golden hour

/usr/include/c++/4.8.2/tr1

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array	6.8 KB
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ccomplex	1.23 KB
cctype	1.38 KB
cfenv	1.96 KB
cfloat	1.35 KB
cinttypes	2.2 KB
climits	1.42 KB
cmath	36.55 KB
complex	12.04 KB
complex.h	1.23 KB
cstdarg	1.22 KB
cstdbool	1.31 KB
cstdint	2.56 KB
cstdio	1.44 KB
cstdlib	1.74 KB
ctgmath	1.22 KB
ctime	1.21 KB
ctype.h	1.18 KB
cwchar	1.67 KB
cwctype	1.42 KB
ell_integral.tcc	26.85 KB
exp_integral.tcc	15.41 KB
fenv.h	1.18 KB
float.h	1.18 KB
functional	69.15 KB
functional_hash.h	5.7 KB
gamma.tcc	13.97 KB
hashtable.h	40.56 KB
hashtable_policy.h	24.64 KB
hypergeometric.tcc	27.07 KB
inttypes.h	1.24 KB
legendre_function.tcc	10.32 KB
limits.h	1.19 KB
math.h	4.45 KB
memory	1.75 KB
modified_bessel_func.tcc	15.35 KB
poly_hermite.tcc	3.61 KB
poly_laguerre.tcc	11.08 KB
random	1.55 KB
random.h	71.48 KB
random.tcc	52.73 KB
regex	90.77 KB
riemann_zeta.tcc	13.34 KB
shared_ptr.h	31.91 KB
special_function_util.h	4.71 KB
stdarg.h	1.19 KB
stdbool.h	1.19 KB
stdint.h	1.19 KB
stdio.h	1.18 KB
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tuple	11.83 KB
type_traits	18.57 KB
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unordered_set	1.54 KB
unordered_set.h	9.32 KB
utility	3.15 KB
wchar.h	1.22 KB
wctype.h	1.23 KB

Edit: poly_laguerre.tcc

// Special functions -*- C++ -*-

// Copyright (C) 2006-2013 Free Software Foundation, Inc.
//
// This file is part of the GNU ISO C++ Library.  This library is free
// software; you can redistribute it and/or modify it under the
// terms of the GNU General Public License as published by the
// Free Software Foundation; either version 3, or (at your option)
// any later version.
//
// This library is distributed in the hope that it will be useful,
// but WITHOUT ANY WARRANTY; without even the implied warranty of
// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
// GNU General Public License for more details.
//
// Under Section 7 of GPL version 3, you are granted additional
// permissions described in the GCC Runtime Library Exception, version
// 3.1, as published by the Free Software Foundation.

// You should have received a copy of the GNU General Public License and
// a copy of the GCC Runtime Library Exception along with this program;
// see the files COPYING3 and COPYING.RUNTIME respectively.  If not, see
// <http://www.gnu.org/licenses/>.

/** @file tr1/poly_laguerre.tcc
 *  This is an internal header file, included by other library headers.
 *  Do not attempt to use it directly. @headername{tr1/cmath}
 */

//
// ISO C++ 14882 TR1: 5.2  Special functions
//

// Written by Edward Smith-Rowland based on:
//   (1) Handbook of Mathematical Functions,
//       Ed. Milton Abramowitz and Irene A. Stegun,
//       Dover Publications,
//       Section 13, pp. 509-510, Section 22 pp. 773-802
//   (2) The Gnu Scientific Library, http://www.gnu.org/software/gsl

#ifndef _GLIBCXX_TR1_POLY_LAGUERRE_TCC
#define _GLIBCXX_TR1_POLY_LAGUERRE_TCC 1

namespace std _GLIBCXX_VISIBILITY(default)
{
namespace tr1
{
  // [5.2] Special functions

// Implementation-space details.
  namespace __detail
  {
  _GLIBCXX_BEGIN_NAMESPACE_VERSION

/**
     *   @brief This routine returns the associated Laguerre polynomial 
     *          of order @f$ n @f$, degree @f$ \alpha @f$ for large n.
     *   Abramowitz & Stegun, 13.5.21
     *
     *   @param __n The order of the Laguerre function.
     *   @param __alpha The degree of the Laguerre function.
     *   @param __x The argument of the Laguerre function.
     *   @return The value of the Laguerre function of order n,
     *           degree @f$ \alpha @f$, and argument x.
     *
     *  This is from the GNU Scientific Library.
     */
    template<typename _Tpa, typename _Tp>
    _Tp
    __poly_laguerre_large_n(unsigned __n, _Tpa __alpha1, _Tp __x)
    {
      const _Tp __a = -_Tp(__n);
      const _Tp __b = _Tp(__alpha1) + _Tp(1);
      const _Tp __eta = _Tp(2) * __b - _Tp(4) * __a;
      const _Tp __cos2th = __x / __eta;
      const _Tp __sin2th = _Tp(1) - __cos2th;
      const _Tp __th = std::acos(std::sqrt(__cos2th));
      const _Tp __pre_h = __numeric_constants<_Tp>::__pi_2()
                        * __numeric_constants<_Tp>::__pi_2()
                        * __eta * __eta * __cos2th * __sin2th;

#if _GLIBCXX_USE_C99_MATH_TR1
      const _Tp __lg_b = std::tr1::lgamma(_Tp(__n) + __b);
      const _Tp __lnfact = std::tr1::lgamma(_Tp(__n + 1));
#else
      const _Tp __lg_b = __log_gamma(_Tp(__n) + __b);
      const _Tp __lnfact = __log_gamma(_Tp(__n + 1));
#endif

_Tp __pre_term1 = _Tp(0.5L) * (_Tp(1) - __b)
                      * std::log(_Tp(0.25L) * __x * __eta);
      _Tp __pre_term2 = _Tp(0.25L) * std::log(__pre_h);
      _Tp __lnpre = __lg_b - __lnfact + _Tp(0.5L) * __x
                      + __pre_term1 - __pre_term2;
      _Tp __ser_term1 = std::sin(__a * __numeric_constants<_Tp>::__pi());
      _Tp __ser_term2 = std::sin(_Tp(0.25L) * __eta
                              * (_Tp(2) * __th
                               - std::sin(_Tp(2) * __th))
                               + __numeric_constants<_Tp>::__pi_4());
      _Tp __ser = __ser_term1 + __ser_term2;

return std::exp(__lnpre) * __ser;
    }

/**
     *  @brief  Evaluate the polynomial based on the confluent hypergeometric
     *          function in a safe way, with no restriction on the arguments.
     *
     *   The associated Laguerre function is defined by
     *   @f[
     *       L_n^\alpha(x) = \frac{(\alpha + 1)_n}{n!}
     *                       _1F_1(-n; \alpha + 1; x)
     *   @f]
     *   where @f$ (\alpha)_n @f$ is the Pochhammer symbol and
     *   @f$ _1F_1(a; c; x) @f$ is the confluent hypergeometric function.
     *
     *  This function assumes x != 0.
     *
     *  This is from the GNU Scientific Library.
     */
    template<typename _Tpa, typename _Tp>
    _Tp
    __poly_laguerre_hyperg(unsigned int __n, _Tpa __alpha1, _Tp __x)
    {
      const _Tp __b = _Tp(__alpha1) + _Tp(1);
      const _Tp __mx = -__x;
      const _Tp __tc_sgn = (__x < _Tp(0) ? _Tp(1)
                         : ((__n % 2 == 1) ? -_Tp(1) : _Tp(1)));
      //  Get |x|^n/n!
      _Tp __tc = _Tp(1);
      const _Tp __ax = std::abs(__x);
      for (unsigned int __k = 1; __k <= __n; ++__k)
        __tc *= (__ax / __k);

_Tp __term = __tc * __tc_sgn;
      _Tp __sum = __term;
      for (int __k = int(__n) - 1; __k >= 0; --__k)
        {
          __term *= ((__b + _Tp(__k)) / _Tp(int(__n) - __k))
                  * _Tp(__k + 1) / __mx;
          __sum += __term;
        }

return __sum;
    }

/**
     *   @brief This routine returns the associated Laguerre polynomial 
     *          of order @f$ n @f$, degree @f$ \alpha @f$: @f$ L_n^\alpha(x) @f$
     *          by recursion.
     *
     *   The associated Laguerre function is defined by
     *   @f[
     *       L_n^\alpha(x) = \frac{(\alpha + 1)_n}{n!}
     *                       _1F_1(-n; \alpha + 1; x)
     *   @f]
     *   where @f$ (\alpha)_n @f$ is the Pochhammer symbol and
     *   @f$ _1F_1(a; c; x) @f$ is the confluent hypergeometric function.
     *
     *   The associated Laguerre polynomial is defined for integral
     *   @f$ \alpha = m @f$ by:
     *   @f[
     *       L_n^m(x) = (-1)^m \frac{d^m}{dx^m} L_{n + m}(x)
     *   @f]
     *   where the Laguerre polynomial is defined by:
     *   @f[
     *       L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n} (x^ne^{-x})
     *   @f]
     *
     *   @param __n The order of the Laguerre function.
     *   @param __alpha The degree of the Laguerre function.
     *   @param __x The argument of the Laguerre function.
     *   @return The value of the Laguerre function of order n,
     *           degree @f$ \alpha @f$, and argument x.
     */
    template<typename _Tpa, typename _Tp>
    _Tp
    __poly_laguerre_recursion(unsigned int __n, _Tpa __alpha1, _Tp __x)
    {
      //   Compute l_0.
      _Tp __l_0 = _Tp(1);
      if  (__n == 0)
        return __l_0;

//  Compute l_1^alpha.
      _Tp __l_1 = -__x + _Tp(1) + _Tp(__alpha1);
      if  (__n == 1)
        return __l_1;

//  Compute l_n^alpha by recursion on n.
      _Tp __l_n2 = __l_0;
      _Tp __l_n1 = __l_1;
      _Tp __l_n = _Tp(0);
      for  (unsigned int __nn = 2; __nn <= __n; ++__nn)
        {
            __l_n = (_Tp(2 * __nn - 1) + _Tp(__alpha1) - __x)
                  * __l_n1 / _Tp(__nn)
                  - (_Tp(__nn - 1) + _Tp(__alpha1)) * __l_n2 / _Tp(__nn);
            __l_n2 = __l_n1;
            __l_n1 = __l_n;
        }

return __l_n;
    }

/**
     *   @brief This routine returns the associated Laguerre polynomial
     *          of order n, degree @f$ \alpha @f$: @f$ L_n^alpha(x) @f$.
     *
     *   The associated Laguerre function is defined by
     *   @f[
     *       L_n^\alpha(x) = \frac{(\alpha + 1)_n}{n!}
     *                       _1F_1(-n; \alpha + 1; x)
     *   @f]
     *   where @f$ (\alpha)_n @f$ is the Pochhammer symbol and
     *   @f$ _1F_1(a; c; x) @f$ is the confluent hypergeometric function.
     *
     *   The associated Laguerre polynomial is defined for integral
     *   @f$ \alpha = m @f$ by:
     *   @f[
     *       L_n^m(x) = (-1)^m \frac{d^m}{dx^m} L_{n + m}(x)
     *   @f]
     *   where the Laguerre polynomial is defined by:
     *   @f[
     *       L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n} (x^ne^{-x})
     *   @f]
     *
     *   @param __n The order of the Laguerre function.
     *   @param __alpha The degree of the Laguerre function.
     *   @param __x The argument of the Laguerre function.
     *   @return The value of the Laguerre function of order n,
     *           degree @f$ \alpha @f$, and argument x.
     */
    template<typename _Tpa, typename _Tp>
    _Tp
    __poly_laguerre(unsigned int __n, _Tpa __alpha1, _Tp __x)
    {
      if (__x < _Tp(0))
        std::__throw_domain_error(__N("Negative argument "
                                      "in __poly_laguerre."));
      //  Return NaN on NaN input.
      else if (__isnan(__x))
        return std::numeric_limits<_Tp>::quiet_NaN();
      else if (__n == 0)
        return _Tp(1);
      else if (__n == 1)
        return _Tp(1) + _Tp(__alpha1) - __x;
      else if (__x == _Tp(0))
        {
          _Tp __prod = _Tp(__alpha1) + _Tp(1);
          for (unsigned int __k = 2; __k <= __n; ++__k)
            __prod *= (_Tp(__alpha1) + _Tp(__k)) / _Tp(__k);
          return __prod;
        }
      else if (__n > 10000000 && _Tp(__alpha1) > -_Tp(1)
            && __x < _Tp(2) * (_Tp(__alpha1) + _Tp(1)) + _Tp(4 * __n))
        return __poly_laguerre_large_n(__n, __alpha1, __x);
      else if (_Tp(__alpha1) >= _Tp(0)
           || (__x > _Tp(0) && _Tp(__alpha1) < -_Tp(__n + 1)))
        return __poly_laguerre_recursion(__n, __alpha1, __x);
      else
        return __poly_laguerre_hyperg(__n, __alpha1, __x);
    }

/**
     *   @brief This routine returns the associated Laguerre polynomial
     *          of order n, degree m: @f$ L_n^m(x) @f$.
     *
     *   The associated Laguerre polynomial is defined for integral
     *   @f$ \alpha = m @f$ by:
     *   @f[
     *       L_n^m(x) = (-1)^m \frac{d^m}{dx^m} L_{n + m}(x)
     *   @f]
     *   where the Laguerre polynomial is defined by:
     *   @f[
     *       L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n} (x^ne^{-x})
     *   @f]
     *
     *   @param __n The order of the Laguerre polynomial.
     *   @param __m The degree of the Laguerre polynomial.
     *   @param __x The argument of the Laguerre polynomial.
     *   @return The value of the associated Laguerre polynomial of order n,
     *           degree m, and argument x.
     */
    template<typename _Tp>
    inline _Tp
    __assoc_laguerre(unsigned int __n, unsigned int __m, _Tp __x)
    { return __poly_laguerre<unsigned int, _Tp>(__n, __m, __x); }

/**
     *   @brief This routine returns the Laguerre polynomial
     *          of order n: @f$ L_n(x) @f$.
     *
     *   The Laguerre polynomial is defined by:
     *   @f[
     *       L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n} (x^ne^{-x})
     *   @f]
     *
     *   @param __n The order of the Laguerre polynomial.
     *   @param __x The argument of the Laguerre polynomial.
     *   @return The value of the Laguerre polynomial of order n
     *           and argument x.
     */
    template<typename _Tp>
    inline _Tp
    __laguerre(unsigned int __n, _Tp __x)
    { return __poly_laguerre<unsigned int, _Tp>(__n, 0, __x); }

_GLIBCXX_END_NAMESPACE_VERSION
  } // namespace std::tr1::__detail
}
}

#endif // _GLIBCXX_TR1_POLY_LAGUERRE_TCC