poly_laguerre.tcc

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00001 // Special functions -*- C++ -*-
00002 
00003 // Copyright (C) 2006, 2007, 2008, 2009
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 3, 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 // Under Section 7 of GPL version 3, you are granted additional
00018 // permissions described in the GCC Runtime Library Exception, version
00019 // 3.1, as published by the Free Software Foundation.
00020 
00021 // You should have received a copy of the GNU General Public License and
00022 // a copy of the GCC Runtime Library Exception along with this program;
00023 // see the files COPYING3 and COPYING.RUNTIME respectively.  If not, see
00024 // <http://www.gnu.org/licenses/>.
00025 
00026 /** @file tr1/poly_laguerre.tcc
00027  *  This is an internal header file, included by other library headers.
00028  *  You should not attempt to use it directly.
00029  */
00030 
00031 //
00032 // ISO C++ 14882 TR1: 5.2  Special functions
00033 //
00034 
00035 // Written by Edward Smith-Rowland based on:
00036 //   (1) Handbook of Mathematical Functions,
00037 //       Ed. Milton Abramowitz and Irene A. Stegun,
00038 //       Dover Publications,
00039 //       Section 13, pp. 509-510, Section 22 pp. 773-802
00040 //   (2) The Gnu Scientific Library, http://www.gnu.org/software/gsl
00041 
00042 #ifndef _GLIBCXX_TR1_POLY_LAGUERRE_TCC
00043 #define _GLIBCXX_TR1_POLY_LAGUERRE_TCC 1
00044 
00045 namespace std
00046 {
00047 namespace tr1
00048 {
00049 
00050   // [5.2] Special functions
00051 
00052   // Implementation-space details.
00053   namespace __detail
00054   {
00055 
00056 
00057     /**
00058      *   @brief This routine returns the associated Laguerre polynomial 
00059      *          of order @f$ n @f$, degree @f$ \alpha @f$ for large n.
00060      *   Abramowitz & Stegun, 13.5.21
00061      *
00062      *   @param __n The order of the Laguerre function.
00063      *   @param __alpha The degree of the Laguerre function.
00064      *   @param __x The argument of the Laguerre function.
00065      *   @return The value of the Laguerre function of order n,
00066      *           degree @f$ \alpha @f$, and argument x.
00067      *
00068      *  This is from the GNU Scientific Library.
00069      */
00070     template<typename _Tpa, typename _Tp>
00071     _Tp
00072     __poly_laguerre_large_n(const unsigned __n, const _Tpa __alpha1,
00073                             const _Tp __x)
00074     {
00075       const _Tp __a = -_Tp(__n);
00076       const _Tp __b = _Tp(__alpha1) + _Tp(1);
00077       const _Tp __eta = _Tp(2) * __b - _Tp(4) * __a;
00078       const _Tp __cos2th = __x / __eta;
00079       const _Tp __sin2th = _Tp(1) - __cos2th;
00080       const _Tp __th = std::acos(std::sqrt(__cos2th));
00081       const _Tp __pre_h = __numeric_constants<_Tp>::__pi_2()
00082                         * __numeric_constants<_Tp>::__pi_2()
00083                         * __eta * __eta * __cos2th * __sin2th;
00084 
00085 #if _GLIBCXX_USE_C99_MATH_TR1
00086       const _Tp __lg_b = std::tr1::lgamma(_Tp(__n) + __b);
00087       const _Tp __lnfact = std::tr1::lgamma(_Tp(__n + 1));
00088 #else
00089       const _Tp __lg_b = __log_gamma(_Tp(__n) + __b);
00090       const _Tp __lnfact = __log_gamma(_Tp(__n + 1));
00091 #endif
00092 
00093       _Tp __pre_term1 = _Tp(0.5L) * (_Tp(1) - __b)
00094                       * std::log(_Tp(0.25L) * __x * __eta);
00095       _Tp __pre_term2 = _Tp(0.25L) * std::log(__pre_h);
00096       _Tp __lnpre = __lg_b - __lnfact + _Tp(0.5L) * __x
00097                       + __pre_term1 - __pre_term2;
00098       _Tp __ser_term1 = std::sin(__a * __numeric_constants<_Tp>::__pi());
00099       _Tp __ser_term2 = std::sin(_Tp(0.25L) * __eta
00100                               * (_Tp(2) * __th
00101                                - std::sin(_Tp(2) * __th))
00102                                + __numeric_constants<_Tp>::__pi_4());
00103       _Tp __ser = __ser_term1 + __ser_term2;
00104 
00105       return std::exp(__lnpre) * __ser;
00106     }
00107 
00108 
00109     /**
00110      *  @brief  Evaluate the polynomial based on the confluent hypergeometric
00111      *          function in a safe way, with no restriction on the arguments.
00112      *
00113      *   The associated Laguerre function is defined by
00114      *   @f[
00115      *       L_n^\alpha(x) = \frac{(\alpha + 1)_n}{n!}
00116      *                       _1F_1(-n; \alpha + 1; x)
00117      *   @f]
00118      *   where @f$ (\alpha)_n @f$ is the Pochhammer symbol and
00119      *   @f$ _1F_1(a; c; x) @f$ is the confluent hypergeometric function.
00120      *
00121      *  This function assumes x != 0.
00122      *
00123      *  This is from the GNU Scientific Library.
00124      */
00125     template<typename _Tpa, typename _Tp>
00126     _Tp
00127     __poly_laguerre_hyperg(const unsigned int __n, const _Tpa __alpha1,
00128                const _Tp __x)
00129     {
00130       const _Tp __b = _Tp(__alpha1) + _Tp(1);
00131       const _Tp __mx = -__x;
00132       const _Tp __tc_sgn = (__x < _Tp(0) ? _Tp(1)
00133                          : ((__n % 2 == 1) ? -_Tp(1) : _Tp(1)));
00134       //  Get |x|^n/n!
00135       _Tp __tc = _Tp(1);
00136       const _Tp __ax = std::abs(__x);
00137       for (unsigned int __k = 1; __k <= __n; ++__k)
00138         __tc *= (__ax / __k);
00139 
00140       _Tp __term = __tc * __tc_sgn;
00141       _Tp __sum = __term;
00142       for (int __k = int(__n) - 1; __k >= 0; --__k)
00143         {
00144           __term *= ((__b + _Tp(__k)) / _Tp(int(__n) - __k))
00145                   * _Tp(__k + 1) / __mx;
00146           __sum += __term;
00147         }
00148 
00149       return __sum;
00150     }
00151 
00152 
00153     /**
00154      *   @brief This routine returns the associated Laguerre polynomial 
00155      *          of order @f$ n @f$, degree @f$ \alpha @f$: @f$ L_n^\alpha(x) @f$
00156      *          by recursion.
00157      *
00158      *   The associated Laguerre function is defined by
00159      *   @f[
00160      *       L_n^\alpha(x) = \frac{(\alpha + 1)_n}{n!}
00161      *                       _1F_1(-n; \alpha + 1; x)
00162      *   @f]
00163      *   where @f$ (\alpha)_n @f$ is the Pochhammer symbol and
00164      *   @f$ _1F_1(a; c; x) @f$ is the confluent hypergeometric function.
00165      *
00166      *   The associated Laguerre polynomial is defined for integral
00167      *   @f$ \alpha = m @f$ by:
00168      *   @f[
00169      *       L_n^m(x) = (-1)^m \frac{d^m}{dx^m} L_{n + m}(x)
00170      *   @f]
00171      *   where the Laguerre polynomial is defined by:
00172      *   @f[
00173      *       L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n} (x^ne^{-x})
00174      *   @f]
00175      *
00176      *   @param __n The order of the Laguerre function.
00177      *   @param __alpha The degree of the Laguerre function.
00178      *   @param __x The argument of the Laguerre function.
00179      *   @return The value of the Laguerre function of order n,
00180      *           degree @f$ \alpha @f$, and argument x.
00181      */
00182     template<typename _Tpa, typename _Tp>
00183     _Tp
00184     __poly_laguerre_recursion(const unsigned int __n,
00185                               const _Tpa __alpha1, const _Tp __x)
00186     {
00187       //   Compute l_0.
00188       _Tp __l_0 = _Tp(1);
00189       if  (__n == 0)
00190         return __l_0;
00191 
00192       //  Compute l_1^alpha.
00193       _Tp __l_1 = -__x + _Tp(1) + _Tp(__alpha1);
00194       if  (__n == 1)
00195         return __l_1;
00196 
00197       //  Compute l_n^alpha by recursion on n.
00198       _Tp __l_n2 = __l_0;
00199       _Tp __l_n1 = __l_1;
00200       _Tp __l_n = _Tp(0);
00201       for  (unsigned int __nn = 2; __nn <= __n; ++__nn)
00202         {
00203             __l_n = (_Tp(2 * __nn - 1) + _Tp(__alpha1) - __x)
00204                   * __l_n1 / _Tp(__nn)
00205                   - (_Tp(__nn - 1) + _Tp(__alpha1)) * __l_n2 / _Tp(__nn);
00206             __l_n2 = __l_n1;
00207             __l_n1 = __l_n;
00208         }
00209 
00210       return __l_n;
00211     }
00212 
00213 
00214     /**
00215      *   @brief This routine returns the associated Laguerre polynomial
00216      *          of order n, degree @f$ \alpha @f$: @f$ L_n^alpha(x) @f$.
00217      *
00218      *   The associated Laguerre function is defined by
00219      *   @f[
00220      *       L_n^\alpha(x) = \frac{(\alpha + 1)_n}{n!}
00221      *                       _1F_1(-n; \alpha + 1; x)
00222      *   @f]
00223      *   where @f$ (\alpha)_n @f$ is the Pochhammer symbol and
00224      *   @f$ _1F_1(a; c; x) @f$ is the confluent hypergeometric function.
00225      *
00226      *   The associated Laguerre polynomial is defined for integral
00227      *   @f$ \alpha = m @f$ by:
00228      *   @f[
00229      *       L_n^m(x) = (-1)^m \frac{d^m}{dx^m} L_{n + m}(x)
00230      *   @f]
00231      *   where the Laguerre polynomial is defined by:
00232      *   @f[
00233      *       L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n} (x^ne^{-x})
00234      *   @f]
00235      *
00236      *   @param __n The order of the Laguerre function.
00237      *   @param __alpha The degree of the Laguerre function.
00238      *   @param __x The argument of the Laguerre function.
00239      *   @return The value of the Laguerre function of order n,
00240      *           degree @f$ \alpha @f$, and argument x.
00241      */
00242     template<typename _Tpa, typename _Tp>
00243     inline _Tp
00244     __poly_laguerre(const unsigned int __n, const _Tpa __alpha1,
00245                     const _Tp __x)
00246     {
00247       if (__x < _Tp(0))
00248         std::__throw_domain_error(__N("Negative argument "
00249                                       "in __poly_laguerre."));
00250       //  Return NaN on NaN input.
00251       else if (__isnan(__x))
00252         return std::numeric_limits<_Tp>::quiet_NaN();
00253       else if (__n == 0)
00254         return _Tp(1);
00255       else if (__n == 1)
00256         return _Tp(1) + _Tp(__alpha1) - __x;
00257       else if (__x == _Tp(0))
00258         {
00259           _Tp __prod = _Tp(__alpha1) + _Tp(1);
00260           for (unsigned int __k = 2; __k <= __n; ++__k)
00261             __prod *= (_Tp(__alpha1) + _Tp(__k)) / _Tp(__k);
00262           return __prod;
00263         }
00264       else if (__n > 10000000 && _Tp(__alpha1) > -_Tp(1)
00265             && __x < _Tp(2) * (_Tp(__alpha1) + _Tp(1)) + _Tp(4 * __n))
00266         return __poly_laguerre_large_n(__n, __alpha1, __x);
00267       else if (_Tp(__alpha1) >= _Tp(0)
00268            || (__x > _Tp(0) && _Tp(__alpha1) < -_Tp(__n + 1)))
00269         return __poly_laguerre_recursion(__n, __alpha1, __x);
00270       else
00271         return __poly_laguerre_hyperg(__n, __alpha1, __x);
00272     }
00273 
00274 
00275     /**
00276      *   @brief This routine returns the associated Laguerre polynomial
00277      *          of order n, degree m: @f$ L_n^m(x) @f$.
00278      *
00279      *   The associated Laguerre polynomial is defined for integral
00280      *   @f$ \alpha = m @f$ by:
00281      *   @f[
00282      *       L_n^m(x) = (-1)^m \frac{d^m}{dx^m} L_{n + m}(x)
00283      *   @f]
00284      *   where the Laguerre polynomial is defined by:
00285      *   @f[
00286      *       L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n} (x^ne^{-x})
00287      *   @f]
00288      *
00289      *   @param __n The order of the Laguerre polynomial.
00290      *   @param __m The degree of the Laguerre polynomial.
00291      *   @param __x The argument of the Laguerre polynomial.
00292      *   @return The value of the associated Laguerre polynomial of order n,
00293      *           degree m, and argument x.
00294      */
00295     template<typename _Tp>
00296     inline _Tp
00297     __assoc_laguerre(const unsigned int __n, const unsigned int __m,
00298                      const _Tp __x)
00299     {
00300       return __poly_laguerre<unsigned int, _Tp>(__n, __m, __x);
00301     }
00302 
00303 
00304     /**
00305      *   @brief This routine returns the Laguerre polynomial
00306      *          of order n: @f$ L_n(x) @f$.
00307      *
00308      *   The Laguerre polynomial is defined by:
00309      *   @f[
00310      *       L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n} (x^ne^{-x})
00311      *   @f]
00312      *
00313      *   @param __n The order of the Laguerre polynomial.
00314      *   @param __x The argument of the Laguerre polynomial.
00315      *   @return The value of the Laguerre polynomial of order n
00316      *           and argument x.
00317      */
00318     template<typename _Tp>
00319     inline _Tp
00320     __laguerre(const unsigned int __n, const _Tp __x)
00321     {
00322       return __poly_laguerre<unsigned int, _Tp>(__n, 0, __x);
00323     }
00324 
00325   } // namespace std::tr1::__detail
00326 }
00327 }
00328 
00329 #endif // _GLIBCXX_TR1_POLY_LAGUERRE_TCC

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