 1 < z < 4 the approximating form is:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../../equations/zeta5.png"></span>
</p>
<p>
For a rational approximation R(n-z) and a constant C and integer n.
</p>
<p>
For z > 4 the approximating form is:
</p>
<p>
ζ(z) = 1 + e<sup>R(z - n)</sup>
</p>
<p>
For a rational approximation R(z-n) and integer n, note that the accuracy
required for R(z-n) is not full machine precision, but an absolute error
of: ε/R(0). This saves us quite a few digits when dealing with large z,
especially when ε is small.
</p>
</div>
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Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani
and Thijs van den Berg<p>
Distributed under the Boost Software License, Version 1.0. (See accompanying
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
</p>
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