Reference documentation for deal.II version 9.3.3

#include <deal.II/lac/solver_qmrs.h>
Classes  
struct  AdditionalData 
struct  IterationResult 
Public Types  
using  vector_type = Vector< double > 
Public Member Functions  
SolverQMRS (SolverControl &cn, VectorMemory< VectorType > &mem, const AdditionalData &data=AdditionalData())  
SolverQMRS (SolverControl &cn, const AdditionalData &data=AdditionalData())  
template<typename MatrixType , typename PreconditionerType >  
void  solve (const MatrixType &A, VectorType &x, const VectorType &b, const PreconditionerType &preconditioner) 
virtual void  print_vectors (const unsigned int step, const VectorType &x, const VectorType &r, const VectorType &d) const 
boost::signals2::connection  connect (const std::function< SolverControl::State(const unsigned int iteration, const double check_value, const Vector< double > ¤t_iterate)> &slot) 
Protected Attributes  
AdditionalData  additional_data 
GrowingVectorMemory< Vector< double > >  static_vector_memory 
VectorMemory< Vector< double > > &  memory 
boost::signals2::signal< SolverControl::State(const unsigned int iteration, const double check_value, const Vector< double > ¤t_iterate), StateCombiner >  iteration_status 
Private Member Functions  
template<typename MatrixType , typename PreconditionerType >  
IterationResult  iterate (const MatrixType &A, VectorType &x, const VectorType &b, const PreconditionerType &preconditioner, VectorType &r, VectorType &u, VectorType &q, VectorType &t, VectorType &d) 
Private Attributes  
unsigned int  step 
Subscriptor functionality  
Classes derived from Subscriptor provide a facility to subscribe to this object. This is mostly used by the SmartPointer class.  
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 
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) 
void  check_no_subscribers () const noexcept 
using  map_value_type = decltype(counter_map)::value_type 
using  map_iterator = decltype(counter_map)::iterator 
static std::mutex  mutex 
static ::ExceptionBase &  ExcInUse (int arg1, std::string arg2, std::string arg3) 
static ::ExceptionBase &  ExcNoSubscriber (std::string arg1, std::string arg2) 
The SQMR (symmetric quasiminimal residual) method is supposed to solve symmetric indefinite linear systems with symmetric, not necessarily definite preconditioners. It is a variant of the original quasiminimal residual method (QMR) and produces the same iterative solution. This version of SQMR is adapted from the respective symmetric QMRfromBiCG algorithm given by both Freund/Nachtigal: A new Krylovsubspace method for symmetric indefinite linear systems, NASA STI/Recon Technical Report N, 95 (1994) and Freund/Nachtigal: Software for simplified Lanczos and QMR algorithms, Appl. Num. Math. 19 (1995), pp. 319341 and provides both right and left (but not split) preconditioning.
Note, that the QMR implementation that the given algorithm is based on is derived from classical BiCG. It can be shown (Freund/Szeto: A transposefree quasiminimal residual squared algorithm for nonHermitian linear systems, Advances in Computer Methods for Partial Differential Equations VII (IMACS, New Brunswick, NJ, 1992) pp. 258264) that the QMR iterates can be generated from the BiCG iteration through one additional vector and some scalar updates. Possible breakdowns (or precisely, divisions by zero) of BiCG therefore obviously transfer to this simple nolookahead algorithm.
In return the algorithm is cheap compared to classical QMR or BiCGStab, using only one matrixvector product with the system matrix and one application of the preconditioner per iteration respectively.
The residual used for measuring convergence is only approximately calculated by an upper bound. If this value comes below a threshold prescribed within the AdditionalData struct, then the exact residual of the current QMR iterate will be calculated using another multiplication with the system matrix. By experience (according to Freund and Nachtigal) this technique is useful for a threshold that is ten times the solving tolerance, and in that case will be only used in the last one or two steps of the complete iteration.
For the requirements on matrices and vectors in order to work with this class, see the documentation of the Solver base class.
Like all other solver classes, this class has a local structure called AdditionalData
which is used to pass additional parameters to the solver, like damping parameters or the number of temporary vectors. We use this additional structure instead of passing these values directly to the constructor because this makes the use of the SolverSelector
and other classes much easier and guarantees that these will continue to work even if number or type of the additional parameters for a certain solver changes.
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.
Definition at line 92 of file solver_qmrs.h.

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

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

) 
Constructor. Use an object of type GrowingVectorMemory as a default to allocate memory.
void SolverQMRS< VectorType >::solve  (  const MatrixType &  A, 
VectorType &  x,  
const VectorType &  b,  
const PreconditionerType &  preconditioner  
) 
Solve the linear system \(Ax=b\) for x.

virtual 
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.

private 
The iteration loop itself. The function returns a structure indicating what happened in this function.

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 197 of file solver_qmrs.h.

private 
Number of the current iteration (accumulated over restarts)
Definition at line 232 of file solver_qmrs.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.