unsigned long CSysSolve::FGMRES_LinSolver(const CSysVector & b, CSysVector & x, CMatrixVectorProduct & mat_vec,
CPreconditioner & precond, su2double tol, unsigned long m, su2double *residual, bool monitoring) {
int rank = 0;
#ifdef HAVE_MPI
MPI_Comm_rank(MPI_COMM_WORLD, &rank);
#endif
/*--- Check the subspace size ---*/
if (m < 1) {
if (rank == MASTER_NODE) cerr << "CSysSolve::FGMRES: illegal value for subspace size, m = " << m << endl;
#ifndef HAVE_MPI
exit(EXIT_FAILURE);
#else
MPI_Abort(MPI_COMM_WORLD,1);
MPI_Finalize();
#endif
}
/*--- Check the subspace size ---*/
if (m > 1000) {
if (rank == MASTER_NODE) cerr << "CSysSolve::FGMRES: illegal value for subspace size (too high), m = " << m << endl;
#ifndef HAVE_MPI
exit(EXIT_FAILURE);
#else
MPI_Abort(MPI_COMM_WORLD,1);
MPI_Finalize();
#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<su2double> g(m+1, 0.0);
vector<su2double> sn(m+1, 0.0);
vector<su2double> cs(m+1, 0.0);
vector<su2double> y(m, 0.0);
vector<vector<su2double> > H(m+1, vector<su2double>(m, 0.0));
/*--- Calculate the norm of the rhs vector ---*/
su2double norm0 = b.norm();
/*--- Calculate the initial residual (actually the negative residual)
and compute its norm ---*/
mat_vec(x, w[0]);
w[0] -= b;
su2double beta = w[0].norm();
if ( (beta < tol*norm0) || (beta < eps) ) {
/*--- System is already solved ---*/
if (rank == MASTER_NODE) 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 == MASTER_NODE)) {
WriteHeader("FGMRES", tol, beta);
WriteHistory(i, beta, norm0);
}
/*--- Loop over all search directions ---*/
for (i = 0; i < (int)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 == MASTER_NODE)) && ((i+1) % 50 == 0)) && (rank == MASTER_NODE)) 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 == MASTER_NODE)) {
cout << "# FGMRES final (true) residual:" << endl;
cout << "# Iteration = " << i << ": |res|/|res0| = " << beta/norm0 << ".\n" << endl;
}
// /*--- Recalculate final (neg.) residual (this should be optional) ---*/
// mat_vec(x, w[0]);
// w[0] -= b;
// su2double res = w[0].norm();
//
// if (fabs(res - beta) > tol*10) {
// if (rank == MASTER_NODE) {
// cout << "# WARNING in CSysSolve::FGMRES(): " << endl;
// cout << "# true residual norm and calculated residual norm do not agree." << endl;
// cout << "# res - beta = " << res - beta << endl;
// }
// }
(*residual) = beta;
return i;
}
Matrix DivideAndSolve(Matrix A,int p)
{
double a,b,d,e,s,c,mu,i;
int h,j,k,l,m,o,rowstartt,colstartt,rowendt,colendt;
Vector v,u,x,y,z;
Matrix B,C,D,E,Q,U,H;
U = newIdMatrix();
H = newMatrix();
v = newVector();
i = p+1;
rowstartt = i;
colstartt = 0;
while (rowstartt<n)
{
rowendt = (rowstartt+i-1<n-1)?rowstartt+i-1:n-1;
colstartt = rowstartt-i;
colendt = colstartt+i-1;
/* Now T_i has dimension (p-h)x(p-h) */
/* First zero all but row 1 in T_i */
for (m=colstartt;m<=colendt;m++)
{
if (norm(A,m,rowstartt,rowendt)!=0.0)
{
/* Find Householder(A(h:p,m)) */
House(A,v,m,rowstartt,rowendt);
for (o=0;o<rowstartt;o++)
v[o]=0.0;
for (o=rowendt+1;o<n;o++)
v[o]=0.0;
ApplyHouse(A,v,rowstartt,rowendt);
}
printf("m=%i, rowstart=%i, rowend=%i\n",m,rowstartt,rowendt);
printVector(v);
printMatrix(A);
}
/* Now zero all but the last entry in row 1 */
WeirdHouse(A,v,rowstartt,colstartt,colendt);
/* Apply the HouseHolder */
ApplyHouse(A,v,colstartt,colendt);
/* Now T_i has one single non-zero entry */
/* Iterate with explicit shift to zero the last
non-zero entry */
while (A[rowstartt][colendt]>
(A[rowstartt-1][colendt]-A[rowstartt][colendt+1])*epsilon)
{
printMatrix(A);
/* Wilkonson Shift?? */
d =(A[rowstartt-1][colendt]-A[rowstartt][colendt+1])/2.0;
b = A[rowstartt][colendt];
mu = A[rowstartt][colendt+1]+d-sign(d)*sqrt(d*d+b*b);
/* Determine the givens */
Givens(A[rowstartt-1][colendt]-mu,
A[rowstartt][colendt],&s,&c);
/* Apply the givens */
ApplyGivens(A,s,c,rowstartt-1,rowstartt,0,n-1);
printf("%e\n",A[colstartt][colendt]);
}
rowstartt+=i;
colstartt+=i;
}
}