| // 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_BESSEL_FUNCTIONS_H |
| #define EIGEN_BESSEL_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 |
| |
| |
| /**************************************************************************** |
| * Implementation of Bessel function, based on Cephes * |
| ****************************************************************************/ |
| |
| template <typename Scalar> |
| struct bessel_i0e_retval { |
| typedef Scalar type; |
| }; |
| |
| template <typename T, typename ScalarType = typename unpacket_traits<T>::type> |
| struct generic_i0e { |
| EIGEN_DEVICE_FUNC |
| static EIGEN_STRONG_INLINE T run(const T&) { |
| EIGEN_STATIC_ASSERT((internal::is_same<T, T>::value == false), |
| THIS_TYPE_IS_NOT_SUPPORTED); |
| return ScalarType(0); |
| } |
| }; |
| |
| template <typename T> |
| struct generic_i0e<T, float> { |
| EIGEN_DEVICE_FUNC |
| static EIGEN_STRONG_INLINE T run(const T& x) { |
| /* i0ef.c |
| * |
| * Modified Bessel function of order zero, |
| * exponentially scaled |
| * |
| * |
| * |
| * SYNOPSIS: |
| * |
| * float x, y, i0ef(); |
| * |
| * y = i0ef( x ); |
| * |
| * |
| * |
| * DESCRIPTION: |
| * |
| * Returns exponentially scaled modified Bessel function |
| * of order zero of the argument. |
| * |
| * The function is defined as i0e(x) = exp(-|x|) j0( ix ). |
| * |
| * |
| * |
| * ACCURACY: |
| * |
| * Relative error: |
| * arithmetic domain # trials peak rms |
| * IEEE 0,30 100000 3.7e-7 7.0e-8 |
| * See i0f(). |
| * |
| */ |
| |
| const float A[] = {-1.30002500998624804212E-8f, 6.04699502254191894932E-8f, |
| -2.67079385394061173391E-7f, 1.11738753912010371815E-6f, |
| -4.41673835845875056359E-6f, 1.64484480707288970893E-5f, |
| -5.75419501008210370398E-5f, 1.88502885095841655729E-4f, |
| -5.76375574538582365885E-4f, 1.63947561694133579842E-3f, |
| -4.32430999505057594430E-3f, 1.05464603945949983183E-2f, |
| -2.37374148058994688156E-2f, 4.93052842396707084878E-2f, |
| -9.49010970480476444210E-2f, 1.71620901522208775349E-1f, |
| -3.04682672343198398683E-1f, 6.76795274409476084995E-1f}; |
| |
| const float B[] = {3.39623202570838634515E-9f, 2.26666899049817806459E-8f, |
| 2.04891858946906374183E-7f, 2.89137052083475648297E-6f, |
| 6.88975834691682398426E-5f, 3.36911647825569408990E-3f, |
| 8.04490411014108831608E-1f}; |
| T y = pabs(x); |
| T y_le_eight = internal::pchebevl<T, 18>::run( |
| pmadd(pset1<T>(0.5f), y, pset1<T>(-2.0f)), A); |
| T y_gt_eight = pmul( |
| internal::pchebevl<T, 7>::run( |
| psub(pdiv(pset1<T>(32.0f), y), pset1<T>(2.0f)), B), |
| prsqrt(y)); |
| // TODO: Perhaps instead check whether all packet elements are in |
| // [-8, 8] and evaluate a branch based off of that. It's possible |
| // in practice most elements are in this region. |
| return pselect(pcmp_le(y, pset1<T>(8.0f)), y_le_eight, y_gt_eight); |
| } |
| }; |
| |
| template <typename T> |
| struct generic_i0e<T, double> { |
| EIGEN_DEVICE_FUNC |
| static EIGEN_STRONG_INLINE T run(const T& x) { |
| /* i0e.c |
| * |
| * Modified Bessel function of order zero, |
| * exponentially scaled |
| * |
| * |
| * |
| * SYNOPSIS: |
| * |
| * double x, y, i0e(); |
| * |
| * y = i0e( x ); |
| * |
| * |
| * |
| * DESCRIPTION: |
| * |
| * Returns exponentially scaled modified Bessel function |
| * of order zero of the argument. |
| * |
| * The function is defined as i0e(x) = exp(-|x|) j0( ix ). |
| * |
| * |
| * |
| * ACCURACY: |
| * |
| * Relative error: |
| * arithmetic domain # trials peak rms |
| * IEEE 0,30 30000 5.4e-16 1.2e-16 |
| * See i0(). |
| * |
| */ |
| |
| const double A[] = {-4.41534164647933937950E-18, 3.33079451882223809783E-17, |
| -2.43127984654795469359E-16, 1.71539128555513303061E-15, |
| -1.16853328779934516808E-14, 7.67618549860493561688E-14, |
| -4.85644678311192946090E-13, 2.95505266312963983461E-12, |
| -1.72682629144155570723E-11, 9.67580903537323691224E-11, |
| -5.18979560163526290666E-10, 2.65982372468238665035E-9, |
| -1.30002500998624804212E-8, 6.04699502254191894932E-8, |
| -2.67079385394061173391E-7, 1.11738753912010371815E-6, |
| -4.41673835845875056359E-6, 1.64484480707288970893E-5, |
| -5.75419501008210370398E-5, 1.88502885095841655729E-4, |
| -5.76375574538582365885E-4, 1.63947561694133579842E-3, |
| -4.32430999505057594430E-3, 1.05464603945949983183E-2, |
| -2.37374148058994688156E-2, 4.93052842396707084878E-2, |
| -9.49010970480476444210E-2, 1.71620901522208775349E-1, |
| -3.04682672343198398683E-1, 6.76795274409476084995E-1}; |
| const double B[] = { |
| -7.23318048787475395456E-18, -4.83050448594418207126E-18, |
| 4.46562142029675999901E-17, 3.46122286769746109310E-17, |
| -2.82762398051658348494E-16, -3.42548561967721913462E-16, |
| 1.77256013305652638360E-15, 3.81168066935262242075E-15, |
| -9.55484669882830764870E-15, -4.15056934728722208663E-14, |
| 1.54008621752140982691E-14, 3.85277838274214270114E-13, |
| 7.18012445138366623367E-13, -1.79417853150680611778E-12, |
| -1.32158118404477131188E-11, -3.14991652796324136454E-11, |
| 1.18891471078464383424E-11, 4.94060238822496958910E-10, |
| 3.39623202570838634515E-9, 2.26666899049817806459E-8, |
| 2.04891858946906374183E-7, 2.89137052083475648297E-6, |
| 6.88975834691682398426E-5, 3.36911647825569408990E-3, |
| 8.04490411014108831608E-1}; |
| T y = pabs(x); |
| T y_le_eight = internal::pchebevl<T, 30>::run( |
| pmadd(pset1<T>(0.5), y, pset1<T>(-2.0)), A); |
| T y_gt_eight = pmul( |
| internal::pchebevl<T, 25>::run( |
| psub(pdiv(pset1<T>(32.0), y), pset1<T>(2.0)), B), |
| prsqrt(y)); |
| // TODO: Perhaps instead check whether all packet elements are in |
| // [-8, 8] and evaluate a branch based off of that. It's possible |
| // in practice most elements are in this region. |
| return pselect(pcmp_le(y, pset1<T>(8.0)), y_le_eight, y_gt_eight); |
| } |
| }; |
| |
| template <typename T> |
| struct bessel_i0e_impl { |
| EIGEN_DEVICE_FUNC |
| static EIGEN_STRONG_INLINE T run(const T x) { |
| return generic_i0e<T>::run(x); |
| } |
| }; |
| |
| template <typename Scalar> |
| struct bessel_i0_retval { |
| typedef Scalar type; |
| }; |
| |
| template <typename T, typename ScalarType = typename unpacket_traits<T>::type> |
| struct generic_i0 { |
| EIGEN_DEVICE_FUNC |
| static EIGEN_STRONG_INLINE T run(const T& x) { |
| return pmul( |
| pexp(pabs(x)), |
| generic_i0e<T, ScalarType>::run(x)); |
| } |
| }; |
| |
| template <typename T> |
| struct bessel_i0_impl { |
| EIGEN_DEVICE_FUNC |
| static EIGEN_STRONG_INLINE T run(const T x) { |
| return generic_i0<T>::run(x); |
| } |
| }; |
| |
| template <typename Scalar> |
| struct bessel_i1e_retval { |
| typedef Scalar type; |
| }; |
| |
| template <typename T, typename ScalarType = typename unpacket_traits<T>::type > |
| struct generic_i1e { |
| EIGEN_DEVICE_FUNC |
| static EIGEN_STRONG_INLINE T run(const T&) { |
| EIGEN_STATIC_ASSERT((internal::is_same<T, T>::value == false), |
| THIS_TYPE_IS_NOT_SUPPORTED); |
| return ScalarType(0); |
| } |
| }; |
| |
| template <typename T> |
| struct generic_i1e<T, float> { |
| EIGEN_DEVICE_FUNC |
| static EIGEN_STRONG_INLINE T run(const T& x) { |
| /* i1ef.c |
| * |
| * Modified Bessel function of order one, |
| * exponentially scaled |
| * |
| * |
| * |
| * SYNOPSIS: |
| * |
| * float x, y, i1ef(); |
| * |
| * y = i1ef( x ); |
| * |
| * |
| * |
| * DESCRIPTION: |
| * |
| * Returns exponentially scaled modified Bessel function |
| * of order one of the argument. |
| * |
| * The function is defined as i1(x) = -i exp(-|x|) j1( ix ). |
| * |
| * |
| * |
| * ACCURACY: |
| * |
| * Relative error: |
| * arithmetic domain # trials peak rms |
| * IEEE 0, 30 30000 1.5e-6 1.5e-7 |
| * See i1(). |
| * |
| */ |
| const float A[] = {9.38153738649577178388E-9f, -4.44505912879632808065E-8f, |
| 2.00329475355213526229E-7f, -8.56872026469545474066E-7f, |
| 3.47025130813767847674E-6f, -1.32731636560394358279E-5f, |
| 4.78156510755005422638E-5f, -1.61760815825896745588E-4f, |
| 5.12285956168575772895E-4f, -1.51357245063125314899E-3f, |
| 4.15642294431288815669E-3f, -1.05640848946261981558E-2f, |
| 2.47264490306265168283E-2f, -5.29459812080949914269E-2f, |
| 1.02643658689847095384E-1f, -1.76416518357834055153E-1f, |
| 2.52587186443633654823E-1f}; |
| |
| const float B[] = {-3.83538038596423702205E-9f, -2.63146884688951950684E-8f, |
| -2.51223623787020892529E-7f, -3.88256480887769039346E-6f, |
| -1.10588938762623716291E-4f, -9.76109749136146840777E-3f, |
| 7.78576235018280120474E-1f}; |
| |
| |
| T y = pabs(x); |
| T y_le_eight = pmul(y, internal::pchebevl<T, 17>::run( |
| pmadd(pset1<T>(0.5f), y, pset1<T>(-2.0f)), A)); |
| T y_gt_eight = pmul( |
| internal::pchebevl<T, 7>::run( |
| psub(pdiv(pset1<T>(32.0f), y), |
| pset1<T>(2.0f)), B), |
| prsqrt(y)); |
| // TODO: Perhaps instead check whether all packet elements are in |
| // [-8, 8] and evaluate a branch based off of that. It's possible |
| // in practice most elements are in this region. |
| y = pselect(pcmp_le(y, pset1<T>(8.0f)), y_le_eight, y_gt_eight); |
| return pselect(pcmp_lt(x, pset1<T>(0.0f)), pnegate(y), y); |
| } |
| }; |
| |
| template <typename T> |
| struct generic_i1e<T, double> { |
| EIGEN_DEVICE_FUNC |
| static EIGEN_STRONG_INLINE T run(const T& x) { |
| /* i1e.c |
| * |
| * Modified Bessel function of order one, |
| * exponentially scaled |
| * |
| * |
| * |
| * SYNOPSIS: |
| * |
| * double x, y, i1e(); |
| * |
| * y = i1e( x ); |
| * |
| * |
| * |
| * DESCRIPTION: |
| * |
| * Returns exponentially scaled modified Bessel function |
| * of order one of the argument. |
| * |
| * The function is defined as i1(x) = -i exp(-|x|) j1( ix ). |
| * |
| * |
| * |
| * ACCURACY: |
| * |
| * Relative error: |
| * arithmetic domain # trials peak rms |
| * IEEE 0, 30 30000 2.0e-15 2.0e-16 |
| * See i1(). |
| * |
| */ |
| const double A[] = {2.77791411276104639959E-18, -2.11142121435816608115E-17, |
| 1.55363195773620046921E-16, -1.10559694773538630805E-15, |
| 7.60068429473540693410E-15, -5.04218550472791168711E-14, |
| 3.22379336594557470981E-13, -1.98397439776494371520E-12, |
| 1.17361862988909016308E-11, -6.66348972350202774223E-11, |
| 3.62559028155211703701E-10, -1.88724975172282928790E-9, |
| 9.38153738649577178388E-9, -4.44505912879632808065E-8, |
| 2.00329475355213526229E-7, -8.56872026469545474066E-7, |
| 3.47025130813767847674E-6, -1.32731636560394358279E-5, |
| 4.78156510755005422638E-5, -1.61760815825896745588E-4, |
| 5.12285956168575772895E-4, -1.51357245063125314899E-3, |
| 4.15642294431288815669E-3, -1.05640848946261981558E-2, |
| 2.47264490306265168283E-2, -5.29459812080949914269E-2, |
| 1.02643658689847095384E-1, -1.76416518357834055153E-1, |
| 2.52587186443633654823E-1}; |
| const double B[] = { |
| 7.51729631084210481353E-18, 4.41434832307170791151E-18, |
| -4.65030536848935832153E-17, -3.20952592199342395980E-17, |
| 2.96262899764595013876E-16, 3.30820231092092828324E-16, |
| -1.88035477551078244854E-15, -3.81440307243700780478E-15, |
| 1.04202769841288027642E-14, 4.27244001671195135429E-14, |
| -2.10154184277266431302E-14, -4.08355111109219731823E-13, |
| -7.19855177624590851209E-13, 2.03562854414708950722E-12, |
| 1.41258074366137813316E-11, 3.25260358301548823856E-11, |
| -1.89749581235054123450E-11, -5.58974346219658380687E-10, |
| -3.83538038596423702205E-9, -2.63146884688951950684E-8, |
| -2.51223623787020892529E-7, -3.88256480887769039346E-6, |
| -1.10588938762623716291E-4, -9.76109749136146840777E-3, |
| 7.78576235018280120474E-1}; |
| T y = pabs(x); |
| T y_le_eight = pmul(y, internal::pchebevl<T, 29>::run( |
| pmadd(pset1<T>(0.5), y, pset1<T>(-2.0)), A)); |
| T y_gt_eight = pmul( |
| internal::pchebevl<T, 25>::run( |
| psub(pdiv(pset1<T>(32.0), y), |
| pset1<T>(2.0)), B), |
| prsqrt(y)); |
| // TODO: Perhaps instead check whether all packet elements are in |
| // [-8, 8] and evaluate a branch based off of that. It's possible |
| // in practice most elements are in this region. |
| y = pselect(pcmp_le(y, pset1<T>(8.0)), y_le_eight, y_gt_eight); |
| return pselect(pcmp_lt(x, pset1<T>(0.0)), pnegate(y), y); |
| } |
| }; |
| |
| template <typename T> |
| struct bessel_i1e_impl { |
| EIGEN_DEVICE_FUNC |
| static EIGEN_STRONG_INLINE T run(const T x) { |
| return generic_i1e<T>::run(x); |
| } |
| }; |
| |
| template <typename T> |
| struct bessel_i1_retval { |
| typedef T type; |
| }; |
| |
| template <typename T, typename ScalarType = typename unpacket_traits<T>::type> |
| struct generic_i1 { |
| EIGEN_DEVICE_FUNC |
| static EIGEN_STRONG_INLINE T run(const T& x) { |
| return pmul( |
| pexp(pabs(x)), |
| generic_i1e<T, ScalarType>::run(x)); |
| } |
| }; |
| |
| template <typename T> |
| struct bessel_i1_impl { |
| EIGEN_DEVICE_FUNC |
| static EIGEN_STRONG_INLINE T run(const T x) { |
| return generic_i1<T>::run(x); |
| } |
| }; |
| |
| template <typename T> |
| struct bessel_k0e_retval { |
| typedef T type; |
| }; |
| |
| template <typename T, typename ScalarType = typename unpacket_traits<T>::type> |
| struct generic_k0e { |
| EIGEN_DEVICE_FUNC |
| static EIGEN_STRONG_INLINE T run(const T&) { |
| EIGEN_STATIC_ASSERT((internal::is_same<T, T>::value == false), |
| THIS_TYPE_IS_NOT_SUPPORTED); |
| return ScalarType(0); |
| } |
| }; |
| |
| template <typename T> |
| struct generic_k0e<T, float> { |
| EIGEN_DEVICE_FUNC |
| static EIGEN_STRONG_INLINE T run(const T& x) { |
| /* k0ef.c |
| * Modified Bessel function, third kind, order zero, |
| * exponentially scaled |
| * |
| * |
| * |
| * SYNOPSIS: |
| * |
| * float x, y, k0ef(); |
| * |
| * y = k0ef( x ); |
| * |
| * |
| * |
| * DESCRIPTION: |
| * |
| * Returns exponentially scaled modified Bessel function |
| * of the third kind of order zero of the argument. |
| * |
| * |
| * |
| * ACCURACY: |
| * |
| * Relative error: |
| * arithmetic domain # trials peak rms |
| * IEEE 0, 30 30000 8.1e-7 7.8e-8 |
| * See k0(). |
| * |
| */ |
| |
| const float A[] = {1.90451637722020886025E-9f, 2.53479107902614945675E-7f, |
| 2.28621210311945178607E-5f, 1.26461541144692592338E-3f, |
| 3.59799365153615016266E-2f, 3.44289899924628486886E-1f, |
| -5.35327393233902768720E-1f}; |
| |
| const float B[] = {-1.69753450938905987466E-9f, 8.57403401741422608519E-9f, |
| -4.66048989768794782956E-8f, 2.76681363944501510342E-7f, |
| -1.83175552271911948767E-6f, 1.39498137188764993662E-5f, |
| -1.28495495816278026384E-4f, 1.56988388573005337491E-3f, |
| -3.14481013119645005427E-2f, 2.44030308206595545468E0f}; |
| const T MAXNUM = pset1<T>(NumTraits<float>::infinity()); |
| const T two = pset1<T>(2.0); |
| T x_le_two = internal::pchebevl<T, 7>::run( |
| pmadd(x, x, pset1<T>(-2.0)), A); |
| x_le_two = pmadd( |
| generic_i0<T, float>::run(x), pnegate( |
| plog(pmul(pset1<T>(0.5), x))), x_le_two); |
| x_le_two = pmul(pexp(x), x_le_two); |
| T x_gt_two = pmul( |
| internal::pchebevl<T, 10>::run( |
| psub(pdiv(pset1<T>(8.0), x), two), B), |
| prsqrt(x)); |
| return pselect( |
| pcmp_le(x, pset1<T>(0.0)), |
| MAXNUM, |
| pselect(pcmp_le(x, two), x_le_two, x_gt_two)); |
| } |
| }; |
| |
| template <typename T> |
| struct generic_k0e<T, double> { |
| EIGEN_DEVICE_FUNC |
| static EIGEN_STRONG_INLINE T run(const T& x) { |
| /* k0e.c |
| * Modified Bessel function, third kind, order zero, |
| * exponentially scaled |
| * |
| * |
| * |
| * SYNOPSIS: |
| * |
| * double x, y, k0e(); |
| * |
| * y = k0e( x ); |
| * |
| * |
| * |
| * DESCRIPTION: |
| * |
| * Returns exponentially scaled modified Bessel function |
| * of the third kind of order zero of the argument. |
| * |
| * |
| * |
| * ACCURACY: |
| * |
| * Relative error: |
| * arithmetic domain # trials peak rms |
| * IEEE 0, 30 30000 1.4e-15 1.4e-16 |
| * See k0(). |
| * |
| */ |
| |
| const double A[] = { |
| 1.37446543561352307156E-16, |
| 4.25981614279661018399E-14, |
| 1.03496952576338420167E-11, |
| 1.90451637722020886025E-9, |
| 2.53479107902614945675E-7, |
| 2.28621210311945178607E-5, |
| 1.26461541144692592338E-3, |
| 3.59799365153615016266E-2, |
| 3.44289899924628486886E-1, |
| -5.35327393233902768720E-1}; |
| const double B[] = { |
| 5.30043377268626276149E-18, -1.64758043015242134646E-17, |
| 5.21039150503902756861E-17, -1.67823109680541210385E-16, |
| 5.51205597852431940784E-16, -1.84859337734377901440E-15, |
| 6.34007647740507060557E-15, -2.22751332699166985548E-14, |
| 8.03289077536357521100E-14, -2.98009692317273043925E-13, |
| 1.14034058820847496303E-12, -4.51459788337394416547E-12, |
| 1.85594911495471785253E-11, -7.95748924447710747776E-11, |
| 3.57739728140030116597E-10, -1.69753450938905987466E-9, |
| 8.57403401741422608519E-9, -4.66048989768794782956E-8, |
| 2.76681363944501510342E-7, -1.83175552271911948767E-6, |
| 1.39498137188764993662E-5, -1.28495495816278026384E-4, |
| 1.56988388573005337491E-3, -3.14481013119645005427E-2, |
| 2.44030308206595545468E0 |
| }; |
| const T MAXNUM = pset1<T>(NumTraits<double>::infinity()); |
| const T two = pset1<T>(2.0); |
| T x_le_two = internal::pchebevl<T, 10>::run( |
| pmadd(x, x, pset1<T>(-2.0)), A); |
| x_le_two = pmadd( |
| generic_i0<T, double>::run(x), pmul( |
| pset1<T>(-1.0), plog(pmul(pset1<T>(0.5), x))), x_le_two); |
| x_le_two = pmul(pexp(x), x_le_two); |
| x_le_two = pselect(pcmp_le(x, pset1<T>(0.0)), MAXNUM, x_le_two); |
| T x_gt_two = pmul( |
| internal::pchebevl<T, 25>::run( |
| psub(pdiv(pset1<T>(8.0), x), two), B), |
| prsqrt(x)); |
| return pselect(pcmp_le(x, two), x_le_two, x_gt_two); |
| } |
| }; |
| |
| template <typename T> |
| struct bessel_k0e_impl { |
| EIGEN_DEVICE_FUNC |
| static EIGEN_STRONG_INLINE T run(const T x) { |
| return generic_k0e<T>::run(x); |
| } |
| }; |
| |
| template <typename T> |
| struct bessel_k0_retval { |
| typedef T type; |
| }; |
| |
| template <typename T, typename ScalarType = typename unpacket_traits<T>::type> |
| struct generic_k0 { |
| EIGEN_DEVICE_FUNC |
| static EIGEN_STRONG_INLINE T run(const T&) { |
| EIGEN_STATIC_ASSERT((internal::is_same<T, T>::value == false), |
| THIS_TYPE_IS_NOT_SUPPORTED); |
| return ScalarType(0); |
| } |
| }; |
| |
| template <typename T> |
| struct generic_k0<T, float> { |
| EIGEN_DEVICE_FUNC |
| static EIGEN_STRONG_INLINE T run(const T& x) { |
| /* k0f.c |
| * Modified Bessel function, third kind, order zero |
| * |
| * |
| * |
| * SYNOPSIS: |
| * |
| * float x, y, k0f(); |
| * |
| * y = k0f( x ); |
| * |
| * |
| * |
| * DESCRIPTION: |
| * |
| * Returns modified Bessel function of the third kind |
| * of order zero of the argument. |
| * |
| * The range is partitioned into the two intervals [0,8] and |
| * (8, infinity). Chebyshev polynomial expansions are employed |
| * in each interval. |
| * |
| * |
| * |
| * ACCURACY: |
| * |
| * Tested at 2000 random points between 0 and 8. Peak absolute |
| * error (relative when K0 > 1) was 1.46e-14; rms, 4.26e-15. |
| * Relative error: |
| * arithmetic domain # trials peak rms |
| * IEEE 0, 30 30000 7.8e-7 8.5e-8 |
| * |
| * ERROR MESSAGES: |
| * |
| * message condition value returned |
| * K0 domain x <= 0 MAXNUM |
| * |
| */ |
| |
| const float A[] = {1.90451637722020886025E-9f, 2.53479107902614945675E-7f, |
| 2.28621210311945178607E-5f, 1.26461541144692592338E-3f, |
| 3.59799365153615016266E-2f, 3.44289899924628486886E-1f, |
| -5.35327393233902768720E-1f}; |
| |
| const float B[] = {-1.69753450938905987466E-9f, 8.57403401741422608519E-9f, |
| -4.66048989768794782956E-8f, 2.76681363944501510342E-7f, |
| -1.83175552271911948767E-6f, 1.39498137188764993662E-5f, |
| -1.28495495816278026384E-4f, 1.56988388573005337491E-3f, |
| -3.14481013119645005427E-2f, 2.44030308206595545468E0f}; |
| const T MAXNUM = pset1<T>(NumTraits<float>::infinity()); |
| const T two = pset1<T>(2.0); |
| T x_le_two = internal::pchebevl<T, 7>::run( |
| pmadd(x, x, pset1<T>(-2.0)), A); |
| x_le_two = pmadd( |
| generic_i0<T, float>::run(x), pnegate( |
| plog(pmul(pset1<T>(0.5), x))), x_le_two); |
| x_le_two = pselect(pcmp_le(x, pset1<T>(0.0)), MAXNUM, x_le_two); |
| T x_gt_two = pmul( |
| pmul( |
| pexp(pnegate(x)), |
| internal::pchebevl<T, 10>::run( |
| psub(pdiv(pset1<T>(8.0), x), two), B)), |
| prsqrt(x)); |
| return pselect(pcmp_le(x, two), x_le_two, x_gt_two); |
| } |
| }; |
| |
| template <typename T> |
| struct generic_k0<T, double> { |
| EIGEN_DEVICE_FUNC |
| static EIGEN_STRONG_INLINE T run(const T& x) { |
| /* |
| * |
| * Modified Bessel function, third kind, order zero, |
| * exponentially scaled |
| * |
| * |
| * |
| * SYNOPSIS: |
| * |
| * double x, y, k0(); |
| * |
| * y = k0( x ); |
| * |
| * |
| * |
| * DESCRIPTION: |
| * |
| * Returns exponentially scaled modified Bessel function |
| * of the third kind of order zero of the argument. |
| * |
| * |
| * |
| * ACCURACY: |
| * |
| * Relative error: |
| * arithmetic domain # trials peak rms |
| * IEEE 0, 30 30000 1.4e-15 1.4e-16 |
| * See k0(). |
| * |
| */ |
| const double A[] = { |
| 1.37446543561352307156E-16, |
| 4.25981614279661018399E-14, |
| 1.03496952576338420167E-11, |
| 1.90451637722020886025E-9, |
| 2.53479107902614945675E-7, |
| 2.28621210311945178607E-5, |
| 1.26461541144692592338E-3, |
| 3.59799365153615016266E-2, |
| 3.44289899924628486886E-1, |
| -5.35327393233902768720E-1}; |
| const double B[] = { |
| 5.30043377268626276149E-18, -1.64758043015242134646E-17, |
| 5.21039150503902756861E-17, -1.67823109680541210385E-16, |
| 5.51205597852431940784E-16, -1.84859337734377901440E-15, |
| 6.34007647740507060557E-15, -2.22751332699166985548E-14, |
| 8.03289077536357521100E-14, -2.98009692317273043925E-13, |
| 1.14034058820847496303E-12, -4.51459788337394416547E-12, |
| 1.85594911495471785253E-11, -7.95748924447710747776E-11, |
| 3.57739728140030116597E-10, -1.69753450938905987466E-9, |
| 8.57403401741422608519E-9, -4.66048989768794782956E-8, |
| 2.76681363944501510342E-7, -1.83175552271911948767E-6, |
| 1.39498137188764993662E-5, -1.28495495816278026384E-4, |
| 1.56988388573005337491E-3, -3.14481013119645005427E-2, |
| 2.44030308206595545468E0 |
| }; |
| const T MAXNUM = pset1<T>(NumTraits<double>::infinity()); |
| const T two = pset1<T>(2.0); |
| T x_le_two = internal::pchebevl<T, 10>::run( |
| pmadd(x, x, pset1<T>(-2.0)), A); |
| x_le_two = pmadd( |
| generic_i0<T, double>::run(x), pnegate( |
| plog(pmul(pset1<T>(0.5), x))), x_le_two); |
| x_le_two = pselect(pcmp_le(x, pset1<T>(0.0)), MAXNUM, x_le_two); |
| T x_gt_two = pmul( |
| pmul( |
| pexp(-x), |
| internal::pchebevl<T, 25>::run( |
| psub(pdiv(pset1<T>(8.0), x), two), B)), |
| prsqrt(x)); |
| return pselect(pcmp_le(x, two), x_le_two, x_gt_two); |
| } |
| }; |
| |
| template <typename T> |
| struct bessel_k0_impl { |
| EIGEN_DEVICE_FUNC |
| static EIGEN_STRONG_INLINE T run(const T x) { |
| return generic_k0<T>::run(x); |
| } |
| }; |
| |
| template <typename T> |
| struct bessel_k1e_retval { |
| typedef T type; |
| }; |
| |
| template <typename T, typename ScalarType = typename unpacket_traits<T>::type> |
| struct generic_k1e { |
| EIGEN_DEVICE_FUNC |
| static EIGEN_STRONG_INLINE T run(const T&) { |
| EIGEN_STATIC_ASSERT((internal::is_same<T, T>::value == false), |
| THIS_TYPE_IS_NOT_SUPPORTED); |
| return ScalarType(0); |
| } |
| }; |
| |
| template <typename T> |
| struct generic_k1e<T, float> { |
| EIGEN_DEVICE_FUNC |
| static EIGEN_STRONG_INLINE T run(const T& x) { |
| /* k1ef.c |
| * |
| * Modified Bessel function, third kind, order one, |
| * exponentially scaled |
| * |
| * |
| * |
| * SYNOPSIS: |
| * |
| * float x, y, k1ef(); |
| * |
| * y = k1ef( x ); |
| * |
| * |
| * |
| * DESCRIPTION: |
| * |
| * Returns exponentially scaled modified Bessel function |
| * of the third kind of order one of the argument: |
| * |
| * k1e(x) = exp(x) * k1(x). |
| * |
| * |
| * |
| * ACCURACY: |
| * |
| * Relative error: |
| * arithmetic domain # trials peak rms |
| * IEEE 0, 30 30000 4.9e-7 6.7e-8 |
| * See k1(). |
| * |
| */ |
| |
| const float A[] = {-2.21338763073472585583E-8f, -2.43340614156596823496E-6f, |
| -1.73028895751305206302E-4f, -6.97572385963986435018E-3f, |
| -1.22611180822657148235E-1f, -3.53155960776544875667E-1f, |
| 1.52530022733894777053E0f}; |
| const float B[] = {2.01504975519703286596E-9f, -1.03457624656780970260E-8f, |
| 5.74108412545004946722E-8f, -3.50196060308781257119E-7f, |
| 2.40648494783721712015E-6f, -1.93619797416608296024E-5f, |
| 1.95215518471351631108E-4f, -2.85781685962277938680E-3f, |
| 1.03923736576817238437E-1f, 2.72062619048444266945E0f}; |
| const T MAXNUM = pset1<T>(NumTraits<float>::infinity()); |
| const T two = pset1<T>(2.0); |
| T x_le_two = pdiv(internal::pchebevl<T, 7>::run( |
| pmadd(x, x, pset1<T>(-2.0)), A), x); |
| x_le_two = pmadd( |
| generic_i1<T, float>::run(x), plog(pmul(pset1<T>(0.5), x)), x_le_two); |
| x_le_two = pmul(x_le_two, pexp(x)); |
| x_le_two = pselect(pcmp_le(x, pset1<T>(0.0)), MAXNUM, x_le_two); |
| T x_gt_two = pmul( |
| internal::pchebevl<T, 10>::run( |
| psub(pdiv(pset1<T>(8.0), x), two), B), |
| prsqrt(x)); |
| return pselect(pcmp_le(x, two), x_le_two, x_gt_two); |
| } |
| }; |
| |
| template <typename T> |
| struct generic_k1e<T, double> { |
| EIGEN_DEVICE_FUNC |
| static EIGEN_STRONG_INLINE T run(const T& x) { |
| /* k1e.c |
| * |
| * Modified Bessel function, third kind, order one, |
| * exponentially scaled |
| * |
| * |
| * |
| * SYNOPSIS: |
| * |
| * double x, y, k1e(); |
| * |
| * y = k1e( x ); |
| * |
| * |
| * |
| * DESCRIPTION: |
| * |
| * Returns exponentially scaled modified Bessel function |
| * of the third kind of order one of the argument: |
| * |
| * k1e(x) = exp(x) * k1(x). |
| * |
| * |
| * |
| * ACCURACY: |
| * |
| * Relative error: |
| * arithmetic domain # trials peak rms |
| * IEEE 0, 30 30000 7.8e-16 1.2e-16 |
| * See k1(). |
| * |
| */ |
| const double A[] = {-7.02386347938628759343E-18, -2.42744985051936593393E-15, |
| -6.66690169419932900609E-13, -1.41148839263352776110E-10, |
| -2.21338763073472585583E-8, -2.43340614156596823496E-6, |
| -1.73028895751305206302E-4, -6.97572385963986435018E-3, |
| -1.22611180822657148235E-1, -3.53155960776544875667E-1, |
| 1.52530022733894777053E0}; |
| const double B[] = {-5.75674448366501715755E-18, 1.79405087314755922667E-17, |
| -5.68946255844285935196E-17, 1.83809354436663880070E-16, |
| -6.05704724837331885336E-16, 2.03870316562433424052E-15, |
| -7.01983709041831346144E-15, 2.47715442448130437068E-14, |
| -8.97670518232499435011E-14, 3.34841966607842919884E-13, |
| -1.28917396095102890680E-12, 5.13963967348173025100E-12, |
| -2.12996783842756842877E-11, 9.21831518760500529508E-11, |
| -4.19035475934189648750E-10, 2.01504975519703286596E-9, |
| -1.03457624656780970260E-8, 5.74108412545004946722E-8, |
| -3.50196060308781257119E-7, 2.40648494783721712015E-6, |
| -1.93619797416608296024E-5, 1.95215518471351631108E-4, |
| -2.85781685962277938680E-3, 1.03923736576817238437E-1, |
| 2.72062619048444266945E0}; |
| const T MAXNUM = pset1<T>(NumTraits<double>::infinity()); |
| const T two = pset1<T>(2.0); |
| T x_le_two = pdiv(internal::pchebevl<T, 11>::run( |
| pmadd(x, x, pset1<T>(-2.0)), A), x); |
| x_le_two = pmadd( |
| generic_i1<T, double>::run(x), plog(pmul(pset1<T>(0.5), x)), x_le_two); |
| x_le_two = pmul(x_le_two, pexp(x)); |
| x_le_two = pselect(pcmp_le(x, pset1<T>(0.0)), MAXNUM, x_le_two); |
| T x_gt_two = pmul( |
| internal::pchebevl<T, 25>::run( |
| psub(pdiv(pset1<T>(8.0), x), two), B), |
| prsqrt(x)); |
| return pselect(pcmp_le(x, two), x_le_two, x_gt_two); |
| } |
| }; |
| |
| template <typename T> |
| struct bessel_k1e_impl { |
| EIGEN_DEVICE_FUNC |
| static EIGEN_STRONG_INLINE T run(const T x) { |
| return generic_k1e<T>::run(x); |
| } |
| }; |
| |
| template <typename T> |
| struct bessel_k1_retval { |
| typedef T type; |
| }; |
| |
| template <typename T, typename ScalarType = typename unpacket_traits<T>::type> |
| struct generic_k1 { |
| EIGEN_DEVICE_FUNC |
| static EIGEN_STRONG_INLINE T run(const T&) { |
| EIGEN_STATIC_ASSERT((internal::is_same<T, T>::value == false), |
| THIS_TYPE_IS_NOT_SUPPORTED); |
| return ScalarType(0); |
| } |
| }; |
| |
| template <typename T> |
| struct generic_k1<T, float> { |
| EIGEN_DEVICE_FUNC |
| static EIGEN_STRONG_INLINE T run(const T& x) { |
| /* k1f.c |
| * Modified Bessel function, third kind, order one |
| * |
| * |
| * |
| * SYNOPSIS: |
| * |
| * float x, y, k1f(); |
| * |
| * y = k1f( x ); |
| * |
| * |
| * |
| * DESCRIPTION: |
| * |
| * Computes the modified Bessel function of the third kind |
| * of order one of the argument. |
| * |
| * The range is partitioned into the two intervals [0,2] and |
| * (2, infinity). Chebyshev polynomial expansions are employed |
| * in each interval. |
| * |
| * |
| * |
| * ACCURACY: |
| * |
| * Relative error: |
| * arithmetic domain # trials peak rms |
| * IEEE 0, 30 30000 4.6e-7 7.6e-8 |
| * |
| * ERROR MESSAGES: |
| * |
| * message condition value returned |
| * k1 domain x <= 0 MAXNUM |
| * |
| */ |
| |
| const float A[] = {-2.21338763073472585583E-8f, -2.43340614156596823496E-6f, |
| -1.73028895751305206302E-4f, -6.97572385963986435018E-3f, |
| -1.22611180822657148235E-1f, -3.53155960776544875667E-1f, |
| 1.52530022733894777053E0f}; |
| const float B[] = {2.01504975519703286596E-9f, -1.03457624656780970260E-8f, |
| 5.74108412545004946722E-8f, -3.50196060308781257119E-7f, |
| 2.40648494783721712015E-6f, -1.93619797416608296024E-5f, |
| 1.95215518471351631108E-4f, -2.85781685962277938680E-3f, |
| 1.03923736576817238437E-1f, 2.72062619048444266945E0f}; |
| const T MAXNUM = pset1<T>(NumTraits<float>::infinity()); |
| const T two = pset1<T>(2.0); |
| T x_le_two = pdiv(internal::pchebevl<T, 7>::run( |
| pmadd(x, x, pset1<T>(-2.0)), A), x); |
| x_le_two = pmadd( |
| generic_i1<T, float>::run(x), plog(pmul(pset1<T>(0.5), x)), x_le_two); |
| x_le_two = pselect(pcmp_le(x, pset1<T>(0.0)), MAXNUM, x_le_two); |
| T x_gt_two = pmul( |
| pexp(pnegate(x)), |
| pmul( |
| internal::pchebevl<T, 10>::run( |
| psub(pdiv(pset1<T>(8.0), x), two), B), |
| prsqrt(x))); |
| return pselect(pcmp_le(x, two), x_le_two, x_gt_two); |
| } |
| }; |
| |
| template <typename T> |
| struct generic_k1<T, double> { |
| EIGEN_DEVICE_FUNC |
| static EIGEN_STRONG_INLINE T run(const T& x) { |
| /* k1.c |
| * Modified Bessel function, third kind, order one |
| * |
| * |
| * |
| * SYNOPSIS: |
| * |
| * float x, y, k1f(); |
| * |
| * y = k1f( x ); |
| * |
| * |
| * |
| * DESCRIPTION: |
| * |
| * Computes the modified Bessel function of the third kind |
| * of order one of the argument. |
| * |
| * The range is partitioned into the two intervals [0,2] and |
| * (2, infinity). Chebyshev polynomial expansions are employed |
| * in each interval. |
| * |
| * |
| * |
| * ACCURACY: |
| * |
| * Relative error: |
| * arithmetic domain # trials peak rms |
| * IEEE 0, 30 30000 4.6e-7 7.6e-8 |
| * |
| * ERROR MESSAGES: |
| * |
| * message condition value returned |
| * k1 domain x <= 0 MAXNUM |
| * |
| */ |
| const double A[] = {-7.02386347938628759343E-18, -2.42744985051936593393E-15, |
| -6.66690169419932900609E-13, -1.41148839263352776110E-10, |
| -2.21338763073472585583E-8, -2.43340614156596823496E-6, |
| -1.73028895751305206302E-4, -6.97572385963986435018E-3, |
| -1.22611180822657148235E-1, -3.53155960776544875667E-1, |
| 1.52530022733894777053E0}; |
| const double B[] = {-5.75674448366501715755E-18, 1.79405087314755922667E-17, |
| -5.68946255844285935196E-17, 1.83809354436663880070E-16, |
| -6.05704724837331885336E-16, 2.03870316562433424052E-15, |
| -7.01983709041831346144E-15, 2.47715442448130437068E-14, |
| -8.97670518232499435011E-14, 3.34841966607842919884E-13, |
| -1.28917396095102890680E-12, 5.13963967348173025100E-12, |
| -2.12996783842756842877E-11, 9.21831518760500529508E-11, |
| -4.19035475934189648750E-10, 2.01504975519703286596E-9, |
| -1.03457624656780970260E-8, 5.74108412545004946722E-8, |
| -3.50196060308781257119E-7, 2.40648494783721712015E-6, |
| -1.93619797416608296024E-5, 1.95215518471351631108E-4, |
| -2.85781685962277938680E-3, 1.03923736576817238437E-1, |
| 2.72062619048444266945E0}; |
| const T MAXNUM = pset1<T>(NumTraits<double>::infinity()); |
| const T two = pset1<T>(2.0); |
| T x_le_two = pdiv(internal::pchebevl<T, 11>::run( |
| pmadd(x, x, pset1<T>(-2.0)), A), x); |
| x_le_two = pmadd( |
| generic_i1<T, double>::run(x), plog(pmul(pset1<T>(0.5), x)), x_le_two); |
| x_le_two = pselect(pcmp_le(x, pset1<T>(0.0)), MAXNUM, x_le_two); |
| T x_gt_two = pmul( |
| pexp(-x), |
| pmul( |
| internal::pchebevl<T, 25>::run( |
| psub(pdiv(pset1<T>(8.0), x), two), B), |
| prsqrt(x))); |
| return pselect(pcmp_le(x, two), x_le_two, x_gt_two); |
| } |
| }; |
| |
| template <typename T> |
| struct bessel_k1_impl { |
| EIGEN_DEVICE_FUNC |
| static EIGEN_STRONG_INLINE T run(const T x) { |
| return generic_k1<T>::run(x); |
| } |
| }; |
| |
| template <typename T> |
| struct bessel_j0_retval { |
| typedef T type; |
| }; |
| |
| template <typename T, typename ScalarType = typename unpacket_traits<T>::type> |
| struct generic_j0 { |
| EIGEN_DEVICE_FUNC |
| static EIGEN_STRONG_INLINE T run(const T&) { |
| EIGEN_STATIC_ASSERT((internal::is_same<T, T>::value == false), |
| THIS_TYPE_IS_NOT_SUPPORTED); |
| return ScalarType(0); |
| } |
| }; |
| |
| template <typename T> |
| struct generic_j0<T, float> { |
| EIGEN_DEVICE_FUNC |
| static EIGEN_STRONG_INLINE T run(const T& x) { |
| /* j0f.c |
| * Bessel function of order zero |
| * |
| * |
| * |
| * SYNOPSIS: |
| * |
| * float x, y, j0f(); |
| * |
| * y = j0f( x ); |
| * |
| * |
| * |
| * DESCRIPTION: |
| * |
| * Returns Bessel function of order zero of the argument. |
| * |
| * The domain is divided into the intervals [0, 2] and |
| * (2, infinity). In the first interval the following polynomial |
| * approximation is used: |
| * |
| * |
| * 2 2 2 |
| * (w - r ) (w - r ) (w - r ) P(w) |
| * 1 2 3 |
| * |
| * 2 |
| * where w = x and the three r's are zeros of the function. |
| * |
| * In the second interval, the modulus and phase are approximated |
| * by polynomials of the form Modulus(x) = sqrt(1/x) Q(1/x) |
| * and Phase(x) = x + 1/x R(1/x^2) - pi/4. The function is |
| * |
| * j0(x) = Modulus(x) cos( Phase(x) ). |
| * |
| * |
| * |
| * ACCURACY: |
| * |
| * Absolute error: |
| * arithmetic domain # trials peak rms |
| * IEEE 0, 2 100000 1.3e-7 3.6e-8 |
| * IEEE 2, 32 100000 1.9e-7 5.4e-8 |
| * |
| */ |
| |
| const float JP[] = {-6.068350350393235E-008f, 6.388945720783375E-006f, |
| -3.969646342510940E-004f, 1.332913422519003E-002f, |
| -1.729150680240724E-001f}; |
| const float MO[] = {-6.838999669318810E-002f, 1.864949361379502E-001f, |
| -2.145007480346739E-001f, 1.197549369473540E-001f, |
| -3.560281861530129E-003f, -4.969382655296620E-002f, |
| -3.355424622293709E-006f, 7.978845717621440E-001f}; |
| const float PH[] = {3.242077816988247E+001f, -3.630592630518434E+001f, |
| 1.756221482109099E+001f, -4.974978466280903E+000f, |
| 1.001973420681837E+000f, -1.939906941791308E-001f, |
| 6.490598792654666E-002f, -1.249992184872738E-001f}; |
| const T DR1 = pset1<T>(5.78318596294678452118f); |
| const T NEG_PIO4F = pset1<T>(-0.7853981633974483096f); /* -pi / 4 */ |
| T y = pabs(x); |
| T z = pmul(y, y); |
| T y_le_two = pselect( |
| pcmp_lt(y, pset1<T>(1.0e-3f)), |
| pmadd(z, pset1<T>(-0.25f), pset1<T>(1.0f)), |
| pmul(psub(z, DR1), internal::ppolevl<T, 4>::run(z, JP))); |
| T q = pdiv(pset1<T>(1.0f), y); |
| T w = prsqrt(y); |
| T p = pmul(w, internal::ppolevl<T, 7>::run(q, MO)); |
| w = pmul(q, q); |
| T yn = pmadd(q, internal::ppolevl<T, 7>::run(w, PH), NEG_PIO4F); |
| T y_gt_two = pmul(p, pcos(padd(yn, y))); |
| return pselect(pcmp_le(y, pset1<T>(2.0)), y_le_two, y_gt_two); |
| } |
| }; |
| |
| template <typename T> |
| struct generic_j0<T, double> { |
| EIGEN_DEVICE_FUNC |
| static EIGEN_STRONG_INLINE T run(const T& x) { |
| /* j0.c |
| * Bessel function of order zero |
| * |
| * |
| * |
| * SYNOPSIS: |
| * |
| * double x, y, j0(); |
| * |
| * y = j0( x ); |
| * |
| * |
| * |
| * DESCRIPTION: |
| * |
| * Returns Bessel function of order zero of the argument. |
| * |
| * The domain is divided into the intervals [0, 5] and |
| * (5, infinity). In the first interval the following rational |
| * approximation is used: |
| * |
| * |
| * 2 2 |
| * (w - r ) (w - r ) P (w) / Q (w) |
| * 1 2 3 8 |
| * |
| * 2 |
| * where w = x and the two r's are zeros of the function. |
| * |
| * In the second interval, the Hankel asymptotic expansion |
| * is employed with two rational functions of degree 6/6 |
| * and 7/7. |
| * |
| * |
| * |
| * ACCURACY: |
| * |
| * Absolute error: |
| * arithmetic domain # trials peak rms |
| * DEC 0, 30 10000 4.4e-17 6.3e-18 |
| * IEEE 0, 30 60000 4.2e-16 1.1e-16 |
| * |
| */ |
| const double PP[] = {7.96936729297347051624E-4, 8.28352392107440799803E-2, |
| 1.23953371646414299388E0, 5.44725003058768775090E0, |
| 8.74716500199817011941E0, 5.30324038235394892183E0, |
| 9.99999999999999997821E-1}; |
| const double PQ[] = {9.24408810558863637013E-4, 8.56288474354474431428E-2, |
| 1.25352743901058953537E0, 5.47097740330417105182E0, |
| 8.76190883237069594232E0, 5.30605288235394617618E0, |
| 1.00000000000000000218E0}; |
| const double QP[] = {-1.13663838898469149931E-2, -1.28252718670509318512E0, |
| -1.95539544257735972385E1, -9.32060152123768231369E1, |
| -1.77681167980488050595E2, -1.47077505154951170175E2, |
| -5.14105326766599330220E1, -6.05014350600728481186E0}; |
| const double QQ[] = {1.00000000000000000000E0, 6.43178256118178023184E1, |
| 8.56430025976980587198E2, 3.88240183605401609683E3, |
| 7.24046774195652478189E3, 5.93072701187316984827E3, |
| 2.06209331660327847417E3, 2.42005740240291393179E2}; |
| const double RP[] = {-4.79443220978201773821E9, 1.95617491946556577543E12, |
| -2.49248344360967716204E14, 9.70862251047306323952E15}; |
| const double RQ[] = {1.00000000000000000000E0, 4.99563147152651017219E2, |
| 1.73785401676374683123E5, 4.84409658339962045305E7, |
| 1.11855537045356834862E10, 2.11277520115489217587E12, |
| 3.10518229857422583814E14, 3.18121955943204943306E16, |
| 1.71086294081043136091E18}; |
| const T DR1 = pset1<T>(5.78318596294678452118E0); |
| const T DR2 = pset1<T>(3.04712623436620863991E1); |
| const T SQ2OPI = pset1<T>(7.9788456080286535587989E-1); /* sqrt(2 / pi) */ |
| const T NEG_PIO4 = pset1<T>(-0.7853981633974483096); /* pi / 4 */ |
| |
| T y = pabs(x); |
| T z = pmul(y, y); |
| T y_le_five = pselect( |
| pcmp_lt(y, pset1<T>(1.0e-5)), |
| pmadd(z, pset1<T>(-0.25), pset1<T>(1.0)), |
| pmul(pmul(psub(z, DR1), psub(z, DR2)), |
| pdiv(internal::ppolevl<T, 3>::run(z, RP), |
| internal::ppolevl<T, 8>::run(z, RQ)))); |
| T s = pdiv(pset1<T>(25.0), z); |
| T p = pdiv( |
| internal::ppolevl<T, 6>::run(s, PP), |
| internal::ppolevl<T, 6>::run(s, PQ)); |
| T q = pdiv( |
| internal::ppolevl<T, 7>::run(s, QP), |
| internal::ppolevl<T, 7>::run(s, QQ)); |
| T yn = padd(y, NEG_PIO4); |
| T w = pdiv(pset1<T>(-5.0), y); |
| p = pmadd(p, pcos(yn), pmul(w, pmul(q, psin(yn)))); |
| T y_gt_five = pmul(p, pmul(SQ2OPI, prsqrt(y))); |
| return pselect(pcmp_le(y, pset1<T>(5.0)), y_le_five, y_gt_five); |
| } |
| }; |
| |
| template <typename T> |
| struct bessel_j0_impl { |
| EIGEN_DEVICE_FUNC |
| static EIGEN_STRONG_INLINE T run(const T x) { |
| return generic_j0<T>::run(x); |
| } |
| }; |
| |
| template <typename T> |
| struct bessel_y0_retval { |
| typedef T type; |
| }; |
| |
| template <typename T, typename ScalarType = typename unpacket_traits<T>::type> |
| struct generic_y0 { |
| EIGEN_DEVICE_FUNC |
| static EIGEN_STRONG_INLINE T run(const T&) { |
| EIGEN_STATIC_ASSERT((internal::is_same<T, T>::value == false), |
| THIS_TYPE_IS_NOT_SUPPORTED); |
| return ScalarType(0); |
| } |
| }; |
| |
| template <typename T> |
| struct generic_y0<T, float> { |
| EIGEN_DEVICE_FUNC |
| static EIGEN_STRONG_INLINE T run(const T& x) { |
| /* j0f.c |
| * Bessel function of the second kind, order zero |
| * |
| * |
| * |
| * SYNOPSIS: |
| * |
| * float x, y, y0f(); |
| * |
| * y = y0f( x ); |
| * |
| * |
| * |
| * DESCRIPTION: |
| * |
| * Returns Bessel function of the second kind, of order |
| * zero, of the argument. |
| * |
| * The domain is divided into the intervals [0, 2] and |
| * (2, infinity). In the first interval a rational approximation |
| * R(x) is employed to compute |
| * |
| * 2 2 2 |
| * y0(x) = (w - r ) (w - r ) (w - r ) R(x) + 2/pi ln(x) j0(x). |
| * 1 2 3 |
| * |
| * Thus a call to j0() is required. The three zeros are removed |
| * from R(x) to improve its numerical stability. |
| * |
| * In the second interval, the modulus and phase are approximated |
| * by polynomials of the form Modulus(x) = sqrt(1/x) Q(1/x) |
| * and Phase(x) = x + 1/x S(1/x^2) - pi/4. Then the function is |
| * |
| * y0(x) = Modulus(x) sin( Phase(x) ). |
| * |
| * |
| * |
| * |
| * ACCURACY: |
| * |
| * Absolute error, when y0(x) < 1; else relative error: |
| * |
| * arithmetic domain # trials peak rms |
| * IEEE 0, 2 100000 2.4e-7 3.4e-8 |
| * IEEE 2, 32 100000 1.8e-7 5.3e-8 |
| * |
| */ |
| |
| const float YP[] = {9.454583683980369E-008f, -9.413212653797057E-006f, |
| 5.344486707214273E-004f, -1.584289289821316E-002f, |
| 1.707584643733568E-001f}; |
| const float MO[] = {-6.838999669318810E-002f, 1.864949361379502E-001f, |
| -2.145007480346739E-001f, 1.197549369473540E-001f, |
| -3.560281861530129E-003f, -4.969382655296620E-002f, |
| -3.355424622293709E-006f, 7.978845717621440E-001f}; |
| const float PH[] = {3.242077816988247E+001f, -3.630592630518434E+001f, |
| 1.756221482109099E+001f, -4.974978466280903E+000f, |
| 1.001973420681837E+000f, -1.939906941791308E-001f, |
| 6.490598792654666E-002f, -1.249992184872738E-001f}; |
| const T YZ1 = pset1<T>(0.43221455686510834878f); |
| const T TWOOPI = pset1<T>(0.636619772367581343075535f); /* 2 / pi */ |
| const T NEG_PIO4F = pset1<T>(-0.7853981633974483096f); /* -pi / 4 */ |
| const T NEG_MAXNUM = pset1<T>(-NumTraits<float>::infinity()); |
| T z = pmul(x, x); |
| T x_le_two = pmul(TWOOPI, pmul(plog(x), generic_j0<T, float>::run(x))); |
| x_le_two = pmadd( |
| psub(z, YZ1), internal::ppolevl<T, 4>::run(z, YP), x_le_two); |
| x_le_two = pselect(pcmp_le(x, pset1<T>(0.0)), NEG_MAXNUM, x_le_two); |
| T q = pdiv(pset1<T>(1.0), x); |
| T w = prsqrt(x); |
| T p = pmul(w, internal::ppolevl<T, 7>::run(q, MO)); |
| T u = pmul(q, q); |
| T xn = pmadd(q, internal::ppolevl<T, 7>::run(u, PH), NEG_PIO4F); |
| T x_gt_two = pmul(p, psin(padd(xn, x))); |
| return pselect(pcmp_le(x, pset1<T>(2.0)), x_le_two, x_gt_two); |
| } |
| }; |
| |
| template <typename T> |
| struct generic_y0<T, double> { |
| EIGEN_DEVICE_FUNC |
| static EIGEN_STRONG_INLINE T run(const T& x) { |
| /* j0.c |
| * Bessel function of the second kind, order zero |
| * |
| * |
| * |
| * SYNOPSIS: |
| * |
| * double x, y, y0(); |
| * |
| * y = y0( x ); |
| * |
| * |
| * |
| * DESCRIPTION: |
| * |
| * Returns Bessel function of the second kind, of order |
| * zero, of the argument. |
| * |
| * The domain is divided into the intervals [0, 5] and |
| * (5, infinity). In the first interval a rational approximation |
| * R(x) is employed to compute |
| * y0(x) = R(x) + 2 * log(x) * j0(x) / PI. |
| * Thus a call to j0() is required. |
| * |
| * In the second interval, the Hankel asymptotic expansion |
| * is employed with two rational functions of degree 6/6 |
| * and 7/7. |
| * |
| * |
| * |
| * ACCURACY: |
| * |
| * Absolute error, when y0(x) < 1; else relative error: |
| * |
| * arithmetic domain # trials peak rms |
| * DEC 0, 30 9400 7.0e-17 7.9e-18 |
| * IEEE 0, 30 30000 1.3e-15 1.6e-16 |
| * |
| */ |
| const double PP[] = {7.96936729297347051624E-4, 8.28352392107440799803E-2, |
| 1.23953371646414299388E0, 5.44725003058768775090E0, |
| 8.74716500199817011941E0, 5.30324038235394892183E0, |
| 9.99999999999999997821E-1}; |
| const double PQ[] = {9.24408810558863637013E-4, 8.56288474354474431428E-2, |
| 1.25352743901058953537E0, 5.47097740330417105182E0, |
| 8.76190883237069594232E0, 5.30605288235394617618E0, |
| 1.00000000000000000218E0}; |
| const double QP[] = {-1.13663838898469149931E-2, -1.28252718670509318512E0, |
| -1.95539544257735972385E1, -9.32060152123768231369E1, |
| -1.77681167980488050595E2, -1.47077505154951170175E2, |
| -5.14105326766599330220E1, -6.05014350600728481186E0}; |
| const double QQ[] = {1.00000000000000000000E0, 6.43178256118178023184E1, |
| 8.56430025976980587198E2, 3.88240183605401609683E3, |
| 7.24046774195652478189E3, 5.93072701187316984827E3, |
| 2.06209331660327847417E3, 2.42005740240291393179E2}; |
| const double YP[] = {1.55924367855235737965E4, -1.46639295903971606143E7, |
| 5.43526477051876500413E9, -9.82136065717911466409E11, |
| 8.75906394395366999549E13, -3.46628303384729719441E15, |
| 4.42733268572569800351E16, -1.84950800436986690637E16}; |
| const double YQ[] = {1.00000000000000000000E0, 1.04128353664259848412E3, |
| 6.26107330137134956842E5, 2.68919633393814121987E8, |
| 8.64002487103935000337E10, 2.02979612750105546709E13, |
| 3.17157752842975028269E15, 2.50596256172653059228E17}; |
| const T SQ2OPI = pset1<T>(7.9788456080286535587989E-1); /* sqrt(2 / pi) */ |
| const T TWOOPI = pset1<T>(0.636619772367581343075535); /* 2 / pi */ |
| const T NEG_PIO4 = pset1<T>(-0.7853981633974483096); /* -pi / 4 */ |
| const T NEG_MAXNUM = pset1<T>(-NumTraits<double>::infinity()); |
| |
| T z = pmul(x, x); |
| T x_le_five = pdiv(internal::ppolevl<T, 7>::run(z, YP), |
| internal::ppolevl<T, 7>::run(z, YQ)); |
| x_le_five = pmadd( |
| pmul(TWOOPI, plog(x)), generic_j0<T, double>::run(x), x_le_five); |
| x_le_five = pselect(pcmp_le(x, pset1<T>(0.0)), NEG_MAXNUM, x_le_five); |
| T s = pdiv(pset1<T>(25.0), z); |
| T p = pdiv( |
| internal::ppolevl<T, 6>::run(s, PP), |
| internal::ppolevl<T, 6>::run(s, PQ)); |
| T q = pdiv( |
| internal::ppolevl<T, 7>::run(s, QP), |
| internal::ppolevl<T, 7>::run(s, QQ)); |
| T xn = padd(x, NEG_PIO4); |
| T w = pdiv(pset1<T>(5.0), x); |
| p = pmadd(p, psin(xn), pmul(w, pmul(q, pcos(xn)))); |
| T x_gt_five = pmul(p, pmul(SQ2OPI, prsqrt(x))); |
| return pselect(pcmp_le(x, pset1<T>(5.0)), x_le_five, x_gt_five); |
| } |
| }; |
| |
| template <typename T> |
| struct bessel_y0_impl { |
| EIGEN_DEVICE_FUNC |
| static EIGEN_STRONG_INLINE T run(const T x) { |
| return generic_y0<T>::run(x); |
| } |
| }; |
| |
| template <typename T> |
| struct bessel_j1_retval { |
| typedef T type; |
| }; |
| |
| template <typename T, typename ScalarType = typename unpacket_traits<T>::type> |
| struct generic_j1 { |
| EIGEN_DEVICE_FUNC |
| static EIGEN_STRONG_INLINE T run(const T&) { |
| EIGEN_STATIC_ASSERT((internal::is_same<T, T>::value == false), |
| THIS_TYPE_IS_NOT_SUPPORTED); |
| return ScalarType(0); |
| } |
| }; |
| |
| template <typename T> |
| struct generic_j1<T, float> { |
| EIGEN_DEVICE_FUNC |
| static EIGEN_STRONG_INLINE T run(const T& x) { |
| /* j1f.c |
| * Bessel function of order one |
| * |
| * |
| * |
| * SYNOPSIS: |
| * |
| * float x, y, j1f(); |
| * |
| * y = j1f( x ); |
| * |
| * |
| * |
| * DESCRIPTION: |
| * |
| * Returns Bessel function of order one of the argument. |
| * |
| * The domain is divided into the intervals [0, 2] and |
| * (2, infinity). In the first interval a polynomial approximation |
| * 2 |
| * (w - r ) x P(w) |
| * 1 |
| * 2 |
| * is used, where w = x and r is the first zero of the function. |
| * |
| * In the second interval, the modulus and phase are approximated |
| * by polynomials of the form Modulus(x) = sqrt(1/x) Q(1/x) |
| * and Phase(x) = x + 1/x R(1/x^2) - 3pi/4. The function is |
| * |
| * j0(x) = Modulus(x) cos( Phase(x) ). |
| * |
| * |
| * |
| * ACCURACY: |
| * |
| * Absolute error: |
| * arithmetic domain # trials peak rms |
| * IEEE 0, 2 100000 1.2e-7 2.5e-8 |
| * IEEE 2, 32 100000 2.0e-7 5.3e-8 |
| * |
| * |
| */ |
| |
| const float JP[] = {-4.878788132172128E-009f, 6.009061827883699E-007f, |
| -4.541343896997497E-005f, 1.937383947804541E-003f, |
| -3.405537384615824E-002f}; |
| const float MO1[] = {6.913942741265801E-002f, -2.284801500053359E-001f, |
| 3.138238455499697E-001f, -2.102302420403875E-001f, |
| 5.435364690523026E-003f, 1.493389585089498E-001f, |
| 4.976029650847191E-006f, 7.978845453073848E-001f}; |
| const float PH1[] = {-4.497014141919556E+001f, 5.073465654089319E+001f, |
| -2.485774108720340E+001f, 7.222973196770240E+000f, |
| -1.544842782180211E+000f, 3.503787691653334E-001f, |
| -1.637986776941202E-001f, 3.749989509080821E-001f}; |
| const T Z1 = pset1<T>(1.46819706421238932572E1f); |
| const T NEG_THPIO4F = pset1<T>(-2.35619449019234492885f); /* -3*pi/4 */ |
| |
| T y = pabs(x); |
| T z = pmul(y, y); |
| T y_le_two = pmul( |
| psub(z, Z1), |
| pmul(x, internal::ppolevl<T, 4>::run(z, JP))); |
| T q = pdiv(pset1<T>(1.0f), y); |
| T w = prsqrt(y); |
| T p = pmul(w, internal::ppolevl<T, 7>::run(q, MO1)); |
| w = pmul(q, q); |
| T yn = pmadd(q, internal::ppolevl<T, 7>::run(w, PH1), NEG_THPIO4F); |
| T y_gt_two = pmul(p, pcos(padd(yn, y))); |
| // j1 is an odd function. This implementation differs from cephes to |
| // take this fact in to account. Cephes returns -j1(x) for y > 2 range. |
| y_gt_two = pselect( |
| pcmp_lt(x, pset1<T>(0.0f)), pnegate(y_gt_two), y_gt_two); |
| return pselect(pcmp_le(y, pset1<T>(2.0f)), y_le_two, y_gt_two); |
| } |
| }; |
| |
| template <typename T> |
| struct generic_j1<T, double> { |
| EIGEN_DEVICE_FUNC |
| static EIGEN_STRONG_INLINE T run(const T& x) { |
| /* j1.c |
| * Bessel function of order one |
| * |
| * |
| * |
| * SYNOPSIS: |
| * |
| * double x, y, j1(); |
| * |
| * y = j1( x ); |
| * |
| * |
| * |
| * DESCRIPTION: |
| * |
| * Returns Bessel function of order one of the argument. |
| * |
| * The domain is divided into the intervals [0, 8] and |
| * (8, infinity). In the first interval a 24 term Chebyshev |
| * expansion is used. In the second, the asymptotic |
| * trigonometric representation is employed using two |
| * rational functions of degree 5/5. |
| * |
| * |
| * |
| * ACCURACY: |
| * |
| * Absolute error: |
| * arithmetic domain # trials peak rms |
| * DEC 0, 30 10000 4.0e-17 1.1e-17 |
| * IEEE 0, 30 30000 2.6e-16 1.1e-16 |
| * |
| */ |
| const double PP[] = {7.62125616208173112003E-4, 7.31397056940917570436E-2, |
| 1.12719608129684925192E0, 5.11207951146807644818E0, |
| 8.42404590141772420927E0, 5.21451598682361504063E0, |
| 1.00000000000000000254E0}; |
| const double PQ[] = {5.71323128072548699714E-4, 6.88455908754495404082E-2, |
| 1.10514232634061696926E0, 5.07386386128601488557E0, |
| 8.39985554327604159757E0, 5.20982848682361821619E0, |
| 9.99999999999999997461E-1}; |
| const double QP[] = {5.10862594750176621635E-2, 4.98213872951233449420E0, |
| 7.58238284132545283818E1, 3.66779609360150777800E2, |
| 7.10856304998926107277E2, 5.97489612400613639965E2, |
| 2.11688757100572135698E2, 2.52070205858023719784E1}; |
| const double QQ[] = {1.00000000000000000000E0, 7.42373277035675149943E1, |
| 1.05644886038262816351E3, 4.98641058337653607651E3, |
| 9.56231892404756170795E3, 7.99704160447350683650E3, |
| 2.82619278517639096600E3, 3.36093607810698293419E2}; |
| const double RP[] = {-8.99971225705559398224E8, 4.52228297998194034323E11, |
| -7.27494245221818276015E13, 3.68295732863852883286E15}; |
| const double RQ[] = {1.00000000000000000000E0, 6.20836478118054335476E2, |
| 2.56987256757748830383E5, 8.35146791431949253037E7, |
| 2.21511595479792499675E10, 4.74914122079991414898E12, |
| 7.84369607876235854894E14, 8.95222336184627338078E16, |
| 5.32278620332680085395E18}; |
| const T Z1 = pset1<T>(1.46819706421238932572E1); |
| const T Z2 = pset1<T>(4.92184563216946036703E1); |
| const T NEG_THPIO4 = pset1<T>(-2.35619449019234492885); /* -3*pi/4 */ |
| const T SQ2OPI = pset1<T>(7.9788456080286535587989E-1); /* sqrt(2 / pi) */ |
| T y = pabs(x); |
| T z = pmul(y, y); |
| T y_le_five = pdiv(internal::ppolevl<T, 3>::run(z, RP), |
| internal::ppolevl<T, 8>::run(z, RQ)); |
| y_le_five = pmul(pmul(pmul(y_le_five, x), psub(z, Z1)), psub(z, Z2)); |
| T s = pdiv(pset1<T>(25.0), z); |
| T p = pdiv( |
| internal::ppolevl<T, 6>::run(s, PP), |
| internal::ppolevl<T, 6>::run(s, PQ)); |
| T q = pdiv( |
| internal::ppolevl<T, 7>::run(s, QP), |
| internal::ppolevl<T, 7>::run(s, QQ)); |
| T yn = padd(y, NEG_THPIO4); |
| T w = pdiv(pset1<T>(-5.0), y); |
| p = pmadd(p, pcos(yn), pmul(w, pmul(q, psin(yn)))); |
| T y_gt_five = pmul(p, pmul(SQ2OPI, prsqrt(y))); |
| // j1 is an odd function. This implementation differs from cephes to |
| // take this fact in to account. Cephes returns -j1(x) for y > 5 range. |
| y_gt_five = pselect( |
| pcmp_lt(x, pset1<T>(0.0)), pnegate(y_gt_five), y_gt_five); |
| return pselect(pcmp_le(y, pset1<T>(5.0)), y_le_five, y_gt_five); |
| } |
| }; |
| |
| template <typename T> |
| struct bessel_j1_impl { |
| EIGEN_DEVICE_FUNC |
| static EIGEN_STRONG_INLINE T run(const T x) { |
| return generic_j1<T>::run(x); |
| } |
| }; |
| |
| template <typename T> |
| struct bessel_y1_retval { |
| typedef T type; |
| }; |
| |
| template <typename T, typename ScalarType = typename unpacket_traits<T>::type> |
| struct generic_y1 { |
| EIGEN_DEVICE_FUNC |
| static EIGEN_STRONG_INLINE T run(const T&) { |
| EIGEN_STATIC_ASSERT((internal::is_same<T, T>::value == false), |
| THIS_TYPE_IS_NOT_SUPPORTED); |
| return ScalarType(0); |
| } |
| }; |
| |
| template <typename T> |
| struct generic_y1<T, float> { |
| EIGEN_DEVICE_FUNC |
| static EIGEN_STRONG_INLINE T run(const T& x) { |
| /* j1f.c |
| * Bessel function of second kind of order one |
| * |
| * |
| * |
| * SYNOPSIS: |
| * |
| * double x, y, y1(); |
| * |
| * y = y1( x ); |
| * |
| * |
| * |
| * DESCRIPTION: |
| * |
| * Returns Bessel function of the second kind of order one |
| * of the argument. |
| * |
| * The domain is divided into the intervals [0, 2] and |
| * (2, infinity). In the first interval a rational approximation |
| * R(x) is employed to compute |
| * |
| * 2 |
| * y0(x) = (w - r ) x R(x^2) + 2/pi (ln(x) j1(x) - 1/x) . |
| * 1 |
| * |
| * Thus a call to j1() is required. |
| * |
| * In the second interval, the modulus and phase are approximated |
| * by polynomials of the form Modulus(x) = sqrt(1/x) Q(1/x) |
| * and Phase(x) = x + 1/x S(1/x^2) - 3pi/4. Then the function is |
| * |
| * y0(x) = Modulus(x) sin( Phase(x) ). |
| * |
| * |
| * |
| * |
| * ACCURACY: |
| * |
| * Absolute error: |
| * arithmetic domain # trials peak rms |
| * IEEE 0, 2 100000 2.2e-7 4.6e-8 |
| * IEEE 2, 32 100000 1.9e-7 5.3e-8 |
| * |
| * (error criterion relative when |y1| > 1). |
| * |
| */ |
| |
| const float YP[] = {8.061978323326852E-009f, -9.496460629917016E-007f, |
| 6.719543806674249E-005f, -2.641785726447862E-003f, |
| 4.202369946500099E-002f}; |
| const float MO1[] = {6.913942741265801E-002f, -2.284801500053359E-001f, |
| 3.138238455499697E-001f, -2.102302420403875E-001f, |
| 5.435364690523026E-003f, 1.493389585089498E-001f, |
| 4.976029650847191E-006f, 7.978845453073848E-001f}; |
| const float PH1[] = {-4.497014141919556E+001f, 5.073465654089319E+001f, |
| -2.485774108720340E+001f, 7.222973196770240E+000f, |
| -1.544842782180211E+000f, 3.503787691653334E-001f, |
| -1.637986776941202E-001f, 3.749989509080821E-001f}; |
| const T YO1 = pset1<T>(4.66539330185668857532f); |
| const T NEG_THPIO4F = pset1<T>(-2.35619449019234492885f); /* -3*pi/4 */ |
| const T TWOOPI = pset1<T>(0.636619772367581343075535f); /* 2/pi */ |
| const T NEG_MAXNUM = pset1<T>(-NumTraits<float>::infinity()); |
| |
| T z = pmul(x, x); |
| T x_le_two = pmul(psub(z, YO1), internal::ppolevl<T, 4>::run(z, YP)); |
| x_le_two = pmadd( |
| x_le_two, x, |
| pmul(TWOOPI, pmadd( |
| generic_j1<T, float>::run(x), plog(x), |
| pdiv(pset1<T>(-1.0f), x)))); |
| x_le_two = pselect(pcmp_lt(x, pset1<T>(0.0f)), NEG_MAXNUM, x_le_two); |
| |
| T q = pdiv(pset1<T>(1.0), x); |
| T w = prsqrt(x); |
| T p = pmul(w, internal::ppolevl<T, 7>::run(q, MO1)); |
| w = pmul(q, q); |
| T xn = pmadd(q, internal::ppolevl<T, 7>::run(w, PH1), NEG_THPIO4F); |
| T x_gt_two = pmul(p, psin(padd(xn, x))); |
| return pselect(pcmp_le(x, pset1<T>(2.0)), x_le_two, x_gt_two); |
| } |
| }; |
| |
| template <typename T> |
| struct generic_y1<T, double> { |
| EIGEN_DEVICE_FUNC |
| static EIGEN_STRONG_INLINE T run(const T& x) { |
| /* j1.c |
| * Bessel function of second kind of order one |
| * |
| * |
| * |
| * SYNOPSIS: |
| * |
| * double x, y, y1(); |
| * |
| * y = y1( x ); |
| * |
| * |
| * |
| * DESCRIPTION: |
| * |
| * Returns Bessel function of the second kind of order one |
| * of the argument. |
| * |
| * The domain is divided into the intervals [0, 8] and |
| * (8, infinity). In the first interval a 25 term Chebyshev |
| * expansion is used, and a call to j1() is required. |
| * In the second, the asymptotic trigonometric representation |
| * is employed using two rational functions of degree 5/5. |
| * |
| * |
| * |
| * ACCURACY: |
| * |
| * Absolute error: |
| * arithmetic domain # trials peak rms |
| * DEC 0, 30 10000 8.6e-17 1.3e-17 |
| * IEEE 0, 30 30000 1.0e-15 1.3e-16 |
| * |
| * (error criterion relative when |y1| > 1). |
| * |
| */ |
| const double PP[] = {7.62125616208173112003E-4, 7.31397056940917570436E-2, |
| 1.12719608129684925192E0, 5.11207951146807644818E0, |
| 8.42404590141772420927E0, 5.21451598682361504063E0, |
| 1.00000000000000000254E0}; |
| const double PQ[] = {5.71323128072548699714E-4, 6.88455908754495404082E-2, |
| 1.10514232634061696926E0, 5.07386386128601488557E0, |
| 8.39985554327604159757E0, 5.20982848682361821619E0, |
| 9.99999999999999997461E-1}; |
| const double QP[] = {5.10862594750176621635E-2, 4.98213872951233449420E0, |
| 7.58238284132545283818E1, 3.66779609360150777800E2, |
| 7.10856304998926107277E2, 5.97489612400613639965E2, |
| 2.11688757100572135698E2, 2.52070205858023719784E1}; |
| const double QQ[] = {1.00000000000000000000E0, 7.42373277035675149943E1, |
| 1.05644886038262816351E3, 4.98641058337653607651E3, |
| 9.56231892404756170795E3, 7.99704160447350683650E3, |
| 2.82619278517639096600E3, 3.36093607810698293419E2}; |
| const double YP[] = {1.26320474790178026440E9, -6.47355876379160291031E11, |
| 1.14509511541823727583E14, -8.12770255501325109621E15, |
| 2.02439475713594898196E17, -7.78877196265950026825E17}; |
| const double YQ[] = {1.00000000000000000000E0, 5.94301592346128195359E2, |
| 2.35564092943068577943E5, 7.34811944459721705660E7, |
| 1.87601316108706159478E10, 3.88231277496238566008E12, |
| 6.20557727146953693363E14, 6.87141087355300489866E16, |
| 3.97270608116560655612E18}; |
| const T SQ2OPI = pset1<T>(.79788456080286535588); |
| const T NEG_THPIO4 = pset1<T>(-2.35619449019234492885); /* -3*pi/4 */ |
| const T TWOOPI = pset1<T>(0.636619772367581343075535); /* 2/pi */ |
| const T NEG_MAXNUM = pset1<T>(-NumTraits<double>::infinity()); |
| |
| T z = pmul(x, x); |
| T x_le_five = pdiv(internal::ppolevl<T, 5>::run(z, YP), |
| internal::ppolevl<T, 8>::run(z, YQ)); |
| x_le_five = pmadd( |
| x_le_five, x, pmul( |
| TWOOPI, pmadd(generic_j1<T, double>::run(x), plog(x), |
| pdiv(pset1<T>(-1.0), x)))); |
| |
| x_le_five = pselect(pcmp_le(x, pset1<T>(0.0)), NEG_MAXNUM, x_le_five); |
| T s = pdiv(pset1<T>(25.0), z); |
| T p = pdiv( |
| internal::ppolevl<T, 6>::run(s, PP), |
| internal::ppolevl<T, 6>::run(s, PQ)); |
| T q = pdiv( |
| internal::ppolevl<T, 7>::run(s, QP), |
| internal::ppolevl<T, 7>::run(s, QQ)); |
| T xn = padd(x, NEG_THPIO4); |
| T w = pdiv(pset1<T>(5.0), x); |
| p = pmadd(p, psin(xn), pmul(w, pmul(q, pcos(xn)))); |
| T x_gt_five = pmul(p, pmul(SQ2OPI, prsqrt(x))); |
| return pselect(pcmp_le(x, pset1<T>(5.0)), x_le_five, x_gt_five); |
| } |
| }; |
| |
| template <typename T> |
| struct bessel_y1_impl { |
| EIGEN_DEVICE_FUNC |
| static EIGEN_STRONG_INLINE T run(const T x) { |
| return generic_y1<T>::run(x); |
| } |
| }; |
| |
| } // end namespace internal |
| |
| namespace numext { |
| |
| template <typename Scalar> |
| EIGEN_DEVICE_FUNC inline EIGEN_MATHFUNC_RETVAL(bessel_i0, Scalar) |
| bessel_i0(const Scalar& x) { |
| return EIGEN_MATHFUNC_IMPL(bessel_i0, Scalar)::run(x); |
| } |
| |
| template <typename Scalar> |
| EIGEN_DEVICE_FUNC inline EIGEN_MATHFUNC_RETVAL(bessel_i0e, Scalar) |
| bessel_i0e(const Scalar& x) { |
| return EIGEN_MATHFUNC_IMPL(bessel_i0e, Scalar)::run(x); |
| } |
| |
| template <typename Scalar> |
| EIGEN_DEVICE_FUNC inline EIGEN_MATHFUNC_RETVAL(bessel_i1, Scalar) |
| bessel_i1(const Scalar& x) { |
| return EIGEN_MATHFUNC_IMPL(bessel_i1, Scalar)::run(x); |
| } |
| |
| template <typename Scalar> |
| EIGEN_DEVICE_FUNC inline EIGEN_MATHFUNC_RETVAL(bessel_i1e, Scalar) |
| bessel_i1e(const Scalar& x) { |
| return EIGEN_MATHFUNC_IMPL(bessel_i1e, Scalar)::run(x); |
| } |
| |
| template <typename Scalar> |
| EIGEN_DEVICE_FUNC inline EIGEN_MATHFUNC_RETVAL(bessel_k0, Scalar) |
| bessel_k0(const Scalar& x) { |
| return EIGEN_MATHFUNC_IMPL(bessel_k0, Scalar)::run(x); |
| } |
| |
| template <typename Scalar> |
| EIGEN_DEVICE_FUNC inline EIGEN_MATHFUNC_RETVAL(bessel_k0e, Scalar) |
| bessel_k0e(const Scalar& x) { |
| return EIGEN_MATHFUNC_IMPL(bessel_k0e, Scalar)::run(x); |
| } |
| |
| template <typename Scalar> |
| EIGEN_DEVICE_FUNC inline EIGEN_MATHFUNC_RETVAL(bessel_k1, Scalar) |
| bessel_k1(const Scalar& x) { |
| return EIGEN_MATHFUNC_IMPL(bessel_k1, Scalar)::run(x); |
| } |
| |
| template <typename Scalar> |
| EIGEN_DEVICE_FUNC inline EIGEN_MATHFUNC_RETVAL(bessel_k1e, Scalar) |
| bessel_k1e(const Scalar& x) { |
| return EIGEN_MATHFUNC_IMPL(bessel_k1e, Scalar)::run(x); |
| } |
| |
| template <typename Scalar> |
| EIGEN_DEVICE_FUNC inline EIGEN_MATHFUNC_RETVAL(bessel_j0, Scalar) |
| bessel_j0(const Scalar& x) { |
| return EIGEN_MATHFUNC_IMPL(bessel_j0, Scalar)::run(x); |
| } |
| |
| template <typename Scalar> |
| EIGEN_DEVICE_FUNC inline EIGEN_MATHFUNC_RETVAL(bessel_y0, Scalar) |
| bessel_y0(const Scalar& x) { |
| return EIGEN_MATHFUNC_IMPL(bessel_y0, Scalar)::run(x); |
| } |
| |
| template <typename Scalar> |
| EIGEN_DEVICE_FUNC inline EIGEN_MATHFUNC_RETVAL(bessel_j1, Scalar) |
| bessel_j1(const Scalar& x) { |
| return EIGEN_MATHFUNC_IMPL(bessel_j1, Scalar)::run(x); |
| } |
| |
| template <typename Scalar> |
| EIGEN_DEVICE_FUNC inline EIGEN_MATHFUNC_RETVAL(bessel_y1, Scalar) |
| bessel_y1(const Scalar& x) { |
| return EIGEN_MATHFUNC_IMPL(bessel_y1, Scalar)::run(x); |
| } |
| |
| } // end namespace numext |
| |
| } // end namespace Eigen |
| |
| #endif // EIGEN_BESSEL_FUNCTIONS_H |
