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\defmodule {num2}
This module provides procedures to compute a few numerical
quantities such as factorials, combinations, Stirling numbers,
Bessel functions, gamma functions, and so on.
These functions are more esoteric than those provided by {\tt num}.
\bigskip\hrule
\code\hide
/* num2.h for ANSI C */
#ifndef NUM2_H
#define NUM2_H
\endhide
#include <testu01/gdef.h>
#include <math.h>
\endcode
%%%%%%%%%%%%%%%%%%%%
\guisec{Prototypes}
\code
double num2_Factorial (int n);
\endcode
\tab The factorial function. Returns the value of $n!$
\endtab
\code
double num2_LnFactorial (int n);
\endcode
\tab Returns the value of $\ln (n!)$, the natural logarithm of the
factorial of $n$. Gives at least 16 decimal digits of precision
(relative error $< 0.5\times 10^{-15}$)
\endtab
\code
double num2_Combination (int n, int s);
\endcode
\tab Returns the value of ${n \choose s}$, the number of different combinations
of $s$ objects amongst $n$. % Uses an algorithm that prevents overflows
% (when computing factorials), if possible.
\endtab
\code
#ifdef HAVE_LGAMMA
#define num2_LnGamma lgamma
#else
double num2_LnGamma (double x);
#endif
\endcode
\tab Calculates the natural logarithm of the gamma function $\Gamma(x)$
at {\tt x}. Our {\tt num2\_LnGamma} gives 16 decimal digits
of precision, but is implemented only for $x>0$.
The function {\tt lgamma} is from the {\tt ISO C99} standard math library.
\endtab
\code
double num2_Digamma (double x);
\endcode
\tab Returns the value of the logarithmic derivative of the Gamma function
$\psi(x) = \Gamma'(x) / \Gamma(x)$.
\endtab
\code
#ifdef HAVE_LOG1P
#define num2_log1p log1p
#else
double num2_log1p (double x);
#endif
\endcode
\tab Returns a value equivalent to {\tt log}$(1 + x)$ accurate also for small
$x$. The function {\tt log1p} is from the {\tt ISO C99} standard math library.
\endtab
\code
void num2_CalcMatStirling (double *** M, int m, int n);
\endcode
\tab Calculates the Stirling numbers of the second kind,
\eq
M[i,j] = \left\{\begin{array}{c}j \\ i\end{array}\right\}
\quad \mbox { for $0\le i\le m$ and $0\le i\le j\le n$}.
\label{Stirling2}
\endeq
See D. E. Knuth, {\em The Art of Computer Programming\/}, vol.~1,
second ed., 1973, Section 1.2.6.
The matrix $M$ is the transpose of Knuth's (1973).
This procedure allocates memory for the 2-dimensionnal matrix $M$,
and fills it with the values of Stirling numbers;
the memory should be freed
later with the function {\tt num2\_FreeMatStirling}.
\endtab
\code
void num2_FreeMatStirling (double *** M, int m);
\endcode
\tab Frees the memory space used by the Stirling matrix created by calling
{\tt num2\_CalcMatStirling}. The parameter {\tt m}
must be the same as the {\tt m} in {\tt num2\_CalcMatStirling}.
\endtab
\code
double num2_VolumeSphere (double p, int t);
\endcode
\tab Calculates the volume $V$ of a sphere of radius 1 in $t$ dimensions
using the norm $L_p$, according to the formula
$$
V = \frac{\left[2 \Gamma(1 + 1/p)\right]^t}
{\Gamma\left(1 + t/p\right)}, \qquad p > 0,
$$
where $\Gamma$ is the well-known gamma function.
The case of the sup norm $L_\infty$ is
obtained by choosing $p=0$.
Restrictions: $p\ge 0$ and $t\ge 1$.
\endtab
\code
double num2_EvalCheby (const double A[], int N, double x);
\endcode
\tab Evaluates a series of Chebyshev polynomials $T_j$, at point
$x \in [-1, \;1]$, using the method of Clenshaw \cite{mCLE62a},
i.e. calculates and returns
$$
y = \frac{A_0}2 + \sum_{j=1}^N A_j T_j(x).
$$
\endtab
\code
double num2_BesselK025 (double x);
\endcode
\tab Returns the value of $K_{1/4}(x)$, where $K_{\nu}$ is the modified
Bessel's
function of the second kind.
The relative error on the returned value is less than
$0.5\times 10^{-6}$ for $x > 10^{-300}$.
\endtab
\code
\hide
#endif
\endhide
\endcode
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