diff options
Diffstat (limited to 'libstdc++-v3/include/bits/specfun.h')
| -rw-r--r-- | libstdc++-v3/include/bits/specfun.h | 297 |
1 files changed, 267 insertions, 30 deletions
diff --git a/libstdc++-v3/include/bits/specfun.h b/libstdc++-v3/include/bits/specfun.h index 8dbfe6086af..2cda681165f 100644 --- a/libstdc++-v3/include/bits/specfun.h +++ b/libstdc++-v3/include/bits/specfun.h @@ -228,6 +228,10 @@ _GLIBCXX_BEGIN_NAMESPACE_VERSION * - @ref __gnu_cxx::jacobi_sn "jacobi_sn - Jacobi sine amplitude functions" * - @ref __gnu_cxx::jacobi_cn "jacobi_cn - Jacobi cosine amplitude functions" * - @ref __gnu_cxx::jacobi_dn "jacobi_dn - Jacobi delta amplitude functions" + * - @ref __gnu_cxx::jacobi_theta_1 "theta_1 - Jacobi theta function 1" + * - @ref __gnu_cxx::jacobi_theta_2 "theta_2 - Jacobi theta function 2" + * - @ref __gnu_cxx::jacobi_theta_3 "theta_3 - Jacobi theta function 3" + * - @ref __gnu_cxx::jacobi_theta_4 "theta_4 - Jacobi theta function 4" * - @ref __gnu_cxx::jacobi_zeta "jacobi_zeta - Jacobi zeta functions" * - @ref __gnu_cxx::lbinomial "lbinomial - Log binomial coefficients" * - @ref __gnu_cxx::ldouble_factorial "ldouble_factorial - Log double factorials" @@ -1048,7 +1052,7 @@ _GLIBCXX_BEGIN_NAMESPACE_VERSION */ inline float hermitef(unsigned int __n, float __x) - { return __detail::__poly_hermite<float>(__n, __x); } + { return __detail::__hermite<float>(__n, __x); } /** * Return the Hermite polynomial @f$ H_n(x) @f$ of nonnegative order n @@ -1058,7 +1062,7 @@ _GLIBCXX_BEGIN_NAMESPACE_VERSION */ inline long double hermitel(unsigned int __n, long double __x) - { return __detail::__poly_hermite<long double>(__n, __x); } + { return __detail::__hermite<long double>(__n, __x); } /** * Return the Hermite polynomial @f$ H_n(x) @f$ of order n @@ -1083,7 +1087,7 @@ _GLIBCXX_BEGIN_NAMESPACE_VERSION hermite(unsigned int __n, _Tp __x) { using __type = __gnu_cxx::__promote_fp_t<_Tp>; - return __detail::__poly_hermite<__type>(__n, __x); + return __detail::__hermite<__type>(__n, __x); } // Laguerre polynomials @@ -1140,7 +1144,7 @@ _GLIBCXX_BEGIN_NAMESPACE_VERSION */ inline float legendref(unsigned int __l, float __x) - { return __detail::__poly_legendre_p<float>(__l, __x); } + { return __detail::__legendre_p<float>(__l, __x).__P_l; } /** * Return the Legendre polynomial @f$ P_l(x) @f$ of nonnegative @@ -1150,7 +1154,7 @@ _GLIBCXX_BEGIN_NAMESPACE_VERSION */ inline long double legendrel(unsigned int __l, long double __x) - { return __detail::__poly_legendre_p<long double>(__l, __x); } + { return __detail::__legendre_p<long double>(__l, __x).__P_l; } /** * Return the Legendre polynomial @f$ P_l(x) @f$ of nonnegative @@ -1172,7 +1176,7 @@ _GLIBCXX_BEGIN_NAMESPACE_VERSION legendre(unsigned int __l, _Tp __x) { using __type = __gnu_cxx::__promote_fp_t<_Tp>; - return __detail::__poly_legendre_p<__type>(__l, __x); + return __detail::__legendre_p<__type>(__l, __x).__P_l; } // Riemann zeta functions @@ -1864,7 +1868,7 @@ _GLIBCXX_BEGIN_NAMESPACE_VERSION inline float jacobi_snf(float __k, float __u) { - return std::__detail::__jacobi_sncndn<float>(__k, __u).__sn_value; + return std::__detail::__jacobi_ellint<float>(__k, __u).__sn_value; } /** @@ -1876,7 +1880,7 @@ _GLIBCXX_BEGIN_NAMESPACE_VERSION inline long double jacobi_snl(long double __k, long double __u) { - return std::__detail::__jacobi_sncndn<long double>(__k, __u).__sn_value; + return std::__detail::__jacobi_ellint<long double>(__k, __u).__sn_value; } /** @@ -1900,7 +1904,7 @@ _GLIBCXX_BEGIN_NAMESPACE_VERSION jacobi_sn(_Kp __k, _Up __u) { using __type = __gnu_cxx::__promote_fp_t<_Kp, _Up>; - return std::__detail::__jacobi_sncndn<__type>(__k, __u).__sn_value; + return std::__detail::__jacobi_ellint<__type>(__k, __u).__sn_value; } // Jacobi elliptic cosine amplitude functions. @@ -1914,7 +1918,7 @@ _GLIBCXX_BEGIN_NAMESPACE_VERSION inline float jacobi_cnf(float __k, float __u) { - return std::__detail::__jacobi_sncndn<float>(__k, __u).__cn_value; + return std::__detail::__jacobi_ellint<float>(__k, __u).__cn_value; } /** @@ -1926,7 +1930,7 @@ _GLIBCXX_BEGIN_NAMESPACE_VERSION inline long double jacobi_cnl(long double __k, long double __u) { - return std::__detail::__jacobi_sncndn<long double>(__k, __u).__cn_value; + return std::__detail::__jacobi_ellint<long double>(__k, __u).__cn_value; } /** @@ -1950,7 +1954,7 @@ _GLIBCXX_BEGIN_NAMESPACE_VERSION jacobi_cn(_Kp __k, _Up __u) { using __type = __gnu_cxx::__promote_fp_t<_Kp, _Up>; - return std::__detail::__jacobi_sncndn<__type>(__k, __u).__cn_value; + return std::__detail::__jacobi_ellint<__type>(__k, __u).__cn_value; } // Jacobi elliptic delta amplitude functions. @@ -1964,7 +1968,7 @@ _GLIBCXX_BEGIN_NAMESPACE_VERSION inline float jacobi_dnf(float __k, float __u) { - return std::__detail::__jacobi_sncndn<float>(__k, __u).__dn_value; + return std::__detail::__jacobi_ellint<float>(__k, __u).__dn_value; } /** @@ -1976,7 +1980,7 @@ _GLIBCXX_BEGIN_NAMESPACE_VERSION inline long double jacobi_dnl(long double __k, long double __u) { - return std::__detail::__jacobi_sncndn<long double>(__k, __u).__dn_value; + return std::__detail::__jacobi_ellint<long double>(__k, __u).__dn_value; } /** @@ -2000,7 +2004,7 @@ _GLIBCXX_BEGIN_NAMESPACE_VERSION jacobi_dn(_Kp __k, _Up __u) { using __type = __gnu_cxx::__promote_fp_t<_Kp, _Up>; - return std::__detail::__jacobi_sncndn<__type>(__k, __u).__dn_value; + return std::__detail::__jacobi_ellint<__type>(__k, __u).__dn_value; } // Chebyshev polynomials of the first kind @@ -2013,7 +2017,7 @@ _GLIBCXX_BEGIN_NAMESPACE_VERSION */ inline float chebyshev_tf(unsigned int __n, float __x) - { return std::__detail::__chebyshev_t<float>(__n, __x); } + { return std::__detail::__chebyshev_t<float>(__n, __x).__T_n; } /** * Return the Chebyshev polynomials of the first kind @f$ T_n(x) @f$ @@ -2023,7 +2027,7 @@ _GLIBCXX_BEGIN_NAMESPACE_VERSION */ inline long double chebyshev_tl(unsigned int __n, long double __x) - { return std::__detail::__chebyshev_t<long double>(__n, __x); } + { return std::__detail::__chebyshev_t<long double>(__n, __x).__T_n; } /** * Return the Chebyshev polynomial of the first kind @f$ T_n(x) @f$ @@ -2044,7 +2048,7 @@ _GLIBCXX_BEGIN_NAMESPACE_VERSION chebyshev_t(unsigned int __n, _Tp __x) { using __type = __gnu_cxx::__promote_fp_t<_Tp>; - return std::__detail::__chebyshev_t<__type>(__n, __x); + return std::__detail::__chebyshev_t<__type>(__n, __x).__T_n; } // Chebyshev polynomials of the second kind @@ -2057,7 +2061,7 @@ _GLIBCXX_BEGIN_NAMESPACE_VERSION */ inline float chebyshev_uf(unsigned int __n, float __x) - { return std::__detail::__chebyshev_u<float>(__n, __x); } + { return std::__detail::__chebyshev_u<float>(__n, __x).__U_n; } /** * Return the Chebyshev polynomials of the second kind @f$ U_n(x) @f$ @@ -2067,7 +2071,7 @@ _GLIBCXX_BEGIN_NAMESPACE_VERSION */ inline long double chebyshev_ul(unsigned int __n, long double __x) - { return std::__detail::__chebyshev_u<long double>(__n, __x); } + { return std::__detail::__chebyshev_u<long double>(__n, __x).__U_n; } /** * Return the Chebyshev polynomial of the second kind @f$ U_n(x) @f$ @@ -2088,7 +2092,7 @@ _GLIBCXX_BEGIN_NAMESPACE_VERSION chebyshev_u(unsigned int __n, _Tp __x) { using __type = __gnu_cxx::__promote_fp_t<_Tp>; - return std::__detail::__chebyshev_u<__type>(__n, __x); + return std::__detail::__chebyshev_u<__type>(__n, __x).__U_n; } // Chebyshev polynomials of the third kind @@ -2192,18 +2196,25 @@ _GLIBCXX_BEGIN_NAMESPACE_VERSION */ inline float jacobif(unsigned __n, float __alpha, float __beta, float __x) - { return std::__detail::__poly_jacobi<float>(__n, __alpha, __beta, __x); } + { + return std::__detail::__jacobi_recur<float>(__n, __alpha, __beta, __x) + .__P_n; + } /** * Return the Jacobi polynomial @f$ P_n^{(\alpha,\beta)}(x) @f$ - * of degree @f$ n @f$ and @c <tt>long double</tt> orders @f$ \alpha, \beta > -1 @f$ - * and argument @f$ x @f$. + * of degree @f$ n @f$ and @c <tt>long double</tt> orders + * @f$ \alpha, \beta > -1 @f$ and argument @f$ x @f$. * * @see jacobi for details. */ inline long double - jacobil(unsigned __n, long double __alpha, long double __beta, long double __x) - { return std::__detail::__poly_jacobi<long double>(__n, __alpha, __beta, __x); } + jacobil(unsigned __n, long double __alpha, long double __beta, + long double __x) + { + return std::__detail::__jacobi_recur<long double>(__n, __alpha, __beta, __x) + .__P_n; + } /** * Return the Jacobi polynomial @f$ P_n^{(\alpha,\beta)}(x) @f$ @@ -2238,14 +2249,16 @@ _GLIBCXX_BEGIN_NAMESPACE_VERSION jacobi(unsigned __n, _Talpha __alpha, _Tbeta __beta, _Tp __x) { using __type = __gnu_cxx::__promote_fp_t<_Talpha, _Tbeta, _Tp>; - return std::__detail::__poly_jacobi<__type>(__n, __alpha, __beta, __x); + return std::__detail::__jacobi_recur<__type>(__n, __alpha, __beta, + __x).__P_n; } // Gegenbauer polynomials /** * Return the Gegenbauer polynomial @f$ C_n^{\alpha}(x) @f$ of degree @c n - * and @c float order @f$ \alpha > -1/2, \alpha \neq 0 @f$ and argument @f$ x @f$. + * and @c float order @f$ \alpha > -1/2, \alpha \neq 0 @f$ + * and argument @f$ x @f$. * * @see gegenbauer for details. */ @@ -2268,7 +2281,7 @@ _GLIBCXX_BEGIN_NAMESPACE_VERSION * Return the Gegenbauer polynomial @f$ C_n^{\alpha}(x) @f$ of degree @c n * and real order @f$ \alpha > -1/2, \alpha \neq 0 @f$ and argument @f$ x @f$. * - * The Gegenbauer polynomials are generated by a three-term recursion relation: + * The Gegenbauer polynomial is generated by a three-term recursion relation: * @f[ * C_n^{\alpha}(x) = \frac{1}{n}\left[ 2x(n+\alpha-1)C_{n-1}^{\alpha}(x) * - (n+2\alpha-2)C_{n-2}^{\alpha}(x) \right] @@ -5357,7 +5370,7 @@ _GLIBCXX_BEGIN_NAMESPACE_VERSION * Return the exponential theta-1 function @f$ \theta_1(\nu,x) @f$ * of period @f$ \nu @f$ and argument @f$ x @f$. * - * The Neville theta-1 function is defined by + * The exponential theta-1 function is defined by * @f[ * \theta_1(\nu,x) = \frac{1}{\sqrt{\pi x}} \sum_{j=-\infty}^{+\infty} * (-1)^j \exp\left( \frac{-(\nu + j - 1/2)^2}{x} \right) @@ -5733,6 +5746,178 @@ _GLIBCXX_BEGIN_NAMESPACE_VERSION return std::__detail::__theta_n<__type>(__k, __x); } + // Jacobi theta_1 functions. + + /** + * Return the Jacobi theta-1 function @f$ \theta_1(q,x) @f$ + * of nome @f$ q @f$ and argument @f$ x @f$. + * + * @see jacobi_theta_1 for details. + */ + inline float + jacobi_theta_1f(float __q, float __x) + { return std::__detail::__jacobi_theta_1<float>(__q, __x); } + + /** + * Return the Jacobi theta-1 function @f$ \theta_1(q,x) @f$ + * of nome @f$ q @f$ and argument @f$ x @f$. + * + * @see jacobi_theta_1 for details. + */ + inline long double + jacobi_theta_1l(long double __q, long double __x) + { return std::__detail::__jacobi_theta_1<long double>(__q, __x); } + + /** + * Return the Jacobi theta-1 function @f$ \theta_1(q,x) @f$ + * of nome @f$ q @f$ and argument @f$ x @f$. + * + * The Jacobi theta-1 function is defined by + * @f[ + * \theta_1(q,x) = \frac{1}{\sqrt{\pi x}} \sum_{j=-\infty}^{+\infty} + * (-1)^j \exp\left( \frac{-(q + j - 1/2)^2}{x} \right) + * @f] + * + * @param __q The periodic (period = 2) argument + * @param __x The argument + */ + template<typename _Tpq, typename _Tp> + inline __gnu_cxx::__promote_fp_t<_Tpq, _Tp> + jacobi_theta_1(_Tpq __q, _Tp __x) + { + using __type = __gnu_cxx::__promote_fp_t<_Tpq, _Tp>; + return std::__detail::__jacobi_theta_1<__type>(__q, __x); + } + + // Jacobi theta_2 functions. + + /** + * Return the Jacobi theta-2 function @f$ \theta_2(q,x) @f$ + * of nome @f$ q @f$ and argument @f$ x @f$. + * + * @see jacobi_theta_2 for details. + */ + inline float + jacobi_theta_2f(float __q, float __x) + { return std::__detail::__jacobi_theta_2<float>(__q, __x); } + + /** + * Return the Jacobi theta-2 function @f$ \theta_2(q,x) @f$ + * of nome @f$ q @f$ and argument @f$ x @f$. + * + * @see jacobi_theta_2 for details. + */ + inline long double + jacobi_theta_2l(long double __q, long double __x) + { return std::__detail::__jacobi_theta_2<long double>(__q, __x); } + + /** + * Return the Jacobi theta-2 function @f$ \theta_2(q,x) @f$ + * of nome @f$ q @f$ and argument @f$ x @f$. + * + * The Jacobi theta-2 function is defined by + * @f[ + * \theta_2(q,x) = \frac{1}{\sqrt{\pi x}} \sum_{j=-\infty}^{+\infty} + * (-1)^j \exp\left( \frac{-(q + j)^2}{x} \right) + * @f] + * + * @param __q The periodic (period = 2) argument + * @param __x The argument + */ + template<typename _Tpq, typename _Tp> + inline __gnu_cxx::__promote_fp_t<_Tpq, _Tp> + jacobi_theta_2(_Tpq __q, _Tp __x) + { + using __type = __gnu_cxx::__promote_fp_t<_Tpq, _Tp>; + return std::__detail::__jacobi_theta_2<__type>(__q, __x); + } + + // Jacobi theta_3 functions. + + /** + * Return the Jacobi theta-3 function @f$ \theta_3(q,x) @f$ + * of nome @f$ q @f$ and argument @f$ x @f$. + * + * @see jacobi_theta_3 for details. + */ + inline float + jacobi_theta_3f(float __q, float __x) + { return std::__detail::__jacobi_theta_3<float>(__q, __x); } + + /** + * Return the Jacobi theta-3 function @f$ \theta_3(q,x) @f$ + * of nome @f$ q @f$ and argument @f$ x @f$. + * + * @see jacobi_theta_3 for details. + */ + inline long double + jacobi_theta_3l(long double __q, long double __x) + { return std::__detail::__jacobi_theta_3<long double>(__q, __x); } + + /** + * Return the Jacobi theta-3 function @f$ \theta_3(q,x) @f$ + * of nome @f$ q @f$ and argument @f$ x @f$. + * + * The Jacobi theta-3 function is defined by + * @f[ + * \theta_3(q,x) = \frac{1}{\sqrt{\pi x}} \sum_{j=-\infty}^{+\infty} + * \exp\left( \frac{-(q+j)^2}{x} \right) + * @f] + * + * @param __q The elliptic nome + * @param __x The argument + */ + template<typename _Tpq, typename _Tp> + inline __gnu_cxx::__promote_fp_t<_Tpq, _Tp> + jacobi_theta_3(_Tpq __q, _Tp __x) + { + using __type = __gnu_cxx::__promote_fp_t<_Tpq, _Tp>; + return std::__detail::__jacobi_theta_3<__type>(__q, __x); + } + + // Jacobi theta_4 functions. + + /** + * Return the Jacobi theta-4 function @f$ \theta_4(q,x) @f$ + * of nome @f$ q @f$ and argument @f$ x @f$. + * + * @see jacobi_theta_4 for details. + */ + inline float + jacobi_theta_4f(float __q, float __x) + { return std::__detail::__jacobi_theta_4<float>(__q, __x); } + + /** + * Return the Jacobi theta-4 function @f$ \theta_4(q,x) @f$ + * of nome @f$ q @f$ and argument @f$ x @f$. + * + * @see jacobi_theta_4 for details. + */ + inline long double + jacobi_theta_4l(long double __q, long double __x) + { return std::__detail::__jacobi_theta_4<long double>(__q, __x); } + + /** + * Return the Jacobi theta-4 function @f$ \theta_4(q,x) @f$ + * of nome @f$ q @f$ and argument @f$ x @f$. + * + * The Jacobi theta-4 function is defined by + * @f[ + * \theta_4(q,x) = \frac{1}{\sqrt{\pi x}} \sum_{j=-\infty}^{+\infty} + * \exp\left( \frac{-(q + j + 1/2)^2}{x} \right) + * @f] + * + * @param __q The elliptic nome + * @param __x The argument + */ + template<typename _Tpq, typename _Tp> + inline __gnu_cxx::__promote_fp_t<_Tpq, _Tp> + jacobi_theta_4(_Tpq __q, _Tp __x) + { + using __type = __gnu_cxx::__promote_fp_t<_Tpq, _Tp>; + return std::__detail::__jacobi_theta_4<__type>(__q, __x); + } + // Owens T functions. /** @@ -5778,14 +5963,40 @@ _GLIBCXX_BEGIN_NAMESPACE_VERSION // Fermi-Dirac integrals. + /** + * Return the Fermi-Dirac integral of @c float order s and argument x. + * + * @see fermi_dirac for details. + */ inline float fermi_diracf(float __s, float __x) { return std::__detail::__fermi_dirac<float>(__s, __x); } + /** + * Return the Fermi-Dirac integral of <tt> long double </tt> + * order s and argument x. + * + * @see fermi_dirac for details. + */ inline long double fermi_diracl(long double __s, long double __x) { return std::__detail::__fermi_dirac<long double>(__s, __x); } + /** + * Return the Fermi-Dirac integral of integer or real order s + * and real argument x. + * @see https://en.wikipedia.org/wiki/Clausen_function + * @see http://dlmf.nist.gov/25.12.16 + * + * @f[ + * F_s(x) = \frac{1}{\Gamma(s+1)}\int_0^\infty \frac{t^s}{e^{t-x} + 1}dt + * = -Li_{s+1}(-e^x) + * @f] + * + * @param __s The order s > -1. + * @param __x The real argument. + * @return The real Fermi-Dirac integral F_s(x), + */ template<typename _Tps, typename _Tp> inline __gnu_cxx::__promote_fp_t<_Tps, _Tp> fermi_dirac(_Tps __s, _Tp __x) @@ -5796,14 +6007,40 @@ _GLIBCXX_BEGIN_NAMESPACE_VERSION // Bose-Einstein integrals. + /** + * Return the Bose-Einstein integral of @c float order s and argument x. + * + * @see bose_einstein for details. + */ inline float bose_einsteinf(float __s, float __x) { return std::__detail::__bose_einstein<float>(__s, __x); } + /** + * Return the Bose-Einstein integral of <tt> long double </tt> + * order s and argument x. + * + * @see bose_einstein for details. + */ inline long double bose_einsteinl(long double __s, long double __x) { return std::__detail::__bose_einstein<long double>(__s, __x); } + /** + * Return the Bose-Einstein integral of integer or real order s + * and real argument x. + * @see https://en.wikipedia.org/wiki/Clausen_function + * @see http://dlmf.nist.gov/25.12.16 + * + * @f[ + * G_s(x) = \frac{1}{\Gamma(s+1)}\int_0^\infty \frac{t^s}{e^{t-x} - 1}dt + * = Li_{s+1}(e^x) + * @f] + * + * @param __s The order s >= 0. + * @param __x The real argument. + * @return The real Bose-Einstein integral G_s(x), + */ template<typename _Tps, typename _Tp> inline __gnu_cxx::__promote_fp_t<_Tps, _Tp> bose_einstein(_Tps __s, _Tp __x) |