NM_Status
SubspaceIteration :: solve(SparseMtrx &a, SparseMtrx &b, FloatArray &_eigv, FloatMatrix &_r, double rtol, int nroot)
//
// this function solve the generalized eigenproblem using the Generalized
// jacobi iteration
//
{
if ( a.giveNumberOfColumns() != b.giveNumberOfColumns() ) {
OOFEM_ERROR("matrices size mismatch");
}
FloatArray temp, w, d, tt, f, rtolv, eigv;
FloatMatrix r;
int nn, nc1, ij = 0, is;
double rt, art, brt, eigvt;
FloatMatrix ar, br, vec;
std :: unique_ptr< SparseLinearSystemNM > solver( GiveClassFactory().createSparseLinSolver(ST_Direct, domain, engngModel) );
GJacobi mtd(domain, engngModel);
int nc = min(2 * nroot, nroot + 8);
nn = a.giveNumberOfColumns();
if ( nc > nn ) {
nc = nn;
}
ar.resize(nc, nc);
ar.zero();
br.resize(nc, nc);
br.zero();
//
// creation of initial iteration vectors
//
nc1 = nc - 1;
w.resize(nn);
w.zero();
d.resize(nc);
d.zero();
tt.resize(nn);
tt.zero();
rtolv.resize(nc);
rtolv.zero();
vec.resize(nc, nc);
vec.zero(); // eigen vectors of reduced problem
//
// create work arrays
//
r.resize(nn, nc);
r.zero();
eigv.resize(nc);
eigv.zero();
FloatArray h(nn);
for ( int i = 1; i <= nn; i++ ) {
h.at(i) = 1.0;
w.at(i) = b.at(i, i) / a.at(i, i);
}
b.times(h, tt);
r.setColumn(tt, 1);
for ( int j = 2; j <= nc; j++ ) {
rt = 0.0;
for ( int i = 1; i <= nn; i++ ) {
if ( fabs( w.at(i) ) >= rt ) {
rt = fabs( w.at(i) );
ij = i;
}
}
tt.at(j) = ij;
w.at(ij) = 0.;
for ( int i = 1; i <= nn; i++ ) {
if ( i == ij ) {
h.at(i) = 1.0;
} else {
h.at(i) = 0.0;
}
}
b.times(h, tt);
r.setColumn(tt, j);
} // (r = z)
# ifdef DETAILED_REPORT
OOFEM_LOG_INFO("SubspaceIteration :: solveYourselfAt: Degrees of freedom invoked by initial vectors :\n");
tt.printYourself();
OOFEM_LOG_INFO("SubspaceIteration :: solveYourselfAt: initial vectors for iteration:\n");
r.printYourself();
# endif
//ish = 0;
a.factorized();
//
// start of iteration loop
//
for ( int nite = 0; ; ++nite ) { // label 100
# ifdef DETAILED_REPORT
printf("SubspaceIteration :: solveYourselfAt: Iteration loop no. %d\n", nite);
# endif
//
// compute projection ar and br of matrices a , b
//
for ( int j = 1; j <= nc; j++ ) {
f.beColumnOf(r, j);
solver->solve(a, f, tt);
for ( int i = j; i <= nc; i++ ) {
art = 0.;
for ( int k = 1; k <= nn; k++ ) {
art += r.at(k, i) * tt.at(k);
}
ar.at(j, i) = art;
}
r.setColumn(tt, j); // (r = xbar)
}
ar.symmetrized(); // label 110
#ifdef DETAILED_REPORT
OOFEM_LOG_INFO("SubspaceIteration :: solveYourselfAt: Printing projection matrix ar\n");
ar.printYourself();
#endif
//
for ( int j = 1; j <= nc; j++ ) {
tt.beColumnOf(r, j);
b.times(tt, temp);
for ( int i = j; i <= nc; i++ ) {
brt = 0.;
for ( int k = 1; k <= nn; k++ ) {
brt += r.at(k, i) * temp.at(k);
}
br.at(j, i) = brt;
} // label 180
r.setColumn(temp, j); // (r=zbar)
} // label 160
br.symmetrized();
#ifdef DETAILED_REPORT
OOFEM_LOG_INFO("SubspaceIteration :: solveYourselfAt: Printing projection matrix br\n");
br.printYourself();
#endif
//
// solution of reduced eigenvalue problem
//
mtd.solve(ar, br, eigv, vec);
// START EXPERIMENTAL
#if 0
// solve the reduced problem by Inverse iteration
{
FloatMatrix x(nc,nc), z(nc,nc), zz(nc,nc), arinv;
FloatArray w(nc), ww(nc), tt(nc), t(nc);
double c;
// initial setting
for ( int i = 1;i <= nc; i++ ) {
ww.at(i)=1.0;
}
for ( int i = 1;i <= nc; i++ )
for ( int j = 1; j <= nc;j++ )
z.at(i,j)=1.0;
arinv.beInverseOf (ar);
for ( int i = 0;i < nitem; i++ ) {
// copy zz=z
zz = z;
// solve matrix equation K.X = M.X
x.beProductOf(arinv, z);
// evaluation of Rayleigh quotients
for ( int j = 1;j <= nc; j++ ) {
w.at(j) = 0.0;
for (k = 1; k<= nc; k++) w.at(j) += zz.at(k,j) * x.at(k,j);
}
z.beProductOf (br, x);
for ( int j = 1;j <= nc; j++ ) {
c = 0;
for ( int k = 1; k<= nc; k++ ) c += z.at(k,j) * x.at(k,j);
w.at(j) /= c;
}
// check convergence
int ac = 0;
for ( int j = 1;j <= nc; j++ ) {
if (fabs((ww.at(j)-w.at(j))/w.at(j))< rtol) ac++;
ww.at(j) = w.at(j);
}
//printf ("\n iterace cislo %d %d",i,ac);
//w.printYourself();
// Gramm-Schmidt ortogonalization
for ( int j = 1;j <= nc;j++ ) {
for ( int k = 1; k<= nc; k++ ) tt.at(k) = x.at(k,j);
t.beProductOf(br,tt) ;
for ( int ii = 1;ii < j; ii++ ) {
c = 0.0;
for ( int k = 1; k<= nc; k++ ) c += x.at(k,ii) * t.at(k);
for ( int k = 1; k<= nc; k++ ) x.at(k,j) -= x.at(k,ii) * c;
}
for ( int k = 1; k<= nc; k++) tt.at(k) = x.at(k,j);
t.beProductOf(br, tt);
c = 0.0;
for ( int k = 1; k<= nc; k++) c += x.at(k,j)*t.at(k);
for ( int k = 1; k<= nc; k++) x.at(k,j) /= sqrt(c);
}
if ( ac > nroot ) {
break;
}
// compute new approximation of Z
z.beProductOf(br,x);
}
eigv = w;
vec = x;
}
#endif
//
// sorting eigenvalues according to their values
//
do {
is = 0; // label 350
for ( int i = 1; i <= nc1; i++ ) {
if ( fabs( eigv.at(i + 1) ) < fabs( eigv.at(i) ) ) {
is++;
eigvt = eigv.at(i + 1);
eigv.at(i + 1) = eigv.at(i);
eigv.at(i) = eigvt;
for ( int k = 1; k <= nc; k++ ) {
rt = vec.at(k, i + 1);
vec.at(k, i + 1) = vec.at(k, i);
vec.at(k, i) = rt;
}
}
} // label 360
} while ( is != 0 );
# ifdef DETAILED_REPORT
OOFEM_LOG_INFO("SubspaceIteration :: solveYourselfAt: current eigen values of reduced problem \n");
eigv.printYourself();
OOFEM_LOG_INFO("SubspaceIteration :: solveYourselfAt: current eigen vectors of reduced problem \n");
vec.printYourself();
# endif
//
// compute eigenvectors
//
for ( int i = 1; i <= nn; i++ ) { // label 375
for ( int j = 1; j <= nc; j++ ) {
tt.at(j) = r.at(i, j);
}
for ( int k = 1; k <= nc; k++ ) {
rt = 0.;
for ( int j = 1; j <= nc; j++ ) {
rt += tt.at(j) * vec.at(j, k);
}
r.at(i, k) = rt;
}
} // label 420 (r = z)
//
// convergency check
//
for ( int i = 1; i <= nc; i++ ) {
double dif = ( eigv.at(i) - d.at(i) );
rtolv.at(i) = fabs( dif / eigv.at(i) );
}
# ifdef DETAILED_REPORT
OOFEM_LOG_INFO("SubspaceIteration :: solveYourselfAt: Reached precision of eigenvalues:\n");
rtolv.printYourself();
# endif
for ( int i = 1; i <= nroot; i++ ) {
if ( rtolv.at(i) > rtol ) {
goto label400;
}
}
OOFEM_LOG_INFO("SubspaceIteration :: solveYourselfAt: Convergence reached for RTOL=%20.15f\n", rtol);
break;
label400:
if ( nite >= nitem ) {
OOFEM_WARNING("SubspaceIteration :: solveYourselfAt: Convergence not reached in %d iteration - using current values", nitem);
break;
}
d = eigv; // label 410 and 440
continue;
}
// compute eigenvectors
for ( int j = 1; j <= nc; j++ ) {
tt.beColumnOf(r, j);
a.backSubstitutionWith(tt);
r.setColumn(tt, j); // r = xbar
}
// one cad add a normalization of eigen-vectors here
// initialize original index locations
_r.resize(nn, nroot);
_eigv.resize(nroot);
for ( int i = 1; i <= nroot; i++ ) {
_eigv.at(i) = eigv.at(i);
for ( int j = 1; j <= nn; j++ ) {
_r.at(j, i) = r.at(j, i);
}
}
return NM_Success;
}
NM_Status
SubspaceIteration :: solve(SparseMtrx &a, SparseMtrx &b, FloatArray &_eigv, FloatMatrix &_r, double rtol, int nroot)
//
// this function solve the generalized eigenproblem using the Generalized
// jacobi iteration
//
//
{
FILE *outStream;
FloatArray temp, w, d, tt, rtolv, eigv;
FloatMatrix r;
int nn, nc1, i, j, k, ij = 0, nite, is;
double rt, art, brt, eigvt, dif;
FloatMatrix ar, br, vec;
GJacobi mtd(domain, engngModel);
outStream = domain->giveEngngModel()->giveOutputStream();
nc = min(2 * nroot, nroot + 8);
//
// check matrix size
//
if ( a.giveNumberOfColumns() != b.giveNumberOfColumns() ) {
OOFEM_ERROR("matrices size mismatch");
}
// check matrix for factorization support
if ( !a.canBeFactorized() ) {
OOFEM_ERROR("a matrix not support factorization");
}
//
// check for wery small problem
//
nn = a.giveNumberOfColumns();
if ( nc > nn ) {
nc = nn;
}
ar.resize(nc, nc);
ar.zero();
br.resize(nc, nc);
br.zero();
//
// creation of initial iteration vectors
//
nc1 = nc - 1;
w.resize(nn);
w.zero();
d.resize(nc);
d.zero();
tt.resize(nn);
tt.zero();
rtolv.resize(nc);
rtolv.zero();
vec.resize(nc, nc);
vec.zero(); // eigen vectors of reduced problem
_r.resize(nn, nroot);
_eigv.resize(nroot);
//
// create work arrays
//
r.resize(nn, nc);
r.zero();
eigv.resize(nc);
eigv.zero();
FloatArray h(nn);
for ( i = 1; i <= nn; i++ ) {
h.at(i) = 1.0;
w.at(i) = b.at(i, i) / a.at(i, i);
}
b.times(h, tt);
for ( i = 1; i <= nn; i++ ) {
r.at(i, 1) = tt.at(i);
}
for ( j = 2; j <= nc; j++ ) {
rt = 0.0;
for ( i = 1; i <= nn; i++ ) {
if ( fabs( w.at(i) ) >= rt ) {
rt = fabs( w.at(i) );
ij = i;
}
}
tt.at(j) = ij;
w.at(ij) = 0.;
for ( i = 1; i <= nn; i++ ) {
if ( i == ij ) {
h.at(i) = 1.0;
} else {
h.at(i) = 0.0;
}
}
b.times(h, tt);
for ( i = 1; i <= nn; i++ ) {
r.at(i, j) = tt.at(i);
}
} // (r = z)
# ifdef DETAILED_REPORT
printf("SubspaceIteration :: solveYourselfAt: Degrees of freedom invoked by initial vectors :\n");
tt.printYourself();
printf("SubspaceIteration :: solveYourselfAt: initial vectors for iteration:\n");
r.printYourself();
# endif
//ish = 0;
a.factorized();
//
// start of iteration loop
//
nite = 0;
do { // label 100
nite++;
# ifdef DETAILED_REPORT
printf("SubspaceIteration :: solveYourselfAt: Iteration loop no. %d\n", nite);
# endif
//
// compute projection ar and br of matrices a , b
//
for ( j = 1; j <= nc; j++ ) {
for ( k = 1; k <= nn; k++ ) {
tt.at(k) = r.at(k, j);
}
//a. forwardReductionWith(&tt) -> diagonalScalingWith (&tt)
// -> backSubstitutionWithtt) ;
a.backSubstitutionWith(tt);
for ( i = j; i <= nc; i++ ) {
art = 0.;
for ( k = 1; k <= nn; k++ ) {
art += r.at(k, i) * tt.at(k);
}
ar.at(j, i) = art;
}
for ( k = 1; k <= nn; k++ ) {
r.at(k, j) = tt.at(k); // (r = xbar)
}
}
ar.symmetrized(); // label 110
#ifdef DETAILED_REPORT
printf("SubspaceIteration :: solveYourselfAt: Printing projection matrix ar\n");
ar.printYourself();
#endif
//
for ( j = 1; j <= nc; j++ ) {
for ( k = 1; k <= nn; k++ ) {
tt.at(k) = r.at(k, j);
}
b.times(tt, temp);
for ( i = j; i <= nc; i++ ) {
brt = 0.;
for ( k = 1; k <= nn; k++ ) {
brt += r.at(k, i) * temp.at(k);
}
br.at(j, i) = brt;
} // label 180
for ( k = 1; k <= nn; k++ ) {
r.at(k, j) = temp.at(k); // (r=zbar)
}
} // label 160
br.symmetrized();
#ifdef DETAILED_REPORT
printf("SubspaceIteration :: solveYourselfAt: Printing projection matrix br\n");
br.printYourself();
#endif
//
// solution of reduced eigenvalue problem
//
mtd.solve(ar, br, eigv, vec);
/// START EXPERIMENTAL
// solve the reduced problem by Inverse iteration
/*
* {
* FloatMatrix x(nc,nc), z(nc,nc), zz(nc,nc), arinv;
* FloatArray w(nc), ww(nc), tt(nc), t(nc);
* double c;
* int ii, i,j,k,ac;
*
*
* // initial setting
* for (i=1;i<=nc;i++){
* ww.at(i)=1.0;
* }
*
* for (i=1;i<=nc;i++)
* for (j=1; j<=nc;j++)
* z.at(i,j)=1.0;
*
* arinv.beInverseOf (ar);
*
* for (i=0;i<nitem;i++) {
*
* // copy zz=z
* zz = z;
*
* // solve matrix equation K.X = M.X
* x.beProductOf (arinv, z);
* // evaluation of Rayleigh quotients
* for (j=1;j<=nc;j++){
* w.at(j) = 0.0;
* for (k = 1; k<= nc; k++) w.at(j) += zz.at(k,j)*x.at(k,j);
* }
*
* z.beProductOf (br, x);
*
* for (j=1;j<=nc;j++){
* c = 0;
* for (k = 1; k<= nc; k++) c+= z.at(k,j)*x.at(k,j);
* w.at(j) /= c;
* }
*
* // check convergence
* ac=0;
* for (j=1;j<=nc;j++){
* if (fabs((ww.at(j)-w.at(j))/w.at(j))< rtol) ac++;
* ww.at(j)=w.at(j);
* }
*
* printf ("\n iterace cislo %d %d",i,ac);
* w.printYourself();
*
* // Gramm-Schmidt ortogonalization
* for (j=1;j<=nc;j++) {
* for (k = 1; k<= nc; k++) tt.at(k) = x.at(k,j) ;
* t.beProductOf(br,tt) ;
* for (ii=1;ii<j;ii++) {
* c = 0.0;
* for (k = 1; k<= nc; k++) c += x.at(k,ii)*t.at(k);
* for (k = 1; k<= nc; k++) x.at(k,j) -= x.at(k,ii)*c;
* }
* for (k = 1; k<= nc; k++) tt.at(k) = x.at(k,j) ;
* t.beProductOf (br,tt) ;
* c = 0.0;
* for (k = 1; k<= nc; k++) c += x.at(k,j)*t.at(k);
* for (k = 1; k<= nc; k++) x.at(k,j) /= sqrt(c);
* }
*
* if (ac>nroot){
* break;
* }
*
* // compute new approximation of Z
* z.beProductOf (br,x);
* }
*
* eigv = w;
* vec = x;
* }
* /// END EXPERIMANTAL
*/
//
// sorting eigenvalues according to their values
//
do {
is = 0; // label 350
for ( i = 1; i <= nc1; i++ ) {
if ( fabs( eigv.at(i + 1) ) < fabs( eigv.at(i) ) ) {
is++;
eigvt = eigv.at(i + 1);
eigv.at(i + 1) = eigv.at(i);
eigv.at(i) = eigvt;
for ( k = 1; k <= nc; k++ ) {
rt = vec.at(k, i + 1);
vec.at(k, i + 1) = vec.at(k, i);
vec.at(k, i) = rt;
}
}
} // label 360
} while ( is != 0 );
# ifdef DETAILED_REPORT
printf("SubspaceIteration :: solveYourselfAt: current eigen values of reduced problem \n");
eigv.printYourself();
printf("SubspaceIteration :: solveYourselfAt: current eigen vectors of reduced problem \n");
vec.printYourself();
# endif
//
// compute eigenvectors
//
for ( i = 1; i <= nn; i++ ) { // label 375
for ( j = 1; j <= nc; j++ ) {
tt.at(j) = r.at(i, j);
}
for ( k = 1; k <= nc; k++ ) {
rt = 0.;
for ( j = 1; j <= nc; j++ ) {
rt += tt.at(j) * vec.at(j, k);
}
r.at(i, k) = rt;
}
} // label 420 (r = z)
//
// convergency check
//
for ( i = 1; i <= nc; i++ ) {
dif = ( eigv.at(i) - d.at(i) );
rtolv.at(i) = fabs( dif / eigv.at(i) );
}
# ifdef DETAILED_REPORT
printf("SubspaceIteration :: solveYourselfAt: Reached precision of eigenvalues:\n");
rtolv.printYourself();
# endif
for ( i = 1; i <= nroot; i++ ) {
if ( rtolv.at(i) > rtol ) {
goto label400;
}
}
fprintf(outStream,
"SubspaceIteration :: solveYourselfAt: Convergence reached for RTOL=%20.15f",
rtol);
break;
label400:
if ( nite >= nitem ) {
fprintf(outStream, " SubspaceIteration :: solveYourselfAt: Convergence not reached in %d iteration - using current values", nitem);
break;
}
for ( i = 1; i <= nc; i++ ) {
d.at(i) = eigv.at(i); // label 410 and 440
}
continue;
} while ( 1 );
// compute eigenvectors
for ( j = 1; j <= nc; j++ ) {
for ( k = 1; k <= nn; k++ ) {
tt.at(k) = r.at(k, j);
}
a.backSubstitutionWith(tt);
for ( k = 1; k <= nn; k++ ) {
r.at(k, j) = tt.at(k); // (r = xbar)
}
}
// one cad add a normalization of eigen-vectors here
for ( i = 1; i <= nroot; i++ ) {
_eigv.at(i) = eigv.at(i);
for ( j = 1; j <= nn; j++ ) {
_r.at(j, i) = r.at(j, i);
}
}
solved = 1;
return NM_Success;
}