int bicgstab(const LinearOperator& A, HilbertSpaceX& x, const HilbertSpaceB& b,
const Preconditioner& M, Iteration& iter)
{
typedef typename mtl::Collection<HilbertSpaceX>::value_type Scalar;
typedef HilbertSpaceX Vector;
Scalar rho_1(0), rho_2(0), alpha(0), beta(0), gamma, omega(0);
Vector p(size(x)), phat(size(x)), s(size(x)), shat(size(x)),
t(size(x)), v(size(x)), r(size(x)), rtilde(size(x));
r = b - A * x;
rtilde = r;
while (! iter.finished(r)) {
rho_1 = dot(rtilde, r);
MTL_THROW_IF(rho_1 == 0.0, unexpected_orthogonality());
if (iter.first())
p = r;
else {
MTL_THROW_IF(omega == 0.0, unexpected_orthogonality());
beta = (rho_1 / rho_2) * (alpha / omega);
p = r + beta * (p - omega * v);
}
phat = solve(M, p);
v = A * phat;
gamma = dot(rtilde, v);
MTL_THROW_IF(gamma == 0.0, unexpected_orthogonality());
alpha = rho_1 / gamma;
s = r - alpha * v;
if (iter.finished(s)) {
x += alpha * phat;
break;
}
shat = solve(M, s);
t = A * shat;
omega = dot(t, s) / dot(t, t);
x += omega * shat + alpha * phat;
r = s - omega * t;
rho_2 = rho_1;
++iter;
}
return iter;
}
unsigned long CSysSolve::BCGSTAB_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::BCGSTAB: illegal value for subspace size, m = " << m << endl;
#ifndef HAVE_MPI
exit(EXIT_FAILURE);
#else
MPI_Abort(MPI_COMM_WORLD,1);
MPI_Finalize();
#endif
}
CSysVector r(b);
CSysVector r_0(b);
CSysVector p(b);
CSysVector v(b);
CSysVector s(b);
CSysVector t(b);
CSysVector phat(b);
CSysVector shat(b);
CSysVector A_x(b);
/*--- Calculate the initial residual, compute norm, and check if system is already solved ---*/
mat_vec(x, A_x);
r -= A_x; r_0 = r; // recall, r holds b initially
su2double norm_r = r.norm();
su2double norm0 = b.norm();
if ( (norm_r < tol*norm0) || (norm_r < eps) ) {
if (rank == MASTER_NODE) cout << "CSysSolve::BCGSTAB(): system solved by initial guess." << endl;
return 0;
}
/*--- Initialization ---*/
su2double alpha = 1.0, beta = 1.0, omega = 1.0, rho = 1.0, rho_prime = 1.0;
/*--- Set the norm to the initial initial residual value ---*/
norm0 = norm_r;
/*--- Output header information including initial residual ---*/
int i = 0;
if ((monitoring) && (rank == MASTER_NODE)) {
WriteHeader("BCGSTAB", tol, norm_r);
WriteHistory(i, norm_r, norm0);
}
/*--- Loop over all search directions ---*/
for (i = 0; i < (int)m; i++) {
/*--- Compute rho_prime ---*/
rho_prime = rho;
/*--- Compute rho_i ---*/
rho = dotProd(r, r_0);
/*--- Compute beta ---*/
beta = (rho / rho_prime) * (alpha /omega);
/*--- p_{i} = r_{i-1} + beta * p_{i-1} - beta * omega * v_{i-1} ---*/
su2double beta_omega = -beta*omega;
p.Equals_AX_Plus_BY(beta, p, beta_omega, v);
p.Plus_AX(1.0, r);
/*--- Preconditioning step ---*/
precond(p, phat);
mat_vec(phat, v);
/*--- Calculate step-length alpha ---*/
su2double r_0_v = dotProd(r_0, v);
alpha = rho / r_0_v;
/*--- s_{i} = r_{i-1} - alpha * v_{i} ---*/
s.Equals_AX_Plus_BY(1.0, r, -alpha, v);
/*--- Preconditioning step ---*/
precond(s, shat);
mat_vec(shat, t);
/*--- Calculate step-length omega ---*/
omega = dotProd(t, s) / dotProd(t, t);
/*--- Update solution and residual: ---*/
x.Plus_AX(alpha, phat); x.Plus_AX(omega, shat);
r.Equals_AX_Plus_BY(1.0, s, -omega, t);
/*--- Check if solution has converged, else output the relative residual if necessary ---*/
norm_r = r.norm();
if (norm_r < tol*norm0) break;
if (((monitoring) && (rank == MASTER_NODE)) && ((i+1) % 50 == 0) && (rank == MASTER_NODE)) WriteHistory(i+1, norm_r, norm0);
}
if ((monitoring) && (rank == MASTER_NODE)) {
cout << "# BCGSTAB final (true) residual:" << endl;
cout << "# Iteration = " << i << ": |res|/|res0| = " << norm_r/norm0 << ".\n" << endl;
}
// /*--- Recalculate final residual (this should be optional) ---*/
// mat_vec(x, A_x);
// r = b; r -= A_x;
// su2double true_res = r.norm();
//
// if ((fabs(true_res - norm_r) > tol*10.0) && (rank == MASTER_NODE)) {
// cout << "# WARNING in CSysSolve::BCGSTAB(): " << endl;
// cout << "# true residual norm and calculated residual norm do not agree." << endl;
// cout << "# true_res - calc_res = " << true_res <<" "<< norm_r << endl;
// }
(*residual) = norm_r;
return i;
}
void BiCGSTAB(Epetra_CrsMatrix &A,
Epetra_Vector &x,
Epetra_Vector &b,
Ifpack_CrsRiluk *M,
int Maxiter,
double Tolerance,
double *residual, bool verbose) {
// Allocate vectors needed for iterations
Epetra_Vector phat(x.Map()); phat.SetFlopCounter(x);
Epetra_Vector p(x.Map()); p.SetFlopCounter(x);
Epetra_Vector shat(x.Map()); shat.SetFlopCounter(x);
Epetra_Vector s(x.Map()); s.SetFlopCounter(x);
Epetra_Vector r(x.Map()); r.SetFlopCounter(x);
Epetra_Vector rtilde(x.Map()); rtilde.Random(); rtilde.SetFlopCounter(x);
Epetra_Vector v(x.Map()); v.SetFlopCounter(x);
A.Multiply(false, x, r); // r = A*x
r.Update(1.0, b, -1.0); // r = b - A*x
double r_norm, b_norm, scaled_r_norm, rhon, rhonm1 = 1.0;
double alpha = 1.0, omega = 1.0, sigma;
double omega_num, omega_den;
r.Norm2(&r_norm);
b.Norm2(&b_norm);
scaled_r_norm = r_norm/b_norm;
r.Dot(rtilde,&rhon);
if (verbose) cout << "Initial residual = " << r_norm
<< " Scaled residual = " << scaled_r_norm << endl;
for (int i=0; i<Maxiter; i++) { // Main iteration loop
double beta = (rhon/rhonm1) * (alpha/omega);
rhonm1 = rhon;
/* p = r + beta*(p - omega*v) */
/* phat = M^-1 p */
/* v = A phat */
double dtemp = - beta*omega;
p.Update(1.0, r, dtemp, v, beta);
if (M==0)
phat.Scale(1.0, p);
else
M->Solve(false, p, phat);
A.Multiply(false, phat, v);
rtilde.Dot(v,&sigma);
alpha = rhon/sigma;
/* s = r - alpha*v */
/* shat = M^-1 s */
/* r = A shat (r is a tmp here for t ) */
s.Update(-alpha, v, 1.0, r, 0.0);
if (M==0)
shat.Scale(1.0, s);
else
M->Solve(false, s, shat);
A.Multiply(false, shat, r);
r.Dot(s, &omega_num);
r.Dot(r, &omega_den);
omega = omega_num / omega_den;
/* x = x + alpha*phat + omega*shat */
/* r = s - omega*r */
x.Update(alpha, phat, omega, shat, 1.0);
r.Update(1.0, s, -omega);
r.Norm2(&r_norm);
scaled_r_norm = r_norm/b_norm;
r.Dot(rtilde,&rhon);
if (verbose) cout << "Iter "<< i << " residual = " << r_norm
<< " Scaled residual = " << scaled_r_norm << endl;
if (scaled_r_norm < Tolerance) break;
}
return;
}
