
.. _program_listing_file__Users_robertshaw_devfiles_libecpint_new_src_lib_bessel.cpp:

Program Listing for File bessel.cpp
===================================

|exhale_lsh| :ref:`Return to documentation for file <file__Users_robertshaw_devfiles_libecpint_new_src_lib_bessel.cpp>` (``/Users/robertshaw/devfiles/libecpint_new/src/lib/bessel.cpp``)

.. |exhale_lsh| unicode:: U+021B0 .. UPWARDS ARROW WITH TIP LEFTWARDS

.. code-block:: cpp

   /* 
    *      Copyright (c) 2020 Robert Shaw
    *      This file is a part of Libecpint.
    *
    *      Permission is hereby granted, free of charge, to any person obtaining
    *      a copy of this software and associated documentation files (the
    *      "Software"), to deal in the Software without restriction, including
    *      without limitation the rights to use, copy, modify, merge, publish,
    *      distribute, sublicense, and/or sell copies of the Software, and to
    *      permit persons to whom the Software is furnished to do so, subject to
    *      the following conditions:
    *
    *      The above copyright notice and this permission notice shall be
    *      included in all copies or substantial portions of the Software.
    *
    *      THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
    *      EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF
    *      MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND
    *      NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE
    *      LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION
    *      OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION
    *      WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
    */
   
   #include "bessel.hpp"
   #include "mathutil.hpp"
   #include <cmath>
   #include <cassert>
   #include <iostream>
   
   namespace libecpint {
   
       // Constructor
       BesselFunction::BesselFunction() {}
       BesselFunction::BesselFunction(int _lMax, int _N, int _order, const double accuracy)
       {
           init(_lMax, _N, _order, accuracy);
       }
   
       void BesselFunction::init(int _lMax, int _N, int _order, const double accuracy) {
           // Check parameters
           lMax = _lMax > -1 ? _lMax : 0;
           N = _N > 0 ? _N : 1;
           order = _order > 0 ? _order : 1;
           scale = N/16.0;
       
           // Allocate arrays
           K = new double*[N+1];
           dK = new double**[N+1];
           for (int i = 0; i < N+1; i++) {
               K[i] = new double[lMax + TAYLOR_CUT + 1];
               dK[i] = new double*[TAYLOR_CUT + 1];
               for (int j = 0; j < TAYLOR_CUT + 1; j++)
                   dK[i][j] = new double[lMax + TAYLOR_CUT];
           }
           C = new double[lMax+TAYLOR_CUT];
       
           // Tabulate values
           tabulate(accuracy);
       }
   
       BesselFunction::~BesselFunction() {
           free(K);
           free(dK);
           free(C);
       }
   
       // Tabulate the bessel function values
       int BesselFunction::tabulate(const double accuracy) {
           int retval = 0; // 0 for success, -1 for not converged
           // Series expansion for bessel function, K, is given by:
           // K_l(z) ~ z^l sum_{j=0 to infty} F_j(z) / (2j + 2l + 1)!! 
           // where F_j(z) = e^(-z) * (z^2/2)^j / j!
           int lmax = lMax + TAYLOR_CUT;
       
           double F[order + 1]; // F_j above
       
           K[0][0] = 1.0;
           double z, z2; // z and z^2 / 2
           double ratio; // F_j(z) / (2j+1)!!
           for (int i = 0; i <= N; i++) {
               // Calculate K(z) at equally spaced points z = 16/N to 16
               z = i / (N/16.0);
               z2 = z * z / 2.0;
           
               F[0] = exp(-z);
               ratio = F[0] / DFAC[0];
               K[i][0] = ratio;
           
               // Series expansion for K_0(z)
               int l = order;
               int j;
               for (j = 1; j <= l; j++) {
               
                   if (ratio < accuracy) {
                       // Reached convergence
                       break;
                   } 
               
                   F[j] = F[j-1] * z2 / ((double)j);
                   ratio = F[j] / DFAC[2*j+1];
                   K[i][0] += ratio;
               }
               //if ( ratio > accuracy ) { retval = -1; break; } // Not converged
   
               // Calculate K_l from K_0
               z2 = z;
               for (l=1; l<=lmax; l++) {
                   ratio = 0;
                   for (int m=0; m < j; m++) ratio += F[m]/DFAC[2*l + 2*m + 1]; 
                   K[i][l] = z2 * ratio;
                   z2 *= z; 
               }
       
           }
       
           // Determine coefficients for derivative recurrence
           for (int i = 1; i<lmax; i++) C[i] = i/(2.0*i + 1.0);
           
           // Determine the necessary derivatives from
           // K_l^(n+1) = C_l K_(l-1)^(n) + (C_l + 1/(2l+1))K_(l+1)^(n) - K_l^(n)
           for (int ix = 0; ix < N+1; ix++) {
               // Copy K values into dK
               for (int l = 0; l < lMax+TAYLOR_CUT; l++)
                   dK[ix][0][l] = K[ix][l];
               
               // Then the rest
               for (int n = 1; n < TAYLOR_CUT+1; n++) { 
                   dK[ix][n][0] = dK[ix][n-1][1] - dK[ix][n-1][0];
                   for (int l = 1; l <= lMax + TAYLOR_CUT - n; l++) 
                       dK[ix][n][l] = C[l]*dK[ix][n-1][l-1] + (C[l] + 1.0/(2.0*l + 1.0))*dK[ix][n-1][l+1] - dK[ix][n-1][l];
               }
           }
       
           return retval;
       }   
   
       // Get an upper bound for M_l(z)
       double BesselFunction::upper_bound(const double z, int L) { 
           // find nearest point (on left) in tabulated values
           int ix = std::floor(N*z/16.0);
           int minix = L > 0 ? 1 : 0;
           ix = std::min(N, std::max(minix, ix));
           int lx = std::min(L, lMax);
           return K[ix][lx];
       }
   
       // Calculate modified spherical Bessel function K_l(z), weighted with an exponential factor e^(-z)
       // for l = 0 to lMax. This restricts K(z) to the interval [0,1].
       void BesselFunction::calculate(const double z, int maxL, std::vector<double> &values) {
           if (lMax < maxL) {
               std::cout << "Asked for " << maxL << " but only initialised to maximum L = " << lMax << "\n";
               maxL = lMax;
           }
       
           // Set K_0(z) = 1.0, and K_l(z) = 0.0 (for l != 0) if z <= 0
           if (z <= 0) values[0] = 1.0;
           // Zeroth order case
           // K_l(z) ~ (1-z)*z^l / (2l + 1)!!
           else if (z < SMALL) { 
               values[0] = 1.0 - z;
               for (int l = 1; l <= maxL; l++) values[l] = values[l-1]*z/(2.0*l+1.0);
           } 
           // Large z case
           // K_l(z) ~ R_l(-z)/(2z)
           // where R_l(z) = sum_{k=0 to l} T_l,k(z)
           // where T_l,k(z) = (l+k)!/[k!(l-k)!] * (2z)^{-k}
           else if (z > 16.0) {
               values[0] = 0.5/z;
               for (int l = 1; l <= maxL; l++) {
                   values[l] = values[0];
                   double Rl = 1.0;
                   double Tlk = 1.0;
                   double cof = 1.0;
                   for (int k = 1; k <= l; k++) {
                       cof = (l-k+1)*(l+k)/((double)k);
                       Tlk *= - cof * values[0];
                       Rl += Tlk;
                   }
                   values[l] *= Rl;
               }
           } 
           // SMALL < z < 16 
           // Use Taylor series around pretabulated values in class
           // 5 terms is usually sufficient for machine accuracy
           else {
               // Index of abscissa z in table
               int ix = std::floor(z * scale + 0.5);
               double dz = z - ix/scale; // z - z0
           
               if (fabs(dz) < 1e-12) { // z is one of the tabulated points
                   for (int l = 0; l <= maxL; l++) values[l] = K[ix][l];
               } else {
           
                   // Calculate (dz)^n/n! terms just once
                   double dzn[TAYLOR_CUT+1];
                   dzn[0] = 1.0;
                   for (int n = 1; n < TAYLOR_CUT + 1; n++)
                       dzn[n] = dzn[n-1] * dz / ((double) n);
           
                   // Now tabulate the values through Taylor seris
                   // K(z) ~ sum_{n=0 to 5} K^(n)(z0)(z-z0)^n / n!
                   for (int l = 0; l <= maxL; l++) {
                       values[l] = 0.0;
                       for (int n = 0; n < TAYLOR_CUT+1; n++)
                           values[l] += dzn[n] * dK[ix][n][l]; 
                   }
               }
           }
       }
       
       // Calculate a modified spherical bessel function value at a point for only a single L
       // method the same as in calculate for multiple L, but with efficiencies
       double BesselFunction::calculate(const double z, int L) {
           double value = 0.0;
           
           if (z <= 0) value = 1.0;
           else if (z < SMALL) {
               value = 1.0 - z;
               for (int k = 1; k < L+1; k++)
                   value *= z/(2.0*L+1.0);
           } else if (z > 16.0) {
               double v0 = 0.5/z;
               value = 1.0;
               double Tlk = 1.0;
               for (int k = 1; k < L+1; k++) {
                   Tlk *= -v0 * (L - k +1)*(L+k)/(double(k));
                   value += Tlk;
               }
               value = v0 * value;
           } else {
               int ix = std::floor(z * scale + 0.5);
               double dz = z - ix/scale; // z - z0
               double dzn = 1.0;
               for (int n = 0; n < TAYLOR_CUT+1; n++) {
                   value += dzn * dK[ix][n][L]; 
                   dzn *= dz / (n+1);
               }
           }
           
           return value;
       }
   }
