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JacobiEllipticFunctions.h
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JacobiEllipticFunctions.h
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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)
*
* 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.
*/
#ifndef __JACOBI_ELLIPTIC_FUNCTIONS_H_
#define __JACOBI_ELLIPTIC_FUNCTIONS_H_
#include <shogun/lib/config.h>
#include <shogun/base/SGObject.h>
#include <complex>
#include <limits>
#include <math.h>
#ifdef HAVE_ARPREC
#include <arprec/mp_real.h>
#include <arprec/mp_complex.h>
#endif //HAVE_ARPREC
namespace shogun
{
/** @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;
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;
}
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);
}
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);
/** 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);
#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();
}
/** 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
/** @return object name */
virtual const char* get_name() const
{
return "JacobiEllipticFunctions";
}
};
}
#endif /* __JACOBI_ELLIPTIC_FUNCTIONS_H_ */