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JacobiEllipticFunctions added
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| /* | ||
| * This program is free software; you can redistribute it and/or modify | ||
| * it under the terms of the GNU General Public License as published by | ||
| * the Free Software Foundation; either version 3 of the License, or | ||
| * (at your option) any later version. | ||
| * | ||
| * Written (W) 2013 Soumyajit De | ||
| * | ||
| * KRYLSTAT Copyright 2011 by Erlend Aune <erlenda@math.ntnu.no> under GPL2+ | ||
| * (few parts rewritten and adjusted for shogun) | ||
| */ | ||
|
|
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| #include <shogun/mathematics/JacobiEllipticFunctions.h> | ||
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| using namespace shogun; | ||
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| void CJacobiEllipticFunctions::ellKKP(Real L, Real &K, Real &Kp) | ||
| { | ||
| REQUIRE(L>=0.0, | ||
| "CJacobiEllipticFunctions::ellKKP(): \ | ||
| Parameter L should be non-negative\n"); | ||
| #ifdef HAVE_ARPREC | ||
| const Real eps=Real(std::numeric_limits<float64_t>::epsilon()); | ||
| const Real pi=mp_real::_pi; | ||
| #else | ||
| const Real eps=std::numeric_limits<Real>::epsilon(); | ||
| const Real pi=M_PI; | ||
| #endif //HAVE_ARPREC | ||
| if (L>10.0) | ||
| { | ||
| K=pi*0.5; | ||
| Kp=pi*L+log(4.0); | ||
| } | ||
| else | ||
| { | ||
| Real m=exp(-2.0*pi*L); | ||
| Real mp=1.0-m; | ||
| if (m<eps) | ||
| { | ||
| K=compute_quarter_period(sqrt(mp)); | ||
| Kp=Real(std::numeric_limits<float64_t>::max()); | ||
| } | ||
| else if (mp<eps) | ||
| { | ||
| K=Real(std::numeric_limits<float64_t>::max()); | ||
| Kp=compute_quarter_period(sqrt(m)); | ||
| } | ||
| else | ||
| { | ||
| K=compute_quarter_period(sqrt(mp)); | ||
| Kp=compute_quarter_period(sqrt(m)); | ||
| } | ||
| } | ||
| } | ||
|
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| void CJacobiEllipticFunctions | ||
| ::ellPJC(Complex u, Real m, Complex &sn, Complex &cn, Complex &dn) | ||
| { | ||
| REQUIRE(m>=0.0 && m<=1.0, | ||
| "CJacobiEllipticFunctions::ellPJC(): \ | ||
| Parameter m should be >=0 and <=1\n"); | ||
|
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| #ifdef HAVE_ARPREC | ||
| const Real eps=sqrt(mp_real::_eps); | ||
| #else | ||
| const Real eps=sqrt(std::numeric_limits<Real>::epsilon()); | ||
| #endif //HAVE_ARPREC | ||
| if (m>=(1.0-eps)) | ||
| { | ||
| #ifdef HAVE_ARPREC | ||
| std::complex<float64_t> _u(dble(u.real),dble(u.imag)); | ||
| std::complex<float64_t> t=tanh(_u); | ||
| std::complex<float64_t> b=cosh(_u); | ||
| std::complex<float64_t> twon=b*sinh(_u); | ||
| std::complex<float64_t> ai=0.25*(1.0-dble(m)); | ||
| std::complex<float64_t> _sn=t+ai*(twon-_u)/(b*b); | ||
| std::complex<float64_t> phi=1.0/b; | ||
| std::complex<float64_t> _cn=phi-ai*(twon-_u); | ||
| std::complex<float64_t> _dn=phi+ai*(twon+_u); | ||
| sn=mp_complex(_sn.real(),_sn.imag()); | ||
| cn=mp_complex(_cn.real(),_cn.imag()); | ||
| dn=mp_complex(_dn.real(),_dn.imag()); | ||
| #else | ||
| Complex t=tanh(u); | ||
| Complex b=cosh(u); | ||
| Complex ai=0.25*(1.0-m); | ||
| Complex twon=b*sinh(u); | ||
| sn=t+ai*(twon-u)/(b*b); | ||
| Complex phi=Real(1.0)/b; | ||
| ai*=t*phi; | ||
| cn=phi-ai*(twon-u); | ||
| dn=phi+ai*(twon+u); | ||
| #endif //HAVE_ARPREC | ||
| } | ||
| else | ||
| { | ||
| const Real prec=4.0*eps; | ||
| const index_t MAX_ITER=128; | ||
| index_t i=0; | ||
| Real kappa[MAX_ITER]; | ||
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| while (i<MAX_ITER && m>prec) | ||
| { | ||
| Real k; | ||
| if (m>0.001) | ||
| { | ||
| Real mp=sqrt(1.0-m); | ||
| k=(1.0-mp)/(1.0+mp); | ||
| } | ||
| else | ||
| k=poly_six(m/4.0); | ||
| u/=(1.0+k); | ||
| m=k*k; | ||
| kappa[i++]=k; | ||
| } | ||
| Complex sin_u=sin(u); | ||
| Complex cos_u=cos(u); | ||
| Complex t=Real(0.25*m)*(u-sin_u*cos_u); | ||
| sn=sin_u-t*cos_u; | ||
| cn=cos_u+t*sin_u; | ||
| dn=Real(1.0)+Real(0.5*m)*(cos_u*cos_u); | ||
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| i--; | ||
| while (i>=0) | ||
| { | ||
| Real k=kappa[i--]; | ||
| Complex ksn2=k*(sn*sn); | ||
| Complex d=Real(1.0)+ksn2; | ||
| sn*=(1.0+k)/d; | ||
| cn*=dn/d; | ||
| dn=(Real(1.0)-ksn2)/d; | ||
| } | ||
| } | ||
| } |
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| Original file line number | Diff line number | Diff line change |
|---|---|---|
| @@ -0,0 +1,164 @@ | ||
| /* | ||
| * This program is free software; you can redistribute it and/or modify | ||
| * it under the terms of the GNU General Public License as published by | ||
| * the Free Software Foundation; either version 3 of the License, or | ||
| * (at your option) any later version. | ||
| * | ||
| * Written (W) 2013 Soumyajit De | ||
| * | ||
| * KRYLSTAT Copyright 2011 by Erlend Aune <erlenda@math.ntnu.no> under GPL2+ | ||
| * (few parts rewritten and adjusted for shogun) | ||
| * | ||
| * NOTE: For higher precision, the methods in this class rely on an external | ||
| * library, ARPREC (http://crd-legacy.lbl.gov/~dhbailey/mpdist/), in absense of | ||
| * which they fallback to shogun datatypes. To use it with shogun, configure | ||
| * ARPREC with "./configure 'CXX c++ -fPIC'" in order to link. | ||
| */ | ||
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| #ifndef __JACOBI_ELLIPTIC_FUNCTIONS_H_ | ||
| #define __JACOBI_ELLIPTIC_FUNCTIONS_H_ | ||
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| #include <shogun/lib/config.h> | ||
| #include <shogun/base/SGObject.h> | ||
| #include <complex> | ||
| #include <limits> | ||
| #include <math.h> | ||
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| #ifdef HAVE_ARPREC | ||
| #include <arprec/mp_real.h> | ||
| #include <arprec/mp_complex.h> | ||
| #endif //HAVE_ARPREC | ||
|
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| namespace shogun | ||
| { | ||
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| /** @brief Class that contains methods for computing Jacobi elliptic functions | ||
| * related to complex analysis. These functions are inverse of the elliptic | ||
| * integral of first kind, i.e. | ||
| * \f[ | ||
| * u(k,m)=\int_{0}^{k}\frac{dt}{\sqrt{(1-t^{2})(1-m^{2}t^{2})}} | ||
| * =\int_{0}^{\varphi}\frac{d\theta}{\sqrt{(1-m^{2}sin^{2}\theta)}} | ||
| * \f] | ||
| * where \f$k=sin\varphi\f$, \f$t=sin\theta\f$ and parameter \f$m, 0\le m | ||
| * \le 1\f$ is called modulus. There main Jacobi elliptic functions are defined | ||
| * as \f$sn(u,m)=k=sin\theta\f$, \f$cn(u,m)=cos\theta=\sqrt{1-sn(u,m)^{2}}\f$ | ||
| * and \f$dn(u,m)=\sqrt{1-m^{2}sn(u,m)^{2}}\f$. | ||
| * For \f$k=1\f$, i.e. \f$\varphi=\frac{\pi}{2}\f$, \f$u(1,m)=K(m)\f$ is known | ||
| * as the complete elliptic integral of first kind. Similarly, \f$u(1,m))= | ||
| * K'(m')\f$, \f$m'=\sqrt{1-m^{2}}\f$ is called the complementary complete | ||
| * elliptic integral of first kind. Jacobi functions are double periodic with | ||
| * quardratic periods \f$K\f$ and \f$K'\f$. | ||
| * | ||
| * This class provides two sets of methods for computing \f$K,K'\f$, and | ||
| * \f$sn,cn,dn\f$. Useful for computing rational approximation of matrix | ||
| * functions given by Cauchy's integral formula, etc. | ||
| */ | ||
| class CJacobiEllipticFunctions: public CSGObject | ||
| { | ||
| #ifdef HAVE_ARPREC | ||
| typedef mp_real Real; | ||
| typedef mp_complex Complex; | ||
| #else | ||
| typedef float64_t Real; | ||
| typedef std::complex<Real> Complex; | ||
| #endif //HAVE_ARPREC | ||
| private: | ||
| static inline Real compute_quarter_period(Real b) | ||
| { | ||
| #ifdef HAVE_ARPREC | ||
| const Real eps=mp_real::_eps; | ||
| const Real pi=mp_real::_pi; | ||
| #else | ||
| const Real eps=std::numeric_limits<Real>::epsilon(); | ||
| const Real pi=M_PI; | ||
| #endif //HAVE_ARPREC | ||
| Real a=1.0; | ||
| Real mm=1.0; | ||
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| int64_t p=2; | ||
| do | ||
| { | ||
| Real a_new=(a+b)*0.5; | ||
| Real b_new=sqrt(a*b); | ||
| Real c=(a-b)*0.5; | ||
| mm=Real(p)*c*c; | ||
| p<<=1; | ||
| a=a_new; | ||
| b=b_new; | ||
| } while (mm>eps); | ||
| return pi*0.5/a; | ||
| } | ||
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| static inline Real poly_six(Real x) | ||
| { | ||
| return (132*pow(x,6)+42*pow(x,5)+14*pow(x,4)+5*pow(x,3)+2*pow(x,2)+x); | ||
| } | ||
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| public: | ||
| /** Computes the quarter periods (K and K') of Jacobian elliptic functions | ||
| * (see class description). | ||
| * @param L | ||
| * @param K the quarter period (to be computed) on the Real axis | ||
| * @param Kp the quarter period (to be computed) on the Imaginary axis | ||
| * computed | ||
| */ | ||
| static void ellKKP(Real L, Real &K, Real &Kp); | ||
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| /** Computes three main Jacobi elliptic functions, \f$sn(u,m)\f$, | ||
| * \f$cn(u,m)\f$ and \f$dn(u,m)\f$ (see class description). | ||
| * @param u the elliptic integral of the first kind \f$u(k,m)\f$ | ||
| * @param m the modulus parameter, \f$0\le m \le 1\f$ | ||
| * @param sn Jacobi elliptic function sn(u,m) | ||
| * @param cn Jacobi elliptic function cn(u,m) | ||
| * @param dn Jacobi elliptic function dn(u,m) | ||
| */ | ||
| static void ellPJC(Complex u, Real m, Complex &sn, Complex &cn, Complex &dn); | ||
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| #ifdef HAVE_ARPREC | ||
| /** Wrapper method for ellKKP if ARPREC is present (for high precision) | ||
| * @param L | ||
| * @param K the quarter period (to be computed) on the Real axis | ||
| * @param Kp the quarter period (to be computed) on the Imaginary axis | ||
| * computed | ||
| */ | ||
| static void ellKKP(float64_t L, float64_t &K, float64_t &Kp) | ||
| { | ||
| mp::mp_init(100, NULL, true); | ||
| mp_real _K, _Kp; | ||
| ellKKP(mp_real(L), _K, _Kp); | ||
| K=dble(_K); | ||
| Kp=dble(_Kp); | ||
| mp::mp_finalize(); | ||
| } | ||
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| /** Wrapper method for ellPJC if ARPREC is present (for high precision) | ||
| * @param u the elliptic integral of the first kind \f$u(k,m)\f$ | ||
| * @param m the modulus parameter, \f$0\le m \le 1\f$ | ||
| * @param sn Jacobi elliptic function sn(u,m) | ||
| * @param cn Jacobi elliptic function cn(u,m) | ||
| * @param dn Jacobi elliptic function dn(u,m) | ||
| */ | ||
| static void ellPJC(std::complex<float64_t> u, float64_t m, | ||
| std::complex<float64_t> &sn, std::complex<float64_t> &cn, | ||
| std::complex<float64_t> &dn) | ||
| { | ||
| mp::mp_init(100, NULL, true); | ||
| mp_complex _sn, _cn, _dn; | ||
| ellPJC(mp_complex(u.real(),u.imag()), mp_real(m), _sn, _cn, _dn); | ||
| sn=std::complex<float64_t>(dble(_sn.real),dble(_sn.imag)); | ||
| cn=std::complex<float64_t>(dble(_cn.real),dble(_cn.imag)); | ||
| dn=std::complex<float64_t>(dble(_dn.real),dble(_dn.imag)); | ||
| mp::mp_finalize(); | ||
| } | ||
| #endif //HAVE_ARPREC | ||
|
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| /** @return object name */ | ||
| virtual const char* get_name() const | ||
| { | ||
| return "JacobiEllipticFunctions"; | ||
| } | ||
| }; | ||
|
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| } | ||
|
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| #endif /* __JACOBI_ELLIPTIC_FUNCTIONS_H_ */ |
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