/*maxwell_juettner_rho_Q: Fitting formula for Faraday conversion coefficient
* rho_Q from Dexter (2016) for a Maxwell-Juettner
* distribution. This formula comes from his
* equations B4, B6, B8, and B13.
*
*@params: struct of parameters params
*@returns: fitting formula to the Faraday conversion coefficient
* for a Maxwell-Juettner distribution of electrons.
*/
double maxwell_juettner_rho_Q(struct parameters * params)
{
double omega0 = params->electron_charge*params->magnetic_field
/ (params->mass_electron*params->speed_light);
double wp2 = 4. * params->pi * params->electron_density
* pow(params->electron_charge, 2.) / params->mass_electron;
/* argument for function f(X) (called jffunc) below */
double x = params->theta_e * sqrt(sqrt(2.) * sin(params->observer_angle)
* (1.e3*omega0 / (2. * params->pi * params->nu)));
/* this is the definition of the modified factor f(X) from Dexter (2016) */
double extraterm = (.011*exp(-x/47.2) - pow(2., (-1./3.)) / pow(3., (23./6.))
* params->pi * 1.e4 * pow((x + 1.e-16), (-8./3.)))
* (0.5 + 0.5 * tanh((log(x) - log(120.)) / 0.1));
double jffunc = 2.011 * exp(-pow(x, 1.035)/4.7) - cos(x/2.)
* exp(-pow(x, 1.2)/2.73) - .011 * exp(-x / 47.2) + extraterm;
double eps11m22 = jffunc * wp2 * pow(omega0, 2.)
/ pow(2.*params->pi * params->nu, 4.)
* (gsl_sf_bessel_Kn(1, 1./params->theta_e)
/ gsl_sf_bessel_Kn(2, 1./params->theta_e)
+ 6. * params->theta_e)
* pow(sin(params->observer_angle), 2.);
double rhoq = 2. * params->pi * params->nu /(2. * params->speed_light)
* eps11m22;
return rhoq;
}
/*differential_of_f: term in gamma integrand only for absorptivity calculation;
*it is the differential Df = 2\pi\nu (1/(mc)*d/dgamma + (beta cos(theta)
* -cos(xi))/(p*beta*c) * d/d(cos(xi))) f
*this is eq. 41 of [1]
*below it is applied for the THERMAL, POWER_LAW, and KAPPA_DIST distributions
*@param gamma: Input, Lorentz factor
*@param nu: Input, frequency of emission/absorption
*@returns: Output, Df term in gamma integrand; depends on distribution function
*/
double differential_of_f(double gamma, double nu) {
#if DISTRIBUTION_FUNCTION == THERMAL
double prefactor = (M_PI * nu / (mass_electron*speed_light*speed_light))
* (n_e/(theta_e * gsl_sf_bessel_Kn(2, 1./theta_e)));
double body = (-1./theta_e) * exp(-gamma/theta_e);
double f = prefactor * body;
#elif DISTRIBUTION_FUNCTION == POWER_LAW
double pl_norm = 4.* M_PI/(normalize_f());
double prefactor = (M_PI * nu / (mass_electron*speed_light*speed_light))
* (n_e_NT*(power_law_p-1.))
/((pow(gamma_min, 1.-power_law_p)
- pow(gamma_max, 1.-power_law_p)));
double term1 = ((-power_law_p-1.)*exp(-gamma/gamma_cutoff)
*pow(gamma,-power_law_p-2.)/(sqrt(gamma*gamma - 1.)));
double term2 = (exp(-gamma/gamma_cutoff) * pow(gamma,(-power_law_p-1.))
/(gamma_cutoff * sqrt(gamma*gamma - 1.)));
double term3 = (exp(-gamma/gamma_cutoff) * pow(gamma,-power_law_p))
/pow((gamma*gamma - 1.), (3./2.));
double f = pl_norm * prefactor * (term1 - term2 - term3);
#elif DISTRIBUTION_FUNCTION == KAPPA_DIST
double prefactor = n_e * (1./normalize_f()) * 4. * M_PI*M_PI * nu
* mass_electron*mass_electron * speed_light;
double term1 = ((- kappa - 1.) / (kappa * w)) * pow((1. + (gamma - 1.)
/(kappa * w)), -kappa-2.);
double term2 = pow((1. + (gamma - 1.)/(kappa * w)), (- kappa - 1.))
* (- 1./gamma_cutoff);
double f = prefactor * (term1 + term2) * exp(-gamma/gamma_cutoff);
#endif
return f;
}
/*dMJ_dgamma: returns the value of df/d\gamma, where f is the Maxwell-Juettner
* (relativistic thermal) distribution function, for the parameters in
* the struct p.
*
*@params: pointer to struct of parameters *params
*
*@returns: d/dgamma MJ(params->gamma, params->theta_e)
*/
double dMJ_dgamma(struct parameters * params)
{
double ans = -exp(-params->gamma/params->theta_e)
/ (4. * params->pi * params->theta_e*params->theta_e
* gsl_sf_bessel_Kn(2, 1./params->theta_e));
return ans;
}
/*maxwell_juttner_f: Maxwell-Juttner distribution function in terms of Lorentz
* factor gamma; uses eq. 47, 49, 50 of [1]
*@param gamma: Input, Lorentz factor
*@returns: THERMAL distribution function, which goes into the gamma integrand
*/
double maxwell_juttner_f(double gamma) {
double beta = sqrt(1. - 1./(gamma*gamma));
double d = (n_e * gamma * sqrt(gamma*gamma-1.) * exp(-gamma/theta_e))
/(4. * M_PI * theta_e * gsl_sf_bessel_Kn(2, 1./theta_e));
double ans = 1./(pow(mass_electron, 3.) * pow(speed_light, 3.) * gamma*gamma
* beta) * d;
return ans;
}
static double f(double w, void *p)
{
struct parameters *params = (struct parameters *)p;
const double kx = (params->kx);
double kr = (params->kr);
double x0 = (params->x0);
double r0 = (params->r0);
int index = (params->index);
double result;
if (w == 0) w = zero;
if (fabs(kx - w) > tol){
result = sin((kx - w)*x0)/(kx - w)*
sqrt(pow(w, 2.0*index)*pow(r0, 2.0*index))/(kr*kr + w*w)
*gsl_sf_bessel_Kn(index, r0*fabs(w));
}
else{
result = x0*sqrt(pow(w, 2.0*index))/(kr*kr + w*w)*gsl_sf_bessel_Kn(index, r0*fabs(w));
}
return result;
}
/*maxwell_juettner_rho_V: Fitting formula for Faraday rotation coefficient
* rho_V from Dexter (2016) for a Maxwell-Juettner
* distribution. This formula comes from his
* equations B7, B8, B14, and B15.
*
*@params: struct of parameters params
*@returns: fitting formula to the Faraday rotation coefficient
* for a Maxwell-Juettner distribution of electrons.
*/
double maxwell_juettner_rho_V(struct parameters * params)
{
double omega0 = params->electron_charge*params->magnetic_field
/ (params->mass_electron*params->speed_light);
double wp2 = 4. * params->pi * params->electron_density
* pow(params->electron_charge, 2.) / params->mass_electron;
/* argument for function g(X) (called shgmfunc) below */
double x = params->theta_e * sqrt(sqrt(2.) * sin(params->observer_angle)
* (1.e3*omega0 / (2. * params->pi * params->nu)));
/* this is the definition of the modified factor g(X) from Dexter (2016) */
double shgmfunc = 0.43793091 * log(1. + 0.00185777 * pow(x, 1.50316886));
double eps12 = wp2 * omega0 / pow((2. * params->pi * params->nu), 3.)
* (gsl_sf_bessel_Kn(0, 1./params->theta_e) - shgmfunc)
/ gsl_sf_bessel_Kn(2, 1./params->theta_e)
* cos(params->observer_angle);
double rhov = 2. * params->pi * params->nu / params->speed_light * eps12;
return rhov;
}
int main()
{
#include "bessel_k_int_data.ipp"
add_data(k0_data);
add_data(k1_data);
add_data(kn_data);
add_data(bessel_k_int_data);
unsigned data_total = data.size();
screen_data([](const std::vector<double>& v){ return boost::math::cyl_bessel_k(static_cast<int>(v[0]), v[1]); }, [](const std::vector<double>& v){ return v[2]; });
#if defined(TEST_LIBSTDCXX) && !defined(COMPILER_COMPARISON_TABLES)
screen_data([](const std::vector<double>& v){ return std::tr1::cyl_bessel_k(static_cast<int>(v[0]), v[1]); }, [](const std::vector<double>& v){ return v[2]; });
#endif
#if defined(TEST_GSL) && !defined(COMPILER_COMPARISON_TABLES)
screen_data([](const std::vector<double>& v){ return gsl_sf_bessel_Kn(static_cast<int>(v[0]), v[1]); }, [](const std::vector<double>& v){ return v[2]; });
#endif
#if defined(TEST_RMATH) && !defined(COMPILER_COMPARISON_TABLES)
screen_data([](const std::vector<double>& v){ return bessel_k(v[1], static_cast<int>(v[0]), 1); }, [](const std::vector<double>& v){ return v[2]; });
#endif
unsigned data_used = data.size();
std::string function = "cyl_bessel_k (integer order)[br](" + boost::lexical_cast<std::string>(data_used) + "/" + boost::lexical_cast<std::string>(data_total) + " tests selected)";
std::string function_short = "cyl_bessel_k (integer order)";
double time;
time = exec_timed_test([](const std::vector<double>& v){ return boost::math::cyl_bessel_k(static_cast<int>(v[0]), v[1]); });
std::cout << time << std::endl;
#if !defined(COMPILER_COMPARISON_TABLES) && (defined(TEST_GSL) || defined(TEST_RMATH) || defined(TEST_LIBSTDCXX))
report_execution_time(time, std::string("Library Comparison with ") + std::string(compiler_name()) + std::string(" on ") + platform_name(), function, boost_name());
#endif
report_execution_time(time, std::string("Compiler Comparison on ") + std::string(platform_name()), function_short, compiler_name() + std::string("[br]") + boost_name());
//
// Boost again, but with promotion to long double turned off:
//
#if !defined(COMPILER_COMPARISON_TABLES)
if(sizeof(long double) != sizeof(double))
{
time = exec_timed_test([](const std::vector<double>& v){ return boost::math::cyl_bessel_k(static_cast<int>(v[0]), v[1], boost::math::policies::make_policy(boost::math::policies::promote_double<false>())); });
std::cout << time << std::endl;
#if !defined(COMPILER_COMPARISON_TABLES) && (defined(TEST_GSL) || defined(TEST_RMATH) || defined(TEST_LIBSTDCXX))
report_execution_time(time, std::string("Library Comparison with ") + std::string(compiler_name()) + std::string(" on ") + platform_name(), function, boost_name() + "[br]promote_double<false>");
#endif
report_execution_time(time, std::string("Compiler Comparison on ") + std::string(platform_name()), function_short, compiler_name() + std::string("[br]") + boost_name() + "[br]promote_double<false>");
}
#endif
#if defined(TEST_LIBSTDCXX) && !defined(COMPILER_COMPARISON_TABLES)
time = exec_timed_test([](const std::vector<double>& v){ return std::tr1::cyl_bessel_k(static_cast<int>(v[0]), v[1]); });
std::cout << time << std::endl;
report_execution_time(time, std::string("Library Comparison with ") + std::string(compiler_name()) + std::string(" on ") + platform_name(), function, "tr1/cmath");
#endif
#if defined(TEST_GSL) && !defined(COMPILER_COMPARISON_TABLES)
time = exec_timed_test([](const std::vector<double>& v){ return gsl_sf_bessel_Kn(static_cast<int>(v[0]), v[1]); });
std::cout << time << std::endl;
report_execution_time(time, std::string("Library Comparison with ") + std::string(compiler_name()) + std::string(" on ") + platform_name(), function, "GSL " GSL_VERSION);
#endif
#if defined(TEST_RMATH) && !defined(COMPILER_COMPARISON_TABLES)
time = exec_timed_test([](const std::vector<double>& v){ return bessel_k(v[1], static_cast<int>(v[0]), 1); });
std::cout << time << std::endl;
report_execution_time(time, std::string("Library Comparison with ") + std::string(compiler_name()) + std::string(" on ") + platform_name(), function, "Rmath " R_VERSION_STRING);
#endif
return 0;
}
