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|
/*
* 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
*
*/
char vector_poisson_block_C[] = "$Header: /cvsroot/Lorene/C++/Source/Tensor/vector_poisson_block.C,v 1.9 2014/10/13 08:53:45 j_novak Exp $" ;
/*
* $Id: vector_poisson_block.C,v 1.9 2014/10/13 08:53:45 j_novak Exp $
* $Log: vector_poisson_block.C,v $
* 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 1.9 2014/10/13 08:53:45 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 ;
}
}
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