--- /dev/null
+// Special functions -*- C++ -*-
+
+// Copyright (C) 2006, 2007, 2008, 2009
+// 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.
+ * You should not attempt to use it directly.
+ */
+
+//
+// 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
+{
+namespace tr1
+{
+
+ // [5.2] Special functions
+
+ // Implementation-space details.
+ namespace __detail
+ {
+
+
+ /**
+ * @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$, and argument x.
+ *
+ * This is from the GNU Scientific Library.
+ */
+ template<typename _Tpa, typename _Tp>
+ _Tp
+ __poly_laguerre_large_n(const unsigned __n, const _Tpa __alpha1,
+ const _Tp __x)
+ {
+ const _Tp __a = -_Tp(__n);
+ const _Tp __b = _Tp(__alpha1) + _Tp(1);
+ const _Tp __eta = _Tp(2) * __b - _Tp(4) * __a;
+ const _Tp __cos2th = __x / __eta;
+ const _Tp __sin2th = _Tp(1) - __cos2th;
+ const _Tp __th = std::acos(std::sqrt(__cos2th));
+ const _Tp __pre_h = __numeric_constants<_Tp>::__pi_2()
+ * __numeric_constants<_Tp>::__pi_2()
+ * __eta * __eta * __cos2th * __sin2th;
+
+#if _GLIBCXX_USE_C99_MATH_TR1
+ const _Tp __lg_b = std::tr1::lgamma(_Tp(__n) + __b);
+ const _Tp __lnfact = std::tr1::lgamma(_Tp(__n + 1));
+#else
+ const _Tp __lg_b = __log_gamma(_Tp(__n) + __b);
+ const _Tp __lnfact = __log_gamma(_Tp(__n + 1));
+#endif
+
+ _Tp __pre_term1 = _Tp(0.5L) * (_Tp(1) - __b)
+ * std::log(_Tp(0.25L) * __x * __eta);
+ _Tp __pre_term2 = _Tp(0.25L) * std::log(__pre_h);
+ _Tp __lnpre = __lg_b - __lnfact + _Tp(0.5L) * __x
+ + __pre_term1 - __pre_term2;
+ _Tp __ser_term1 = std::sin(__a * __numeric_constants<_Tp>::__pi());
+ _Tp __ser_term2 = std::sin(_Tp(0.25L) * __eta
+ * (_Tp(2) * __th
+ - std::sin(_Tp(2) * __th))
+ + __numeric_constants<_Tp>::__pi_4());
+ _Tp __ser = __ser_term1 + __ser_term2;
+
+ return std::exp(__lnpre) * __ser;
+ }
+
+
+ /**
+ * @brief Evaluate the polynomial based on the confluent hypergeometric
+ * function in a safe way, with no restriction on the arguments.
+ *
+ * The associated Laguerre function is defined by
+ * @f[
+ * L_n^\alpha(x) = \frac{(\alpha + 1)_n}{n!}
+ * _1F_1(-n; \alpha + 1; x)
+ * @f]
+ * where @f$ (\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(const unsigned int __n, const _Tpa __alpha1,
+ const _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$, and argument x.
+ */
+ template<typename _Tpa, typename _Tp>
+ _Tp
+ __poly_laguerre_recursion(const unsigned int __n,
+ const _Tpa __alpha1, const _Tp __x)
+ {
+ // Compute l_0.
+ _Tp __l_0 = _Tp(1);
+ if (__n == 0)
+ return __l_0;
+
+ // Compute l_1^alpha.
+ _Tp __l_1 = -__x + _Tp(1) + _Tp(__alpha1);
+ if (__n == 1)
+ return __l_1;
+
+ // Compute l_n^alpha by recursion on n.
+ _Tp __l_n2 = __l_0;
+ _Tp __l_n1 = __l_1;
+ _Tp __l_n = _Tp(0);
+ for (unsigned int __nn = 2; __nn <= __n; ++__nn)
+ {
+ __l_n = (_Tp(2 * __nn - 1) + _Tp(__alpha1) - __x)
+ * __l_n1 / _Tp(__nn)
+ - (_Tp(__nn - 1) + _Tp(__alpha1)) * __l_n2 / _Tp(__nn);
+ __l_n2 = __l_n1;
+ __l_n1 = __l_n;
+ }
+
+ return __l_n;
+ }
+
+
+ /**
+ * @brief This routine returns the associated Laguerre polynomial
+ * of order n, degree @f$ \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$, and argument x.
+ */
+ template<typename _Tpa, typename _Tp>
+ inline _Tp
+ __poly_laguerre(const unsigned int __n, const _Tpa __alpha1,
+ const _Tp __x)
+ {
+ if (__x < _Tp(0))
+ std::__throw_domain_error(__N("Negative argument "
+ "in __poly_laguerre."));
+ // Return NaN on NaN input.
+ else if (__isnan(__x))
+ return std::numeric_limits<_Tp>::quiet_NaN();
+ else if (__n == 0)
+ return _Tp(1);
+ else if (__n == 1)
+ return _Tp(1) + _Tp(__alpha1) - __x;
+ else if (__x == _Tp(0))
+ {
+ _Tp __prod = _Tp(__alpha1) + _Tp(1);
+ for (unsigned int __k = 2; __k <= __n; ++__k)
+ __prod *= (_Tp(__alpha1) + _Tp(__k)) / _Tp(__k);
+ return __prod;
+ }
+ else if (__n > 10000000 && _Tp(__alpha1) > -_Tp(1)
+ && __x < _Tp(2) * (_Tp(__alpha1) + _Tp(1)) + _Tp(4 * __n))
+ return __poly_laguerre_large_n(__n, __alpha1, __x);
+ else if (_Tp(__alpha1) >= _Tp(0)
+ || (__x > _Tp(0) && _Tp(__alpha1) < -_Tp(__n + 1)))
+ return __poly_laguerre_recursion(__n, __alpha1, __x);
+ else
+ return __poly_laguerre_hyperg(__n, __alpha1, __x);
+ }
+
+
+ /**
+ * @brief This routine returns the associated Laguerre polynomial
+ * of order n, degree m: @f$ 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(const unsigned int __n, const unsigned int __m,
+ const _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(const unsigned int __n, const _Tp __x)
+ {
+ return __poly_laguerre<unsigned int, _Tp>(__n, 0, __x);
+ }
+
+ } // namespace std::tr1::__detail
+}
+}
+
+#endif // _GLIBCXX_TR1_POLY_LAGUERRE_TCC
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