matrix symmetricQRStep(matrix A, int startIndex, int stopIndex){
/*
Single step of a QR factorization. Uses givens rotations.
Does not explicitly store information on Q.
Performed on the square submatrix from i,j = startIndex ... stopIndex - 1
Input:
matrix A Working matrix.
int startIndex First index to modify.
int stopIndex Upper boundary
Output:
Matrix A Modified matrix.
*/
int k;
double x, z;
double d = 0.5 * (A.values[stopIndex-2][stopIndex-2] - A.values[stopIndex-1][stopIndex-1] ) ;
double mu = A.values[stopIndex-1][stopIndex-1] - pow(A.values[stopIndex-1][stopIndex-2], 2) /
(d + sign(d) * sqrt(d*d + pow(A.values[stopIndex-1][stopIndex-2], 2))) ;
x = A.values[startIndex][startIndex] - mu;
z = A.values[startIndex + 1][startIndex] ;
for(k = startIndex; k<stopIndex-1; k++){
A = applyGivens(A, x, z, k, k+1, startIndex, stopIndex);
if (k < stopIndex-2){
x = A.values[k+1][k];
z = A.values[k+2][k];
}
}
return A;
}
unsigned long CSysSolve::FGMRES(const CSysVector & b, CSysVector & x, CMatrixVectorProduct & mat_vec,
CPreconditioner & precond, double tol, unsigned long m, bool monitoring) {
int rank = 0;
#ifndef NO_MPI
#ifdef WINDOWS
MPI_Comm_rank(MPI_COMM_WORLD, &rank);
#else
rank = MPI::COMM_WORLD.Get_rank();
#endif
#endif
/*--- Check the subspace size ---*/
if (m < 1) {
if (rank == 0) cerr << "CSysSolve::FGMRES: illegal value for subspace size, m = " << m << endl;
#ifdef NO_MPI
exit(1);
#else
#ifdef WINDOWS
MPI_Abort(MPI_COMM_WORLD,1);
MPI_Finalize();
#else
MPI::COMM_WORLD.Abort(1);
MPI::Finalize();
#endif
#endif
}
/*--- Check the subspace size ---*/
if (m > 1000) {
if (rank == 0) cerr << "CSysSolve::FGMRES: illegal value for subspace size (too high), m = " << m << endl;
#ifdef NO_MPI
exit(1);
#else
#ifdef WINDOWS
MPI_Abort(MPI_COMM_WORLD,1);
MPI_Finalize();
#else
MPI::COMM_WORLD.Abort(1);
MPI::Finalize();
#endif
#endif
}
/*--- Define various arrays
Note: elements in w and z are initialized to x to avoid creating
a temporary CSysVector object for the copy constructor ---*/
vector<CSysVector> w(m+1, x);
vector<CSysVector> z(m+1, x);
vector<double> g(m+1, 0.0);
vector<double> sn(m+1, 0.0);
vector<double> cs(m+1, 0.0);
vector<double> y(m, 0.0);
vector<vector<double> > H(m+1, vector<double>(m, 0.0));
/*--- Calculate the norm of the rhs vector ---*/
double norm0 = b.norm();
/*--- Calculate the initial residual (actually the negative residual)
and compute its norm ---*/
mat_vec(x,w[0]);
w[0] -= b;
double beta = w[0].norm();
if ( (beta < tol*norm0) || (beta < eps) ) {
/*--- System is already solved ---*/
if (rank == 0) cout << "CSysSolve::FGMRES(): system solved by initial guess." << endl;
return 0;
}
/*--- Normalize residual to get w_{0} (the negative sign is because w[0]
holds the negative residual, as mentioned above) ---*/
w[0] /= -beta;
/*--- Initialize the RHS of the reduced system ---*/
g[0] = beta;
/*--- Set the norm to the initial residual value ---*/
norm0 = beta;
/*--- Output header information including initial residual ---*/
int i = 0;
if ((monitoring) && (rank == 0)) {
writeHeader("FGMRES", tol, beta);
writeHistory(i, beta, norm0);
}
/*--- Loop over all search directions ---*/
for (i = 0; i < m; i++) {
/*--- Check if solution has converged ---*/
if (beta < tol*norm0) break;
/*--- Precondition the CSysVector w[i] and store result in z[i] ---*/
precond(w[i], z[i]);
/*--- Add to Krylov subspace ---*/
mat_vec(z[i], w[i+1]);
/*--- Modified Gram-Schmidt orthogonalization ---*/
modGramSchmidt(i, H, w);
/*--- Apply old Givens rotations to new column of the Hessenberg matrix
then generate the new Givens rotation matrix and apply it to
the last two elements of H[:][i] and g ---*/
for (int k = 0; k < i; k++)
applyGivens(sn[k], cs[k], H[k][i], H[k+1][i]);
generateGivens(H[i][i], H[i+1][i], sn[i], cs[i]);
applyGivens(sn[i], cs[i], g[i], g[i+1]);
/*--- Set L2 norm of residual and check if solution has converged ---*/
beta = fabs(g[i+1]);
/*--- Output the relative residual if necessary ---*/
if ((((monitoring) && (rank == 0)) && ((i+1) % 100 == 0)) && (rank == 0)) writeHistory(i+1, beta, norm0);
}
/*--- Solve the least-squares system and update solution ---*/
solveReduced(i, H, g, y);
for (int k = 0; k < i; k++) {
x.Plus_AX(y[k], z[k]);
}
if ((monitoring) && (rank == 0)) {
cout << "# FGMRES final (true) residual:" << endl;
cout << "# Iteration = " << i << ": |res|/|res0| = " << beta/norm0 << endl;
}
// /*--- Recalculate final (neg.) residual (this should be optional) ---*/
// mat_vec(x, w[0]);
// w[0] -= b;
// double res = w[0].norm();
//
// if (fabs(res - beta) > tol*10) {
// if (rank == 0) {
// cout << "# WARNING in CSysSolve::FGMRES(): " << endl;
// cout << "# true residual norm and calculated residual norm do not agree." << endl;
// cout << "# res - beta = " << res - beta << endl;
// }
// }
return i;
}