vector_poisson_block.C

/*
 *  Method for vector Poisson equation inverting eqs. for V^r and eta as a block.
 *
 *    (see file vector.h for documentation).
 *
 */

/*
 *   Copyright (c) 2005  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_poisson_block.C,v 1.10 2016/12/05 16:18:18 j_novak Exp $
 * $Log: vector_poisson_block.C,v $
 * Revision 1.10  2016/12/05 16:18:18  j_novak
 * Suppression of some global variables (file names, loch, ...) to prevent redefinitions
 *
 * Revision 1.9  2014/10/13 08:53:45  j_novak
 * Lorene classes and functions now belong to the namespace Lorene.
 *
 * Revision 1.8  2014/10/06 15:13:21  j_novak
 * Modified #include directives to use c++ syntax.
 *
 * Revision 1.7  2012/10/12 11:43:38  j_novak
 * Removed some headers
 *
 * Revision 1.6  2007/12/21 10:45:06  j_novak
 * Better treatment of spherical symmetry
 *
 * Revision 1.5  2007/09/05 12:35:18  j_novak
 * Homogeneous solutions are no longer obtained through the analytic formula, but
 * by solving (again) the operator system with almost zero as r.h.s. This is to
 * lower the condition number of the matching system.
 *
 * Revision 1.4  2007/01/23 17:08:46  j_novak
 * New function pois_vect_r0.C to solve the l=0 part of the vector Poisson
 * equation, which involves only the r-component.
 *
 * Revision 1.3  2005/12/30 13:39:38  j_novak
 * Changed the Galerkin base in the nucleus to (hopefully) stabilise the solver
 * when used in an iteration. Similar changes in the CED too.
 *
 * Revision 1.2  2005/02/15 15:43:18  j_novak
 * First version of the block inversion for the vector Poisson equation (method 6).
 *
 *
 * $Header: /cvsroot/Lorene/C++/Source/Tensor/vector_poisson_block.C,v 1.10 2016/12/05 16:18:18 j_novak Exp $
 *
 */

// C headers
#include <cmath>

// Lorene headers
#include "metric.h"
#include "diff.h"
#include "proto.h"

namespace Lorene {
void Vector::poisson_block(double lam, Vector& resu) 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

     if (fabs(lam + 1.) < 1.e-8) {
    const Metric_flat& mets = mp->flat_met_spher() ;
    Vector_divfree sou(*mp, *triad, mets) ;
    for (int i=1; i<=3; i++) sou.set(i) = *cmp[i-1] ;
    resu = sou.poisson() ;
    return ;
     }

    // Some working objects
    //---------------------
  const Mg3d& mg = *mpaff->get_mg() ;
  int nz = mg.get_nzone() ; int nzm1 = nz - 1;
  int nt = mg.get_nt(0) ;
  int np = mg.get_np(0) ;
  assert(mg.get_type_r(0) == RARE) ;
  assert(mg.get_type_r(nzm1) == UNSURR) ;
  Scalar S_r = *cmp[0] ;
  Scalar S_eta = eta() ;
  Scalar het(*mpaff) ; 
  Scalar vr(*mpaff) ; 
 bool all_zero = false ;
 if (S_r.get_etat() == ETATZERO) {
     if (S_eta.get_etat() == ETATZERO) {
     vr.set_etat_zero() ;
     het.set_etat_zero() ;
     all_zero = true ;
     }
     else {
     S_r.annule_hard() ;
     S_r.set_spectral_base(S_eta.get_spectral_base()) ;
     }
 }
 if ((S_eta.get_etat() == ETATZERO)&&(!all_zero)) {
     S_eta.annule_hard() ;
     S_eta.set_spectral_base(S_r.get_spectral_base()) ;
 }
 if (!all_zero) {
  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 sol_hom_un_eta(mg, base) ; sol_hom_un_eta.annule_hard() ;
  Mtbl_cf sol_hom_deux_eta(mg, base) ; sol_hom_deux_eta.annule_hard() ;
  Mtbl_cf sol_hom_trois_eta(mg, base) ; sol_hom_trois_eta.annule_hard() ;
  Mtbl_cf sol_hom_quatre_eta(mg, base) ; sol_hom_quatre_eta.annule_hard() ;
  Mtbl_cf sol_hom_un_vr(mg, base) ; sol_hom_un_vr.annule_hard() ;
  Mtbl_cf sol_hom_deux_vr(mg, base) ; sol_hom_deux_vr.annule_hard() ;
  Mtbl_cf sol_hom_trois_vr(mg, base) ; sol_hom_trois_vr.annule_hard() ;
  Mtbl_cf sol_hom_quatre_vr(mg, base) ; sol_hom_quatre_vr.annule_hard() ;
 
  // Solution of the l=0 part for Vr
  //---------------------------------
  Scalar sou_l0 = (*cmp[0]) / (1. + lam) ;
  sou_l0.set_spectral_va().ylm() ;
  Scalar vrl0 = pois_vect_r0(sou_l0) ;

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

  // Nucleus
  //--------
  int nr = mg.get_nr(0) ;
  double alpha = mpaff->get_alpha()[0] ; double alp2 = alpha*alpha ;
  double beta = mpaff->get_beta()[0] ;
  int l_q = 0 ; int m_q = 0; int base_r = 0 ;

  // 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 aa = 0 ; int bb = 0 ;  int nr0 = 0 ;
          if (base_r == R_CHEBP) {
          nr0 = nr - 1 ;
          aa = 0 ; bb = 1 ;
          }
          else {
          assert (base_r == R_CHEBI) ;
          nr0 = nr - 2 ;
          aa = 2 ; bb = 1 ;
          }
          int d0 = nr - nr0 ;
          int nrtot = 2*nr0 ;
          Matrice oper(2*nr, 2*nr) ; oper.set_etat_qcq() ;
          Tbl sec_membre(nrtot) ; sec_membre.set_etat_qcq() ;
          Diff_dsdx2 d2(base_r, nr) ; const Matrice& md2 = d2.get_matrice() ;
          Diff_sxdsdx xd(base_r, nr) ; const Matrice& mxd = xd.get_matrice() ;
          Diff_sx2 s2(base_r, nr) ; const Matrice& ms2 = s2.get_matrice() ;

          // Building the operator
          //----------------------
          for (int lin=0; lin<nr; lin++) { //eq.1 
          for (int col=0; col<nr; col++) 
              oper.set(lin,col) = 
              (md2(lin,col) + 2*mxd(lin,col) 
               -(lam+1)*l_q*(l_q+1)*ms2(lin,col)) / alp2 ;
          for (int col=0; col<nr; col++) 
              oper.set(lin,col+nr) = 
              (lam*mxd(lin,col) + 2*(1+lam)*ms2(lin,col)) / alp2 ;
          }
          for (int lin=0; lin<nr; lin++) { //eq.2
          for (int col=0; col<nr; col++)
              oper.set(lin+nr,col) = 
              (-lam*l_q*(l_q+1)*mxd(lin,col) 
               +(lam+2)*l_q*(l_q+1)*ms2(lin,col)) / alp2 ;
          for (int col=0; col<nr; col++)
              oper.set(lin+nr, col+nr) = 
              ((lam+1)*(md2(lin,col) + 2*mxd(lin,col)) 
               - (2*(lam+1) + l_q*(l_q+1))*ms2(lin,col)) / alp2 ;
          }
          bool pb_eta = ( ( fabs( lam*double(l_q+3) + 2 ) < 0.01) && (l_q <=2) ) ;
          if (!pb_eta) {
          for (int col=0; col<nr; col++) 
              oper.set(nr0-1, col) = 1 ;
          for (int col=0; col<nr; col++) 
              oper.set(nr0-1,nr+col) = 0 ;        
          }
          if ((l_q > 2)||pb_eta) {
          for (int col=0; col<nr; col++) 
              oper.set(nr+nr0-1, col) = 0 ;
          for (int col=0; col<nr; col++) 
              oper.set(nr+nr0-1,nr+col) = 1 ;         
          }

          Matrice op2(nrtot, nrtot) ;
          op2.set_etat_qcq() ;
          for (int i=0; i<nr0; i++) {
          for (int col=0; col<nr0;col++) 
              op2.set(i,col) = (aa*col+bb)*oper(i,col+1) 
              + (aa*(col+1)+bb)*oper(i,col) ;
          for (int col=0; col<nr0;col++) 
              op2.set(i,col+nr0) = (aa*col+bb)*oper(i,col+nr+1)
              + (aa*(col+1)+bb)*oper(i,col+nr) ;
          }
          
          for (int i=nr0; i<nrtot; i++) {
          for (int col=0; col<nr0;col++) 
              op2.set(i,col) = (aa*col+bb)*oper(i+d0,col+1) 
              + (aa*(col+1)+bb)*oper(i+d0,col) ;
          for (int col=0; col<nr0;col++) 
              op2.set(i,col+nr0) = (aa*col+bb)*oper(i+d0,col+nr+1)
              + (aa*(col+1)+bb)*oper(i+d0,col+nr) ;
          }
          op2.set_lu() ;

          // Filling the r.h.s
          //------------------
          for (int i=0; i<nr0; i++)  //eq.1
          sec_membre.set(i) = (*S_eta.get_spectral_va().c_cf)(0, k, j, i) ;
          if (!pb_eta) sec_membre.set(nr0-1) = 0 ;
          for (int i=0; i<nr0; i++) //eq.2
          sec_membre.set(i+nr0) 
              = (*S_r.get_spectral_va().c_cf)(0, k, j, i) ;
          if ((l_q > 2)||pb_eta) sec_membre.set(nrtot-1) = 0 ;

          // Inversion of the "big" operator
          //--------------------------------
          Tbl big_res = op2.inverse(sec_membre) ;
          
          // Putting coefficients of eta and Vr to individual arrays
          //--------------------------------------------------------
          sol_part_eta.set(0, k, j, 0) = (aa+bb)*big_res(0) ;
          for (int i=1; i<nr0; i++)
          sol_part_eta.set(0, k, j, i) = (aa*(i-1)+bb)*big_res(i-1) 
              + (aa*(i+1)+bb)*big_res(i);
          sol_part_eta.set(0, k, j, nr0) = big_res(nr0-1)*(aa*(nr0-1) + bb) ;
          sol_part_vr.set(0, k, j, 0) = (aa+bb)*big_res(nr0) ;
          for (int i=1; i<nr0; i++)
          sol_part_vr.set(0, k, j, i) = (aa*(i-1)+bb)*big_res(nr0+i-1) 
              + (aa*(i+1)+bb)*big_res(nr0+i);
          sol_part_vr.set(0, k, j, nr0) = big_res(nrtot-1)*(aa*(nr0-1)+bb) ;
          if (base_r == R_CHEBI) {
          sol_part_eta.set(0, k, j, nr-1) = 0 ;
          sol_part_vr.set(0, k, j, nr-1) = 0 ;
          }

          // Homogeneous solutions (only r^(l-1) and r^(l+1) in the nucleus)
          if (l_q<=2) {
          double fac_eta = lam*double(l_q+3) + 2. ;
          double fac_vr = double(l_q+1)*(lam*l_q - 2.) ;
          Tbl sol_hom1 = solh(nr, l_q-1, 0., base_r) ;
          Tbl sol_hom2 = solh(nr, l_q+1, 0., base_r) ;
          for (int i=0 ; i<nr ; i++) {
              sol_hom_un_eta.set(0, k, j, i) = sol_hom1(i) ;
              sol_hom_un_vr.set(0, k, j, i) = l_q*sol_hom1(i) ;
              sol_hom_trois_eta.set(0, k, j, i) = fac_eta*sol_hom2(i) ; 
              sol_hom_trois_vr.set(0, k, j, i) = fac_vr*sol_hom2(i) ; 
          }
          }
          else {
          for (int i=0; i<nrtot; i++) sec_membre.set(i) = 0 ;
          // First homogeneous sol.
          sec_membre.set(nr0-1) = 1 ;
          big_res = op2.inverse(sec_membre) ;

          sol_hom_un_eta.set(0, k, j, 0) = (aa+bb)*big_res(0) ;
          for (int i=1; i<nr0; i++)
              sol_hom_un_eta.set(0, k, j, i) = (aa*(i-1)+bb)*big_res(i-1) 
              + (aa*(i+1)+bb)*big_res(i);
          sol_hom_un_eta.set(0, k, j, nr0) = big_res(nr0-1)*(aa*(nr0-1) + bb) ;
          sol_hom_un_vr.set(0, k, j, 0) = (aa+bb)*big_res(nr0) ;
          for (int i=1; i<nr0; i++)
              sol_hom_un_vr.set(0, k, j, i) = (aa*(i-1)+bb)*big_res(nr0+i-1) 
              + (aa*(i+1)+bb)*big_res(nr0+i);
          sol_hom_un_vr.set(0, k, j, nr0) = big_res(nrtot-1)*(aa*(nr0-1)+bb) ;
          if (base_r == R_CHEBI) {
              sol_hom_un_eta.set(0, k, j, nr-1) = 0 ;
              sol_hom_un_vr.set(0, k, j, nr-1) = 0 ;
          }

          sec_membre.set(nr0-1) = 0 ;
          sec_membre.set(nrtot-1) = 1 ;
          big_res = op2.inverse(sec_membre) ;

          sol_hom_trois_eta.set(0, k, j, 0) = (aa+bb)*big_res(0) ;
          for (int i=1; i<nr0; i++)
              sol_hom_trois_eta.set(0, k, j, i) = (aa*(i-1)+bb)*big_res(i-1) 
              + (aa*(i+1)+bb)*big_res(i);
          sol_hom_trois_eta.set(0, k, j, nr0) = big_res(nr0-1)*(aa*(nr0-1)+bb) ;
          sol_hom_trois_vr.set(0, k, j, 0) = (aa+bb)*big_res(nr0) ;
          for (int i=1; i<nr0; i++)
              sol_hom_trois_vr.set(0, k, j, i) = (aa*(i-1)+bb)*big_res(nr0+i-1) 
              + (aa*(i+1)+bb)*big_res(nr0+i);
          sol_hom_trois_vr.set(0, k, j, nr0) = big_res(nrtot-1)*(aa*(nr0-1)+bb) ;
          if (base_r == R_CHEBI) {
              sol_hom_trois_eta.set(0, k, j, nr-1) = 0 ;
              sol_hom_trois_vr.set(0, k, j, nr-1) = 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 = 2 ;  //degeneracy of eq.2
          int nr_eq1 = nr - dege1 ; //Eq.1 is for eta
          int nr_eq2 = nr - dege2 ; //Eq.2 is for V^r
          int nrtot = 2*nr ;
          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_xdsdx2 xd2(base_r, nr) ; const Matrice& mxd2 = xd2.get_matrice() ;
          Diff_dsdx2 d2(base_r, nr) ; const Matrice& md2 = d2.get_matrice() ;
          Diff_xdsdx xd(base_r, nr) ; const Matrice& mxd = xd.get_matrice() ;
          Diff_dsdx d1(base_r, nr) ; const Matrice& md = d1.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=0; col<nr; col++) 
              oper.set(lin,col) 
              = m2d2(lin,col) + 2*ech*mxd2(lin,col) + ech*ech*md2(lin,col) 
              + 2*(mxd(lin,col) + ech*md(lin,col)) 
              - (lam+1)*l_q*(l_q+1)*mid(lin,col) ;
          for (int col=0; col<nr; col++) 
              oper.set(lin,col+nr) 
              = lam*(mxd(lin,col) + ech*md(lin,col)) 
              + 2*(1+lam)*mid(lin,col) ; 
          }
          oper.set(nr_eq1, 0) = 1 ;
          for (int col=1; col<2*nr; col++) 
          oper.set(nr_eq1, col) = 0 ;
          oper.set(nr_eq1+1, 0) = 0 ;
          oper.set(nr_eq1+1, 1) = 1 ;
          for (int col=2; col<2*nr; col++) 
          oper.set(nr_eq1+1, col) = 0 ;
          for (int lin=0; lin<nr_eq2; lin++) {
          for (int col=0; col<nr; col++)
              oper.set(lin+nr,col) 
              = -l_q*(l_q+1)*(lam*(mxd(lin,col) + ech*md(lin,col))
                      - (lam+2)*mid(lin,col)) ;
          for (int col=0; col<nr; col++)
              oper.set(lin+nr, col+nr) 
              = (lam+1)*(m2d2(lin,col) + 2*ech*mxd2(lin,col) 
                     + ech*ech*md2(lin,col) 
                     + 2*(mxd(lin,col) + ech*md(lin,col)))
              -(2*(lam+1)+l_q*(l_q+1))*mid(lin,col) ;         
          }
          for (int col=0; col<nr; col++) 
          oper.set(nr+nr_eq2, col) = 0 ;
          oper.set(nr+nr_eq2, nr) = 1 ;
          for (int col=nr+1; col<2*nr; col++) 
          oper.set(nr+nr_eq2, col) = 0 ;
          for (int col=0; col<nr+1; col++) 
          oper.set(nr+nr_eq2+1, col) = 0 ;
          oper.set(nr+nr_eq2+1, nr+1) = 1 ;
          for (int col=nr+2; col<2*nr; col++) 
          oper.set(nr+nr_eq2+1, col) = 0 ;

          oper.set_lu() ;
          
          // Filling the r.h.s
          //------------------
          Tbl sr(nr) ; sr.set_etat_qcq() ; 
          Tbl seta(nr) ; seta.set_etat_qcq() ;
          for (int i=0; i<nr; 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) ;
          }
          Tbl xsr= sr ;  Tbl x2sr= sr ;
          Tbl xseta= seta ; Tbl x2seta = seta ;
          multx2_1d(nr, &x2sr.t, base_r) ; multx_1d(nr, &xsr.t, base_r) ;
          multx2_1d(nr, &x2seta.t, base_r) ; multx_1d(nr, &xseta.t, base_r) ;
          
          for (int i=0; i<nr_eq1; i++) 
          sec_membre.set(i) = alpha*(alpha*x2seta(i) + 2*beta*xseta(i))
              + beta*beta*seta(i);
          sec_membre.set(nr_eq1) = 0 ;
          sec_membre.set(nr_eq1+1) = 0 ;
          for (int i=0; i<nr_eq2; i++)
          sec_membre.set(i+nr) = beta*beta*sr(i) 
              + alpha*(alpha*x2sr(i) + 2*beta*xsr(i)) ;
          sec_membre.set(nr+nr_eq2) = 0 ;
          sec_membre.set(nr+nr_eq2+1) = 0 ;

          // Inversion of the "big" operator
          //--------------------------------
          Tbl big_res = oper.inverse(sec_membre) ;
          for (int i=0; i<nr; i++) {
          sol_part_eta.set(zone, k, j, i) = big_res(i) ;
          sol_part_vr.set(zone, k, j, i) = big_res(nr+i) ;        
          }
          
          // Getting the homogeneous solutions
          //----------------------------------
          sec_membre.annule_hard() ;
          sec_membre.set(nr_eq1) = 1 ;
          big_res = oper.inverse(sec_membre) ;
          for (int i=0 ; i<nr ; i++) {
          sol_hom_un_eta.set(zone, k, j, i) = big_res(i) ;
          sol_hom_un_vr.set(zone, k, j, i) = big_res(nr+i) ;
          }
          sec_membre.set(nr_eq1) = 0 ;
          sec_membre.set(nr_eq1+1) = 1 ;
          big_res = oper.inverse(sec_membre) ;
          for (int i=0 ; i<nr ; i++) {
          sol_hom_deux_eta.set(zone, k, j, i) = big_res(i) ;
          sol_hom_deux_vr.set(zone, k, j, i) = big_res(nr+i) ;
          }
          sec_membre.set(nr_eq1+1) = 0 ;
          sec_membre.set(nr+nr_eq2) = 1 ;
          big_res = oper.inverse(sec_membre) ;
          for (int i=0 ; i<nr ; i++) {
          sol_hom_trois_eta.set(zone, k, j, i) = big_res(i) ;
          sol_hom_trois_vr.set(zone, k, j, i) = big_res(nr+i) ;
          }
          sec_membre.set(nr+nr_eq2) = 0 ;
          sec_membre.set(nr+nr_eq2+1) = 1 ;
          big_res = oper.inverse(sec_membre) ;
          for (int i=0 ; i<nr ; i++) {
          sol_hom_quatre_eta.set(zone, k, j, i) = big_res(i) ;
          sol_hom_quatre_vr.set(zone, k, j, i) = big_res(nr+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] ; alp2 = alpha*alpha ;
  assert (nr > 4) ;
  
  // 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 = 3; //degeneracy of eq.1
          int dege2 = (l_q == 1 ? 2 : 3); //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 = 2*nr ;
          Matrice oper(nrtot, nrtot) ; oper.set_etat_qcq() ;
          Tbl sec_membre(nrtot) ; sec_membre.set_etat_qcq() ;
          Diff_dsdx2 d2(base_r, nr) ; const Matrice& md2 = d2.get_matrice() ;
          Diff_sxdsdx sxd(base_r, nr) ; const Matrice& mxd = sxd.get_matrice() ;
          Diff_sx2 sx2(base_r, nr) ; const Matrice& ms2 = sx2.get_matrice() ;

          // Building the operator
          //----------------------
          for (int lin=0; lin<nr_eq1; lin++) { 
          for (int col=0; col<nr; col++) 
              oper.set(lin,col) 
              = (md2(lin,col) - (lam+1)*l_q*(l_q+1)*ms2(lin,col))/alp2 ;
              for (int col=0; col<nr; col++) 
                  oper.set(lin,col+nr) = 
                  (-lam*mxd(lin,col) + 2*(1+lam)*ms2(lin,col)) / alp2 ;
          }
          for (int col=0; col<nr; col++)
          oper.set(nr_eq1, col) = 1 ;
          for (int col=nr; col<nrtot; col++)
          oper.set(nr_eq1, col) = 0 ;
          int pari = -1 ;
          for (int col=0; col<nr; col++) {
          oper.set(nr_eq1+1, col) = pari*col*col ;
          pari = pari ;
          }
          for (int col=nr; col<nrtot; col++)
          oper.set(nr_eq1+1, col) = 0 ;
          oper.set(nr_eq1+2, 0) = 1 ;
          for (int col=1; col<nrtot; col++)
          oper.set(nr_eq1+2, col) = 0 ;
          for (int lin=0; lin<nr_eq2; lin++) {
          for (int col=0; col<nr; col++)
              oper.set(lin+nr,col) = (l_q*(l_q+1)*(lam*mxd(lin,col) 
                   + (lam+2)*ms2(lin,col))) / alp2 ;
          for (int col=0; col<nr; col++)
              oper.set(lin+nr, col+nr) = ((lam+1)*md2(lin,col) 
                 - (2*(lam+1) + l_q*(l_q+1))*ms2(lin,col)) / alp2 ;
          }
          for (int col=0; col<nr; col++)
          oper.set(nr+nr_eq2, col) = 0 ;
          for (int col=nr; col<nrtot; col++)
          oper.set(nr+nr_eq2, col) = 1 ;
          for (int col=0; col<nr; col++)
          oper.set(nr+nr_eq2+1, col) = 0 ;
          pari = -1 ;
          for (int col=0; col<nr; col++) {
          oper.set(nr+nr_eq2+1, nr+col) = pari*col*col ;
          pari = pari ;
          }
          if (l_q > 1) {
          for (int col=0; col<nr; col++)
              oper.set(nr+nr_eq2+2, col) = 0 ;
          oper.set(nr+nr_eq2+2, nr) = 1 ;
          for (int col=nr+1; col<nrtot; col++)
              oper.set(nr+nr_eq2+2, col) = 0 ;
          }
          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=nr_eq1; i<nr; i++)
          sec_membre.set(i) = 0 ;
          for (int i=0; i<nr_eq2; i++)
          sec_membre.set(i+nr) =(*S_r.get_spectral_va().c_cf)(nzm1, k, j, i);
          for (int i=nr_eq2; i<nr; i++)
          sec_membre.set(nr+i) = 0 ;
          Tbl big_res = oper.inverse(sec_membre) ;
          
          // Putting coefficients of h and v to individual arrays
          //-----------------------------------------------------
          for (int i=0; i<nr; i++) {
          sol_part_eta.set(nzm1, k, j, i) = big_res(i) ;
          sol_part_vr.set(nzm1, k, j, i) = big_res(i+nr) ;
          }

          // Homogeneous solutions (only 1/r^(l+2) and 1/r^l in the CED)
          //------------------------------------------------------------
          if (l_q == 1) {         
          Tbl sol_hom1 = solh(nr, 0, 0., base_r) ;
          Tbl sol_hom2 = solh(nr, 2, 0., base_r) ;
          double fac_eta = lam + 2. ;
          double fac_vr = 2*lam + 2. ;
          for (int i=0 ; i<nr ; i++) {
              sol_hom_deux_eta.set(nzm1, k, j, i) = sol_hom2(i) ;
              sol_hom_quatre_eta.set(nzm1, k, j, i) = fac_eta*sol_hom1(i) ;
              sol_hom_deux_vr.set(nzm1, k, j, i) = -2*sol_hom2(i) ;
              sol_hom_quatre_vr.set(nzm1, k, j, i) = fac_vr*sol_hom1(i) ;
          }
          }
          else {
          sec_membre.annule_hard() ;
          sec_membre.set(nr-1) = 1 ;
          big_res = oper.inverse(sec_membre) ;

          for (int i=0; i<nr; i++) {
              sol_hom_deux_eta.set(nzm1, k, j, i) = big_res(i) ;
              sol_hom_deux_vr.set(nzm1, k, j, i) = big_res(nr+i) ;
          }
          sec_membre.set(nr-1) = 0 ;
          sec_membre.set(2*nr-1) = 1 ;
          big_res = oper.inverse(sec_membre) ;
          for (int i=0; i<nr; i++) {
              sol_hom_quatre_eta.set(nzm1, k, j, i) = big_res(i) ;
              sol_hom_quatre_vr.set(nzm1, k, j, i) = big_res(nr+i) ;
          }
          }
      } 
      }
  }

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

  // Resulting V^r & eta
  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() ;
  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 = 4*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 ;
          systeme.annule_hard() ;
          sec_membre.annule_hard() ;

          //Nucleus 
          nr = mg.get_nr(0) ;
          alpha = mpaff->get_alpha()[0] ;
          // value of at x=1 of eta ...
          systeme.set(ligne, colonne) = sol_hom_un_eta.val_out_bound_jk(0, j, k) ;
          systeme.set(ligne, colonne+1) = sol_hom_trois_eta.val_out_bound_jk(0, j, k) ;
          sec_membre.set(ligne) = -sol_part_eta.val_out_bound_jk(0, j, k) ;
          ligne++ ;
          // ... and of its couterpart for V^r
          systeme.set(ligne, colonne) = sol_hom_un_vr.val_out_bound_jk(0, j, k) ;
          systeme.set(ligne, colonne+1) = sol_hom_trois_vr.val_out_bound_jk(0, j, k) ;
          sec_membre.set(ligne) = -sol_part_vr.val_out_bound_jk(0,j,k) ; 
          ligne++ ; 

              //derivatives
          int pari = (base_r == R_CHEBP ? 0 : 1) ;
          for (int i=0; i<nr; i++) {
          systeme.set(ligne, colonne) 
              += (2*i+pari)*(2*i+pari)*sol_hom_un_eta(0, k, j, i)/alpha ;
          systeme.set(ligne, colonne+1) 
              += (2*i+pari)*(2*i+pari)*sol_hom_trois_eta(0, k, j, i)/alpha ;
          sec_membre.set(ligne) 
              -= (2*i+pari)*(2*i+pari)* sol_part_eta(0, k, j, i)/alpha ;
          }
          ligne++ ;
          // ... and of its couterpart for V^r
          for (int i=0; i<nr; i++) {
          systeme.set(ligne, colonne) 
              += (2*i+pari)*(2*i+pari)*sol_hom_un_vr(0, k, j, i)/alpha ;
          systeme.set(ligne, colonne+1) 
              += (2*i+pari)*(2*i+pari)*sol_hom_trois_vr(0, k, j, i)/alpha ;
          sec_membre.set(ligne) 
              -= (2*i+pari)*(2*i+pari)* sol_part_vr(0, k, j, i)/alpha ;
          }
          colonne += 2 ; 

              //shells
          for (int zone=1 ; zone<nzm1 ; zone++) {
          nr = mg.get_nr(zone) ;
          alpha = mpaff->get_alpha()[zone] ;
          ligne -= 3 ;
          systeme.set(ligne, colonne) 
              = -sol_hom_un_eta.val_in_bound_jk(zone, j, k)  ;
          systeme.set(ligne, colonne+1) 
              = -sol_hom_deux_eta.val_in_bound_jk(zone, j, k)  ;
          systeme.set(ligne, colonne+2) 
              = -sol_hom_trois_eta.val_in_bound_jk(zone, j, k)  ;
          systeme.set(ligne, colonne+3) 
              = -sol_hom_quatre_eta.val_in_bound_jk(zone, j, k)  ;
          sec_membre.set(ligne) += sol_part_eta.val_in_bound_jk(zone, j, k) ;
          ligne++ ;
          // ... and their couterparts for V^r
          systeme.set(ligne, colonne) 
              = -sol_hom_un_vr.val_in_bound_jk(zone, j, k)  ;
          systeme.set(ligne, colonne+1) 
              = -sol_hom_deux_vr.val_in_bound_jk(zone, j, k)  ;
          systeme.set(ligne, colonne+2) 
              = -sol_hom_trois_vr.val_in_bound_jk(zone, j, k)  ;
          systeme.set(ligne, colonne+3) 
              = -sol_hom_quatre_vr.val_in_bound_jk(zone, j, k)  ;
          sec_membre.set(ligne) += sol_part_vr.val_in_bound_jk(zone, j, k) ;
          ligne++ ;

          //derivative of (x+echelle)^(l-1) at -1 
          pari = -1 ;
          for (int i=0; i<nr; i++) {
              systeme.set(ligne, colonne) 
              -= pari*i*i*sol_hom_un_eta(zone, k, j, i)/alpha ;
              systeme.set(ligne, colonne+1) 
              -= pari*i*i*sol_hom_deux_eta(zone, k, j, i)/alpha ;
              systeme.set(ligne, colonne+2) 
              -= pari*i*i*sol_hom_trois_eta(zone, k, j, i)/alpha ;
              systeme.set(ligne, colonne+3) 
              -= pari*i*i*sol_hom_quatre_eta(zone, k, j, i)/alpha ;
              sec_membre.set(ligne) 
              += pari*i*i* sol_part_eta(zone, k, j, i)/alpha ;
              pari = -pari ;
          }
          ligne++ ;
          // ... and their couterparts for V^r
          pari = -1 ;
          for (int i=0; i<nr; i++) {
              systeme.set(ligne, colonne) 
              -= pari*i*i*sol_hom_un_vr(zone, k, j, i)/alpha ;
              systeme.set(ligne, colonne+1) 
              -= pari*i*i*sol_hom_deux_vr(zone, k, j, i)/alpha ;
              systeme.set(ligne, colonne+2) 
              -= pari*i*i*sol_hom_trois_vr(zone, k, j, i)/alpha ;
              systeme.set(ligne, colonne+3) 
              -= pari*i*i*sol_hom_quatre_vr(zone, k, j, i)/alpha ;
              sec_membre.set(ligne) 
              += pari*i*i* sol_part_vr(zone, k, j, i)/alpha ;
              pari = -pari ;
          }
          ligne++ ;
            
          systeme.set(ligne, colonne) 
              += sol_hom_un_eta.val_out_bound_jk(zone, j, k)  ;
          systeme.set(ligne, colonne+1) 
              += sol_hom_deux_eta.val_out_bound_jk(zone, j, k)  ;
          systeme.set(ligne, colonne+2) 
              += sol_hom_trois_eta.val_out_bound_jk(zone, j, k)  ;
          systeme.set(ligne, colonne+3) 
              += sol_hom_quatre_eta.val_out_bound_jk(zone, j, k)  ;
          sec_membre.set(ligne) -= sol_part_eta.val_out_bound_jk(zone, j, k) ;
          ligne++ ;
          // ... and their couterparts for V^r
          systeme.set(ligne, colonne) 
              += sol_hom_un_vr.val_out_bound_jk(zone, j, k)  ;
          systeme.set(ligne, colonne+1) 
              += sol_hom_deux_vr.val_out_bound_jk(zone, j, k)  ;
          systeme.set(ligne, colonne+2) 
              += sol_hom_trois_vr.val_out_bound_jk(zone, j, k)  ;
          systeme.set(ligne, colonne+3) 
              += sol_hom_quatre_vr.val_out_bound_jk(zone, j, k)  ;
          sec_membre.set(ligne) -= sol_part_vr.val_out_bound_jk(zone, j, k) ;
          ligne++ ;

          //derivative at 1 
          pari = 1 ;
          for (int i=0; i<nr; i++) {
              systeme.set(ligne, colonne) 
              += pari*i*i*sol_hom_un_eta(zone, k, j, i)/alpha ;
              systeme.set(ligne, colonne+1) 
              += pari*i*i*sol_hom_deux_eta(zone, k, j, i)/alpha ;
              systeme.set(ligne, colonne+2) 
              += pari*i*i*sol_hom_trois_eta(zone, k, j, i)/alpha ;
              systeme.set(ligne, colonne+3) 
              += pari*i*i*sol_hom_quatre_eta(zone, k, j, i)/alpha ;
              sec_membre.set(ligne) 
              -= pari*i*i* sol_part_eta(zone, k, j, i)/alpha ;
              pari = pari ;
          }
          ligne++ ;
          // ... and their couterparts for V^r
          pari = 1 ;
          for (int i=0; i<nr; i++) {
              systeme.set(ligne, colonne) 
              += pari*i*i*sol_hom_un_vr(zone, k, j, i)/alpha ;
              systeme.set(ligne, colonne+1) 
              += pari*i*i*sol_hom_deux_vr(zone, k, j, i)/alpha ;
              systeme.set(ligne, colonne+2) 
              += pari*i*i*sol_hom_trois_vr(zone, k, j, i)/alpha ;
              systeme.set(ligne, colonne+3) 
              += pari*i*i*sol_hom_quatre_vr(zone, k, j, i)/alpha ;
              sec_membre.set(ligne) 
              -= pari*i*i* sol_part_vr(zone, k, j, i)/alpha ;
              pari = pari ;
          }
          colonne += 4 ;
          }         
          //Compactified external domain
          nr = mg.get_nr(nzm1) ;

          alpha = mpaff->get_alpha()[nzm1] ;
          ligne -= 3 ;
          //value of (x-1)^(l+2) at -1 :
          systeme.set(ligne, colonne) 
          -= sol_hom_deux_eta.val_in_bound_jk(nzm1, j, k) ;
          systeme.set(ligne, colonne+1) 
          -= sol_hom_quatre_eta.val_in_bound_jk(nzm1, j, k) ;
          sec_membre.set(ligne) += sol_part_eta.val_in_bound_jk(nzm1, j, k) ;
          //... and of its couterpart for V^r
          systeme.set(ligne+1, colonne) 
          -= sol_hom_deux_vr.val_in_bound_jk(nzm1, j, k) ;
          systeme.set(ligne+1, colonne+1) 
          -= sol_hom_quatre_vr.val_in_bound_jk(nzm1, j, k) ;
          sec_membre.set(ligne+1) += sol_part_vr.val_in_bound_jk(nzm1, j, k) ;
            
          ligne += 2 ;
          //derivative of (x-1)^(l+2) at -1 :
          pari = 1 ;
          for (int i=0; i<nr; i++) {
              systeme.set(ligne, colonne) 
              -= 4*alpha*pari*i*i*sol_hom_deux_eta(nzm1, k, j, i) ;
              systeme.set(ligne, colonne+1) 
              -= 4*alpha*pari*i*i*sol_hom_quatre_eta(nzm1, k, j, i) ;
              sec_membre.set(ligne) 
              += 4*alpha*pari*i*i* sol_part_eta(nzm1, k, j, i) ;
              pari = -pari ;
          }
          ligne++ ;
          // ... and their couterparts for V^r
          pari = 1 ;
          for (int i=0; i<nr; i++) {
              systeme.set(ligne, colonne) 
              -= 4*alpha*pari*i*i*sol_hom_deux_vr(nzm1, k, j, i) ;
              systeme.set(ligne, colonne+1) 
              -= 4*alpha*pari*i*i*sol_hom_quatre_vr(nzm1, k, j, i) ;
              sec_membre.set(ligne) 
              += 4*alpha*pari*i*i* sol_part_vr(nzm1, k, j, i) ;
              pari = -pari ;
          }
            
          // Solution of the system giving the coefficients for the homogeneous 
          // solutions
          //-------------------------------------------------------------------
          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)*sol_hom_un_eta(0, k, j, i) 
              +facteurs(conte+1)*sol_hom_trois_eta(0, k, j, i) ;
          cf_vr.set(0, k, j, i) = sol_part_vr(0, k, j, i)
              +facteurs(conte)*sol_hom_un_vr(0, k, j, i) 
              +facteurs(conte+1)*sol_hom_trois_vr(0, k, j, i) ;
          }
          conte += 2 ;
          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)*sol_hom_un_eta(zone, k, j, i) 
              +facteurs(conte+1)*sol_hom_deux_eta(zone, k, j, i) 
              +facteurs(conte+2)*sol_hom_trois_eta(zone, k, j, i) 
              +facteurs(conte+3)*sol_hom_quatre_eta(zone, k, j, i) ;
              cf_vr.set(zone, k, j, i) = sol_part_vr(zone, k, j, i)
              +facteurs(conte)*sol_hom_un_vr(zone, k, j, i) 
              +facteurs(conte+1)*sol_hom_deux_vr(zone, k, j, i) 
              +facteurs(conte+2)*sol_hom_trois_vr(zone, k, j, i) 
              +facteurs(conte+3)*sol_hom_quatre_vr(zone, k, j, i) ;
          }
          conte+=4 ;
          }
          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)*sol_hom_deux_eta(nzm1, k, j, i) 
              +facteurs(conte+1)*sol_hom_quatre_eta(nzm1, k, j, i) ;
          cf_vr.set(nzm1, k, j, i) = sol_part_vr(nzm1, k, j, i)
              +facteurs(conte)*sol_hom_deux_vr(nzm1, k, j, i) 
              +facteurs(conte+1)*sol_hom_quatre_vr(nzm1, k, j, i) ;

          }
      } // End of nullite_plm  
      } //End of loop on theta
  vr.set_spectral_va().ylm_i() ;
  vr += vrl0 ;
  het.set_spectral_va().ylm_i() ;
 }
 resu.set_vr_eta_mu(vr, het, mu().poisson()) ;
 if ((nt==1)&&(np==1)) {
     resu.set(2).set_etat_zero() ;
     resu.set(3).set_etat_zero() ;
 }
 
 return ;
  
}
}