// Special functions -*- C++ -*-
-// Copyright (C) 2006-2014 Free Software Foundation, Inc.
+// Copyright (C) 2006-2020 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
#ifndef _GLIBCXX_TR1_LEGENDRE_FUNCTION_TCC
#define _GLIBCXX_TR1_LEGENDRE_FUNCTION_TCC 1
-#include "special_function_util.h"
+#include <tr1/special_function_util.h>
namespace std _GLIBCXX_VISIBILITY(default)
{
+_GLIBCXX_BEGIN_NAMESPACE_VERSION
+
+#if _GLIBCXX_USE_STD_SPEC_FUNCS
+# define _GLIBCXX_MATH_NS ::std
+#elif defined(_GLIBCXX_TR1_CMATH)
namespace tr1
{
+# define _GLIBCXX_MATH_NS ::std::tr1
+#else
+# error do not include this header directly, use <cmath> or <tr1/cmath>
+#endif
// [5.2] Special functions
// Implementation-space details.
namespace __detail
{
- _GLIBCXX_BEGIN_NAMESPACE_VERSION
-
/**
- * @brief Return the Legendre polynomial by recursion on order
+ * @brief Return the Legendre polynomial by recursion on degree
* @f$ l @f$.
*
* The Legendre function of @f$ l @f$ and @f$ x @f$,
* P_l(x) = \frac{1}{2^l l!}\frac{d^l}{dx^l}(x^2 - 1)^{l}
* @f]
*
- * @param l The order of the Legendre polynomial. @f$l >= 0@f$.
+ * @param l The degree of the Legendre polynomial. @f$l >= 0@f$.
* @param x The argument of the Legendre polynomial. @f$|x| <= 1@f$.
*/
template<typename _Tp>
__poly_legendre_p(unsigned int __l, _Tp __x)
{
- if ((__x < _Tp(-1)) || (__x > _Tp(+1)))
- std::__throw_domain_error(__N("Argument out of range"
- " in __poly_legendre_p."));
- else if (__isnan(__x))
+ if (__isnan(__x))
return std::numeric_limits<_Tp>::quiet_NaN();
else if (__x == +_Tp(1))
return +_Tp(1);
* @f[
* P_l^m(x) = (1 - x^2)^{m/2}\frac{d^m}{dx^m}P_l(x)
* @f]
+ * @note @f$ P_l^m(x) = 0 @f$ if @f$ m > l @f$.
*
- * @param l The order of the associated Legendre function.
+ * @param l The degree of the associated Legendre function.
* @f$ l >= 0 @f$.
* @param m The order of the associated Legendre function.
- * @f$ m <= l @f$.
* @param x The argument of the associated Legendre function.
* @f$ |x| <= 1 @f$.
+ * @param phase The phase of the associated Legendre function.
+ * Use -1 for the Condon-Shortley phase convention.
*/
template<typename _Tp>
_Tp
- __assoc_legendre_p(unsigned int __l, unsigned int __m, _Tp __x)
+ __assoc_legendre_p(unsigned int __l, unsigned int __m, _Tp __x,
+ _Tp __phase = _Tp(+1))
{
- if (__x < _Tp(-1) || __x > _Tp(+1))
- std::__throw_domain_error(__N("Argument out of range"
- " in __assoc_legendre_p."));
- else if (__m > __l)
- std::__throw_domain_error(__N("Degree out of range"
- " in __assoc_legendre_p."));
+ if (__m > __l)
+ return _Tp(0);
else if (__isnan(__x))
return std::numeric_limits<_Tp>::quiet_NaN();
else if (__m == 0)
_Tp __fact = _Tp(1);
for (unsigned int __i = 1; __i <= __m; ++__i)
{
- __p_mm *= -__fact * __root;
+ __p_mm *= __phase * __fact * __root;
__fact += _Tp(2);
}
}
* but this factor is rather large for large @f$ l @f$ and @f$ m @f$
* and so this function is stable for larger differences of @f$ l @f$
* and @f$ m @f$.
+ * @note Unlike the case for __assoc_legendre_p the Condon-Shortley
+ * phase factor @f$ (-1)^m @f$ is present here.
+ * @note @f$ Y_l^m(\theta) = 0 @f$ if @f$ m > l @f$.
*
- * @param l The order of the spherical associated Legendre function.
+ * @param l The degree of the spherical associated Legendre function.
* @f$ l >= 0 @f$.
* @param m The order of the spherical associated Legendre function.
- * @f$ m <= l @f$.
* @param theta The radian angle argument of the spherical associated
* Legendre function.
*/
const _Tp __x = std::cos(__theta);
- if (__l < __m)
- {
- std::__throw_domain_error(__N("Bad argument "
- "in __sph_legendre."));
- }
+ if (__m > __l)
+ return _Tp(0);
else if (__m == 0)
{
_Tp __P = __poly_legendre_p(__l, __x);
const _Tp __sgn = ( __m % 2 == 1 ? -_Tp(1) : _Tp(1));
const _Tp __y_mp1m_factor = __x * std::sqrt(_Tp(2 * __m + 3));
#if _GLIBCXX_USE_C99_MATH_TR1
- const _Tp __lncirc = std::tr1::log1p(-__x * __x);
+ const _Tp __lncirc = _GLIBCXX_MATH_NS::log1p(-__x * __x);
#else
const _Tp __lncirc = std::log(_Tp(1) - __x * __x);
#endif
// Gamma(m+1/2) / Gamma(m)
#if _GLIBCXX_USE_C99_MATH_TR1
- const _Tp __lnpoch = std::tr1::lgamma(_Tp(__m + _Tp(0.5L)))
- - std::tr1::lgamma(_Tp(__m));
+ const _Tp __lnpoch = _GLIBCXX_MATH_NS::lgamma(_Tp(__m + _Tp(0.5L)))
+ - _GLIBCXX_MATH_NS::lgamma(_Tp(__m));
#else
const _Tp __lnpoch = __log_gamma(_Tp(__m + _Tp(0.5L)))
- __log_gamma(_Tp(__m));
const _Tp __lnpre_val =
-_Tp(0.25L) * __numeric_constants<_Tp>::__lnpi()
+ _Tp(0.5L) * (__lnpoch + __m * __lncirc);
- _Tp __sr = std::sqrt((_Tp(2) + _Tp(1) / __m)
- / (_Tp(4) * __numeric_constants<_Tp>::__pi()));
+ const _Tp __sr = std::sqrt((_Tp(2) + _Tp(1) / __m)
+ / (_Tp(4) * __numeric_constants<_Tp>::__pi()));
_Tp __y_mm = __sgn * __sr * std::exp(__lnpre_val);
_Tp __y_mp1m = __y_mp1m_factor * __y_mm;
if (__l == __m)
- {
- return __y_mm;
- }
+ return __y_mm;
else if (__l == __m + 1)
- {
- return __y_mp1m;
- }
+ return __y_mp1m;
else
{
_Tp __y_lm = _Tp(0);
// Compute Y_l^m, l > m+1, upward recursion on l.
- for ( int __ll = __m + 2; __ll <= __l; ++__ll)
+ for (int __ll = __m + 2; __ll <= __l; ++__ll)
{
const _Tp __rat1 = _Tp(__ll - __m) / _Tp(__ll + __m);
const _Tp __rat2 = _Tp(__ll - __m - 1) / _Tp(__ll + __m - 1);
}
}
}
+ } // namespace __detail
+#undef _GLIBCXX_MATH_NS
+#if ! _GLIBCXX_USE_STD_SPEC_FUNCS && defined(_GLIBCXX_TR1_CMATH)
+} // namespace tr1
+#endif
- _GLIBCXX_END_NAMESPACE_VERSION
- } // namespace std::tr1::__detail
-}
+_GLIBCXX_END_NAMESPACE_VERSION
}
#endif // _GLIBCXX_TR1_LEGENDRE_FUNCTION_TCC
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