int bicg(const LinearOperator &A, Vector &x, const Vector &b,
const Preconditioner &M, Iteration& iter)
{
using mtl::conj;
typedef typename mtl::Collection<Vector>::value_type Scalar;
Scalar rho_1(0), rho_2(0), alpha(0), beta(0);
Vector r(b - A * x), z(size(x)), p(size(x)), q(size(x)),
r_tilde(r), z_tilde(size(x)), p_tilde(size(x)), q_tilde(size(x));
while (! iter.finished(r)) {
z= solve(M, r);
z_tilde= adjoint_solve(M, r_tilde);
rho_1= dot(z_tilde, z);
if (rho_1 == 0.) {
iter.fail(2, "bicg breakdown");
break;
}
if (iter.first()) {
p= z;
p_tilde= z_tilde;
} else {
beta= rho_1 / rho_2;
p= z + beta * p;
p_tilde= z_tilde + conj(beta) * p_tilde;
}
q= A * p;
q_tilde= adjoint(A) * p_tilde;
alpha= rho_1 / dot(p_tilde, q);
x+= alpha * p;
r-= alpha * q;
r_tilde-= conj(alpha) * q_tilde;
rho_2= rho_1;
++iter;
}
return iter.error_code();
}
int qmr(const Matrix& A, Vector& x, const Vector& b, LeftPreconditioner& L,
const RightPreconditioner& R, Iteration& iter)
{
typedef typename mtl::Collection<Vector>::value_type Scalar;
typedef typename mtl::Collection<Vector>::size_type Size;
if (size(b) == 0) throw mtl::logic_error("empty rhs vector");
const Scalar zero= math::zero(b[0]), one= math::one(b[0]);
Scalar rho_1, gamma(one), gamma_1, theta(zero), theta_1,
eta(-one), delta, ep(one), beta;
Size n(size(x));
Vector r(b - A * x), v_tld(r), y(solve(L, v_tld)), w_tld(r), z(adjoint_solve(R,w_tld)), v(n),
w(n), y_tld(n), z_tld, p, q, p_tld, d, s;
if (iter.finished(r))
return iter;
Scalar rho = two_norm(y), xi = two_norm(z);
while(! iter.finished(rho)) {
if (rho == zero)
return iter.fail(1, "qmr breakdown, rho=0 #1");
if (xi == zero)
return iter.fail(2, "qmr breakdown, xi=0 #2");
v= v_tld / rho;
y/= rho;
w= w_tld / xi;
z/= xi;
delta = dot(z,y);
if (delta == zero)
return iter.fail(3, "qmr breakdown, delta=0 #3");
y_tld = solve(R,y);
z_tld = adjoint_solve(L,z);
if (iter.first()) {
p = y_tld;
q = z_tld;
} else {
p = y_tld - ((xi * delta) / ep) * p;
q = z_tld - ((rho* delta) / ep) * q;
}
p_tld = A * p;
ep = dot(q, p_tld);
if (ep == zero)
return iter.fail(4, "qmr breakdown ep=0 #4");
beta= ep / delta;
if (beta == zero)
return iter.fail(5, "qmr breakdown beta=0 #5");
v_tld = p_tld - beta * v;
y = solve(L,v_tld);
rho_1 = rho = two_norm(y);
w_tld= trans(A)*q - beta*w;
z = adjoint_solve(R, w_tld);
xi = two_norm(z);
gamma_1 = gamma;
theta_1 = theta;
theta = rho / (gamma_1 * beta);
gamma = one / (sqrt(one + theta * theta));
if (gamma == zero)
return iter.fail(6, "qmr breakdown gamma=0 #6");
eta= -eta * rho_1 * gamma * gamma / (beta * gamma_1 * gamma_1);
if (iter.first()) {
d= eta * p;
s= eta * p_tld;
} else {
d= eta * p + (theta_1 * theta_1 * gamma * gamma) * d;
s= eta * p_tld + (theta_1 * theta_1 * gamma * gamma) * s;
}
x += d;
r -= s;
++iter;
}
return iter;
}
int bicgstab_ell(const LinearOperator &A, Vector &x, const Vector &b,
const LeftPreconditioner &L, const RightPreconditioner &R,
Iteration& iter, size_t l)
{
mtl::vampir_trace<7006> tracer;
using mtl::size; using mtl::irange; using mtl::imax; using mtl::matrix::strict_upper;
typedef typename mtl::Collection<Vector>::value_type Scalar;
typedef typename mtl::Collection<Vector>::size_type Size;
if (size(b) == 0) throw mtl::logic_error("empty rhs vector");
const Scalar zero= math::zero(Scalar()), one= math::one(Scalar());
Vector x0(resource(x)), y(resource(x));
mtl::vector::dense_vector<Vector> r_hat(l+1,Vector(resource(x))), u_hat(l+1,Vector(resource(x)));
// shift problem
x0= zero;
r_hat[0]= b;
if (two_norm(x) != zero) {
r_hat[0]-= A * x;
x0= x;
x= zero;
}
Vector r0_tilde(r_hat[0]/two_norm(r_hat[0]));
y= solve(L, r_hat[0]);
r_hat[0]= y;
u_hat[0]= zero;
Scalar rho_0(one), rho_1(zero), alpha(zero), Gamma(zero), beta(zero), omega(one);
mtl::matrix::dense2D<Scalar> tau(l+1, l+1);
mtl::vector::dense_vector<Scalar> sigma(l+1), gamma(l+1), gamma_a(l+1), gamma_aa(l+1);
while (! iter.finished(r_hat[0])) {
++iter;
rho_0= -omega * rho_0;
for (Size j= 0; j < l; ++j) {
rho_1= dot(r0_tilde, r_hat[j]);
beta= alpha * rho_1/rho_0; rho_0= rho_1;
for (Size i= 0; i <= j; ++i)
u_hat[i]= r_hat[i] - beta * u_hat[i];
y= A * Vector(solve(R, u_hat[j]));
u_hat[j+1]= solve(L, y);
Gamma= dot(r0_tilde, u_hat[j+1]);
alpha= rho_0 / Gamma;
for (Size i= 0; i <= j; ++i)
r_hat[i]-= alpha * u_hat[i+1];
if (iter.finished(r_hat[j])) {
x= solve(R, x);
x+= x0;
return iter;
}
r_hat[j+1]= solve(R, r_hat[j]);
y= A * r_hat[j+1];
r_hat[j+1]= solve(L, y);
x+= alpha * u_hat[0];
}
// mod GS (MR part)
irange i1m(1, imax);
mtl::vector::dense_vector<Vector> r_hat_tail(r_hat[i1m]);
tau[i1m][i1m]= orthogonalize_factors(r_hat_tail);
for (Size j= 1; j <= l; ++j)
gamma_a[j]= dot(r_hat[j], r_hat[0]) / tau[j][j];
gamma[l]= gamma_a[l]; omega= gamma[l];
if (omega == zero) return iter.fail(3, "bicg breakdown #2");
// is this something like a tri-solve?
for (Size j= l-1; j > 0; --j) {
Scalar sum= zero;
for (Size i=j+1;i<=l;++i)
sum += tau[j][i] * gamma[i];
gamma[j] = gamma_a[j] - sum;
}
gamma_aa[irange(1, l)]= strict_upper(tau[irange(1, l)][irange(1, l)]) * gamma[irange(2, l+1)] + gamma[irange(2, l+1)];
x+= gamma[1] * r_hat[0];
r_hat[0]-= gamma_a[l] * r_hat[l];
u_hat[0]-= gamma[l] * u_hat[l];
for (Size j=1; j < l; ++j) {
u_hat[0] -= gamma[j] * u_hat[j];
x+= gamma_aa[j] * r_hat[j];
r_hat[0] -= gamma_a[j] * r_hat[j];
}
}
x= solve(R, x); x+= x0; // convert to real solution and undo shift
return iter;
}
