VectorType solve(const MatrixType & matrix, VectorType const & rhs, cg_tag const & tag, PreconditionerType const & precond)
{
typedef typename viennacl::result_of::value_type<VectorType>::type ScalarType;
typedef typename viennacl::result_of::cpu_value_type<ScalarType>::type CPU_ScalarType;
unsigned int problem_size = viennacl::traits::size(rhs);
VectorType result(problem_size);
viennacl::traits::clear(result);
VectorType residual = rhs;
VectorType tmp(problem_size);
VectorType z = rhs;
precond.apply(z);
VectorType p = z;
CPU_ScalarType ip_rr = viennacl::linalg::inner_prod(residual, z);
CPU_ScalarType alpha;
CPU_ScalarType new_ip_rr = 0;
CPU_ScalarType beta;
CPU_ScalarType norm_rhs_squared = ip_rr;
CPU_ScalarType new_ipp_rr_over_norm_rhs;
if (norm_rhs_squared == 0) //solution is zero if RHS norm is zero
return result;
for (unsigned int i = 0; i < tag.max_iterations(); ++i)
{
tag.iters(i+1);
tmp = viennacl::linalg::prod(matrix, p);
alpha = ip_rr / viennacl::linalg::inner_prod(tmp, p);
result += alpha * p;
residual -= alpha * tmp;
z = residual;
precond.apply(z);
new_ip_rr = viennacl::linalg::inner_prod(residual, z);
new_ipp_rr_over_norm_rhs = new_ip_rr / norm_rhs_squared;
if (std::fabs(new_ipp_rr_over_norm_rhs) < tag.tolerance() * tag.tolerance()) //squared norms involved here
break;
beta = new_ip_rr / ip_rr;
ip_rr = new_ip_rr;
p = z + beta*p;
}
//store last error estimate:
tag.error(std::sqrt(std::fabs(new_ip_rr / norm_rhs_squared)));
return result;
}
VectorType solve(const MatrixType & matrix, VectorType const & rhs, cg_tag const & tag, PreconditionerType const & precond)
{
typedef typename VectorType::value_type ScalarType;
typedef typename viennacl::tools::CPU_SCALAR_TYPE_DEDUCER<ScalarType>::ResultType CPU_ScalarType;
VectorType result(rhs.size());
result.clear();
VectorType residual = rhs;
VectorType tmp(rhs.size());
VectorType z = rhs;
precond.apply(z);
VectorType p = z;
ScalarType ip_rr = viennacl::linalg::inner_prod(residual, z);
ScalarType alpha;
ScalarType new_ip_rr = 0;
ScalarType beta;
ScalarType norm_rhs = ip_rr;
for (unsigned int i = 0; i < tag.iterations(); ++i)
{
tag.iters(i+1);
tmp = viennacl::linalg::prod(matrix, p);
alpha = ip_rr / viennacl::linalg::inner_prod(tmp, p);
result += alpha * p;
residual -= alpha * tmp;
z = residual;
precond.apply(z);
new_ip_rr = viennacl::linalg::inner_prod(residual, z);
if (std::fabs(new_ip_rr / norm_rhs) < tag.tolerance() * tag.tolerance()) //squared norms involved here
break;
beta = new_ip_rr / ip_rr;
ip_rr = new_ip_rr;
p = z + beta*p;
}
//store last error estimate:
tag.error(sqrt(std::fabs(new_ip_rr / norm_rhs)));
return result;
}
VectorType solve(const MatrixType & matrix, VectorType const & rhs, gmres_tag const & tag, PreconditionerType const & precond)
{
typedef typename viennacl::result_of::value_type<VectorType>::type ScalarType;
typedef typename viennacl::result_of::cpu_value_type<ScalarType>::type CPU_ScalarType;
unsigned int problem_size = viennacl::traits::size(rhs);
VectorType result(problem_size);
viennacl::traits::clear(result);
unsigned int krylov_dim = tag.krylov_dim();
if (problem_size < tag.krylov_dim())
krylov_dim = problem_size; //A Krylov space larger than the matrix would lead to seg-faults (mathematically, error is certain to be zero already)
VectorType res(problem_size);
VectorType v_k_tilde(problem_size);
VectorType v_k_tilde_temp(problem_size);
std::vector< std::vector<CPU_ScalarType> > R(krylov_dim);
std::vector<CPU_ScalarType> projection_rhs(krylov_dim);
std::vector<VectorType> U(krylov_dim);
const CPU_ScalarType gpu_scalar_minus_1 = static_cast<CPU_ScalarType>(-1); //representing the scalar '-1' on the GPU. Prevents blocking write operations
const CPU_ScalarType gpu_scalar_1 = static_cast<CPU_ScalarType>(1); //representing the scalar '1' on the GPU. Prevents blocking write operations
const CPU_ScalarType gpu_scalar_2 = static_cast<CPU_ScalarType>(2); //representing the scalar '2' on the GPU. Prevents blocking write operations
CPU_ScalarType norm_rhs = viennacl::linalg::norm_2(rhs);
if (norm_rhs == 0) //solution is zero if RHS norm is zero
return result;
unsigned int k;
for (k = 0; k < krylov_dim; ++k)
{
R[k].resize(tag.krylov_dim());
viennacl::traits::resize(U[k], problem_size);
}
//std::cout << "Starting GMRES..." << std::endl;
tag.iters(0);
for (unsigned int it = 0; it <= tag.max_restarts(); ++it)
{
//std::cout << "-- GMRES Start " << it << " -- " << std::endl;
res = rhs;
res -= viennacl::linalg::prod(matrix, result); //initial guess zero
precond.apply(res);
//std::cout << "Residual: " << res << std::endl;
CPU_ScalarType rho_0 = viennacl::linalg::norm_2(res);
CPU_ScalarType rho = static_cast<CPU_ScalarType>(1.0);
//std::cout << "rho_0: " << rho_0 << std::endl;
if (rho_0 / norm_rhs < tag.tolerance() || (norm_rhs == CPU_ScalarType(0.0)) )
{
//std::cout << "Allowed Error reached at begin of loop" << std::endl;
tag.error(rho_0 / norm_rhs);
return result;
}
res /= rho_0;
//std::cout << "Normalized Residual: " << res << std::endl;
for (k=0; k<krylov_dim; ++k)
{
viennacl::traits::clear(R[k]);
viennacl::traits::clear(U[k]);
R[k].resize(krylov_dim);
viennacl::traits::resize(U[k], problem_size);
}
for (k = 0; k < krylov_dim; ++k)
{
tag.iters( tag.iters() + 1 ); //increase iteration counter
//compute v_k = A * v_{k-1} via Householder matrices
if (k == 0)
{
v_k_tilde = viennacl::linalg::prod(matrix, res);
precond.apply(v_k_tilde);
}
else
{
viennacl::traits::clear(v_k_tilde);
v_k_tilde[k-1] = gpu_scalar_1;
//Householder rotations part 1
for (int i = k-1; i > -1; --i)
v_k_tilde -= U[i] * (viennacl::linalg::inner_prod(U[i], v_k_tilde) * gpu_scalar_2);
v_k_tilde_temp = viennacl::linalg::prod(matrix, v_k_tilde);
precond.apply(v_k_tilde_temp);
v_k_tilde = v_k_tilde_temp;
//Householder rotations part 2
for (unsigned int i = 0; i < k; ++i)
v_k_tilde -= U[i] * (viennacl::linalg::inner_prod(U[i], v_k_tilde) * gpu_scalar_2);
}
//std::cout << "v_k_tilde: " << v_k_tilde << std::endl;
viennacl::traits::clear(U[k]);
viennacl::traits::resize(U[k], problem_size);
//copy first k entries from v_k_tilde to U[k]:
gmres_copy_helper(v_k_tilde, U[k], k);
U[k][k] = std::sqrt( viennacl::linalg::inner_prod(v_k_tilde, v_k_tilde) - viennacl::linalg::inner_prod(U[k], U[k]) );
if (fabs(U[k][k]) < CPU_ScalarType(10 * std::numeric_limits<CPU_ScalarType>::epsilon()))
break; //Note: Solution is essentially (up to round-off error) already in Krylov space. No need to proceed.
//copy first k+1 entries from U[k] to R[k]
gmres_copy_helper(U[k], R[k], k+1);
U[k] -= v_k_tilde;
//std::cout << "U[k] before normalization: " << U[k] << std::endl;
U[k] *= gpu_scalar_minus_1 / viennacl::linalg::norm_2( U[k] );
//std::cout << "Householder vector U[k]: " << U[k] << std::endl;
//DEBUG: Make sure that P_k v_k_tilde equals (rho_{1,k}, ... , rho_{k,k}, 0, 0 )
#ifdef VIENNACL_GMRES_DEBUG
std::cout << "P_k v_k_tilde: " << (v_k_tilde - 2.0 * U[k] * inner_prod(U[k], v_k_tilde)) << std::endl;
std::cout << "R[k]: [" << R[k].size() << "](";
for (size_t i=0; i<R[k].size(); ++i)
std::cout << R[k][i] << ",";
std::cout << ")" << std::endl;
#endif
//std::cout << "P_k res: " << (res - 2.0 * U[k] * inner_prod(U[k], res)) << std::endl;
res -= U[k] * (viennacl::linalg::inner_prod( U[k], res ) * gpu_scalar_2);
//std::cout << "zeta_k: " << viennacl::linalg::inner_prod( U[k], res ) * gpu_scalar_2 << std::endl;
//std::cout << "Updated res: " << res << std::endl;
#ifdef VIENNACL_GMRES_DEBUG
VectorType v1(U[k].size()); v1.clear(); v1.resize(U[k].size());
v1(0) = 1.0;
v1 -= U[k] * (viennacl::linalg::inner_prod( U[k], v1 ) * gpu_scalar_2);
std::cout << "v1: " << v1 << std::endl;
boost::numeric::ublas::matrix<ScalarType> P = -2.0 * outer_prod(U[k], U[k]);
P(0,0) += 1.0; P(1,1) += 1.0; P(2,2) += 1.0;
std::cout << "P: " << P << std::endl;
#endif
if (res[k] > rho) //machine precision reached
res[k] = rho;
if (res[k] < -1.0 * rho) //machine precision reached
res[k] = -1.0 * rho;
projection_rhs[k] = res[k];
rho *= std::sin( std::acos(projection_rhs[k] / rho) );
#ifdef VIENNACL_GMRES_DEBUG
std::cout << "k-th component of r: " << res[k] << std::endl;
std::cout << "New rho (norm of res): " << rho << std::endl;
#endif
if (std::fabs(rho * rho_0 / norm_rhs) < tag.tolerance())
{
//std::cout << "Krylov space big enough" << endl;
tag.error( std::fabs(rho*rho_0 / norm_rhs) );
++k;
break;
}
//std::cout << "Current residual: " << rho * rho_0 << std::endl;
//std::cout << " - End of Krylov space setup - " << std::endl;
} // for k
#ifdef VIENNACL_GMRES_DEBUG
//inplace solution of the upper triangular matrix:
std::cout << "Upper triangular system:" << std::endl;
std::cout << "Size of Krylov space: " << k << std::endl;
for (size_t i=0; i<k; ++i)
{
for (size_t j=0; j<k; ++j)
{
std::cout << R[j][i] << ", ";
}
std::cout << " | " << projection_rhs[i] << std::endl;
}
#endif
for (int i=k-1; i>-1; --i)
{
for (unsigned int j=i+1; j<k; ++j)
//temp_rhs[i] -= R[i][j] * temp_rhs[j]; //if R is not transposed
projection_rhs[i] -= R[j][i] * projection_rhs[j]; //R is transposed
projection_rhs[i] /= R[i][i];
}
#ifdef VIENNACL_GMRES_DEBUG
std::cout << "Result of triangular solver: ";
for (size_t i=0; i<k; ++i)
std::cout << projection_rhs[i] << ", ";
std::cout << std::endl;
#endif
res *= projection_rhs[0];
if (k > 0)
{
for (unsigned int i = 0; i < k-1; ++i)
{
res[i] += projection_rhs[i+1];
}
}
for (int i = k-1; i > -1; --i)
res -= U[i] * (viennacl::linalg::inner_prod(U[i], res) * gpu_scalar_2);
res *= rho_0;
result += res;
if ( std::fabs(rho*rho_0 / norm_rhs) < tag.tolerance() )
{
//std::cout << "Allowed Error reached at end of loop" << std::endl;
tag.error(std::fabs(rho*rho_0 / norm_rhs));
return result;
}
//res = rhs;
//res -= viennacl::linalg::prod(matrix, result);
//std::cout << "norm_2(r)=" << norm_2(r) << std::endl;
//std::cout << "std::abs(rho*rho_0)=" << std::abs(rho*rho_0) << std::endl;
//std::cout << r << std::endl;
tag.error(std::fabs(rho*rho_0));
}
return result;
}
VectorType solve(const MatrixType & matrix, VectorType const & rhs, bicgstab_tag const & tag, PreconditionerType const & precond)
{
typedef typename viennacl::result_of::value_type<VectorType>::type ScalarType;
typedef typename viennacl::result_of::cpu_value_type<ScalarType>::type CPU_ScalarType;
VectorType result = rhs;
viennacl::traits::clear(result);
VectorType residual = rhs;
VectorType r0star = residual; //can be chosen arbitrarily in fact
VectorType tmp0 = rhs;
VectorType tmp1 = rhs;
VectorType s = rhs;
VectorType p = residual;
CPU_ScalarType ip_rr0star = viennacl::linalg::norm_2(residual);
CPU_ScalarType norm_rhs_host = viennacl::linalg::norm_2(residual);
CPU_ScalarType beta;
CPU_ScalarType alpha;
CPU_ScalarType omega;
CPU_ScalarType new_ip_rr0star = 0;
CPU_ScalarType residual_norm = norm_rhs_host;
if (norm_rhs_host == 0) //solution is zero if RHS norm is zero
return result;
bool restart_flag = true;
vcl_size_t last_restart = 0;
for (unsigned int i = 0; i < tag.max_iterations(); ++i)
{
if (restart_flag)
{
residual = rhs;
residual -= viennacl::linalg::prod(matrix, result);
precond.apply(residual);
p = residual;
r0star = residual;
ip_rr0star = viennacl::linalg::norm_2(residual);
ip_rr0star *= ip_rr0star;
restart_flag = false;
last_restart = i;
}
tag.iters(i+1);
tmp0 = viennacl::linalg::prod(matrix, p);
precond.apply(tmp0);
alpha = ip_rr0star / viennacl::linalg::inner_prod(tmp0, r0star);
s = residual - alpha*tmp0;
tmp1 = viennacl::linalg::prod(matrix, s);
precond.apply(tmp1);
CPU_ScalarType norm_tmp1 = viennacl::linalg::norm_2(tmp1);
omega = viennacl::linalg::inner_prod(tmp1, s) / (norm_tmp1 * norm_tmp1);
result += alpha * p + omega * s;
residual = s - omega * tmp1;
residual_norm = viennacl::linalg::norm_2(residual);
if (residual_norm / norm_rhs_host < tag.tolerance())
break;
new_ip_rr0star = viennacl::linalg::inner_prod(residual, r0star);
beta = new_ip_rr0star / ip_rr0star * alpha/omega;
ip_rr0star = new_ip_rr0star;
if (ip_rr0star == 0 || omega == 0 || i - last_restart > tag.max_iterations_before_restart()) //search direction degenerate. A restart might help
restart_flag = true;
// Execution of
// p = residual + beta * (p - omega*tmp0);
// without introducing temporary vectors:
p -= omega * tmp0;
p = residual + beta * p;
//std::cout << "Rel. Residual in current step: " << std::sqrt(std::fabs(viennacl::linalg::inner_prod(residual, residual) / norm_rhs_host)) << std::endl;
}
//store last error estimate:
tag.error(residual_norm / norm_rhs_host);
return result;
}