/usr/include/viennacl/linalg/gmres.hpp is in libviennacl-dev 1.5.2-2.
This file is owned by root:root, with mode 0o644.
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#define VIENNACL_GMRES_HPP_
/* =========================================================================
Copyright (c) 2010-2014, Institute for Microelectronics,
Institute for Analysis and Scientific Computing,
TU Wien.
Portions of this software are copyright by UChicago Argonne, LLC.
-----------------
ViennaCL - The Vienna Computing Library
-----------------
Project Head: Karl Rupp rupp@iue.tuwien.ac.at
(A list of authors and contributors can be found in the PDF manual)
License: MIT (X11), see file LICENSE in the base directory
============================================================================= */
/** @file gmres.hpp
@brief Implementations of the generalized minimum residual method are in this file.
*/
#include <vector>
#include <cmath>
#include <limits>
#include "viennacl/forwards.h"
#include "viennacl/tools/tools.hpp"
#include "viennacl/linalg/norm_2.hpp"
#include "viennacl/linalg/prod.hpp"
#include "viennacl/linalg/inner_prod.hpp"
#include "viennacl/traits/clear.hpp"
#include "viennacl/traits/size.hpp"
#include "viennacl/meta/result_of.hpp"
namespace viennacl
{
namespace linalg
{
/** @brief A tag for the solver GMRES. Used for supplying solver parameters and for dispatching the solve() function
*/
class gmres_tag //generalized minimum residual
{
public:
/** @brief The constructor
*
* @param tol Relative tolerance for the residual (solver quits if ||r|| < tol * ||r_initial||)
* @param max_iterations The maximum number of iterations (including restarts
* @param krylov_dim The maximum dimension of the Krylov space before restart (number of restarts is found by max_iterations / krylov_dim)
*/
gmres_tag(double tol = 1e-10, unsigned int max_iterations = 300, unsigned int krylov_dim = 20)
: tol_(tol), iterations_(max_iterations), krylov_dim_(krylov_dim), iters_taken_(0) {}
/** @brief Returns the relative tolerance */
double tolerance() const { return tol_; }
/** @brief Returns the maximum number of iterations */
unsigned int max_iterations() const { return iterations_; }
/** @brief Returns the maximum dimension of the Krylov space before restart */
unsigned int krylov_dim() const { return krylov_dim_; }
/** @brief Returns the maximum number of GMRES restarts */
unsigned int max_restarts() const
{
unsigned int ret = iterations_ / krylov_dim_;
if (ret > 0 && (ret * krylov_dim_ == iterations_) )
return ret - 1;
return ret;
}
/** @brief Return the number of solver iterations: */
unsigned int iters() const { return iters_taken_; }
/** @brief Set the number of solver iterations (should only be modified by the solver) */
void iters(unsigned int i) const { iters_taken_ = i; }
/** @brief Returns the estimated relative error at the end of the solver run */
double error() const { return last_error_; }
/** @brief Sets the estimated relative error at the end of the solver run */
void error(double e) const { last_error_ = e; }
private:
double tol_;
unsigned int iterations_;
unsigned int krylov_dim_;
//return values from solver
mutable unsigned int iters_taken_;
mutable double last_error_;
};
namespace detail
{
template <typename SRC_VECTOR, typename DEST_VECTOR>
void gmres_copy_helper(SRC_VECTOR const & src, DEST_VECTOR & dest, vcl_size_t len, vcl_size_t start = 0)
{
for (vcl_size_t i=0; i<len; ++i)
dest[start+i] = src[start+i];
}
template <typename ScalarType, typename DEST_VECTOR>
void gmres_copy_helper(viennacl::vector<ScalarType> const & src, DEST_VECTOR & dest, vcl_size_t len, vcl_size_t start = 0)
{
typedef typename viennacl::vector<ScalarType>::difference_type difference_type;
viennacl::copy( src.begin() + static_cast<difference_type>(start),
src.begin() + static_cast<difference_type>(start + len),
dest.begin() + static_cast<difference_type>(start));
}
/** @brief Computes the householder vector 'hh_vec' which rotates 'input_vec' such that all entries below the j-th entry of 'v' become zero.
*
* @param input_vec The input vector
* @param hh_vec The householder vector defining the relection (I - beta * hh_vec * hh_vec^T)
* @param beta The coefficient beta in (I - beta * hh_vec * hh_vec^T)
* @param mu The norm of the input vector part relevant for the reflection: norm_2(input_vec[j:size])
* @param j Index of the last nonzero index in 'input_vec' after applying the reflection
*/
template <typename VectorType, typename ScalarType>
void gmres_setup_householder_vector(VectorType const & input_vec, VectorType & hh_vec, ScalarType & beta, ScalarType & mu, vcl_size_t j)
{
ScalarType input_j = input_vec(j);
// copy entries from input vector to householder vector:
detail::gmres_copy_helper(input_vec, hh_vec, viennacl::traits::size(hh_vec) - (j+1), j+1);
ScalarType sigma = viennacl::linalg::norm_2(hh_vec);
sigma *= sigma;
if (sigma == 0)
{
beta = 0;
mu = input_j;
}
else
{
mu = std::sqrt(sigma + input_j*input_j);
ScalarType hh_vec_0 = (input_j <= 0) ? (input_j - mu) : (-sigma / (input_j + mu));
beta = ScalarType(2) * hh_vec_0 * hh_vec_0 / (sigma + hh_vec_0 * hh_vec_0);
//divide hh_vec by its diagonal element hh_vec_0
hh_vec /= hh_vec_0;
hh_vec[j] = ScalarType(1);
}
}
// Apply (I - beta h h^T) to x (Householder reflection with Householder vector h)
template <typename VectorType, typename ScalarType>
void gmres_householder_reflect(VectorType & x, VectorType const & h, ScalarType beta)
{
ScalarType hT_in_x = viennacl::linalg::inner_prod(h, x);
x -= (beta * hT_in_x) * h;
}
}
/** @brief Implementation of the GMRES solver.
*
* Following the algorithm proposed by Walker in "A Simpler GMRES"
*
* @param matrix The system matrix
* @param rhs The load vector
* @param tag Solver configuration tag
* @param precond A preconditioner. Precondition operation is done via member function apply()
* @return The result vector
*/
template <typename MatrixType, typename VectorType, typename PreconditionerType>
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 = static_cast<unsigned int>(viennacl::traits::size(rhs));
VectorType result = rhs;
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 = rhs;
VectorType v_k_tilde = rhs;
VectorType v_k_tilde_temp = rhs;
std::vector< std::vector<CPU_ScalarType> > R(krylov_dim, std::vector<CPU_ScalarType>(tag.krylov_dim()));
std::vector<CPU_ScalarType> projection_rhs(krylov_dim);
std::vector<VectorType> householder_reflectors(krylov_dim, rhs);
std::vector<CPU_ScalarType> betas(krylov_dim);
CPU_ScalarType norm_rhs = viennacl::linalg::norm_2(rhs);
if (norm_rhs == 0) //solution is zero if RHS norm is zero
return result;
tag.iters(0);
for (unsigned int it = 0; it <= tag.max_restarts(); ++it)
{
//
// (Re-)Initialize residual: r = b - A*x (without temporary for the result of A*x)
//
res = rhs;
res -= viennacl::linalg::prod(matrix, result); //initial guess zero
precond.apply(res);
CPU_ScalarType rho_0 = viennacl::linalg::norm_2(res);
//
// Check for premature convergence
//
if (rho_0 / norm_rhs < tag.tolerance() ) // norm_rhs is known to be nonzero here
{
tag.error(rho_0 / norm_rhs);
return result;
}
//
// Normalize residual and set 'rho' to 1 as requested in 'A Simpler GMRES' by Walker and Zhou.
//
res /= rho_0;
CPU_ScalarType rho = static_cast<CPU_ScalarType>(1.0);
//
// Iterate up until maximal Krylove space dimension is reached:
//
unsigned int k = 0;
for (k = 0; k < krylov_dim; ++k)
{
tag.iters( tag.iters() + 1 ); //increase iteration counter
// prepare storage:
viennacl::traits::clear(R[k]);
viennacl::traits::clear(householder_reflectors[k]);
//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] = CPU_ScalarType(1);
//Householder rotations, part 1: Compute P_1 * P_2 * ... * P_{k-1} * e_{k-1}
for (int i = k-1; i > -1; --i)
detail::gmres_householder_reflect(v_k_tilde, householder_reflectors[i], betas[i]);
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: Compute P_{k-1} * ... * P_{1} * v_k_tilde
for (unsigned int i = 0; i < k; ++i)
detail::gmres_householder_reflect(v_k_tilde, householder_reflectors[i], betas[i]);
}
//
// Compute Householder reflection for v_k_tilde such that all entries below k-th entry are zero:
//
CPU_ScalarType rho_k_k = 0;
detail::gmres_setup_householder_vector(v_k_tilde, householder_reflectors[k], betas[k], rho_k_k, k);
//
// copy first k entries from v_k_tilde to R[k] in order to fill k-th column with result of
// P_k * v_k_tilde = (v[0], ... , v[k-1], norm(v), 0, 0, ...) =: (rho_{1,k}, rho_{2,k}, ..., rho_{k,k}, 0, ..., 0);
//
detail::gmres_copy_helper(v_k_tilde, R[k], k);
R[k][k] = rho_k_k;
//
// Update residual: r = P_k r
// Set zeta_k = r[k] including machine precision considerations: mathematically we have |r[k]| <= rho
// Set rho *= sin(acos(r[k] / rho))
//
detail::gmres_householder_reflect(res, householder_reflectors[k], betas[k]);
if (res[k] > rho) //machine precision reached
res[k] = rho;
if (res[k] < -rho) //machine precision reached
res[k] = -rho;
projection_rhs[k] = res[k];
rho *= std::sin( std::acos(projection_rhs[k] / rho) );
if (std::fabs(rho * rho_0 / norm_rhs) < tag.tolerance()) // Residual is sufficiently reduced, stop here
{
tag.error( std::fabs(rho*rho_0 / norm_rhs) );
++k;
break;
}
} // for k
//
// Triangular solver stage:
//
for (int i=k-1; i>-1; --i)
{
for (unsigned int j=i+1; j<k; ++j)
projection_rhs[i] -= R[j][i] * projection_rhs[j]; //R is transposed
projection_rhs[i] /= R[i][i];
}
//
// Note: 'projection_rhs' now holds the solution (eta_1, ..., eta_k)
//
res *= projection_rhs[0];
if (k > 0)
{
for (unsigned int i = 0; i < k-1; ++i)
res[i] += projection_rhs[i+1];
}
//
// Form z inplace in 'res' by applying P_1 * ... * P_{k}
//
for (int i=k-1; i>=0; --i)
detail::gmres_householder_reflect(res, householder_reflectors[i], betas[i]);
res *= rho_0;
result += res; // x += rho_0 * z in the paper
//
// Check for convergence:
//
tag.error(std::fabs(rho*rho_0 / norm_rhs));
if ( tag.error() < tag.tolerance() )
return result;
}
return result;
}
/** @brief Convenience overload of the solve() function using GMRES. Per default, no preconditioner is used
*/
template <typename MatrixType, typename VectorType>
VectorType solve(const MatrixType & matrix, VectorType const & rhs, gmres_tag const & tag)
{
return solve(matrix, rhs, tag, no_precond());
}
}
}
#endif
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