| // This file is part of Eigen, a lightweight C++ template library |
| // for linear algebra. |
| // |
| // Copyright (C) 2015 Eugene Brevdo <ebrevdo@gmail.com> |
| // |
| // This Source Code Form is subject to the terms of the Mozilla |
| // Public License v. 2.0. If a copy of the MPL was not distributed |
| // with this file, You can obtain one at http://mozilla.org/MPL/2.0/. |
| |
| #ifndef EIGEN_SPECIAL_FUNCTIONS_H |
| #define EIGEN_SPECIAL_FUNCTIONS_H |
| |
| namespace Eigen { |
| namespace internal { |
| |
| // Parts of this code are based on the Cephes Math Library. |
| // |
| // Cephes Math Library Release 2.8: June, 2000 |
| // Copyright 1984, 1987, 1992, 2000 by Stephen L. Moshier |
| // |
| // Permission has been kindly provided by the original author |
| // to incorporate the Cephes software into the Eigen codebase: |
| // |
| // From: Stephen Moshier |
| // To: Eugene Brevdo |
| // Subject: Re: Permission to wrap several cephes functions in Eigen |
| // |
| // Hello Eugene, |
| // |
| // Thank you for writing. |
| // |
| // If your licensing is similar to BSD, the formal way that has been |
| // handled is simply to add a statement to the effect that you are incorporating |
| // the Cephes software by permission of the author. |
| // |
| // Good luck with your project, |
| // Steve |
| |
| namespace cephes { |
| |
| /* polevl (modified for Eigen) |
| * |
| * Evaluate polynomial |
| * |
| * |
| * |
| * SYNOPSIS: |
| * |
| * int N; |
| * Scalar x, y, coef[N+1]; |
| * |
| * y = polevl<decltype(x), N>( x, coef); |
| * |
| * |
| * |
| * DESCRIPTION: |
| * |
| * Evaluates polynomial of degree N: |
| * |
| * 2 N |
| * y = C + C x + C x +...+ C x |
| * 0 1 2 N |
| * |
| * Coefficients are stored in reverse order: |
| * |
| * coef[0] = C , ..., coef[N] = C . |
| * N 0 |
| * |
| * The function p1evl() assumes that coef[N] = 1.0 and is |
| * omitted from the array. Its calling arguments are |
| * otherwise the same as polevl(). |
| * |
| * |
| * The Eigen implementation is templatized. For best speed, store |
| * coef as a const array (constexpr), e.g. |
| * |
| * const double coef[] = {1.0, 2.0, 3.0, ...}; |
| * |
| */ |
| template <typename Scalar, int N> |
| struct polevl { |
| EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE |
| static Scalar run(const Scalar x, const Scalar coef[]) { |
| EIGEN_STATIC_ASSERT((N > 0), YOU_MADE_A_PROGRAMMING_MISTAKE); |
| |
| return polevl<Scalar, N - 1>::run(x, coef) * x + coef[N]; |
| } |
| }; |
| |
| template <typename Scalar> |
| struct polevl<Scalar, 0> { |
| EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE |
| static Scalar run(const Scalar, const Scalar coef[]) { |
| return coef[0]; |
| } |
| }; |
| |
| } // end namespace cephes |
| |
| /**************************************************************************** |
| * Implementation of lgamma * |
| ****************************************************************************/ |
| |
| template <typename Scalar> |
| struct lgamma_impl { |
| EIGEN_DEVICE_FUNC |
| static EIGEN_STRONG_INLINE Scalar run(const Scalar) { |
| EIGEN_STATIC_ASSERT((internal::is_same<Scalar, Scalar>::value == false), |
| THIS_TYPE_IS_NOT_SUPPORTED); |
| return Scalar(0); |
| } |
| }; |
| |
| template <typename Scalar> |
| struct lgamma_retval { |
| typedef Scalar type; |
| }; |
| |
| #ifdef EIGEN_HAS_C99_MATH |
| template <> |
| struct lgamma_impl<float> { |
| EIGEN_DEVICE_FUNC |
| static EIGEN_STRONG_INLINE float run(float x) { return ::lgammaf(x); } |
| }; |
| |
| template <> |
| struct lgamma_impl<double> { |
| EIGEN_DEVICE_FUNC |
| static EIGEN_STRONG_INLINE double run(double x) { return ::lgamma(x); } |
| }; |
| #endif |
| |
| /**************************************************************************** |
| * Implementation of digamma (psi) * |
| ****************************************************************************/ |
| |
| template <typename Scalar> |
| struct digamma_retval { |
| typedef Scalar type; |
| }; |
| |
| #ifndef EIGEN_HAS_C99_MATH |
| |
| template <typename Scalar> |
| struct digamma_impl { |
| EIGEN_DEVICE_FUNC |
| static Scalar run(Scalar x) { |
| EIGEN_STATIC_ASSERT((internal::is_same<Scalar, Scalar>::value == false), |
| THIS_TYPE_IS_NOT_SUPPORTED); |
| return Scalar(0); |
| } |
| }; |
| |
| #else |
| |
| /* |
| * |
| * Polynomial evaluation helper for the Psi (digamma) function. |
| * |
| * digamma_impl_maybe_poly::run(s) evaluates the asymptotic Psi expansion for |
| * input Scalar s, assuming s is above 10.0. |
| * |
| * If s is above a certain threshold for the given Scalar type, zero |
| * is returned. Otherwise the polynomial is evaluated with enough |
| * coefficients for results matching Scalar machine precision. |
| * |
| * |
| */ |
| template <typename Scalar> |
| struct digamma_impl_maybe_poly { |
| EIGEN_DEVICE_FUNC |
| static EIGEN_STRONG_INLINE Scalar run(const Scalar) { |
| EIGEN_STATIC_ASSERT((internal::is_same<Scalar, Scalar>::value == false), |
| THIS_TYPE_IS_NOT_SUPPORTED); |
| return Scalar(0); |
| } |
| }; |
| |
| |
| template <> |
| struct digamma_impl_maybe_poly<float> { |
| EIGEN_DEVICE_FUNC |
| static EIGEN_STRONG_INLINE float run(const float s) { |
| const float A[] = { |
| -4.16666666666666666667E-3f, |
| 3.96825396825396825397E-3f, |
| -8.33333333333333333333E-3f, |
| 8.33333333333333333333E-2f |
| }; |
| |
| float z; |
| if (s < 1.0e8f) { |
| z = 1.0f / (s * s); |
| return z * cephes::polevl<float, 3>::run(z, A); |
| } else return 0.0f; |
| } |
| }; |
| |
| template <> |
| struct digamma_impl_maybe_poly<double> { |
| EIGEN_DEVICE_FUNC |
| static EIGEN_STRONG_INLINE double run(const double s) { |
| const double A[] = { |
| 8.33333333333333333333E-2, |
| -2.10927960927960927961E-2, |
| 7.57575757575757575758E-3, |
| -4.16666666666666666667E-3, |
| 3.96825396825396825397E-3, |
| -8.33333333333333333333E-3, |
| 8.33333333333333333333E-2 |
| }; |
| |
| double z; |
| if (s < 1.0e17) { |
| z = 1.0 / (s * s); |
| return z * cephes::polevl<double, 6>::run(z, A); |
| } |
| else return 0.0; |
| } |
| }; |
| |
| template <typename Scalar> |
| struct digamma_impl { |
| EIGEN_DEVICE_FUNC |
| static Scalar run(Scalar x) { |
| /* |
| * |
| * Psi (digamma) function (modified for Eigen) |
| * |
| * |
| * SYNOPSIS: |
| * |
| * double x, y, psi(); |
| * |
| * y = psi( x ); |
| * |
| * |
| * DESCRIPTION: |
| * |
| * d - |
| * psi(x) = -- ln | (x) |
| * dx |
| * |
| * is the logarithmic derivative of the gamma function. |
| * For integer x, |
| * n-1 |
| * - |
| * psi(n) = -EUL + > 1/k. |
| * - |
| * k=1 |
| * |
| * If x is negative, it is transformed to a positive argument by the |
| * reflection formula psi(1-x) = psi(x) + pi cot(pi x). |
| * For general positive x, the argument is made greater than 10 |
| * using the recurrence psi(x+1) = psi(x) + 1/x. |
| * Then the following asymptotic expansion is applied: |
| * |
| * inf. B |
| * - 2k |
| * psi(x) = log(x) - 1/2x - > ------- |
| * - 2k |
| * k=1 2k x |
| * |
| * where the B2k are Bernoulli numbers. |
| * |
| * ACCURACY (float): |
| * Relative error (except absolute when |psi| < 1): |
| * arithmetic domain # trials peak rms |
| * IEEE 0,30 30000 1.3e-15 1.4e-16 |
| * IEEE -30,0 40000 1.5e-15 2.2e-16 |
| * |
| * ACCURACY (double): |
| * Absolute error, relative when |psi| > 1 : |
| * arithmetic domain # trials peak rms |
| * IEEE -33,0 30000 8.2e-7 1.2e-7 |
| * IEEE 0,33 100000 7.3e-7 7.7e-8 |
| * |
| * ERROR MESSAGES: |
| * message condition value returned |
| * psi singularity x integer <=0 INFINITY |
| */ |
| |
| Scalar p, q, nz, s, w, y; |
| bool negative; |
| |
| const Scalar maxnum = NumTraits<Scalar>::infinity(); |
| const Scalar m_pi = 3.14159265358979323846; |
| |
| negative = 0; |
| nz = 0.0; |
| |
| const Scalar zero = 0.0; |
| const Scalar one = 1.0; |
| const Scalar half = 0.5; |
| |
| if (x <= zero) { |
| negative = one; |
| q = x; |
| p = ::floor(q); |
| if (p == q) { |
| return maxnum; |
| } |
| /* Remove the zeros of tan(m_pi x) |
| * by subtracting the nearest integer from x |
| */ |
| nz = q - p; |
| if (nz != half) { |
| if (nz > half) { |
| p += one; |
| nz = q - p; |
| } |
| nz = m_pi / ::tan(m_pi * nz); |
| } |
| else { |
| nz = zero; |
| } |
| x = one - x; |
| } |
| |
| /* use the recurrence psi(x+1) = psi(x) + 1/x. */ |
| s = x; |
| w = zero; |
| while (s < Scalar(10)) { |
| w += one / s; |
| s += one; |
| } |
| |
| y = digamma_impl_maybe_poly<Scalar>::run(s); |
| |
| y = ::log(s) - (half / s) - y - w; |
| |
| return (negative) ? y - nz : y; |
| } |
| }; |
| |
| #endif // EIGEN_HAS_C99_MATH |
| |
| /**************************************************************************** |
| * Implementation of erf * |
| ****************************************************************************/ |
| |
| template <typename Scalar> |
| struct erf_impl { |
| EIGEN_DEVICE_FUNC |
| static EIGEN_STRONG_INLINE Scalar run(const Scalar) { |
| EIGEN_STATIC_ASSERT((internal::is_same<Scalar, Scalar>::value == false), |
| THIS_TYPE_IS_NOT_SUPPORTED); |
| return Scalar(0); |
| } |
| }; |
| |
| template <typename Scalar> |
| struct erf_retval { |
| typedef Scalar type; |
| }; |
| |
| #ifdef EIGEN_HAS_C99_MATH |
| template <> |
| struct erf_impl<float> { |
| EIGEN_DEVICE_FUNC |
| static EIGEN_STRONG_INLINE float run(float x) { return ::erff(x); } |
| }; |
| |
| template <> |
| struct erf_impl<double> { |
| EIGEN_DEVICE_FUNC |
| static EIGEN_STRONG_INLINE double run(double x) { return ::erf(x); } |
| }; |
| #endif // EIGEN_HAS_C99_MATH |
| |
| /*************************************************************************** |
| * Implementation of erfc * |
| ****************************************************************************/ |
| |
| template <typename Scalar> |
| struct erfc_impl { |
| EIGEN_DEVICE_FUNC |
| static EIGEN_STRONG_INLINE Scalar run(const Scalar) { |
| EIGEN_STATIC_ASSERT((internal::is_same<Scalar, Scalar>::value == false), |
| THIS_TYPE_IS_NOT_SUPPORTED); |
| return Scalar(0); |
| } |
| }; |
| |
| template <typename Scalar> |
| struct erfc_retval { |
| typedef Scalar type; |
| }; |
| |
| #ifdef EIGEN_HAS_C99_MATH |
| template <> |
| struct erfc_impl<float> { |
| EIGEN_DEVICE_FUNC |
| static EIGEN_STRONG_INLINE float run(const float x) { return ::erfcf(x); } |
| }; |
| |
| template <> |
| struct erfc_impl<double> { |
| EIGEN_DEVICE_FUNC |
| static EIGEN_STRONG_INLINE double run(const double x) { return ::erfc(x); } |
| }; |
| #endif // EIGEN_HAS_C99_MATH |
| |
| /**************************************************************************** |
| * Implementation of igammac (complemented incomplete gamma integral) * |
| ****************************************************************************/ |
| |
| template <typename Scalar> |
| struct igammac_retval { |
| typedef Scalar type; |
| }; |
| |
| #ifndef EIGEN_HAS_C99_MATH |
| |
| template <typename Scalar> |
| struct igammac_impl { |
| EIGEN_DEVICE_FUNC |
| static Scalar run(Scalar a, Scalar x) { |
| EIGEN_STATIC_ASSERT((internal::is_same<Scalar, Scalar>::value == false), |
| THIS_TYPE_IS_NOT_SUPPORTED); |
| return Scalar(0); |
| } |
| }; |
| |
| #else |
| |
| template <typename Scalar> struct igamma_impl; // predeclare igamma_impl |
| |
| template <typename Scalar> |
| struct igammac_impl { |
| EIGEN_DEVICE_FUNC |
| static Scalar run(Scalar a, Scalar x) { |
| /* igamc() |
| * |
| * Incomplete gamma integral (modified for Eigen) |
| * |
| * |
| * |
| * SYNOPSIS: |
| * |
| * double a, x, y, igamc(); |
| * |
| * y = igamc( a, x ); |
| * |
| * DESCRIPTION: |
| * |
| * The function is defined by |
| * |
| * |
| * igamc(a,x) = 1 - igam(a,x) |
| * |
| * inf. |
| * - |
| * 1 | | -t a-1 |
| * = ----- | e t dt. |
| * - | | |
| * | (a) - |
| * x |
| * |
| * |
| * In this implementation both arguments must be positive. |
| * The integral is evaluated by either a power series or |
| * continued fraction expansion, depending on the relative |
| * values of a and x. |
| * |
| * ACCURACY (float): |
| * |
| * Relative error: |
| * arithmetic domain # trials peak rms |
| * IEEE 0,30 30000 7.8e-6 5.9e-7 |
| * |
| * |
| * ACCURACY (double): |
| * |
| * Tested at random a, x. |
| * a x Relative error: |
| * arithmetic domain domain # trials peak rms |
| * IEEE 0.5,100 0,100 200000 1.9e-14 1.7e-15 |
| * IEEE 0.01,0.5 0,100 200000 1.4e-13 1.6e-15 |
| * |
| */ |
| /* |
| Cephes Math Library Release 2.2: June, 1992 |
| Copyright 1985, 1987, 1992 by Stephen L. Moshier |
| Direct inquiries to 30 Frost Street, Cambridge, MA 02140 |
| */ |
| const Scalar zero = 0; |
| const Scalar one = 1; |
| const Scalar two = 2; |
| const Scalar machep = NumTraits<Scalar>::epsilon(); |
| const Scalar maxlog = ::log(NumTraits<Scalar>::highest()); |
| const Scalar big = one / machep; |
| |
| Scalar ans, ax, c, yc, r, t, y, z; |
| Scalar pk, pkm1, pkm2, qk, qkm1, qkm2; |
| |
| if ((x <= zero) || ( a <= zero)) { |
| return one; |
| } |
| |
| if ((x < one) || (x < a)) { |
| return (one - igamma_impl<Scalar>::run(a, x)); |
| } |
| |
| ax = a * ::log(x) - x - lgamma_impl<Scalar>::run(a); |
| if( ax < -maxlog ) { // underflow |
| return zero; |
| } |
| ax = ::exp(ax); |
| |
| // continued fraction |
| y = one - a; |
| z = x + y + one; |
| c = zero; |
| pkm2 = one; |
| qkm2 = x; |
| pkm1 = x + one; |
| qkm1 = z * x; |
| ans = pkm1/qkm1; |
| |
| do { |
| c += one; |
| y += one; |
| z += two; |
| yc = y * c; |
| pk = pkm1 * z - pkm2 * yc; |
| qk = qkm1 * z - qkm2 * yc; |
| if( qk != zero ) { |
| r = pk/qk; |
| t = ::abs( (ans - r)/r ); |
| ans = r; |
| } else { |
| t = one; |
| } |
| pkm2 = pkm1; |
| pkm1 = pk; |
| qkm2 = qkm1; |
| qkm1 = qk; |
| if (::abs(pk) > big) { |
| pkm2 *= machep; |
| pkm1 *= machep; |
| qkm2 *= machep; |
| qkm1 *= machep; |
| } |
| } while( t > machep ); |
| |
| return ( ans * ax ); |
| } |
| }; |
| |
| #endif // EIGEN_HAS_C99_MATH |
| |
| /**************************************************************************** |
| * Implementation of igamma (incomplete gamma integral) * |
| ****************************************************************************/ |
| |
| template <typename Scalar> |
| struct igamma_retval { |
| typedef Scalar type; |
| }; |
| |
| #ifndef EIGEN_HAS_C99_MATH |
| |
| template <typename Scalar> |
| struct igamma_impl { |
| EIGEN_DEVICE_FUNC |
| static Scalar run(Scalar a, Scalar x) { |
| EIGEN_STATIC_ASSERT((internal::is_same<Scalar, Scalar>::value == false), |
| THIS_TYPE_IS_NOT_SUPPORTED); |
| return Scalar(0); |
| } |
| }; |
| |
| #else |
| |
| template <typename Scalar> |
| struct igamma_impl { |
| EIGEN_DEVICE_FUNC |
| static Scalar run(Scalar a, Scalar x) { |
| /* igam() |
| * Incomplete gamma integral |
| * |
| * |
| * |
| * SYNOPSIS: |
| * |
| * double a, x, y, igam(); |
| * |
| * y = igam( a, x ); |
| * |
| * DESCRIPTION: |
| * |
| * The function is defined by |
| * |
| * x |
| * - |
| * 1 | | -t a-1 |
| * igam(a,x) = ----- | e t dt. |
| * - | | |
| * | (a) - |
| * 0 |
| * |
| * |
| * In this implementation both arguments must be positive. |
| * The integral is evaluated by either a power series or |
| * continued fraction expansion, depending on the relative |
| * values of a and x. |
| * |
| * ACCURACY (double): |
| * |
| * Relative error: |
| * arithmetic domain # trials peak rms |
| * IEEE 0,30 200000 3.6e-14 2.9e-15 |
| * IEEE 0,100 300000 9.9e-14 1.5e-14 |
| * |
| * |
| * ACCURACY (float): |
| * |
| * Relative error: |
| * arithmetic domain # trials peak rms |
| * IEEE 0,30 20000 7.8e-6 5.9e-7 |
| * |
| */ |
| /* |
| Cephes Math Library Release 2.2: June, 1992 |
| Copyright 1985, 1987, 1992 by Stephen L. Moshier |
| Direct inquiries to 30 Frost Street, Cambridge, MA 02140 |
| */ |
| |
| |
| /* left tail of incomplete gamma function: |
| * |
| * inf. k |
| * a -x - x |
| * x e > ---------- |
| * - - |
| * k=0 | (a+k+1) |
| * |
| */ |
| const Scalar zero = 0; |
| const Scalar one = 1; |
| const Scalar machep = NumTraits<Scalar>::epsilon(); |
| const Scalar maxlog = ::log(NumTraits<Scalar>::highest()); |
| |
| double ans, ax, c, r; |
| |
| if( (x <= zero) || ( a <= zero) ) { |
| return zero; |
| } |
| |
| if( (x > one) && (x > a ) ) { |
| return (one - igammac_impl<Scalar>::run(a,x)); |
| } |
| |
| /* Compute x**a * exp(-x) / gamma(a) */ |
| ax = a * ::log(x) - x - lgamma_impl<Scalar>::run(a); |
| if( ax < -maxlog ) { |
| // underflow |
| return zero; |
| } |
| ax = ::exp(ax); |
| |
| /* power series */ |
| r = a; |
| c = one; |
| ans = one; |
| |
| do { |
| r += one; |
| c *= x/r; |
| ans += c; |
| } while( c/ans > machep ); |
| |
| return( ans * ax/a ); |
| } |
| }; |
| |
| #endif // EIGEN_HAS_C99_MATH |
| |
| } // end namespace internal |
| |
| namespace numext { |
| |
| template <typename Scalar> |
| EIGEN_DEVICE_FUNC inline EIGEN_MATHFUNC_RETVAL(lgamma, Scalar) |
| lgamma(const Scalar& x) { |
| return EIGEN_MATHFUNC_IMPL(lgamma, Scalar)::run(x); |
| } |
| |
| template <typename Scalar> |
| EIGEN_DEVICE_FUNC inline EIGEN_MATHFUNC_RETVAL(digamma, Scalar) |
| digamma(const Scalar& x) { |
| return EIGEN_MATHFUNC_IMPL(digamma, Scalar)::run(x); |
| } |
| |
| template <typename Scalar> |
| EIGEN_DEVICE_FUNC inline EIGEN_MATHFUNC_RETVAL(erf, Scalar) |
| erf(const Scalar& x) { |
| return EIGEN_MATHFUNC_IMPL(erf, Scalar)::run(x); |
| } |
| |
| template <typename Scalar> |
| EIGEN_DEVICE_FUNC inline EIGEN_MATHFUNC_RETVAL(erfc, Scalar) |
| erfc(const Scalar& x) { |
| return EIGEN_MATHFUNC_IMPL(erfc, Scalar)::run(x); |
| } |
| |
| template <typename Scalar> |
| EIGEN_DEVICE_FUNC inline EIGEN_MATHFUNC_RETVAL(igamma, Scalar) |
| igamma(const Scalar& a, const Scalar& x) { |
| return EIGEN_MATHFUNC_IMPL(igamma, Scalar)::run(a, x); |
| } |
| |
| template <typename Scalar> |
| EIGEN_DEVICE_FUNC inline EIGEN_MATHFUNC_RETVAL(igammac, Scalar) |
| igammac(const Scalar& a, const Scalar& x) { |
| return EIGEN_MATHFUNC_IMPL(igammac, Scalar)::run(a, x); |
| } |
| |
| } // end namespace numext |
| |
| |
| } // end namespace Eigen |
| |
| #endif // EIGEN_SPECIAL_FUNCTIONS_H |
