GModelSpectralExpPlaw.cpp

Go to the documentation of this file.

/***************************************************************************
 *     GModelSpectralExpPlaw.cpp - Exponential cut off power law model     *
 * ----------------------------------------------------------------------- *
 *  copyright (C) 2010-2021 by Juergen Knoedlseder                         *
 * ----------------------------------------------------------------------- *
 *                                                                         *
 *  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.                                    *
 *                                                                         *
 *  This program is distributed in the hope that it will be useful,        *
 *  but WITHOUT ANY WARRANTY; without even the implied warranty of         *
 *  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the          *
 *  GNU General Public License for more details.                           *
 *                                                                         *
 *  You should have received a copy of the GNU General Public License      *
 *  along with this program.  If not, see <http://www.gnu.org/licenses/>.  *
 *                                                                         *
 ***************************************************************************/
/**
 * @file GModelSpectralExpPlaw.cpp
 * @brief Exponential cut off power law spectral class implementation
 * @author Juergen Knoedlseder
 */

/* __ Includes ___________________________________________________________ */
#ifdef HAVE_CONFIG_H
#include <config.h>
#endif
#include <cmath>
#include "GException.hpp"
#include "GTools.hpp"
#include "GRan.hpp"
#include "GIntegral.hpp"
#include "GModelSpectralExpPlaw.hpp"
#include "GModelSpectralRegistry.hpp"

/* __ Constants __________________________________________________________ */

/* __ Globals ____________________________________________________________ */
const GModelSpectralExpPlaw  g_spectral_eplaw_seed1("ExponentialCutoffPowerLaw",
                                                    "Prefactor",
                                                    "Index",
                                                    "PivotEnergy",
                                                    "CutoffEnergy");
const GModelSpectralRegistry g_spectral_eplaw_registry1(&g_spectral_eplaw_seed1);
#if defined(G_LEGACY_XML_FORMAT)
const GModelSpectralExpPlaw  g_spectral_eplaw_seed2("ExpCutoff",
                                                    "Prefactor",
                                                    "Index",
                                                    "Scale",
                                                    "Cutoff");
const GModelSpectralRegistry g_spectral_eplaw_registry2(&g_spectral_eplaw_seed2);
#endif

/* __ Method name definitions ____________________________________________ */
#define G_MC   "GModelSpectralExpPlaw::mc(GEnergy&, GEnergy&, GTime&, GRan&)"
#define G_READ                    "GModelSpectralExpPlaw::read(GXmlElement&)"
#define G_WRITE                  "GModelSpectralExpPlaw::write(GXmlElement&)"

/* __ Macros _____________________________________________________________ */

/* __ Coding definitions _________________________________________________ */

/* __ Debug definitions __________________________________________________ */
//#define G_DEBUG_MC                          //!< Debug Monte-Carlo sampling


/*==========================================================================
 =                                                                         =
 =                        Constructors/destructors                         =
 =                                                                         =
 ==========================================================================*/

/***********************************************************************//**
 * @brief Void constructor
 ***************************************************************************/
GModelSpectralExpPlaw::GModelSpectralExpPlaw(void) : GModelSpectral()
{
    // Initialise members
    init_members();

    // Return
    return;
}


/***********************************************************************//**
 * @brief Model type and parameter name constructor
 *
 * @param[in] type Model type.
 * @param[in] prefactor Name of prefactor parameter.
 * @param[in] index Name of index parameter.
 * @param[in] pivot Name of pivot parameter.
 * @param[in] cutoff Name of cutoff parameter.
 ***************************************************************************/
GModelSpectralExpPlaw::GModelSpectralExpPlaw(const std::string& type,
                                             const std::string& prefactor,
                                             const std::string& index,
                                             const std::string& pivot,
                                             const std::string& cutoff) :
                       GModelSpectral()
{
    // Initialise members
    init_members();

    // Set model type
    m_type = type;

    // Set parameter names
    m_norm.name(prefactor);
    m_index.name(index);
    m_pivot.name(pivot);
    m_ecut.name(cutoff);

    // Return
    return;
}


/***********************************************************************//**
 * @brief Constructor
 *
 * @param[in] prefactor Pre factor normalization (ph/cm2/s/MeV).
 * @param[in] index Power law index.
 * @param[in] pivot Pivot energy.
 * @param[in] cutoff Cut off energy.
 *
 * Construct an exponentially cut off power law from
 * - a prefactor value (in units of ph/cm2/s/MeV)
 * - a spectral index,
 * - a pivot energy, and
 * - a cut off energy.
 ***************************************************************************/
GModelSpectralExpPlaw::GModelSpectralExpPlaw(const double&  prefactor,
                                             const double&  index,
                                             const GEnergy& pivot,
                                             const GEnergy& cutoff) :
                       GModelSpectral()
{
    // Initialise members
    init_members();

    // Set parameters
    m_norm.value(prefactor);
    m_index.value(index);
    m_pivot.value(pivot.MeV()); // Internally stored in MeV
    m_ecut.value(cutoff.MeV()); // Internally stored in MeV

    // Autoscale parameters
    autoscale();

    // Return
    return;
}


/***********************************************************************//**
 * @brief XML constructor
 *
 * @param[in] xml XML element.
 *
 * Constructs an exponentially cut off power law spectral model by extracting
 * information from an XML element. See the read() method for more
 * information about the expected structure of the XML element.
 ***************************************************************************/
GModelSpectralExpPlaw::GModelSpectralExpPlaw(const GXmlElement& xml) :
                       GModelSpectral()
{
    // Initialise members
    init_members();

    // Read information from XML element
    read(xml);

    // Return
    return;
}


/***********************************************************************//**
 * @brief Copy constructor
 *
 * @param[in] model Exponentially cut off power law model.
 ***************************************************************************/
GModelSpectralExpPlaw::GModelSpectralExpPlaw(const GModelSpectralExpPlaw& model) :
                       GModelSpectral(model)
{
    // Initialise members
    init_members();

    // Copy members
    copy_members(model);

    // Return
    return;
}


/***********************************************************************//**
 * @brief Destructor
 ***************************************************************************/
GModelSpectralExpPlaw::~GModelSpectralExpPlaw(void)
{
    // Free members
    free_members();

    // Return
    return;
}


/*==========================================================================
 =                                                                         =
 =                                Operators                                =
 =                                                                         =
 ==========================================================================*/

/***********************************************************************//**
 * @brief Assignment operator
 *
 * @param[in] model Exponentially cut off power law model.
 * @return Exponentially cut off power law model.
 ***************************************************************************/
GModelSpectralExpPlaw& GModelSpectralExpPlaw::operator=(const GModelSpectralExpPlaw& model)
{
    // Execute only if object is not identical
    if (this != &model) {

        // Copy base class members
        this->GModelSpectral::operator=(model);

        // Free members
        free_members();

        // Initialise members
        init_members();

        // Copy members
        copy_members(model);

    } // endif: object was not identical

    // Return
    return *this;
}


/*==========================================================================
 =                                                                         =
 =                              Public methods                             =
 =                                                                         =
 ==========================================================================*/

/***********************************************************************//**
 * @brief Clear exponentially cut off power law model
 ***************************************************************************/
void GModelSpectralExpPlaw::clear(void)
{
    // Free class members (base and derived classes, derived class first)
    free_members();
    this->GModelSpectral::free_members();

    // Initialise members
    this->GModelSpectral::init_members();
    init_members();

    // Return
    return;
}


/***********************************************************************//**
 * @brief Clone exponentially cut off power law model
 *
 * @return Pointer to deep copy of exponentially cut off power law model.
 ***************************************************************************/
GModelSpectralExpPlaw* GModelSpectralExpPlaw::clone(void) const
{
    // Clone exponentially cut off power law model
    return new GModelSpectralExpPlaw(*this);
}


/***********************************************************************//**
 * @brief Evaluate function
 *
 * @param[in] srcEng True photon energy.
 * @param[in] srcTime True photon arrival time.
 * @param[in] gradients Compute gradients?
 * @return Model value (ph/cm2/s/MeV).
 *
 * Evaluates
 *
 * \f[
 *    S_{\rm E}(E | t) = {\tt m\_norm}
 *    \left( \frac{E}{\tt m\_pivot} \right)^{\tt m\_index}
 *    \exp \left( \frac{-E}{\tt m\_ecut} \right)
 * \f]
 *
 * where
 * - \f${\tt m\_norm}\f$ is the normalization or prefactor,
 * - \f${\tt m\_pivot}\f$ is the pivot energy,
 * - \f${\tt m\_index}\f$ is the spectral index, and
 * - \f${\tt m\_ecut}\f$ is the cut off energy.
 *
 * If the @p gradients flag is true the method will also compute the
 * partial derivatives of the model with respect to the parameters using
 *
 * \f[
 *    \frac{\delta S_{\rm E}(E | t)}{\delta {\tt m\_norm}} =
 *      \frac{S_{\rm E}(E | t)}{{\tt m\_norm}}
 * \f]
 *
 * \f[
 *    \frac{\delta S_{\rm E}(E | t)}{\delta {\tt m\_index}} =
 *      S_{\rm E}(E | t) \, \ln(E/{\tt m_pivot})
 * \f]
 *
 * \f[
 *    \frac{\delta S_{\rm E}(E | t)}{\delta {\tt m\_ecut}} =
 *      S_{\rm E}(E | t) \, \left( \frac{E}{{\tt m\_ecut}^2} \right)
 * \f]
 *
 * \f[
 *    \frac{\delta S_{\rm E}(E | t)}{\delta {\tt m\_pivot}} =
 *      -S_{\rm E}(E | t) \,
 *      \left( \frac{{\tt m\_index}}{{\tt m\_pivot}} \right)
 * \f]
 *
 * @todo The method expects that pivot!=0 and ecut!=0. Otherwise Inf or NaN
 *       may result. We should add a test that prevents using invalid
 *       values.
 ***************************************************************************/
double GModelSpectralExpPlaw::eval(const GEnergy& srcEng,
                                   const GTime&   srcTime,
                                   const bool&    gradients) const
{
    // Update the evaluation cache
    update_eval_cache(srcEng);

    // Compute function value
    double value = m_norm.value() * m_last_power;

    // Optionally compute gradients
    if (gradients) {

        // Compute partial derivatives with respect to the parameter factor
        // values. The partial derivatives with respect to the parameter
        // values are obtained by division by the scale factor.
        double g_norm  = (m_norm.is_free())
                         ? m_norm.scale() * m_last_power
                         : 0.0;
        double g_index = (m_index.is_free())
                         ? value * m_index.scale() * std::log(m_last_e_norm)
                         : 0.0;
        double g_ecut  = (m_ecut.is_free() && m_ecut.factor_value() != 0.0)
                         ? value * m_last_e_cut / m_ecut.factor_value()
                         : 0.0;
        double g_pivot = (m_pivot.is_free() && m_pivot.factor_value() != 0.0)
                         ? -value * m_last_index / m_pivot.factor_value()
                         : 0.0;

        // Set gradients
        m_norm.factor_gradient(g_norm);
        m_index.factor_gradient(g_index);
        m_ecut.factor_gradient(g_ecut);
        m_pivot.factor_gradient(g_pivot);

    } // endif: gradient computation was requested

    // Compile option: Check for NaN/Inf
    #if defined(G_NAN_CHECK)
    if (gammalib::is_notanumber(value) || gammalib::is_infinite(value)) {
        std::cout << "*** ERROR: GModelSpectralExpPlaw::eval";
        std::cout << "(srcEng=" << srcEng;
        std::cout << ", srcTime=" << srcTime << "):";
        std::cout << " NaN/Inf encountered";
        std::cout << " (value=" << value;
        std::cout << ", power=" << m_last_power;
        std::cout << ")" << std::endl;
    }
    #endif

    // Return
    return value;
}


/***********************************************************************//**
 * @brief Returns model photon flux between [emin, emax] (units: ph/cm2/s)
 *
 * @param[in] emin Minimum photon energy.
 * @param[in] emax Maximum photon energy.
 * @return Photon flux (ph/cm2/s).
 *
 * Computes
 *
 * \f[
 *    \int_{\tt emin}^{\tt emax} S_{\rm E}(E | t) dE
 * \f]
 *
 * where
 * - [@p emin, @p emax] is an energy interval, and
 * - \f$S_{\rm E}(E | t)\f$ is the spectral model (ph/cm2/s/MeV).
 * The integration is done numerically.
 ***************************************************************************/
double GModelSpectralExpPlaw::flux(const GEnergy& emin,
                                   const GEnergy& emax) const
{
    // Initialise flux
    double flux = 0.0;

    // Compute only if integration range is valid
    if (emin < emax) {

        // Setup integration kernel
        flux_kernel integrand(m_norm.value(),  m_index.value(),
                              m_pivot.value(), m_ecut.value());
        GIntegral integral(&integrand);

        // Get integration boundaries in MeV
        double e_min = emin.MeV();
        double e_max = emax.MeV();

        // Perform integration
        flux = integral.romberg(e_min, e_max);

    } // endif: integration range was valid

    // Return
    return flux;
}


/***********************************************************************//**
 * @brief Returns model energy flux between [emin, emax] (units: erg/cm2/s)
 *
 * @param[in] emin Minimum photon energy.
 * @param[in] emax Maximum photon energy.
 * @return Energy flux (erg/cm2/s).
 *
 * Computes
 *
 * \f[
 *    \int_{\tt emin}^{\tt emax} S_{\rm E}(E | t) E \, dE
 * \f]
 *
 * where
 * - [@p emin, @p emax] is an energy interval, and
 * - \f$S_{\rm E}(E | t)\f$ is the spectral model (ph/cm2/s/MeV).
 * The integration is done numerically.
 ***************************************************************************/
double GModelSpectralExpPlaw::eflux(const GEnergy& emin,
                                    const GEnergy& emax) const
{
    // Initialise flux
    double eflux = 0.0;

    // Compute only if integration range is valid
    if (emin < emax) {

        // Setup integration kernel
        eflux_kernel integrand(m_norm.value(),  m_index.value(),
                               m_pivot.value(), m_ecut.value());
        GIntegral integral(&integrand);

        // Get integration boundaries in MeV
        double e_min = emin.MeV();
        double e_max = emax.MeV();

        // Perform integration
        eflux = integral.romberg(e_min, e_max);

        // Convert from MeV/cm2/s to erg/cm2/s
        eflux *= gammalib::MeV2erg;

    } // endif: integration range was valid

    // Return
    return eflux;
}


/***********************************************************************//**
 * @brief Returns MC energy between [emin, emax]
 *
 * @param[in] emin Minimum photon energy.
 * @param[in] emax Maximum photon energy.
 * @param[in] time True photon arrival time.
 * @param[in,out] ran Random number generator.
 * @return Energy.
 *
 * Simulates a random energy in the interval [emin, emax] for an
 * exponentially cut off power law. The simulation is done using a rejection
 * method. First, a random energy within [emin, emax] is drawn from an
 * power law distribution. Then the energy is accepted or rejected based
 * on an acceptance fraction that is computed from the exponential cut off.
 ***************************************************************************/
GEnergy GModelSpectralExpPlaw::mc(const GEnergy& emin,
                                  const GEnergy& emax,
                                  const GTime&   time,
                                  GRan&          ran) const
{
    // Check energy interval
    gammalib::check_energy_interval(G_MC, emin, emax);

    // Allocate energy
    GEnergy energy;

    // Update cache
    update_mc_cache(emin, emax);

    // Initialise energy
    double eng;

    // Initialise acceptance fraction
    double acceptance_fraction;
    double inv_ecut = 1.0 / m_ecut.value();

    // Compute random number generator normalisation for speed-up of the
    // computations. Normally it should be sufficient to always use the
    // minimum energy for the normalisation, yet for the case that the model
    // grows with energy we better also test the maximum energy, and we
    // just use the larger of both normalisations
    double norm_emin = std::exp(-emin.MeV() * inv_ecut);
    double norm_emax = std::exp(-emax.MeV() * inv_ecut);
    double norm      = (norm_emin > norm_emax) ? norm_emin : norm_emax;

    // Debug option: initialise number of samples
    #if defined(G_DEBUG_MC)
    int samples = 0;
    #endif

    // Use rejection method to draw a random energy. We first draw
    // analytically from a power law, and then compare the power law
    // at the drawn energy to the exponentially cut off function. This
    // gives an acceptance fraction, and we accept the energy only if
    // a uniform random number is <= the acceptance fraction.
    do {

        // Debug option: increment number of samples
        #if defined(G_DEBUG_MC)
        samples++;
        #endif

        // Get uniform random number
        double u = ran.uniform();

        // Case A: Index is not -1
        if (index() != -1.0) {
            if (u > 0.0) {
                eng = std::exp(std::log(u * m_mc_pow_ewidth + m_mc_pow_emin) /
                               m_mc_exponent);
            }
            else {
                eng = 0.0;
            }
        }

        // Case B: Index is -1
        else {
            eng = std::exp(u * m_mc_pow_ewidth + m_mc_pow_emin);
        }

        // Compute acceptance fraction
        acceptance_fraction = std::exp(-eng * inv_ecut);

    } while (ran.uniform() * norm > acceptance_fraction);

    // Set energy
    energy.MeV(eng);

    // Debug option: write result
    #if defined(G_DEBUG_MC)
    std::cout << "GModelSpectralExpPlaw::mc(";
    std::cout << emin.print() << "," << emax.print() << "," << time.print() << "):";
    std::cout << " energy=" << energy.print();
    std::cout << " samples=" << samples << std::endl;
    #endif

    // Return energy
    return energy;
}


/***********************************************************************//**
 * @brief Read model from XML element
 *
 * @param[in] xml XML element.
 *
 * Reads the spectral information from an XML element.
 ***************************************************************************/
void GModelSpectralExpPlaw::read(const GXmlElement& xml)
{
    // Verify number of model parameters
    gammalib::xml_check_parnum(G_READ, xml, 4);

    // Get parameter pointers
    const GXmlElement* norm  = gammalib::xml_get_par(G_READ, xml, m_norm.name());
    const GXmlElement* index = gammalib::xml_get_par(G_READ, xml, m_index.name());
    const GXmlElement* ecut  = gammalib::xml_get_par(G_READ, xml, m_ecut.name());
    const GXmlElement* pivot = gammalib::xml_get_par(G_READ, xml, m_pivot.name());

    // Read parameters
    m_norm.read(*norm);
    m_index.read(*index);
    m_ecut.read(*ecut);
    m_pivot.read(*pivot);

    // Return
    return;
}


/***********************************************************************//**
 * @brief Write model into XML element
 *
 * @param[in] xml XML element.
 *
 * Writes the spectral information into an XML element.
 ***************************************************************************/
void GModelSpectralExpPlaw::write(GXmlElement& xml) const
{
    // Verify model type
    gammalib::xml_check_type(G_WRITE, xml, type());

    // Get XML parameters
    GXmlElement* norm  = gammalib::xml_need_par(G_WRITE, xml, m_norm.name());
    GXmlElement* index = gammalib::xml_need_par(G_WRITE, xml, m_index.name());
    GXmlElement* ecut  = gammalib::xml_need_par(G_WRITE, xml, m_ecut.name());
    GXmlElement* pivot = gammalib::xml_need_par(G_WRITE, xml, m_pivot.name());

    // Write parameters
    m_norm.write(*norm);
    m_index.write(*index);
    m_ecut.write(*ecut);
    m_pivot.write(*pivot);

    // Return
    return;
}


/***********************************************************************//**
 * @brief Print model information
 *
 * @param[in] chatter Chattiness.
 * @return String containing model information.
 ***************************************************************************/
std::string GModelSpectralExpPlaw::print(const GChatter& chatter) const
{
    // Initialise result string
    std::string result;

    // Continue only if chatter is not silent
    if (chatter != SILENT) {

        // Append header
        result.append("=== GModelSpectralExpPlaw ===");

        // Append information
        result.append("\n"+gammalib::parformat("Number of parameters"));
        result.append(gammalib::str(size()));
        for (int i = 0; i < size(); ++i) {
            result.append("\n"+m_pars[i]->print(chatter));
        }

    } // endif: chatter was not silent

    // Return result
    return result;
}


/*==========================================================================
 =                                                                         =
 =                             Private methods                             =
 =                                                                         =
 ==========================================================================*/

/***********************************************************************//**
 * @brief Initialise class members
 ***************************************************************************/
void GModelSpectralExpPlaw::init_members(void)
{
    // Initialise model type
    m_type = "ExponentialCutoffPowerLaw";

    // Initialise powerlaw normalisation
    m_norm.clear();
    m_norm.name("Prefactor");
    m_norm.unit("ph/cm2/s/MeV");
    m_norm.scale(1.0);
    m_norm.value(1.0);          // default: 1.0
    m_norm.min(0.0);            // min:     0.0
    m_norm.free();
    m_norm.gradient(0.0);
    m_norm.has_grad(true);

    // Initialise powerlaw index
    m_index.clear();
    m_index.name("Index");
    m_index.scale(1.0);
    m_index.value(-2.0);        // default: -2.0
    m_index.range(-10.0,+10.0); // range:   [-10,+10]
    m_index.free();
    m_index.gradient(0.0);
    m_index.has_grad(true);

    // Initialise cut off energy
    m_ecut.clear();
    m_ecut.name("CutoffEnergy");
    m_ecut.unit("MeV");
    m_ecut.scale(1.0);
    m_ecut.value(1000.0);       // default: 1000.0
    m_ecut.min(0.1);            // min:     0.1
    m_ecut.free();
    m_ecut.gradient(0.0);
    m_ecut.has_grad(true);

    // Initialise pivot energy
    m_pivot.clear();
    m_pivot.name("PivotEnergy");
    m_pivot.unit("MeV");
    m_pivot.scale(1.0);
    m_pivot.value(100.0);
    m_pivot.fix();
    m_pivot.gradient(0.0);
    m_pivot.has_grad(true);

    // Set parameter pointer(s)
    m_pars.clear();
    m_pars.push_back(&m_norm);
    m_pars.push_back(&m_index);
    m_pars.push_back(&m_ecut);
    m_pars.push_back(&m_pivot);

    // Initialise eval cache
    m_last_energy.clear();
    m_last_index  = 1.0e30;
    m_last_ecut   = 1.0e30;
    m_last_pivot  = 1.0e30;
    m_last_e_norm = 0.0;
    m_last_e_cut  = 0.0;
    m_last_power  = 0.0;

    // Initialise MC cache
    m_mc_emin       = 0.0;
    m_mc_emax       = 0.0;
    m_mc_exponent   = 0.0;
    m_mc_pow_emin   = 0.0;
    m_mc_pow_ewidth = 0.0;

    // Return
    return;
}


/***********************************************************************//**
 * @brief Copy class members
 *
 * @param[in] model Exponential cut off power law model.
 ***************************************************************************/
void GModelSpectralExpPlaw::copy_members(const GModelSpectralExpPlaw& model)
{
    // Copy members
    m_type  = model.m_type;
    m_norm  = model.m_norm;
    m_index = model.m_index;
    m_ecut  = model.m_ecut;
    m_pivot = model.m_pivot;

    // Set parameter pointer(s)
    m_pars.clear();
    m_pars.push_back(&m_norm);
    m_pars.push_back(&m_index);
    m_pars.push_back(&m_ecut);
    m_pars.push_back(&m_pivot);

    // Copy eval cache
    m_last_energy = model.m_last_energy;
    m_last_index  = model.m_last_index;
    m_last_ecut   = model.m_last_ecut;
    m_last_pivot  = model.m_last_pivot;
    m_last_e_norm = model.m_last_e_norm;
    m_last_e_cut  = model.m_last_e_cut;
    m_last_power  = model.m_last_power;

    // Copy MC cache
    m_mc_emin       = model.m_mc_emin;
    m_mc_emax       = model.m_mc_emax;
    m_mc_exponent   = model.m_mc_exponent;
    m_mc_pow_emin   = model.m_mc_pow_emin;
    m_mc_pow_ewidth = model.m_mc_pow_ewidth;

    // Return
    return;
}


/***********************************************************************//**
 * @brief Delete class members
 ***************************************************************************/
void GModelSpectralExpPlaw::free_members(void)
{
    // Return
    return;
}


/***********************************************************************//**
 * @brief Update eval precomputation cache
 *
 * @param[in] energy Energy.
 *
 * Updates the precomputation cache for eval() method.
 ***************************************************************************/
void GModelSpectralExpPlaw::update_eval_cache(const GEnergy& energy) const
{
    // Get parameter values (takes 3 multiplications which are difficult
    // to avoid)
    double index = m_index.value();
    double ecut  = m_ecut.value();
    double pivot = m_pivot.value();

    // If the energy or one of the parameters index, cut-off or pivot
    // energy has changed then recompute the cache
    if ((m_last_energy != energy) ||
        (m_last_index  != index)  ||
        (m_last_ecut   != ecut)   ||
        (m_last_pivot  != pivot)) {

        // Store actual energy and parameter values
        m_last_energy = energy;
        m_last_index  = index;
        m_last_ecut   = ecut;
        m_last_pivot  = pivot;

        // Compute and store value
        double eng    = energy.MeV();
        m_last_e_norm = eng / m_last_pivot;
        m_last_e_cut  = eng / m_last_ecut;
        m_last_power  = std::pow(m_last_e_norm, m_last_index) *
                        std::exp(-m_last_e_cut);
    } // endif: recomputation was required

    // Return
    return;
}


/***********************************************************************//**
 * @brief Update Monte Carlo pre computation cache
 *
 * @param[in] emin Minimum photon energy.
 * @param[in] emax Maximum photon energy.
 *
 * Updates the precomputation cache for Monte Carlo simulations.
 ***************************************************************************/
void GModelSpectralExpPlaw::update_mc_cache(const GEnergy& emin,
                                            const GEnergy& emax) const

{
    // Case A: Index is not -1
    if (index() != -1.0) {

        // Change in energy boundaries?
        if (emin.MeV() != m_mc_emin || emax.MeV() != m_mc_emax) {
            m_mc_emin       = emin.MeV();
            m_mc_emax       = emax.MeV();
            m_mc_exponent   = index() + 1.0;
            m_mc_pow_emin   = std::pow(m_mc_emin, m_mc_exponent);
            m_mc_pow_ewidth = std::pow(m_mc_emax, m_mc_exponent) - m_mc_pow_emin;
        }

    }

    // Case B: Index is -1
    else {

        // Change in energy boundaries?
        if (emin.MeV() != m_mc_emin || emax.MeV() != m_mc_emax) {
            m_mc_emin       = emin.MeV();
            m_mc_emax       = emax.MeV();
            m_mc_exponent   = 0.0;
            m_mc_pow_emin   = std::log(m_mc_emin);
            m_mc_pow_ewidth = std::log(m_mc_emax) - m_mc_pow_emin;
        }

    }

    // Return
    return;
}


/***********************************************************************//**
 * @brief Kernel for photon flux integration
 *
 * @param[in] energy Energy (MeV).
 ***************************************************************************/
double GModelSpectralExpPlaw::flux_kernel::eval(const double& energy)
{
    // Evaluate function value
    double e_norm = energy * m_inv_pivot;
    double e_cut  = energy * m_inv_ecut;
    double power  = std::pow(e_norm, m_index) * std::exp(-e_cut);
    double value  = m_norm * power;

    // Return value
    return value;
}


/***********************************************************************//**
 * @brief Kernel for energy flux integration
 *
 * @param[in] energy Energy (MeV).
 ***************************************************************************/
double GModelSpectralExpPlaw::eflux_kernel::eval(const double& energy)
{
    // Evaluate function value
    double e_norm = energy * m_inv_pivot;
    double e_cut  = energy * m_inv_ecut;
    double power  = std::pow(e_norm, m_index) * std::exp(-e_cut);
    double value  = m_norm * power * energy;

    // Return value
    return value;
}