poly_laguerre.tcc

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00001 // Special functions -*- C++ -*-
00002 
00003 // Copyright (C) 2006, 2007, 2008
00004 // Free Software Foundation, Inc.
00005 //
00006 // This file is part of the GNU ISO C++ Library.  This library is free
00007 // software; you can redistribute it and/or modify it under the
00008 // terms of the GNU General Public License as published by the
00009 // Free Software Foundation; either version 2, or (at your option)
00010 // any later version.
00011 //
00012 // This library is distributed in the hope that it will be useful,
00013 // but WITHOUT ANY WARRANTY; without even the implied warranty of
00014 // MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
00015 // GNU General Public License for more details.
00016 //
00017 // You should have received a copy of the GNU General Public License along
00018 // with this library; see the file COPYING.  If not, write to the Free
00019 // Software Foundation, 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301,
00020 // USA.
00021 //
00022 // As a special exception, you may use this file as part of a free software
00023 // library without restriction.  Specifically, if other files instantiate
00024 // templates or use macros or inline functions from this file, or you compile
00025 // this file and link it with other files to produce an executable, this
00026 // file does not by itself cause the resulting executable to be covered by
00027 // the GNU General Public License.  This exception does not however
00028 // invalidate any other reasons why the executable file might be covered by
00029 // the GNU General Public License.
00030 
00031 /** @file tr1/poly_laguerre.tcc
00032  *  This is an internal header file, included by other library headers.
00033  *  You should not attempt to use it directly.
00034  */
00035 
00036 //
00037 // ISO C++ 14882 TR1: 5.2  Special functions
00038 //
00039 
00040 // Written by Edward Smith-Rowland based on:
00041 //   (1) Handbook of Mathematical Functions,
00042 //       Ed. Milton Abramowitz and Irene A. Stegun,
00043 //       Dover Publications,
00044 //       Section 13, pp. 509-510, Section 22 pp. 773-802
00045 //   (2) The Gnu Scientific Library, http://www.gnu.org/software/gsl
00046 
00047 #ifndef _GLIBCXX_TR1_POLY_LAGUERRE_TCC
00048 #define _GLIBCXX_TR1_POLY_LAGUERRE_TCC 1
00049 
00050 namespace std
00051 {
00052 namespace tr1
00053 {
00054 
00055   // [5.2] Special functions
00056 
00057   // Implementation-space details.
00058   namespace __detail
00059   {
00060 
00061 
00062     /**
00063      *   @brief This routine returns the associated Laguerre polynomial 
00064      *          of order @f$ n @f$, degree @f$ \alpha @f$ for large n.
00065      *   Abramowitz & Stegun, 13.5.21
00066      *
00067      *   @param __n The order of the Laguerre function.
00068      *   @param __alpha The degree of the Laguerre function.
00069      *   @param __x The argument of the Laguerre function.
00070      *   @return The value of the Laguerre function of order n,
00071      *           degree @f$ \alpha @f$, and argument x.
00072      *
00073      *  This is from the GNU Scientific Library.
00074      */
00075     template<typename _Tpa, typename _Tp>
00076     _Tp
00077     __poly_laguerre_large_n(const unsigned __n, const _Tpa __alpha1,
00078                             const _Tp __x)
00079     {
00080       const _Tp __a = -_Tp(__n);
00081       const _Tp __b = _Tp(__alpha1) + _Tp(1);
00082       const _Tp __eta = _Tp(2) * __b - _Tp(4) * __a;
00083       const _Tp __cos2th = __x / __eta;
00084       const _Tp __sin2th = _Tp(1) - __cos2th;
00085       const _Tp __th = std::acos(std::sqrt(__cos2th));
00086       const _Tp __pre_h = __numeric_constants<_Tp>::__pi_2()
00087                         * __numeric_constants<_Tp>::__pi_2()
00088                         * __eta * __eta * __cos2th * __sin2th;
00089 
00090 #if _GLIBCXX_USE_C99_MATH_TR1
00091       const _Tp __lg_b = std::tr1::lgamma(_Tp(__n) + __b);
00092       const _Tp __lnfact = std::tr1::lgamma(_Tp(__n + 1));
00093 #else
00094       const _Tp __lg_b = __log_gamma(_Tp(__n) + __b);
00095       const _Tp __lnfact = __log_gamma(_Tp(__n + 1));
00096 #endif
00097 
00098       _Tp __pre_term1 = _Tp(0.5L) * (_Tp(1) - __b)
00099                       * std::log(_Tp(0.25L) * __x * __eta);
00100       _Tp __pre_term2 = _Tp(0.25L) * std::log(__pre_h);
00101       _Tp __lnpre = __lg_b - __lnfact + _Tp(0.5L) * __x
00102                       + __pre_term1 - __pre_term2;
00103       _Tp __ser_term1 = std::sin(__a * __numeric_constants<_Tp>::__pi());
00104       _Tp __ser_term2 = std::sin(_Tp(0.25L) * __eta
00105                               * (_Tp(2) * __th
00106                                - std::sin(_Tp(2) * __th))
00107                                + __numeric_constants<_Tp>::__pi_4());
00108       _Tp __ser = __ser_term1 + __ser_term2;
00109 
00110       return std::exp(__lnpre) * __ser;
00111     }
00112 
00113 
00114     /**
00115      *  @brief  Evaluate the polynomial based on the confluent hypergeometric
00116      *          function in a safe way, with no restriction on the arguments.
00117      *
00118      *   The associated Laguerre function is defined by
00119      *   @f[
00120      *       L_n^\alpha(x) = \frac{(\alpha + 1)_n}{n!}
00121      *                       _1F_1(-n; \alpha + 1; x)
00122      *   @f]
00123      *   where @f$ (\alpha)_n @f$ is the Pochhammer symbol and
00124      *   @f$ _1F_1(a; c; x) @f$ is the confluent hypergeometric function.
00125      *
00126      *  This function assumes x != 0.
00127      *
00128      *  This is from the GNU Scientific Library.
00129      */
00130     template<typename _Tpa, typename _Tp>
00131     _Tp
00132     __poly_laguerre_hyperg(const unsigned int __n, const _Tpa __alpha1,
00133                const _Tp __x)
00134     {
00135       const _Tp __b = _Tp(__alpha1) + _Tp(1);
00136       const _Tp __mx = -__x;
00137       const _Tp __tc_sgn = (__x < _Tp(0) ? _Tp(1)
00138                          : ((__n % 2 == 1) ? -_Tp(1) : _Tp(1)));
00139       //  Get |x|^n/n!
00140       _Tp __tc = _Tp(1);
00141       const _Tp __ax = std::abs(__x);
00142       for (unsigned int __k = 1; __k <= __n; ++__k)
00143         __tc *= (__ax / __k);
00144 
00145       _Tp __term = __tc * __tc_sgn;
00146       _Tp __sum = __term;
00147       for (int __k = int(__n) - 1; __k >= 0; --__k)
00148         {
00149           __term *= ((__b + _Tp(__k)) / _Tp(int(__n) - __k))
00150                   * _Tp(__k + 1) / __mx;
00151           __sum += __term;
00152         }
00153 
00154       return __sum;
00155     }
00156 
00157 
00158     /**
00159      *   @brief This routine returns the associated Laguerre polynomial 
00160      *          of order @f$ n @f$, degree @f$ \alpha @f$: @f$ L_n^\alpha(x) @f$
00161      *          by recursion.
00162      *
00163      *   The associated Laguerre function is defined by
00164      *   @f[
00165      *       L_n^\alpha(x) = \frac{(\alpha + 1)_n}{n!}
00166      *                       _1F_1(-n; \alpha + 1; x)
00167      *   @f]
00168      *   where @f$ (\alpha)_n @f$ is the Pochhammer symbol and
00169      *   @f$ _1F_1(a; c; x) @f$ is the confluent hypergeometric function.
00170      *
00171      *   The associated Laguerre polynomial is defined for integral
00172      *   @f$ \alpha = m @f$ by:
00173      *   @f[
00174      *       L_n^m(x) = (-1)^m \frac{d^m}{dx^m} L_{n + m}(x)
00175      *   @f]
00176      *   where the Laguerre polynomial is defined by:
00177      *   @f[
00178      *       L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n} (x^ne^{-x})
00179      *   @f]
00180      *
00181      *   @param __n The order of the Laguerre function.
00182      *   @param __alpha The degree of the Laguerre function.
00183      *   @param __x The argument of the Laguerre function.
00184      *   @return The value of the Laguerre function of order n,
00185      *           degree @f$ \alpha @f$, and argument x.
00186      */
00187     template<typename _Tpa, typename _Tp>
00188     _Tp
00189     __poly_laguerre_recursion(const unsigned int __n,
00190                               const _Tpa __alpha1, const _Tp __x)
00191     {
00192       //   Compute l_0.
00193       _Tp __l_0 = _Tp(1);
00194       if  (__n == 0)
00195         return __l_0;
00196 
00197       //  Compute l_1^alpha.
00198       _Tp __l_1 = -__x + _Tp(1) + _Tp(__alpha1);
00199       if  (__n == 1)
00200         return __l_1;
00201 
00202       //  Compute l_n^alpha by recursion on n.
00203       _Tp __l_n2 = __l_0;
00204       _Tp __l_n1 = __l_1;
00205       _Tp __l_n = _Tp(0);
00206       for  (unsigned int __nn = 2; __nn <= __n; ++__nn)
00207         {
00208             __l_n = (_Tp(2 * __nn - 1) + _Tp(__alpha1) - __x)
00209                   * __l_n1 / _Tp(__nn)
00210                   - (_Tp(__nn - 1) + _Tp(__alpha1)) * __l_n2 / _Tp(__nn);
00211             __l_n2 = __l_n1;
00212             __l_n1 = __l_n;
00213         }
00214 
00215       return __l_n;
00216     }
00217 
00218 
00219     /**
00220      *   @brief This routine returns the associated Laguerre polynomial
00221      *          of order n, degree @f$ \alpha @f$: @f$ L_n^alpha(x) @f$.
00222      *
00223      *   The associated Laguerre function is defined by
00224      *   @f[
00225      *       L_n^\alpha(x) = \frac{(\alpha + 1)_n}{n!}
00226      *                       _1F_1(-n; \alpha + 1; x)
00227      *   @f]
00228      *   where @f$ (\alpha)_n @f$ is the Pochhammer symbol and
00229      *   @f$ _1F_1(a; c; x) @f$ is the confluent hypergeometric function.
00230      *
00231      *   The associated Laguerre polynomial is defined for integral
00232      *   @f$ \alpha = m @f$ by:
00233      *   @f[
00234      *       L_n^m(x) = (-1)^m \frac{d^m}{dx^m} L_{n + m}(x)
00235      *   @f]
00236      *   where the Laguerre polynomial is defined by:
00237      *   @f[
00238      *       L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n} (x^ne^{-x})
00239      *   @f]
00240      *
00241      *   @param __n The order of the Laguerre function.
00242      *   @param __alpha The degree of the Laguerre function.
00243      *   @param __x The argument of the Laguerre function.
00244      *   @return The value of the Laguerre function of order n,
00245      *           degree @f$ \alpha @f$, and argument x.
00246      */
00247     template<typename _Tpa, typename _Tp>
00248     inline _Tp
00249     __poly_laguerre(const unsigned int __n, const _Tpa __alpha1,
00250                     const _Tp __x)
00251     {
00252       if (__x < _Tp(0))
00253         std::__throw_domain_error(__N("Negative argument "
00254                                       "in __poly_laguerre."));
00255       //  Return NaN on NaN input.
00256       else if (__isnan(__x))
00257         return std::numeric_limits<_Tp>::quiet_NaN();
00258       else if (__n == 0)
00259         return _Tp(1);
00260       else if (__n == 1)
00261         return _Tp(1) + _Tp(__alpha1) - __x;
00262       else if (__x == _Tp(0))
00263         {
00264           _Tp __prod = _Tp(__alpha1) + _Tp(1);
00265           for (unsigned int __k = 2; __k <= __n; ++__k)
00266             __prod *= (_Tp(__alpha1) + _Tp(__k)) / _Tp(__k);
00267           return __prod;
00268         }
00269       else if (__n > 10000000 && _Tp(__alpha1) > -_Tp(1)
00270             && __x < _Tp(2) * (_Tp(__alpha1) + _Tp(1)) + _Tp(4 * __n))
00271         return __poly_laguerre_large_n(__n, __alpha1, __x);
00272       else if (_Tp(__alpha1) >= _Tp(0)
00273            || (__x > _Tp(0) && _Tp(__alpha1) < -_Tp(__n + 1)))
00274         return __poly_laguerre_recursion(__n, __alpha1, __x);
00275       else
00276         return __poly_laguerre_hyperg(__n, __alpha1, __x);
00277     }
00278 
00279 
00280     /**
00281      *   @brief This routine returns the associated Laguerre polynomial
00282      *          of order n, degree m: @f$ L_n^m(x) @f$.
00283      *
00284      *   The associated Laguerre polynomial is defined for integral
00285      *   @f$ \alpha = m @f$ by:
00286      *   @f[
00287      *       L_n^m(x) = (-1)^m \frac{d^m}{dx^m} L_{n + m}(x)
00288      *   @f]
00289      *   where the Laguerre polynomial is defined by:
00290      *   @f[
00291      *       L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n} (x^ne^{-x})
00292      *   @f]
00293      *
00294      *   @param __n The order of the Laguerre polynomial.
00295      *   @param __m The degree of the Laguerre polynomial.
00296      *   @param __x The argument of the Laguerre polynomial.
00297      *   @return The value of the associated Laguerre polynomial of order n,
00298      *           degree m, and argument x.
00299      */
00300     template<typename _Tp>
00301     inline _Tp
00302     __assoc_laguerre(const unsigned int __n, const unsigned int __m,
00303                      const _Tp __x)
00304     {
00305       return __poly_laguerre<unsigned int, _Tp>(__n, __m, __x);
00306     }
00307 
00308 
00309     /**
00310      *   @brief This routine returns the Laguerre polynomial
00311      *          of order n: @f$ L_n(x) @f$.
00312      *
00313      *   The Laguerre polynomial is defined by:
00314      *   @f[
00315      *       L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n} (x^ne^{-x})
00316      *   @f]
00317      *
00318      *   @param __n The order of the Laguerre polynomial.
00319      *   @param __x The argument of the Laguerre polynomial.
00320      *   @return The value of the Laguerre polynomial of order n
00321      *           and argument x.
00322      */
00323     template<typename _Tp>
00324     inline _Tp
00325     __laguerre(const unsigned int __n, const _Tp __x)
00326     {
00327       return __poly_laguerre<unsigned int, _Tp>(__n, 0, __x);
00328     }
00329 
00330   } // namespace std::tr1::__detail
00331 }
00332 }
00333 
00334 #endif // _GLIBCXX_TR1_POLY_LAGUERRE_TCC

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