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[CI SKIP] corrected accent etc in refs hypergeometric.gbk

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2019-08-09 16:35:25 +01:00
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</div>
</div>
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
<td align="left"><p><small>Last revised: August 09, 2019 at 11:49:38 GMT</small></p></td>
<td align="left"><p><small>Last revised: August 09, 2019 at 15:22:32 GMT</small></p></td>
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<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/sf_implementation.html" title="Additional Implementation Notes"><span class="index-entry-level-1">Additional Implementation Notes</span></a></p></li>
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<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/cubic_b.html" title="Cubic B-spline interpolation"><span class="index-entry-level-1">Cubic B-spline interpolation</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/stat_tut/weg/inverse_chi_squared_eg.html" title="Inverse Chi-Squared Distribution Bayes Example"><span class="index-entry-level-1">Inverse Chi-Squared Distribution Bayes Example</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/polynomials.html" title="Polynomials"><span class="index-entry-level-1">Polynomials</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/stat_tut/weg/st_eg/tut_mean_test.html" title='Testing a sample mean for difference from a "true" mean'><span class="index-entry-level-1">Testing a sample mean for difference from a "true" mean</span></a></p></li>

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<div class="section">
<div class="titlepage"><div><div><h2 class="title" style="clear: both">
<a name="id2001459"></a>Class Index</h2></div></div></div>
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<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/cubic_b.html" title="Cubic B-spline interpolation"><span class="index-entry-level-1">accuracy</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/cubic_b.html" title="Cubic B-spline interpolation"><span class="index-entry-level-1">constants</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/cubic_b.html" title="Cubic B-spline interpolation"><span class="index-entry-level-1">cubic_b_spline</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/cubic_b.html" title="Cubic B-spline interpolation"><span class="index-entry-level-1">data</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/cubic_b.html" title="Cubic B-spline interpolation"><span class="index-entry-level-1">interpolation</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/cubic_b.html" title="Cubic B-spline interpolation"><span class="index-entry-level-1">operator</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/cubic_b.html" title="Cubic B-spline interpolation"><span class="index-entry-level-1">p</span></a></p></li>
@@ -2663,6 +2664,7 @@
<div class="index"><ul class="index" style="list-style-type: none; ">
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/sf_implementation.html" title="Additional Implementation Notes"><span class="index-entry-level-1">Additional Implementation Notes</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/bessel/bessel_first.html" title="Bessel Functions of the First and Second Kinds"><span class="index-entry-level-1">Bessel Functions of the First and Second Kinds</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/cubic_b.html" title="Cubic B-spline interpolation"><span class="index-entry-level-1">Cubic B-spline interpolation</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/stat_tut/weg/inverse_chi_squared_eg.html" title="Inverse Chi-Squared Distribution Bayes Example"><span class="index-entry-level-1">Inverse Chi-Squared Distribution Bayes Example</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/polynomials.html" title="Polynomials"><span class="index-entry-level-1">Polynomials</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/stat_tut/weg/st_eg/tut_mean_test.html" title='Testing a sample mean for difference from a "true" mean'><span class="index-entry-level-1">Testing a sample mean for difference from a "true" mean</span></a></p></li>

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<a name="math_toolkit.conventions"></a><a class="link" href="conventions.html" title="Document Conventions">Document Conventions</a>
</h2></div></div></div>
<p>
<a class="indexterm" name="id1001310"></a>
<a class="indexterm" name="id992320"></a>
</p>
<p>
This documentation aims to use of the following naming and formatting conventions.

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<span class="identifier">Real</span> <span class="keyword">operator</span><span class="special">()(</span><span class="identifier">Real</span> <span class="identifier">x</span><span class="special">)</span> <span class="keyword">const</span><span class="special">;</span>
<span class="identifier">Real</span> <span class="identifier">prime</span><span class="special">(</span><span class="identifier">Real</span> <span class="identifier">x</span><span class="special">)</span> <span class="keyword">const</span><span class="special">;</span>
<span class="identifier">Real</span> <span class="identifier">double_prime</span><span class="special">(</span><span class="identifier">Real</span> <span class="identifier">x</span><span class="special">)</span> <span class="keyword">const</span><span class="special">;</span>
<span class="special">};</span>
<span class="special">}}</span> <span class="comment">// namespaces</span>
@@ -60,7 +62,7 @@
B-Spline Interpolation</a>
</h4>
<p>
The cubic B-spline class provided by boost allows fast and accurate interpolation
The cubic B-spline class provided by Boost allows fast and accurate interpolation
of a function which is known at equally spaced points. The cubic B-spline interpolation
is numerically stable as it uses compactly supported basis functions constructed
via iterative convolution. This is to be contrasted to traditional cubic spline
@@ -131,6 +133,20 @@
order lower than the guarantees on the original function, see <a href="http://www.springer.com/us/book/9780387984087" target="_top">Numerical
Analysis, Graduate Texts in Mathematics, 181, Rainer Kress</a> for details.
</p>
<p>
The last interesting member is the second derivative, evaluated via
</p>
<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">ypp</span> <span class="special">=</span> <span class="identifier">spline</span><span class="special">.</span><span class="identifier">double_prime</span><span class="special">(</span><span class="identifier">x</span><span class="special">);</span>
</pre>
<p>
The basis functions of the spline are cubic polynomials, so the second derivative
is simply linear interpolation. But the interpolation is not constrained by
data (you don't pass in the second derivatives), and hence is less accurate
than would be expected from assuming it is a linear interpolation. The problem
is especially pronounced at the boundaries, where the second derivative is
totally unconstrained. Use the second derivative of the cubic B-spline interpolator
only in desperation.
</p>
<p>
Finally, note that this is an interpolator, not an extrapolator. Therefore,
you should strenuously avoid evaluating the spline outside the endpoints. However,

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<a name="math_toolkit.navigation"></a><a class="link" href="navigation.html" title="Navigation">Navigation</a>
</h2></div></div></div>
<p>
<a class="indexterm" name="id1001129"></a>
<a class="indexterm" name="id992178"></a>
</p>
<p>
Boost.Math documentation is provided in both HTML and PDF formats.

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[section:hypergeometric Hypergeometric Functions]
[section:hyper_geometric_1f0 Hypergeometric [sub 1]/F/[sub 0] ]
@@ -513,11 +512,11 @@ error inherent in calculating the N'th term via logarithms.
# Beals, Richard, and Roderick Wong. ['Special functions: a graduate text.] Vol. 126. Cambridge University Press, 2010.
# Pearson, John W., Sheehan Olver, and Mason A. Porter. ['Numerical methods for the computation of the confluent and Gauss hypergeometric functions.] Numerical Algorithms 74.3 (2017): 821-866.
# Luke, Yudell L. ['Algorithms for Rational Approximations for a Confluent Hypergeometric Function II.] MISSOURI UNIV KANSAS CITY DEPT OF MATHEMATICS, 1976.
# Derezinski, Jan. ['Hypergeometric type functions and their symmetries.] Annales Henri Poincaré. Vol. 15. No. 8. Springer Basel, 2014.
# Derezinski, Jan. ['Hypergeometric type functions and their symmetries.] Annales Henri Poincar[eacute]. Vol. 15. No. 8. Springer Basel, 2014.
# Keith E. Muller ['Computing the confluent hypergeometric function, M(a, b, x)]. Numer. Math. 90: 179-196 (2001).
# Carlo Morosi, Livio Pizzocchero. ['On the expansion of the Kummer function in terms of incomplete Gamma functions.] Arch. Inequal. Appl. 2 (2004), 49-72.
# Jose Luis Lopez, Nico M. Temme. ['Asymptotics and numerics of polynomials used in Tricomi and Buchholz expansions of Kummer functions]. Numerische Mathematik, August 2010.
# Javier Sesma. ['The Temme’s sum rule for confluent hypergeometric functions revisited]. Journal of Computational and Applied Mathematics 163 (2004) 429–431.
# Javier Sesma. ['The Temme's sum rule for confluent hypergeometric functions revisited]. Journal of Computational and Applied Mathematics 163 (2004) 429-431.
# Javier Segura, Nico M. Temme. ['Numerically satisfactory solutions of Kummer recurrence relations]. Numer. Math. (2008) 111:109-119.
# Alfredo Deano, Javier Segura. ['Transitory Minimal Solutions Of Hypergeometric Recursions And Pseudoconvergence of Associated Continued Fractions]. Mathematics of Computation, Volume 76, Number 258, April 2007.
# W. Gautschi. ['Computational aspects of three-term recurrence relations]. SIAM Review 9, no.1 (1967) 24-82.