LauIntegrals.cc

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// Copyright University of Warwick 2004 - 2013.
// Distributed under the Boost Software License, Version 1.0.
// (See accompanying file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)

// Authors:
// Thomas Latham
// John Back
// Paul Harrison

#include <iostream>
using std::cout;
using std::endl;

#include "TMath.h"

#include "LauConstants.hh"
#include "LauIntegrals.hh"


ClassImp(LauIntegrals)


LauIntegrals::LauIntegrals(Double_t weightsPrecision) :
        weightsPrecision_( weightsPrecision )
{
}

LauIntegrals::~LauIntegrals()
{
}

void LauIntegrals::calcGaussLegendreWeights(const Int_t numPoints, std::vector<Double_t>& abscissas, std::vector<Double_t>& weights) 
{
        // Calculate the Gauss-Legendre weights that will be used for the 
        // simple Gaussian integration method given the number of points

        abscissas.clear();
        weights.clear();
        abscissas.resize(numPoints);
        weights.resize(numPoints);

        Int_t m = (numPoints+1)/2;
        Double_t dnumPoints(numPoints);
        Double_t dnumPointsPlusHalf = static_cast<Double_t>(numPoints + 0.5);
        Int_t i(0), j(0);
        Double_t p1(0.0), p2(0.0), p3(0.0), pp(0.0), z(0.0), zSq(0.0), z1(0.0), di(0.0);

        for (i = 1; i <= m; i++){

                di = static_cast<Double_t>(i);
                z = TMath::Cos(LauConstants::pi*(di - 0.25)/dnumPointsPlusHalf);
                zSq = z*z;

                // Starting with the above approximation for the ith root, we enter the 
                // main loop of refinement by Newton's method
                do{
                        p1 = 1.0;
                        p2 = 0.0;

                        // Calculate the Legendre polynomial at z - recurrence relation
                        for (j = 1; j <= numPoints; j++){
                                p3 = p2;
                                p2 = p1;
                                p1 = (( 2.0*j - 1.0) * z * p2 - (j - 1.0)*p3)/static_cast<Double_t>(j);
                        }
                        // p1 = Legendre polynomial. Compute its derivative, pp.

                        pp = dnumPoints * (z*p1 - p2)/(zSq - 1.0);
                        z1 = z;
                        z = z1 - p1/pp; // Newton's method
                } while (TMath::Abs(z-z1) > weightsPrecision_);

                // Scale the root to the desired interval
                // Remember that the vector entries start with 0, hence i-1 (where first i value is 1)
                abscissas[i-1] = z;
                abscissas[numPoints-i] = abscissas[i-1]; // Symmetric abscissa
                weights[i-1] = 2.0/((1.0 - zSq)*pp*pp);
                weights[numPoints-i] = weights[i-1]; // Symmetric weight

        }
}

void LauIntegrals::calcGaussHermiteWeights(const Int_t numPoints, std::vector<Double_t>& abscissas, std::vector<Double_t>& weights) 
{
        // Calculate the Gauss-Hermite weights that will be used for the 
        // simple Gaussian integration method for a function weighted by an exp(-x**2) term
        // These weights are common to all integration intervals - therefore they
        // should only be calculated once - when the constructor is called, for example.

        abscissas.clear();
        weights.clear();
        abscissas.resize(numPoints); weights.resize(numPoints);

        Int_t m = (numPoints+1)/2;
        Double_t dnumPoints(numPoints);
        Int_t i(0), j(0), its(0);
        Double_t p1(0.0), p2(0.0), p3(0.0), pp(0.0), z(0.0), z1(0.0);
        Double_t numPointsTerm(2*numPoints+1);

        // The roots are symmetric about the origin. Therefore, we only need to find half of them
        // Loop over the desired roots.
        for (i = 1; i <= m; i++) {

                if (i == 1) {
                        // Initial guess for largest root
                        z = TMath::Sqrt(numPointsTerm) - 1.85575*TMath::Power(numPointsTerm, -0.16667);
                } else if (i == 2) {
                        // Initial guess for second largest root
                        z -= 1.14*TMath::Power(dnumPoints, 0.426)/z;
                } else if (i == 3) {
                        // Initial guess for third largest root
                        z = 1.86*z - 0.86*abscissas[0];
                } else if (i == 4) {
                        // Initial guess for fourth largest root
                        z = 1.91*z - 0.91*abscissas[1];
                } else {
                        // Initial guess for other roots
                        z = 2.0*z - abscissas[i-3];
                }

                for (its = 1; its <= 10; its++) {
                        p1 = LauConstants::pim4;
                        p2 = 0.0;

                        for (j = 1; j <= numPoints; j++) {
                                Double_t dj(j);
                                p3 = p2;
                                p2 = p1;
                                p1 = z*TMath::Sqrt(2.0/dj)*p2 - TMath::Sqrt((dj-1.0)/dj)*p3;
                        }
                        // p1 is now the desired Hermite polynomial. Compute its derivative, pp
                        pp = TMath::Sqrt(2.0*dnumPoints)*p2;
                        z1 = z;
                        if (pp > 1e-10) {z = z1 - p1/pp;}
                        if (TMath::Abs(z - z1) < weightsPrecision_) break;
                } 

                // Scale the root to the desired interval
                // Remember that the vector entries start with 0, hence i-1 (where first i value is 1)
                abscissas[i-1] = z;
                abscissas[numPoints-i] = -z; // Symmetric abscissa
                weights[i-1] = 2.0/(pp*pp);
                weights[numPoints-i] = weights[i-1]; // Symmetric weight

        }
}