Reference documentation for deal.II version Git 3c0b904 2015-07-28 14:31:11 -0500
Classes | Public Types | Public Member Functions | Protected Member Functions | Static Protected Member Functions | Protected Attributes | Private Member Functions | List of all members
SolverCG< VECTOR > Class Template Reference

#include <deal.II/lac/solver_cg.h>

Inheritance diagram for SolverCG< VECTOR >:
[legend]

Classes

struct  AdditionalData
 

Public Types

typedef types::global_dof_index size_type
 
- Public Types inherited from Solver< VECTOR >
typedef VECTOR vector_type
 

Public Member Functions

 SolverCG (SolverControl &cn, VectorMemory< VECTOR > &mem, const AdditionalData &data=AdditionalData())
 
 SolverCG (SolverControl &cn, const AdditionalData &data=AdditionalData())
 
virtual ~SolverCG ()
 
template<class MATRIX , class PRECONDITIONER >
void solve (const MATRIX &A, VECTOR &x, const VECTOR &b, const PRECONDITIONER &precondition)
 
boost::signals2::connection connect_coefficients_slot (const std_cxx11::function< void(double, double)> &slot)
 
boost::signals2::connection connect_condition_number_slot (const std_cxx11::function< void(double)> &slot, const bool every_iteration=false)
 
boost::signals2::connection connect_eigenvalues_slot (const std_cxx11::function< void(const std::vector< double > &)> &slot, const bool every_iteration=false)
 
- Public Member Functions inherited from Solver< VECTOR >
 Solver (SolverControl &solver_control, VectorMemory< VECTOR > &vector_memory)
 
 Solver (SolverControl &solver_control)
 
boost::signals2::connection connect (const std_cxx11::function< SolverControl::State(const unsigned int iteration, const doublecheck_value, const VECTOR &current_iterate)> &slot)
 
- Public Member Functions inherited from Subscriptor
 Subscriptor ()
 
 Subscriptor (const Subscriptor &)
 
virtual ~Subscriptor ()
 
Subscriptoroperator= (const Subscriptor &)
 
void subscribe (const char *identifier=0) const
 
void unsubscribe (const char *identifier=0) const
 
unsigned int n_subscriptions () const
 
void list_subscribers () const
 
 DeclException3 (ExcInUse, int, char *, std::string &,<< "Object of class "<< arg2<< " is still used by "<< arg1<< " other objects."<< "\n\n"<< "(Additional information: "<< arg3<< ")\n\n"<< "See the entry in the Frequently Asked Questions of "<< "deal.II (linked to from http://www.dealii.org/) for "<< "a lot more information on what this error means and "<< "how to fix programs in which it happens.")
 
 DeclException2 (ExcNoSubscriber, char *, char *,<< "No subscriber with identifier <"<< arg2<< "> subscribes to this object of class "<< arg1<< ". Consequently, it cannot be unsubscribed.")
 
template<class Archive >
void serialize (Archive &ar, const unsigned int version)
 

Protected Member Functions

virtual double criterion ()
 
virtual void print_vectors (const unsigned int step, const VECTOR &x, const VECTOR &r, const VECTOR &d) const
 

Static Protected Member Functions

static void compute_eigs_and_cond (const std::vector< double > &diagonal, const std::vector< double > &offdiagonal, const boost::signals2::signal< void(const std::vector< double > &)> &eigenvalues_signal, const boost::signals2::signal< void(double)> &cond_signal, const bool log_eigenvalues, const bool log_cond)
 

Protected Attributes

VECTOR * Vr
 
VECTOR * Vp
 
VECTOR * Vz
 
double res2
 
AdditionalData additional_data
 
boost::signals2::signal< void(double, double)> 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
 
- Protected Attributes inherited from Solver< VECTOR >
GrowingVectorMemory< VECTOR > static_vector_memory
 
VectorMemory< VECTOR > & memory
 
boost::signals2::signal< SolverControl::State(const unsigned int iteration, const double check_value, const VECTOR &current_iterate), StateCombineriteration_status
 

Private Member Functions

void cleanup ()
 

Detailed Description

template<class VECTOR = Vector<double>>
class SolverCG< VECTOR >

Preconditioned cg method for symmetric positive definite matrices. This class is used first in step-3 and step-4, 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>.

Like all other solver classes, this class has a local structure called AdditionalData which is used to pass additional parameters to the solver. For this class, there is (among other things) a switch allowing for additional output for the computation of eigenvalues of the matrix.

Note
This version of CG is taken from D. Braess's book "Finite Elements". It requires a symmetric preconditioner (i.e., for example, SOR is not a possible choice).

Eigenvalue computation

The cg-method performs an orthogonal projection of the original preconditioned linear system to another system of smaller dimension. Furthermore, the projected matrix T is tri-diagonal. 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 tri-diagonal matrix with diagonal elements 1/alpha_0, 1/alpha_1 + beta_0/alpha_0, ..., 1/alpha_{m-1+beta_{m-2}/alpha_{m-2}} and off-diagonal elements sqrt(beta_0)/alpha_0, ..., sqrt(beta_{m-2)/alpha_{m-2}}. The eigenvalues of this matrix can be computed by postprocessing.

See also
Y. Saad: "Iterative methods for Sparse Linear Systems", section 6.7.3 for details.

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.

Deprecated:
Alternatively these estimates can be written to deallog by setting flags in AdditionalData.

Observing the progress of linear solver iterations

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.

Author
W. Bangerth, G. Kanschat, R. Becker and F.-T. Suttmeier

Definition at line 99 of file solver_cg.h.

Member Typedef Documentation

template<class VECTOR = Vector<double>>
typedef types::global_dof_index SolverCG< VECTOR >::size_type

Declare type for container size.

Definition at line 105 of file solver_cg.h.

Constructor & Destructor Documentation

template<class VECTOR = Vector<double>>
SolverCG< VECTOR >::SolverCG ( SolverControl cn,
VectorMemory< VECTOR > &  mem,
const AdditionalData data = AdditionalData() 
)

Constructor.

template<class VECTOR = Vector<double>>
SolverCG< VECTOR >::SolverCG ( SolverControl cn,
const AdditionalData data = AdditionalData() 
)

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

template<class VECTOR = Vector<double>>
virtual SolverCG< VECTOR >::~SolverCG ( )
virtual

Virtual destructor.

Member Function Documentation

template<class VECTOR = Vector<double>>
template<class MATRIX , class PRECONDITIONER >
void SolverCG< VECTOR >::solve ( const MATRIX A,
VECTOR &  x,
const VECTOR &  b,
const PRECONDITIONER &  precondition 
)

Solve the linear system \(Ax=b\) for x.

template<class VECTOR = Vector<double>>
boost::signals2::connection SolverCG< VECTOR >::connect_coefficients_slot ( const std_cxx11::function< void(double, double)> &  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

template<class VECTOR = Vector<double>>
boost::signals2::connection SolverCG< VECTOR >::connect_condition_number_slot ( const std_cxx11::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).

template<class VECTOR = Vector<double>>
boost::signals2::connection SolverCG< VECTOR >::connect_eigenvalues_slot ( const std_cxx11::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).

template<class VECTOR = Vector<double>>
virtual double SolverCG< VECTOR >::criterion ( )
protectedvirtual

Implementation of the computation of the norm of the residual. This can be replaced by a more problem oriented functional in a derived class.

template<class VECTOR = Vector<double>>
virtual void SolverCG< VECTOR >::print_vectors ( const unsigned int  step,
const VECTOR &  x,
const VECTOR &  r,
const VECTOR &  d 
) const
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 a graphical output of the convergence history.

template<class VECTOR = Vector<double>>
static void SolverCG< VECTOR >::compute_eigs_and_cond ( const std::vector< double > &  diagonal,
const std::vector< double > &  offdiagonal,
const boost::signals2::signal< void(const std::vector< double > &)> &  eigenvalues_signal,
const boost::signals2::signal< void(double)> &  cond_signal,
const bool  log_eigenvalues,
const bool  log_cond 
)
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. Outputs the eigenvalues/condition-number to deallog if log_eigenvalues/log_cond is true.

Member Data Documentation

template<class VECTOR = Vector<double>>
VECTOR* SolverCG< VECTOR >::Vr
protected

Temporary vectors, allocated through the VectorMemory object at the start of the actual solution process and deallocated at the end.

Definition at line 253 of file solver_cg.h.

template<class VECTOR = Vector<double>>
double SolverCG< VECTOR >::res2
protected

Within the iteration loop, the square of the residual vector is stored in this variable. The function criterion uses this variable to compute the convergence value, which in this class is the norm of the residual vector and thus the square root of the res2 value.

Definition at line 263 of file solver_cg.h.

template<class VECTOR = Vector<double>>
AdditionalData SolverCG< VECTOR >::additional_data
protected

Additional parameters.

Definition at line 268 of file solver_cg.h.

template<class VECTOR = Vector<double>>
boost::signals2::signal<void (double,double)> SolverCG< VECTOR >::coefficients_signal
protected

Signal used to retrieve the CG coefficients. Called on each iteration.

Definition at line 274 of file solver_cg.h.

template<class VECTOR = Vector<double>>
boost::signals2::signal<void (double)> SolverCG< VECTOR >::condition_number_signal
protected

Signal used to retrieve the estimated condition number. Called once when all iterations are ended.

Definition at line 280 of file solver_cg.h.

template<class VECTOR = Vector<double>>
boost::signals2::signal<void (double)> SolverCG< VECTOR >::all_condition_numbers_signal
protected

Signal used to retrieve the estimated condition numbers. Called on each iteration.

Definition at line 286 of file solver_cg.h.

template<class VECTOR = Vector<double>>
boost::signals2::signal<void (const std::vector<double> &)> SolverCG< VECTOR >::eigenvalues_signal
protected

Signal used to retrieve the estimated eigenvalues. Called once when all iterations are ended.

Definition at line 292 of file solver_cg.h.

template<class VECTOR = Vector<double>>
boost::signals2::signal<void (const std::vector<double> &)> SolverCG< VECTOR >::all_eigenvalues_signal
protected

Signal used to retrieve the estimated eigenvalues. Called on each iteration.

Definition at line 298 of file solver_cg.h.


The documentation for this class was generated from the following file: