vector_df_poisson.C

/*
 *  Resolution of the divergence-free vector Poisson equation
 *
 *    (see file vector.h for documentation).
 *
 */

/*
 *   Copyright (c) 2003 Eric Gourgoulhon & Jerome Novak
 *
 *   This file is part of LORENE.
 *
 *   LORENE is free software; you can redistribute it and/or modify
 *   it under the terms of the GNU General Public License version 2
 *   as published by the Free Software Foundation.
 *
 *   LORENE 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 LORENE; if not, write to the Free Software
 *   Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA
 *
 */

 

/*
 * $Id: vector_df_poisson.C,v 1.16 2016/12/05 16:18:18 j_novak Exp $
 * $Log: vector_df_poisson.C,v $
 * Revision 1.16  2016/12/05 16:18:18  j_novak
 * Suppression of some global variables (file names, loch, ...) to prevent redefinitions
 *
 * Revision 1.15  2014/10/13 08:53:44  j_novak
 * Lorene classes and functions now belong to the namespace Lorene.
 *
 * Revision 1.14  2014/10/06 15:13:20  j_novak
 * Modified #include directives to use c++ syntax.
 *
 * Revision 1.13  2005/02/15 09:45:22  j_novak
 * Correction of an error in the matching.
 *
 * Revision 1.12  2005/02/09 16:53:11  j_novak
 * Now V^r and eta are matched across domains, but not any of their derivatives.
 *
 * Revision 1.11  2005/02/09 14:52:01  j_novak
 * Better solution in the shells.
 *
 * Revision 1.10  2005/02/09 13:20:27  j_novak
 * Completely new way of solving the vector Poisson equation in the Div_free
 * case: the system is inverted "as a whole" for V^r and eta. This works only
 * with Map_af...
 *
 *
 * $Header: /cvsroot/Lorene/C++/Source/Tensor/vector_df_poisson.C,v 1.16 2016/12/05 16:18:18 j_novak Exp $
 *
 */

// C headers
#include <cassert>
#include <cmath>

// Lorene headers
#include "tensor.h"
#include "diff.h"
#include "proto.h"
#include "param.h"

namespace Lorene {
Vector_divfree Vector_divfree::poisson(Param& par ) const {

   // All this has a meaning only for spherical components:
#ifndef NDEBUG 
    const Base_vect_spher* bvs = dynamic_cast<const Base_vect_spher*>(triad) ;
    assert(bvs != 0x0) ; 
#endif
    
    int nitermax = par.get_int() ; 
    int& niter = par.get_int_mod() ; 
    double relax = par.get_double() ; 
    double precis = par.get_double(1) ;     
    Cmp& ss_khi = par.get_cmp_mod(0) ;
    Cmp& ss_mu = par.get_cmp_mod(1) ;
      
    // Solution for the r-component
    // ----------------------------
    
    Scalar source_r = *(cmp[0]) ; 
    source_r.mult_r() ; 
    
    Param par_khi ;
    par_khi.add_int(nitermax, 0) ;
    par_khi.add_int_mod(niter, 0) ;
    par_khi.add_double(relax, 0) ;
    par_khi.add_double(precis, 1) ;
    par_khi.add_cmp_mod(ss_khi, 0) ;

    Scalar khi (*mp) ;
    khi.set_etat_zero() ;

    source_r.poisson(par_khi, khi) ; 
    khi.div_r() ;   // khi now contains V^r
    
    // Solution for mu
    // ---------------
    
    Param par_mu ;
    par_mu.add_int(nitermax, 0) ;
    par_mu.add_int_mod(niter, 0) ;
    par_mu.add_double(relax, 0) ;
    par_mu.add_double(precis, 1) ;
    par_mu.add_cmp_mod(ss_mu, 0) ;

    Scalar mu_resu (*mp) ;
    mu_resu.set_etat_zero() ;
 
    mu().poisson(par_mu, mu_resu) ;

    // Final result
    // ------------
    
    Vector_divfree resu(*mp, *triad, *met_div) ; 
    
    resu.set_vr_mu(khi, mu_resu) ; 
    
    return resu ;

}

/*
 * In the case without parameters, first is solved the equation for mu and then
 * the system of equations for (eta, V^r) is inverted as a whole:
 * d2 eta / dr2 + 2/r d eta / dr - 1/r d V^r / dr = S(eta)
 * d V^r / dr + 2/r V^r - l(l+1)/r eta = 0 (div free condition)
 * 
 * There is no l=0 contribution (divergence free in all space!).
 * In the nucleus and the CED the system is inverted for h(r) and v(r) , 
 * such that eta = r^2 h and V^r = r^2 v in the nucleus,
 * in the compactified domain one has eta = u^2 h and V^r = u^2 v (where u=1/r); 
 * In the shells, both equations are  multiplied by r.
 * These methods are used only to get particular solutions.
 * 
 * Homogeneous solutions are known analitically: r^(l-1) and/or 1/r^(l+2)
 * It is then only possible to match eta and V^r, but not their derivatives,
 * due to the div-free condition.
 */
Vector_divfree Vector_divfree::poisson() const {

    const Map_af* mpaff = dynamic_cast<const Map_af*>(mp) ;
#ifndef NDEBUG 
    for (int i=0; i<3; i++)
    assert(cmp[i]->check_dzpuis(4)) ;
    // All this has a meaning only for spherical components:
    const Base_vect_spher* bvs = dynamic_cast<const Base_vect_spher*>(triad) ;
    assert(bvs != 0x0) ; 
    //## ... and affine mapping, for the moment!
    assert( mpaff != 0x0) ;
#endif

  // Final result
  // ------------
    Vector_divfree resu(*mpaff, *triad, *met_div) ; 

  // Solution for mu
  // ---------------
    Scalar mu_resu = mu().poisson() ;
    
    Scalar f_r(*mpaff) ;
    if (cmp[0]->get_etat() == ETATZERO) { // no work needed ...
    f_r.set_etat_zero() ;
    resu.set_vr_mu(f_r, mu_resu) ; 
    return resu ;
    }
  
    // Some working objects
    //---------------------
  const Mg3d& mg = *mpaff->get_mg() ;
  int nz = mg.get_nzone() ; int nzm1 = nz - 1;
  assert(mg.get_type_r(0) == RARE) ;
  assert(mg.get_type_r(nzm1) == UNSURR) ;
  Scalar S_r = *cmp[0] ;
  Scalar S_eta = eta() ;
  S_r.set_spectral_va().ylm() ;
  S_eta.set_spectral_va().ylm() ;
  const Base_val& base = S_eta.get_spectral_va().base ;
  Mtbl_cf sol_part_eta(mg, base) ; sol_part_eta.annule_hard() ;
  Mtbl_cf sol_part_vr(mg, base) ; sol_part_vr.annule_hard() ;
  Mtbl_cf solution_hom_un(mg, base) ; solution_hom_un.annule_hard() ;
  Mtbl_cf solution_hom_deux(mg, base) ; solution_hom_deux.annule_hard() ;

  // Build-up & inversion of the system for (eta, V^r) in each domain
  //-----------------------------------------------------------------

  // Nucleus
  //--------
  int nr = mg.get_nr(0) ;
  int nt = mg.get_nt(0) ;
  int np = mg.get_np(0) ;
  double alpha = mpaff->get_alpha()[0] ;
  double beta = mpaff->get_beta()[0] ;
  int l_q = 0 ; int m_q = 0; int base_r = 0 ;
  int nr0 = nr - 1 ; //one degree of freedom less because of division by r^2

  // Loop on l and m
  //----------------
  for (int k=0 ; k<np+1 ; k++) {
      for (int j=0 ; j<nt ; j++) { 
      base.give_quant_numbers(0, k, j, m_q, l_q, base_r) ;
      if ( (nullite_plm(j, nt, k, np, base) == 1) && (l_q != 0) ) {
          int dege1 = (l_q <3 ? 0 : 1) ; //degeneracy of eq.1
          int dege2 = 0 ;  //degeneracy of eq.2
          int nr_eq1 = nr0 - dege1 ; //Eq.1 is for h (eta/r^2)
          int nr_eq2 = nr0 - dege2 ; //Eq.2 is the div-free condition
          int nrtot = nr_eq1 + nr_eq2 ;
          Matrice oper(nrtot, nrtot) ; oper.set_etat_qcq() ;
          Tbl sec_membre(nrtot) ; sec_membre.set_etat_qcq() ;
          Diff_x2dsdx2 d2(base_r, nr) ; const Matrice& md2 = d2.get_matrice() ;
          Diff_xdsdx xd(base_r, nr) ; const Matrice& mxd = xd.get_matrice() ;
          Diff_id id(base_r, nr) ; const Matrice& mid = id.get_matrice() ;

          // Building the operator
          //----------------------
          for (int lin=0; lin<nr_eq1; lin++) { //eq.1 
          for (int col=dege1; col<nr0; col++) 
              oper.set(lin,col-dege1) 
              = md2(lin,col) + 6*mxd(lin,col) + 6*mid(lin,col) ;
          for (int col=dege2; col<nr0; col++) 
              oper.set(lin,col-dege2+nr_eq1) = -mxd(lin,col) -2*mid(lin,col) ;
          }
          for (int lin=0; lin<nr0; lin++) { //eq.2
          for (int col=dege1; col<nr0; col++)
              oper.set(lin+nr_eq1,col-dege1) = -l_q*(l_q+1)*mid(lin,col) ;
          for (int col=dege2; col<nr0; col++)
              oper.set(lin+nr_eq1, col-dege2+nr_eq1) = mxd(lin,col) + 4*mid(lin,col) ;
          }
          oper.set_lu() ;

          // Filling the r.h.s
          //------------------
          for (int i=0; i<nr_eq1; i++)  //eq.1
          sec_membre.set(i) = (*S_eta.get_spectral_va().c_cf)(0, k, j, i) ;
          for (int i=0; i<nr0; i++) //eq.2
          sec_membre.set(i+nr_eq1) = 0 ;

          // Inversion of the "big" operator
          //--------------------------------
          Tbl big_res = oper.inverse(sec_membre) ;
          Tbl res_eta(nr) ;  res_eta.set_etat_qcq() ;
          Tbl res_vr(nr) ;  res_vr.set_etat_qcq() ;
          
          // Putting coefficients of h and v to individual arrays
          //-----------------------------------------------------
          for (int i=0; i<dege1; i++)
          res_eta.set(i) = 0 ;
          for (int i=dege1; i<nr0; i++)
          res_eta.set(i) = big_res(i-dege1) ;
          res_eta.set(nr0) = 0 ;
          for (int i=0; i<dege2; i++)
          res_vr.set(i) = 0 ;
          for (int i=dege2; i<nr0; i++)
          res_vr.set(i) = big_res(i-dege2+nr_eq1) ;
          res_vr.set(nr0) = 0 ;

          // Multiplication by xi^2 (the alpha^2 factor comes soon)
          //-------------------------------------------------------
          multx2_1d(nr, &res_eta.t, base_r) ;
          multx2_1d(nr, &res_vr.t, base_r) ;

          // Homogeneous solution (only r^(l-1) in the nucleus)
          Tbl sol_hom = solh(nr, l_q-1, 0., base_r) ;
          for (int i=0 ; i<nr ; i++) {
          sol_part_eta.set(0, k, j, i) = alpha*alpha*res_eta(i) ;
          sol_part_vr.set(0, k, j, i) = alpha*alpha*res_vr(i) ;
          solution_hom_un.set(0, k, j, i) = sol_hom(i) ;
          solution_hom_deux.set(0, k, j, i) = 0. ; 
          }
      }
      }
  }     


  // Shells
  //-------
  for (int zone=1 ; zone<nzm1 ; zone++) {
      nr = mg.get_nr(zone) ; 
      assert (nr > 5) ;
      assert(nt == mg.get_nt(zone)) ;
      assert(np == mg.get_np(zone)) ;
      alpha = mpaff->get_alpha()[zone] ;
      beta = mpaff->get_beta()[zone] ;
      double ech = beta / alpha ;

  // Loop on l and m
  //----------------
      for (int k=0 ; k<np+1 ; k++) {
      for (int j=0 ; j<nt ; j++) {
      base.give_quant_numbers(zone, k, j, m_q, l_q, base_r) ;
      if ( (nullite_plm(j, nt, k, np, base) == 1) && (l_q != 0) ) {
          int dege1 = 2 ; //degeneracy of eq.1
          int dege2 = 0 ;  //degeneracy of eq.2
          int nr_eq1 = nr - dege1 ; //Eq.1 is for eta
          int nr_eq2 = nr - dege2 ; //Eq.2 is the div-free condition
          int nrtot = nr_eq1 + nr_eq2 + 1;
          Matrice oper(nrtot, nrtot) ; oper.set_etat_qcq() ;
          Tbl sec_membre(nrtot) ; sec_membre.set_etat_qcq() ;
          Diff_x2dsdx2 x2d2(base_r, nr+1); const Matrice& m2d2 = x2d2.get_matrice() ;
          Diff_xdsdx2 xd2(base_r, nr+1) ; const Matrice& mxd2 = xd2.get_matrice() ;
          Diff_dsdx2 d2(base_r, nr+1) ; const Matrice& md2 = d2.get_matrice() ;
          Diff_xdsdx xd(base_r, nr+1) ; const Matrice& mxd = xd.get_matrice() ;
          Diff_dsdx d1(base_r, nr+1) ; const Matrice& md = d1.get_matrice() ;
          Diff_id id(base_r, nr+1) ; const Matrice& mid = id.get_matrice() ;

          // Building the operator
          //----------------------
          for (int lin=0; lin<nr_eq1; lin++) { 
          for (int col=dege1; col<nr; col++) 
              oper.set(lin,col-dege1) 
              = mxd2(lin,col) + ech*md2(lin,col) + 2*md(lin,col) ;
              for (int col=dege2; col<nr+1; col++) 
                  oper.set(lin,col-dege2+nr_eq1) = -md(lin,col) ; 
          }
          for (int lin=0; lin<nr_eq2; lin++) {
          for (int col=dege1; col<nr; col++)
              oper.set(lin+nr_eq1,col-dege1) = -l_q*(l_q+1)*mid(lin,col) ;
          for (int col=dege2; col<nr+1; col++)
              oper.set(lin+nr_eq1, col-dege2+nr_eq1) = 
              mxd(lin,col) + ech*md(lin,col) + 2*mid(lin,col) ;
          }
          //Additional line to avoid spurious homogeneous solutions
          //this line is the first one of the V^r eq.
          for (int col=dege1; col<nr; col++)
          oper.set(nrtot-1, col-dege1) = 0 ;
          for (int col=dege2; col<nr+1; col++)
          oper.set(nrtot-1, col-dege2+nr_eq1) = 
              m2d2(0,col) + ech*(2*mxd2(0,col) + ech*md2(0,col))
              +4*(mxd(0,col) +ech*md(0,col)) 
              +(2 - l_q*(l_q+1))*mid(0,col) ;
          oper.set_lu() ;
          
          // Filling the r.h.s
          //------------------
          Tbl sr(5) ; sr.set_etat_qcq() ; 
          Tbl seta(nr) ; seta.set_etat_qcq() ;
          for (int i=0; i<5; i++) {
          sr.set(i) = (*S_r.get_spectral_va().c_cf)(zone, k, j, i);
          seta.set(i) = (*S_eta.get_spectral_va().c_cf)(zone, k, j, i) ;
          }
          for (int i=5; i<nr; i++) 
          seta.set(i) = (*S_eta.get_spectral_va().c_cf)(zone, k, j, i) ;
          Tbl xsr= sr ;  Tbl x2sr= sr ;
          Tbl xseta= seta ;
          multx2_1d(5, &x2sr.t, base_r) ; multx_1d(5, &xsr.t, base_r) ;
          multx_1d(nr, &xseta.t, base_r) ;
          
          for (int i=0; i<nr_eq1; i++) 
          sec_membre.set(i) = alpha*(alpha*xseta(i) + beta*seta(i)) ;
          for (int i=0; i<nr_eq2; i++)
          sec_membre.set(i+nr_eq1) = 0 ;
          sec_membre.set(nr_eq1+nr_eq2) = alpha*alpha*x2sr(0) + 2*alpha*beta*xsr(0)
          + beta*beta*sr(0) ;

          // Inversion of the "big" operator
          //--------------------------------
          Tbl big_res = oper.inverse(sec_membre) ;
          Tbl res_eta(nr) ;  res_eta.set_etat_qcq() ;
          Tbl res_vr(nr) ;  res_vr.set_etat_qcq() ;
          
          // Putting coefficients of h and v to individual arrays
          //-----------------------------------------------------
          for (int i=0; i<dege1; i++)
          res_eta.set(i) = 0 ;
          for (int i=dege1; i<nr; i++)
          res_eta.set(i) = big_res(i-dege1) ;
          for (int i=0; i<dege2; i++)
          res_vr.set(i) = 0 ;
          for (int i=dege2; i<nr; i++)
          res_vr.set(i) = big_res(i-dege2+nr_eq1) ;

          //homogeneous solutions
          Tbl sol_hom1 = solh(nr, l_q-1, ech, base_r) ;
          Tbl sol_hom2 = solh(nr, l_q+1, ech, base_r) ;
          for (int i=0 ; i<nr ; i++) {
          sol_part_eta.set(zone, k, j, i) = res_eta(i) ;
          sol_part_vr.set(zone, k, j, i) = res_vr(i) ;
          solution_hom_un.set(zone, k, j, i) = sol_hom1(0,i) ;
          solution_hom_deux.set(zone, k, j, i) = sol_hom2(1,i) ;
          }
      } 
      }
      }
  }

  // Compactified external domain
  //-----------------------------
  nr = mg.get_nr(nzm1) ;
  assert(nt == mg.get_nt(nzm1)) ;
  assert(np == mg.get_np(nzm1)) ;
  alpha = mpaff->get_alpha()[nzm1] ;
  assert (nr > 4) ;
  nr0 = nr - 2 ;  //two degrees of freedom less because of division by r^2

  // Loop on l and m
  //----------------
  for (int k=0 ; k<np+1 ; k++) {
      for (int j=0 ; j<nt ; j++) { 
      base.give_quant_numbers(nzm1, k, j, m_q, l_q, base_r) ;
      if ( (nullite_plm(j, nt, k, np, base) == 1) && (l_q != 0) ) {
          int dege1 = 0; //degeneracy of eq.1
          int dege2 = 1; //degeneracy of eq.2
          int nr_eq1 = nr0 - dege1 ; //Eq.1 is for eta
          int nr_eq2 = nr0 - dege2 ; //Eq.2 is the div-free condition
          int nrtot = nr_eq1 + nr_eq2 ;
          Matrice oper(nrtot, nrtot) ; oper.set_etat_qcq() ;
          Tbl sec_membre(nrtot) ; sec_membre.set_etat_qcq() ;
          Diff_x2dsdx2 x2d2(base_r, nr) ; const Matrice& m2d2 = x2d2.get_matrice() ;
          Diff_xdsdx xd(base_r, nr) ; const Matrice& mxd = xd.get_matrice() ;
          Diff_id id(base_r, nr) ; const Matrice& mid = id.get_matrice() ;

          // Building the operator
          //----------------------
          for (int lin=0; lin<nr_eq1; lin++) { 
          for (int col=dege1; col<nr0; col++) 
              oper.set(lin,col-dege1) 
              = m2d2(lin,col) + 4*mxd(lin,col) + 2*mid(lin,col) ;
              for (int col=dege2; col<nr0; col++) 
                  oper.set(lin,col-dege2+nr_eq1) = 
                  mxd(lin,col) + 2*mid(lin,col) ;
          }
          for (int lin=0; lin<nr_eq2; lin++) {
          for (int col=dege1; col<nr0; col++)
              oper.set(lin+nr_eq1,col-dege1) = l_q*(l_q+1)*mid(lin,col) ;
          for (int col=dege2; col<nr0; col++)
              oper.set(lin+nr_eq1, col-dege2+nr_eq1) = mxd(lin,col) ;
          }
          oper.set_lu() ;
          
          // Filling the r.h.s
          //------------------
          for (int i=0; i<nr_eq1; i++) 
          sec_membre.set(i) = (*S_eta.get_spectral_va().c_cf)(nzm1, k, j, i) ;
          for (int i=0; i<nr_eq2; i++)
          sec_membre.set(i+nr_eq1) = 0 ;
          Tbl big_res = oper.inverse(sec_membre) ;
          Tbl res_eta(nr) ;  res_eta.set_etat_qcq() ;
          Tbl res_vr(nr) ;  res_vr.set_etat_qcq() ;
          
          // Putting coefficients of h and v to individual arrays
          //-----------------------------------------------------
          for (int i=0; i<dege1; i++)
          res_eta.set(i) = 0 ;
          for (int i=dege1; i<nr0; i++)
          res_eta.set(i) = big_res(i-dege1) ;
          res_eta.set(nr0) = 0 ;
          res_eta.set(nr0+1) = 0 ;
          for (int i=0; i<dege2; i++)
          res_vr.set(i) = 0 ;
          for (int i=dege2; i<nr0; i++)
          res_vr.set(i) = big_res(i-dege2+nr_eq1) ;
          res_vr.set(nr0) = 0 ;
          res_vr.set(nr0+1) = 0 ;

          // Multiplication by r^2 
          //-----------------------
          Tbl res_v2(nr) ;  res_v2.set_etat_qcq() ;
          Tbl res_e2(nr) ; res_e2.set_etat_qcq() ;
          mult2_xm1_1d_cheb(nr, res_eta.t, res_e2.t) ; 
          mult2_xm1_1d_cheb(nr, res_vr.t, res_v2.t) ; 

          // Homogeneous solution (only 1/r^(l+2) in the CED)
          Tbl sol_hom = solh(nr, l_q+1, 0., base_r) ;
          for (int i=0 ; i<nr ; i++) {
          sol_part_eta.set(nzm1, k, j, i) = alpha*alpha*res_e2(i) ;
          sol_part_vr.set(nzm1, k, j, i) = alpha*alpha*res_v2(i) ;
          solution_hom_un.set(nzm1, k, j, i) = sol_hom(i) ;
          solution_hom_deux.set(nzm1, k, j, i) = 0. ; 
          }
      }     
      }
  }

  // Now let's match everything ...
  //-------------------------------

  // Resulting V^r & eta
  Scalar vr(*mpaff) ; vr.set_etat_qcq() ;
  vr.set_spectral_base(base) ;
  vr.set_spectral_va().set_etat_cf_qcq() ;
  Mtbl_cf& cf_vr = *vr.set_spectral_va().c_cf ;
  cf_vr.annule_hard() ;
  Scalar het(*mpaff) ; het.set_etat_qcq() ;
  het.set_spectral_base(base) ;
  het.set_spectral_va().set_etat_cf_qcq() ;
  Mtbl_cf& cf_eta = *het.set_spectral_va().c_cf ;
  cf_eta.annule_hard() ;
  int taille = 2*nzm1 ;
  Tbl sec_membre(taille) ; 
  Matrice systeme(taille, taille) ; 
  systeme.set_etat_qcq() ;
  int ligne ;  int colonne ;
  
  // Loop on l and m
  //----------------
  for (int k=0 ; k<np+1 ; k++)
      for (int j=0 ; j<nt ; j++) {
      base.give_quant_numbers(0, k, j, m_q, l_q, base_r) ;
      if ((nullite_plm(j, nt, k, np, base) == 1)&&(l_q != 0)) {
        
          ligne = 0 ;
          colonne = 0 ;
          sec_membre.annule_hard() ;
          for (int l=0; l<taille; l++) 
          for (int c=0; c<taille; c++)
              systeme.set(l,c) = 0 ;
          //Nucleus 
          nr = mg.get_nr(0) ;
          alpha = mpaff->get_alpha()[0] ;
          // value of x^(l-1) at 1 ...
          systeme.set(ligne, colonne) = 1. ;
          for (int i=0 ; i<nr ; i++)
          sec_membre.set(ligne) -= sol_part_eta(0, k, j, i) ;
          ligne++ ;
          // ... and of its couterpart for V^r
          systeme.set(ligne, colonne) = l_q;
          for (int i=0; i<nr; i++)
          sec_membre.set(ligne) -= sol_part_vr(0,k,j,i) ; 
          colonne++ ; 
              //shells
          for (int zone=1 ; zone<nzm1 ; zone++) {
          nr = mg.get_nr(zone) ;
          alpha = mpaff->get_alpha()[zone] ;
          double echelle = mpaff->get_beta()[zone]/alpha ;
          ligne -- ;
          //value of (x+echelle)^(l-1) at -1 
          systeme.set(ligne, colonne) = -pow(echelle-1., double(l_q-1)) ;
          // value of 1/(x+echelle) ^(l+2) at -1 
          systeme.set(ligne, colonne+1) = -1/pow(echelle-1., double(l_q+2)) ;
          for (int i=0 ; i<nr ; i++)
              if (i%2 == 0)
              sec_membre.set(ligne) += sol_part_eta(zone, k, j, i) ;
              else sec_membre.set(ligne) -= sol_part_eta(zone, k, j, i) ;
          ligne++ ;
          // ... and their couterparts for V^r
          systeme.set(ligne, colonne) = -l_q*pow(echelle-1., double(l_q-1)) ;
          systeme.set(ligne, colonne+1) = (l_q+1)/pow(echelle-1., double(l_q+2));
          for (int i=0 ; i<nr ; i++)
              if (i%2 == 0)
              sec_membre.set(ligne) += sol_part_vr(zone, k, j, i) ;
              else sec_membre.set(ligne) -= sol_part_vr(zone, k, j, i) ;
          ligne++ ;
            
          //value of (x+echelle)^(l-1) at 1 :
          systeme.set(ligne, colonne) = pow(echelle+1., double(l_q-1)) ;
          // value of 1/(x+echelle)^(l+2) at 1 :
          systeme.set(ligne, colonne+1) = 1./pow(echelle+1., double(l_q+2)) ;
          for (int i=0 ; i<nr ; i++)
              sec_membre.set(ligne) -= sol_part_eta(zone, k, j, i) ;
          ligne ++ ;
          //... and their couterparts for V^r
          systeme.set(ligne, colonne) = l_q*pow(echelle+1., double(l_q-1)) ;
          systeme.set(ligne, colonne+1) = -double(l_q+1)
              / pow(echelle+1., double(l_q+2)) ;
          for (int i=0 ; i<nr ; i++)
              sec_membre.set(ligne) -= sol_part_vr(zone, k, j, i); 
          colonne += 2 ;
          }         
          //Compactified external domain
          nr = mg.get_nr(nzm1) ;

          alpha = mpaff->get_alpha()[nzm1] ;
          ligne -- ;
          //value of (x-1)^(l+2) at -1 :
          systeme.set(ligne, colonne) = -pow(-2, double(l_q+2)) ;
          for (int i=0 ; i<nr ; i++)
          if (i%2 == 0) sec_membre.set(ligne) += sol_part_eta(nzm1, k, j, i) ;
          else sec_membre.set(ligne) -= sol_part_eta(nzm1, k, j, i) ;
          //... and of its couterpart for V^r
          systeme.set(ligne+1, colonne) = double(l_q+1)*pow(-2, double(l_q+2)) ;
          for (int i=0 ; i<nr ; i++)
          if (i%2 == 0) sec_membre.set(ligne+1) += sol_part_vr(nzm1, k, j, i) ;
          else sec_membre.set(ligne+1) -= sol_part_vr(nzm1, k, j, i) ;
            
          // Solution of the system giving the coefficients for the homogeneous 
          // solutions
          //-------------------------------------------------------------------
          if (taille > 2) systeme.set_band(2,2) ;
          else systeme.set_band(1,1) ;
          systeme.set_lu() ;
          Tbl facteurs(systeme.inverse(sec_membre)) ;
          int conte = 0 ;

          // everything is put to the right place, the same combination of hom.
          // solutions (with some l or -(l+1) factors) must be used for V^r
          //-------------------------------------------------------------------
          nr = mg.get_nr(0) ; //nucleus
          for (int i=0 ; i<nr ; i++) {
          cf_eta.set(0, k, j, i) = sol_part_eta(0, k, j, i)
              +facteurs(conte)*solution_hom_un(0, k, j, i) ;
          cf_vr.set(0, k, j, i) = sol_part_vr(0, k, j, i)
              +double(l_q)*facteurs(conte)*solution_hom_un(0, k, j, i) ;
          }
          conte++ ;
          for (int zone=1 ; zone<nzm1 ; zone++) { //shells
          nr = mg.get_nr(zone) ;
          for (int i=0 ; i<nr ; i++) {
              cf_eta.set(zone, k, j, i) = 
              sol_part_eta(zone, k, j, i)
              +facteurs(conte)*solution_hom_un(zone, k, j, i) 
              +facteurs(conte+1)*solution_hom_deux(zone, k, j, i) ;
              cf_vr.set(zone, k, j, i) = sol_part_vr(zone, k, j, i)
              +double(l_q)*facteurs(conte)*solution_hom_un(zone, k, j, i) 
              -double(l_q+1)*facteurs(conte+1)*solution_hom_deux(zone, k, j, i) ;
          }
          conte+=2 ;
          }
          nr = mg.get_nr(nz-1) ; //compactified external domain
          for (int i=0 ; i<nr ; i++) {
          cf_eta.set(nzm1, k, j, i) = sol_part_eta(nzm1, k, j, i)
              +facteurs(conte)*solution_hom_un(nzm1, k, j, i) ;
          cf_vr.set(nzm1, k, j, i) = sol_part_vr(nzm1, k, j, i)
              -double(l_q+1)*facteurs(conte)*solution_hom_un(nzm1, k, j, i) ;
          }
      } // End of nullite_plm  
      } //End of loop on theta

  vr.set_spectral_va().ylm_i() ;
  het.set_spectral_va().ylm_i() ;

  resu.set_vr_eta_mu(vr, het, mu_resu) ;

  return resu ;

}

}