Reference documentation for deal.II version 9.4.1

#include <deal.II/lac/solver_cg.h>
Classes  
struct  AdditionalData 
Public Types  
using  size_type = types::global_dof_index 
using  vector_type = VectorType 
Public Member Functions  
SolverCG (SolverControl &cn, VectorMemory< VectorType > &mem, const AdditionalData &data=AdditionalData())  
SolverCG (SolverControl &cn, const AdditionalData &data=AdditionalData())  
virtual  ~SolverCG () override=default 
template<typename MatrixType , typename PreconditionerType >  
void  solve (const MatrixType &A, VectorType &x, const VectorType &b, const PreconditionerType &preconditioner) 
boost::signals2::connection  connect_coefficients_slot (const std::function< void(typename VectorType::value_type, typename VectorType::value_type)> &slot) 
boost::signals2::connection  connect_condition_number_slot (const std::function< void(double)> &slot, const bool every_iteration=false) 
boost::signals2::connection  connect_eigenvalues_slot (const std::function< void(const std::vector< double > &)> &slot, const bool every_iteration=false) 
boost::signals2::connection  connect (const std::function< SolverControl::State(const unsigned int iteration, const double check_value, const VectorType ¤t_iterate)> &slot) 
Protected Member Functions  
virtual void  print_vectors (const unsigned int step, const VectorType &x, const VectorType &r, const VectorType &d) const 
Static Protected Member Functions  
static void  compute_eigs_and_cond (const std::vector< typename VectorType::value_type > &diagonal, const std::vector< typename VectorType::value_type > &offdiagonal, const boost::signals2::signal< void(const std::vector< double > &)> &eigenvalues_signal, const boost::signals2::signal< void(double)> &cond_signal) 
Protected Attributes  
AdditionalData  additional_data 
boost::signals2::signal< void(typename VectorType::value_type, typename VectorType::value_type)>  coefficients_signal 
boost::signals2::signal< void(double)>  condition_number_signal 
boost::signals2::signal< void(double)>  all_condition_numbers_signal 
boost::signals2::signal< void(const std::vector< double > &)>  eigenvalues_signal 
boost::signals2::signal< void(const std::vector< double > &)>  all_eigenvalues_signal 
bool  determine_beta_by_flexible_formula 
GrowingVectorMemory< VectorType >  static_vector_memory 
VectorMemory< VectorType > &  memory 
boost::signals2::signal< SolverControl::State(const unsigned int iteration, const double check_value, const VectorType ¤t_iterate), StateCombiner >  iteration_status 
Subscriptor functionality  
Classes derived from Subscriptor provide a facility to subscribe to this object. This is mostly used by the SmartPointer class.  
void  subscribe (std::atomic< bool > *const validity, const std::string &identifier="") const 
void  unsubscribe (std::atomic< bool > *const validity, const std::string &identifier="") const 
unsigned int  n_subscriptions () const 
template<typename StreamType >  
void  list_subscribers (StreamType &stream) const 
void  list_subscribers () const 
template<class Archive >  
void  serialize (Archive &ar, const unsigned int version) 
using  map_value_type = decltype(counter_map)::value_type 
using  map_iterator = decltype(counter_map)::iterator 
std::atomic< unsigned int >  counter 
std::map< std::string, unsigned int >  counter_map 
std::vector< std::atomic< bool > * >  validity_pointers 
const std::type_info *  object_info 
static std::mutex  mutex 
static ::ExceptionBase &  ExcInUse (int arg1, std::string arg2, std::string arg3) 
static ::ExceptionBase &  ExcNoSubscriber (std::string arg1, std::string arg2) 
void  check_no_subscribers () const noexcept 
This class implements the preconditioned Conjugate Gradients (CG) method that can be used to solve linear systems with a symmetric positive definite matrix. This class is used first in step3 and step4, but is used in many other tutorial programs as well. Like all other solver classes, it can work on any kind of vector and matrix as long as they satisfy certain requirements (for the requirements on matrices and vectors in order to work with this class, see the documentation of the Solver base class). The type of the solution vector must be passed as template argument, and defaults to Vector<double>.
The cgmethod performs an orthogonal projection of the original preconditioned linear system to another system of smaller dimension. Furthermore, the projected matrix T
is tridiagonal. Since the projection is orthogonal, the eigenvalues of T
approximate those of the original preconditioned matrix PA
. In fact, after n
steps, where n
is the dimension of the original system, the eigenvalues of both matrices are equal. But, even for small numbers of iteration steps, the condition number of T
is a good estimate for the one of PA
.
After m
steps the matrix T_m can be written in terms of the coefficients alpha
and beta
as the tridiagonal matrix with diagonal elements 1/alpha_0
, 1/alpha_1 + beta_0/alpha_0
, ..., 1/alpha_{m1}+beta_{m2}/alpha_{m2}
and offdiagonal elements sqrt(beta_0)/alpha_0
, ..., sqrt(beta_{m2})/alpha_{m2}
. The eigenvalues of this matrix can be computed by postprocessing.
The coefficients, eigenvalues and condition number (computed as the ratio of the largest over smallest eigenvalue) can be obtained by connecting a function as a slot to the solver using one of the functions connect_coefficients_slot
, connect_eigenvalues_slot
and connect_condition_number_slot
. These slots will then be called from the solver with the estimates as argument.
The solve() function of this class uses the mechanism described in the Solver base class to determine convergence. This mechanism can also be used to observe the progress of the iteration.
MatrixType
argumentThis class enables to embed the vector updates into the matrixvector product in case the MatrixType
and PreconditionerType
support such a mode of operation. To this end, the VectorType
needs to be LinearAlgebra::distributed::Vector, the class MatrixType
needs to provide a function with the signature
where the two given functions run before and after the matrixvector product, respectively, and the PreconditionerType
needs to provide a function either the signature
to apply the action of the preconditioner on a single element (effectively being a diagonal preconditioner), or the signature
where the pointers src_ptr_to_subrange
and dst_ptr_to_subrange
point to the location in the vector where the operation should be applied to. If both functions are given, the more optimized apply
path is selected. The functions passed to MatrixType::vmult
take as arguments a subrange among the locally owned elements of the vector, defined as halfopen intervals. The intervals are designed to be scheduled close to the time the matrixvector product touches those entries in the src
and dst
vectors, respectively, with the requirement that
src
or dst
once the operation_before_matrix_vector_product
has been run on that vector entry; operation_after_matrix_vector_product
may run on a range of entries [i,j)
once the matrixvector product does not access the entries [i,j)
in src
and dst
any more. The motivation for this function is to increase data locality and hence cache usage. For the example of a class similar to the one in the step37 tutorial program, the implementation is
In terms of the SolverCG implementation, the operation before the loop will run the updates on the vectors according to a variant presented in Algorithm 2.2 of [44] (but for a preconditioner), whereas the operation after the loop performs a total of 7 reductions in parallel.
Definition at line 177 of file solver_cg.h.
using SolverCG< VectorType >::size_type = types::global_dof_index 
Declare type for container size.
Definition at line 183 of file solver_cg.h.

inherited 
SolverCG< VectorType >::SolverCG  (  SolverControl &  cn, 
VectorMemory< VectorType > &  mem,  
const AdditionalData &  data = AdditionalData() 

) 
Constructor.
SolverCG< VectorType >::SolverCG  (  SolverControl &  cn, 
const AdditionalData &  data = AdditionalData() 

) 
Constructor. Use an object of type GrowingVectorMemory as a default to allocate memory.

overridevirtualdefault 
Virtual destructor.
void SolverCG< VectorType >::solve  (  const MatrixType &  A, 
VectorType &  x,  
const VectorType &  b,  
const PreconditionerType &  preconditioner  
) 
Solve the linear system \(Ax=b\) for x.
boost::signals2::connection SolverCG< VectorType >::connect_coefficients_slot  (  const std::function< void(typename VectorType::value_type, typename VectorType::value_type)> &  slot  ) 
Connect a slot to retrieve the CG coefficients. The slot will be called with alpha as the first argument and with beta as the second argument, where alpha and beta follow the notation in Y. Saad: "Iterative methods for Sparse Linear Systems", section 6.7. Called once per iteration
boost::signals2::connection SolverCG< VectorType >::connect_condition_number_slot  (  const std::function< void(double)> &  slot, 
const bool  every_iteration = false 

) 
Connect a slot to retrieve the estimated condition number. Called on each iteration if every_iteration=true, otherwise called once when iterations are ended (i.e., either because convergence has been achieved, or because divergence has been detected).
boost::signals2::connection SolverCG< VectorType >::connect_eigenvalues_slot  (  const std::function< void(const std::vector< double > &)> &  slot, 
const bool  every_iteration = false 

) 
Connect a slot to retrieve the estimated eigenvalues. Called on each iteration if every_iteration=true, otherwise called once when iterations are ended (i.e., either because convergence has been achieved, or because divergence has been detected).

protectedvirtual 
Interface for derived class. This function gets the current iteration vector, the residual and the update vector in each step. It can be used for graphical output of the convergence history.

staticprotected 
Estimates the eigenvalues from diagonal and offdiagonal. Uses these estimate to compute the condition number. Calls the signals eigenvalues_signal and cond_signal with these estimates as arguments.

inherited 
Connect a function object that will be called periodically within iterative solvers. This function is used to attach monitors to iterative solvers, either to determine when convergence has happened, or simply to observe the progress of an iteration. See the documentation of this class for more information.
slot  A function object specified here will, with each call, receive the number of the current iteration, the value that is used to check for convergence (typically the residual of the current iterate with respect to the linear system to be solved) and the currently best available guess for the current iterate. Note that some solvers do not update the approximate solution in every iteration but only after convergence or failure has been determined (GMRES is an example); in such cases, the vector passed as the last argument to the signal is simply the best approximate at the time the signal is called, but not the vector that will be returned if the signal's return value indicates that the iteration should be terminated. The function object must return a SolverControl::State value that indicates whether the iteration should continue, has failed, or has succeeded. The results of all connected functions will then be combined to determine what should happen with the iteration. 

protected 
Additional parameters.
Definition at line 281 of file solver_cg.h.

protected 
Signal used to retrieve the CG coefficients. Called on each iteration.
Definition at line 288 of file solver_cg.h.

protected 
Signal used to retrieve the estimated condition number. Called once when all iterations are ended.
Definition at line 294 of file solver_cg.h.

protected 
Signal used to retrieve the estimated condition numbers. Called on each iteration.
Definition at line 300 of file solver_cg.h.

protected 
Signal used to retrieve the estimated eigenvalues. Called once when all iterations are ended.
Definition at line 306 of file solver_cg.h.

protected 
Signal used to retrieve the estimated eigenvalues. Called on each iteration.
Definition at line 313 of file solver_cg.h.

protected 
Flag to indicate whether the classical Fletcher–Reeves update formula for the parameter \(\beta_k\) (standard CG algorithm, minimal storage needs) or the flexible conjugate gradient method with PolakRibiere formula for \(\beta_k\) should be used. This base class implementation of SolverCG will always use the former method, whereas the derived class SolverFlexibleCG will use the latter.
Definition at line 323 of file solver_cg.h.

mutableprotectedinherited 

protectedinherited 

protectedinherited 
A signal that iterative solvers can execute at the end of every iteration (or in an otherwise periodic fashion) to find out whether we should continue iterating or not. The signal may call one or more slots that each will make this determination by themselves, and the result over all slots (function calls) will be determined by the StateCombiner object.
The arguments passed to the signal are (i) the number of the current iteration; (ii) the value that is used to determine convergence (oftentimes the residual, but in other cases other quantities may be used as long as they converge to zero as the iterate approaches the solution of the linear system); and (iii) a vector that corresponds to the current best guess for the solution at the point where the signal is called. Note that some solvers do not update the approximate solution in every iteration but only after convergence or failure has been determined (GMRES is an example); in such cases, the vector passed as the last argument to the signal is simply the best approximate at the time the signal is called, but not the vector that will be returned if the signal's return value indicates that the iteration should be terminated.