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Ifpack_Polynomial Class Reference

Ifpack_Polynomial: class for preconditioning with least squares polynomials in Ifpack. More...

#include <Ifpack_Polynomial.h>

Inheritance diagram for Ifpack_Polynomial:
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Public Member Functions

virtual int SetUseTranspose (bool UseTranspose_in)
 
 Ifpack_Polynomial (const Epetra_Operator *Matrix)
 Ifpack_Polynomial constructor with given Epetra_Operator/Epetra_RowMatrix. More...
 
 Ifpack_Polynomial (const Epetra_RowMatrix *Matrix)
 Ifpack_Polynomial constructor with given Epetra_Operator/Epetra_RowMatrix. More...
 
virtual ~Ifpack_Polynomial ()
 Destructor. More...
 
virtual int Apply (const Epetra_MultiVector &X, Epetra_MultiVector &Y) const
 Applies the matrix to an Epetra_MultiVector. More...
 
virtual int ApplyInverse (const Epetra_MultiVector &X, Epetra_MultiVector &Y) const
 Applies the preconditioner to X, returns the result in Y. More...
 
virtual double NormInf () const
 Returns the infinity norm of the global matrix (not implemented) More...
 
virtual const char * Label () const
 
virtual bool UseTranspose () const
 Returns the current UseTranspose setting. More...
 
virtual bool HasNormInf () const
 Returns true if the this object can provide an approximate Inf-norm, false otherwise. More...
 
virtual const Epetra_CommComm () const
 Returns a pointer to the Epetra_Comm communicator associated with this operator. More...
 
virtual const Epetra_MapOperatorDomainMap () const
 Returns the Epetra_Map object associated with the domain of this operator. More...
 
virtual const Epetra_MapOperatorRangeMap () const
 Returns the Epetra_Map object associated with the range of this operator. More...
 
virtual int Initialize ()
 Computes all it is necessary to initialize the preconditioner. More...
 
virtual bool IsInitialized () const
 Returns true if the preconditioner has been successfully initialized, false otherwise. More...
 
virtual bool IsComputed () const
 Returns true if the preconditioner has been successfully computed. More...
 
virtual int Compute ()
 Computes the preconditioners. More...
 
virtual const Epetra_RowMatrixMatrix () const
 Returns a pointer to the matrix to be preconditioned. More...
 
virtual double Condest (const Ifpack_CondestType CT=Ifpack_Cheap, const int MaxIters=1550, const double Tol=1e-9, Epetra_RowMatrix *Matrix_in=0)
 Computes the condition number estimates and returns the value. More...
 
virtual double Condest () const
 Returns the condition number estimate, or -1.0 if not computed. More...
 
virtual int SetParameters (Teuchos::ParameterList &List)
 Sets all the parameters for the preconditioner. More...
 
virtual std::ostream & Print (std::ostream &os) const
 Prints object to an output stream. More...
 
virtual int NumInitialize () const
 Returns the number of calls to Initialize(). More...
 
virtual int NumCompute () const
 Returns the number of calls to Compute(). More...
 
virtual int NumApplyInverse () const
 Returns the number of calls to ApplyInverse(). More...
 
virtual double InitializeTime () const
 Returns the time spent in Initialize(). More...
 
virtual double ComputeTime () const
 Returns the time spent in Compute(). More...
 
virtual double ApplyInverseTime () const
 Returns the time spent in ApplyInverse(). More...
 
virtual double InitializeFlops () const
 Returns the number of flops in the initialization phase. More...
 
virtual double ComputeFlops () const
 Returns the number of flops in the computation phase. More...
 
virtual double ApplyInverseFlops () const
 Returns the number of flops for the application of the preconditioner. More...
 
int GMRES (const Epetra_Operator &Operator, const Epetra_Vector &InvPointDiagonal, const int MaximumIterations, double &lambda_real_min, double &lambda_real_max, double &lambda_imag_min, double &lambda_imag_max)
 Uses AztecOO's GMRES to estimate the height and width of the spectrum. More...
 
static int PowerMethod (const Epetra_Operator &Operator, const Epetra_Vector &InvPointDiagonal, const int MaximumIterations, double &LambdaMax)
 Simple power method to compute lambda_max. More...
 
static int CG (const Epetra_Operator &Operator, const Epetra_Vector &InvPointDiagonal, const int MaximumIterations, double &lambda_min, double &lambda_max)
 Uses AztecOO's CG to estimate lambda_min and lambda_max. More...
 
bool IsInitialized_
 If true, the preconditioner has been computed successfully. More...
 
bool IsComputed_
 If true, the preconditioner has been computed successfully. More...
 
bool IsIndefinite_
 If true, have to compute polynomial for a spectrum with negative eigenvalues. More...
 
bool IsComplex_
 If true, have to compute polynomial for a spectrum with nonzero imaginary part. More...
 
int NumInitialize_
 Contains the number of successful calls to Initialize(). More...
 
int NumCompute_
 Contains the number of successful call to Compute(). More...
 
int NumApplyInverse_
 Contains the number of successful call to ApplyInverse(). More...
 
double InitializeTime_
 Contains the time for all successful calls to Initialize(). More...
 
double ComputeTime_
 Contains the time for all successful calls to Compute(). More...
 
double ApplyInverseTime_
 Contains the time for all successful calls to ApplyInverse(). More...
 
double ComputeFlops_
 Contains the number of flops for Compute(). More...
 
double ApplyInverseFlops_
 Contain sthe number of flops for ApplyInverse(). More...
 
virtual void SetLabel ()
 Sets the label. More...
 
 Ifpack_Polynomial (const Ifpack_Polynomial &)
 Copy constructor (PRIVATE, should not be used) More...
 
Ifpack_Polynomialoperator= (const Ifpack_Polynomial &)
 operator = (PRIVATE, should not be used) More...
 
int PolyDegree_
 Contains the degree of the least squares polynomial. More...
 
int LSPointsReal_
 Contains the number of discretization points of the least squares problem. More...
 
int LSPointsImag_
 
bool UseTranspose_
 If true, use the tranpose of Matrix_. More...
 
double Condest_
 Contains the estimated condition number. More...
 
double RealEigRatio_
 Contains the ratio such that the rectangular domain in the complex plane is [-LambdaRealMax_ / EigRatio_, LambdaRealMax_] x [-LambdaRealMax_ / ImagEigRatio_, LambdaRealMax_ / ImagEigRatio_]. More...
 
double ImagEigRatio_
 
int EigMaxIters_
 Max number of iterations to use in eigenvalue estimation (if automatic). More...
 
std::string Label_
 Contains the label of this object. More...
 
double LambdaRealMin_
 Bounds on the spectrum. More...
 
double LambdaRealMax_
 
double LambdaImagMin_
 
double LambdaImagMax_
 
double MinDiagonalValue_
 Contains the minimum value on the diagonal. More...
 
std::vector< double > coeff_
 coefficients of the polynomial More...
 
int NumMyRows_
 Number of local rows. More...
 
int NumMyNonzeros_
 Number of local nonzeros. More...
 
long long NumGlobalRows_
 Number of global rows. More...
 
long long NumGlobalNonzeros_
 Number of global nonzeros. More...
 
Teuchos::RefCountPtr< const
Epetra_Operator
Operator_
 Pointers to the matrix to be preconditioned as an Epetra_Operator. More...
 
Teuchos::RefCountPtr< const
Epetra_RowMatrix
Matrix_
 Pointers to the matrix to be preconditioned as an Epetra_RowMatrix. More...
 
Teuchos::RefCountPtr
< Epetra_Vector
InvDiagonal_
 Contains the inverse of diagonal elements of Matrix. More...
 
bool UseBlockMode_
 Use Block Preconditioning. More...
 
bool SolveNormalEquations_
 Run on the normal equations. More...
 
bool IsRowMatrix_
 If true, the Operator_ is an Epetra_RowMatrix. More...
 
Teuchos::RefCountPtr< Epetra_TimeTime_
 Time object to track timing. More...
 
bool ZeroStartingSolution_
 If true, the starting solution is always the zero vector. More...
 

Detailed Description

Ifpack_Polynomial: class for preconditioning with least squares polynomials in Ifpack.

The Ifpack_Polynomial class enables the construction of preconditioners based on least squares polynomials for an Epetra_RowMatrix. Similar to Ifpack_Chebyshev, Ifpack_Polynomial is designed for indefinite linear systems.

The list of parameters is

Author
Paul Tsuji, May 2013.

Definition at line 99 of file Ifpack_Polynomial.h.

Constructor & Destructor Documentation

Ifpack_Polynomial::Ifpack_Polynomial ( const Epetra_Operator Matrix)

Ifpack_Polynomial constructor with given Epetra_Operator/Epetra_RowMatrix.

Creates an instance of Ifpack_Polynomial class.

Parameters
Matrix- (In) Pointer to the operator to precondition.

Definition at line 86 of file Ifpack_Polynomial.cpp.

Ifpack_Polynomial::Ifpack_Polynomial ( const Epetra_RowMatrix Matrix)

Ifpack_Polynomial constructor with given Epetra_Operator/Epetra_RowMatrix.

Creates an instance of Ifpack_Polynomial class.

Parameters
Matrix- (In) Pointer to the matrix to precondition.

Definition at line 135 of file Ifpack_Polynomial.cpp.

virtual Ifpack_Polynomial::~Ifpack_Polynomial ( )
inlinevirtual

Destructor.

Definition at line 121 of file Ifpack_Polynomial.h.

Ifpack_Polynomial::Ifpack_Polynomial ( const Ifpack_Polynomial )
inlineprivate

Copy constructor (PRIVATE, should not be used)

Definition at line 340 of file Ifpack_Polynomial.h.

Member Function Documentation

virtual int Ifpack_Polynomial::SetUseTranspose ( bool  UseTranspose_in)
inlinevirtual

This flag can be used to apply the preconditioner to the transpose of the input operator.

Returns
Integer error code, set to 0 if successful. Set to -1 if this implementation does not support transpose.

Implements Epetra_Operator.

Definition at line 131 of file Ifpack_Polynomial.h.

int Ifpack_Polynomial::Apply ( const Epetra_MultiVector X,
Epetra_MultiVector Y 
) const
inlinevirtual

Applies the matrix to an Epetra_MultiVector.

Parameters
X- (In) A Epetra_MultiVector of dimension NumVectors to multiply with matrix.
Y- (Out) A Epetra_MultiVector of dimension NumVectors containing the result.
Returns
Integer error code, set to 0 if successful.

Implements Epetra_Operator.

Definition at line 252 of file Ifpack_Polynomial.cpp.

int Ifpack_Polynomial::ApplyInverse ( const Epetra_MultiVector X,
Epetra_MultiVector Y 
) const
virtual

Applies the preconditioner to X, returns the result in Y.

Parameters
X- (In) A Epetra_MultiVector of dimension NumVectors to be preconditioned.
Y- (InOut) A Epetra_MultiVector of dimension NumVectors containing result.
Returns
Integer error code, set to 0 if successful.
Warning
This routine is NOT AztecOO complaint.

Implements Ifpack_Preconditioner.

Definition at line 628 of file Ifpack_Polynomial.cpp.

virtual double Ifpack_Polynomial::NormInf ( ) const
inlinevirtual

Returns the infinity norm of the global matrix (not implemented)

Implements Epetra_Operator.

Definition at line 166 of file Ifpack_Polynomial.h.

virtual const char* Ifpack_Polynomial::Label ( ) const
inlinevirtual

Implements Epetra_Operator.

Definition at line 174 of file Ifpack_Polynomial.h.

virtual bool Ifpack_Polynomial::UseTranspose ( ) const
inlinevirtual

Returns the current UseTranspose setting.

Implements Epetra_Operator.

Definition at line 180 of file Ifpack_Polynomial.h.

virtual bool Ifpack_Polynomial::HasNormInf ( ) const
inlinevirtual

Returns true if the this object can provide an approximate Inf-norm, false otherwise.

Implements Epetra_Operator.

Definition at line 186 of file Ifpack_Polynomial.h.

const Epetra_Comm & Ifpack_Polynomial::Comm ( ) const
virtual

Returns a pointer to the Epetra_Comm communicator associated with this operator.

Implements Epetra_Operator.

Definition at line 233 of file Ifpack_Polynomial.cpp.

const Epetra_Map & Ifpack_Polynomial::OperatorDomainMap ( ) const
virtual

Returns the Epetra_Map object associated with the domain of this operator.

Implements Epetra_Operator.

Definition at line 239 of file Ifpack_Polynomial.cpp.

const Epetra_Map & Ifpack_Polynomial::OperatorRangeMap ( ) const
virtual

Returns the Epetra_Map object associated with the range of this operator.

Implements Epetra_Operator.

Definition at line 245 of file Ifpack_Polynomial.cpp.

int Ifpack_Polynomial::Initialize ( )
virtual

Computes all it is necessary to initialize the preconditioner.

Implements Ifpack_Preconditioner.

Definition at line 273 of file Ifpack_Polynomial.cpp.

virtual bool Ifpack_Polynomial::IsInitialized ( ) const
inlinevirtual

Returns true if the preconditioner has been successfully initialized, false otherwise.

Implements Ifpack_Preconditioner.

Definition at line 202 of file Ifpack_Polynomial.h.

virtual bool Ifpack_Polynomial::IsComputed ( ) const
inlinevirtual

Returns true if the preconditioner has been successfully computed.

Implements Ifpack_Preconditioner.

Definition at line 208 of file Ifpack_Polynomial.h.

int Ifpack_Polynomial::Compute ( )
virtual

Computes the preconditioners.

Implements Ifpack_Preconditioner.

Definition at line 307 of file Ifpack_Polynomial.cpp.

virtual const Epetra_RowMatrix& Ifpack_Polynomial::Matrix ( ) const
inlinevirtual

Returns a pointer to the matrix to be preconditioned.

Implements Ifpack_Preconditioner.

Definition at line 218 of file Ifpack_Polynomial.h.

double Ifpack_Polynomial::Condest ( const Ifpack_CondestType  CT = Ifpack_Cheap,
const int  MaxIters = 1550,
const double  Tol = 1e-9,
Epetra_RowMatrix Matrix_in = 0 
)
virtual

Computes the condition number estimates and returns the value.

Implements Ifpack_Preconditioner.

Definition at line 606 of file Ifpack_Polynomial.cpp.

virtual double Ifpack_Polynomial::Condest ( ) const
inlinevirtual

Returns the condition number estimate, or -1.0 if not computed.

Implements Ifpack_Preconditioner.

Definition at line 230 of file Ifpack_Polynomial.h.

int Ifpack_Polynomial::SetParameters ( Teuchos::ParameterList List)
virtual

Sets all the parameters for the preconditioner.

Implements Ifpack_Preconditioner.

Definition at line 177 of file Ifpack_Polynomial.cpp.

std::ostream & Ifpack_Polynomial::Print ( std::ostream &  os) const
virtual

Prints object to an output stream.

Implements Ifpack_Preconditioner.

Definition at line 548 of file Ifpack_Polynomial.cpp.

virtual int Ifpack_Polynomial::NumInitialize ( ) const
inlinevirtual

Returns the number of calls to Initialize().

Implements Ifpack_Preconditioner.

Definition at line 246 of file Ifpack_Polynomial.h.

virtual int Ifpack_Polynomial::NumCompute ( ) const
inlinevirtual

Returns the number of calls to Compute().

Implements Ifpack_Preconditioner.

Definition at line 252 of file Ifpack_Polynomial.h.

virtual int Ifpack_Polynomial::NumApplyInverse ( ) const
inlinevirtual

Returns the number of calls to ApplyInverse().

Implements Ifpack_Preconditioner.

Definition at line 258 of file Ifpack_Polynomial.h.

virtual double Ifpack_Polynomial::InitializeTime ( ) const
inlinevirtual

Returns the time spent in Initialize().

Implements Ifpack_Preconditioner.

Definition at line 264 of file Ifpack_Polynomial.h.

virtual double Ifpack_Polynomial::ComputeTime ( ) const
inlinevirtual

Returns the time spent in Compute().

Implements Ifpack_Preconditioner.

Definition at line 270 of file Ifpack_Polynomial.h.

virtual double Ifpack_Polynomial::ApplyInverseTime ( ) const
inlinevirtual

Returns the time spent in ApplyInverse().

Implements Ifpack_Preconditioner.

Definition at line 276 of file Ifpack_Polynomial.h.

virtual double Ifpack_Polynomial::InitializeFlops ( ) const
inlinevirtual

Returns the number of flops in the initialization phase.

Implements Ifpack_Preconditioner.

Definition at line 282 of file Ifpack_Polynomial.h.

virtual double Ifpack_Polynomial::ComputeFlops ( ) const
inlinevirtual

Returns the number of flops in the computation phase.

Implements Ifpack_Preconditioner.

Definition at line 288 of file Ifpack_Polynomial.h.

virtual double Ifpack_Polynomial::ApplyInverseFlops ( ) const
inlinevirtual

Returns the number of flops for the application of the preconditioner.

Implements Ifpack_Preconditioner.

Definition at line 294 of file Ifpack_Polynomial.h.

int Ifpack_Polynomial::PowerMethod ( const Epetra_Operator Operator,
const Epetra_Vector InvPointDiagonal,
const int  MaximumIterations,
double &  LambdaMax 
)
static

Simple power method to compute lambda_max.

Definition at line 672 of file Ifpack_Polynomial.cpp.

int Ifpack_Polynomial::CG ( const Epetra_Operator Operator,
const Epetra_Vector InvPointDiagonal,
const int  MaximumIterations,
double &  lambda_min,
double &  lambda_max 
)
static

Uses AztecOO's CG to estimate lambda_min and lambda_max.

Definition at line 705 of file Ifpack_Polynomial.cpp.

int Ifpack_Polynomial::GMRES ( const Epetra_Operator Operator,
const Epetra_Vector InvPointDiagonal,
const int  MaximumIterations,
double &  lambda_real_min,
double &  lambda_real_max,
double &  lambda_imag_min,
double &  lambda_imag_max 
)

Uses AztecOO's GMRES to estimate the height and width of the spectrum.

Definition at line 836 of file Ifpack_Polynomial.cpp.

void Ifpack_Polynomial::SetLabel ( )
privatevirtual

Sets the label.

Definition at line 621 of file Ifpack_Polynomial.cpp.

Ifpack_Polynomial& Ifpack_Polynomial::operator= ( const Ifpack_Polynomial )
inlineprivate

operator = (PRIVATE, should not be used)

Definition at line 344 of file Ifpack_Polynomial.h.

Member Data Documentation

bool Ifpack_Polynomial::IsInitialized_
private

If true, the preconditioner has been computed successfully.

Definition at line 351 of file Ifpack_Polynomial.h.

bool Ifpack_Polynomial::IsComputed_
private

If true, the preconditioner has been computed successfully.

Definition at line 353 of file Ifpack_Polynomial.h.

bool Ifpack_Polynomial::IsIndefinite_
private

If true, have to compute polynomial for a spectrum with negative eigenvalues.

Definition at line 355 of file Ifpack_Polynomial.h.

bool Ifpack_Polynomial::IsComplex_
private

If true, have to compute polynomial for a spectrum with nonzero imaginary part.

Definition at line 357 of file Ifpack_Polynomial.h.

int Ifpack_Polynomial::NumInitialize_
private

Contains the number of successful calls to Initialize().

Definition at line 359 of file Ifpack_Polynomial.h.

int Ifpack_Polynomial::NumCompute_
private

Contains the number of successful call to Compute().

Definition at line 361 of file Ifpack_Polynomial.h.

int Ifpack_Polynomial::NumApplyInverse_
mutableprivate

Contains the number of successful call to ApplyInverse().

Definition at line 363 of file Ifpack_Polynomial.h.

double Ifpack_Polynomial::InitializeTime_
private

Contains the time for all successful calls to Initialize().

Definition at line 365 of file Ifpack_Polynomial.h.

double Ifpack_Polynomial::ComputeTime_
private

Contains the time for all successful calls to Compute().

Definition at line 367 of file Ifpack_Polynomial.h.

double Ifpack_Polynomial::ApplyInverseTime_
mutableprivate

Contains the time for all successful calls to ApplyInverse().

Definition at line 369 of file Ifpack_Polynomial.h.

double Ifpack_Polynomial::ComputeFlops_
private

Contains the number of flops for Compute().

Definition at line 371 of file Ifpack_Polynomial.h.

double Ifpack_Polynomial::ApplyInverseFlops_
mutableprivate

Contain sthe number of flops for ApplyInverse().

Definition at line 373 of file Ifpack_Polynomial.h.

int Ifpack_Polynomial::PolyDegree_
private

Contains the degree of the least squares polynomial.

Definition at line 378 of file Ifpack_Polynomial.h.

int Ifpack_Polynomial::LSPointsReal_
private

Contains the number of discretization points of the least squares problem.

Definition at line 380 of file Ifpack_Polynomial.h.

int Ifpack_Polynomial::LSPointsImag_
private

Definition at line 380 of file Ifpack_Polynomial.h.

bool Ifpack_Polynomial::UseTranspose_
private

If true, use the tranpose of Matrix_.

Definition at line 382 of file Ifpack_Polynomial.h.

double Ifpack_Polynomial::Condest_
private

Contains the estimated condition number.

Definition at line 384 of file Ifpack_Polynomial.h.

double Ifpack_Polynomial::RealEigRatio_
private

Contains the ratio such that the rectangular domain in the complex plane is [-LambdaRealMax_ / EigRatio_, LambdaRealMax_] x [-LambdaRealMax_ / ImagEigRatio_, LambdaRealMax_ / ImagEigRatio_].

Definition at line 393 of file Ifpack_Polynomial.h.

double Ifpack_Polynomial::ImagEigRatio_
private

Definition at line 393 of file Ifpack_Polynomial.h.

int Ifpack_Polynomial::EigMaxIters_
private

Max number of iterations to use in eigenvalue estimation (if automatic).

Definition at line 395 of file Ifpack_Polynomial.h.

std::string Ifpack_Polynomial::Label_
private

Contains the label of this object.

Definition at line 397 of file Ifpack_Polynomial.h.

double Ifpack_Polynomial::LambdaRealMin_
private

Bounds on the spectrum.

Definition at line 399 of file Ifpack_Polynomial.h.

double Ifpack_Polynomial::LambdaRealMax_
private

Definition at line 399 of file Ifpack_Polynomial.h.

double Ifpack_Polynomial::LambdaImagMin_
private

Definition at line 399 of file Ifpack_Polynomial.h.

double Ifpack_Polynomial::LambdaImagMax_
private

Definition at line 399 of file Ifpack_Polynomial.h.

double Ifpack_Polynomial::MinDiagonalValue_
private

Contains the minimum value on the diagonal.

Definition at line 401 of file Ifpack_Polynomial.h.

std::vector<double> Ifpack_Polynomial::coeff_
private

coefficients of the polynomial

Definition at line 403 of file Ifpack_Polynomial.h.

int Ifpack_Polynomial::NumMyRows_
private

Number of local rows.

Definition at line 407 of file Ifpack_Polynomial.h.

int Ifpack_Polynomial::NumMyNonzeros_
private

Number of local nonzeros.

Definition at line 409 of file Ifpack_Polynomial.h.

long long Ifpack_Polynomial::NumGlobalRows_
private

Number of global rows.

Definition at line 411 of file Ifpack_Polynomial.h.

long long Ifpack_Polynomial::NumGlobalNonzeros_
private

Number of global nonzeros.

Definition at line 413 of file Ifpack_Polynomial.h.

Teuchos::RefCountPtr<const Epetra_Operator> Ifpack_Polynomial::Operator_
private

Pointers to the matrix to be preconditioned as an Epetra_Operator.

Definition at line 415 of file Ifpack_Polynomial.h.

Teuchos::RefCountPtr<const Epetra_RowMatrix> Ifpack_Polynomial::Matrix_
private

Pointers to the matrix to be preconditioned as an Epetra_RowMatrix.

Definition at line 417 of file Ifpack_Polynomial.h.

Teuchos::RefCountPtr<Epetra_Vector> Ifpack_Polynomial::InvDiagonal_
mutableprivate

Contains the inverse of diagonal elements of Matrix.

Definition at line 419 of file Ifpack_Polynomial.h.

bool Ifpack_Polynomial::UseBlockMode_
private

Use Block Preconditioning.

Definition at line 421 of file Ifpack_Polynomial.h.

bool Ifpack_Polynomial::SolveNormalEquations_
private

Run on the normal equations.

Definition at line 430 of file Ifpack_Polynomial.h.

bool Ifpack_Polynomial::IsRowMatrix_
private

If true, the Operator_ is an Epetra_RowMatrix.

Definition at line 433 of file Ifpack_Polynomial.h.

Teuchos::RefCountPtr<Epetra_Time> Ifpack_Polynomial::Time_
private

Time object to track timing.

Definition at line 435 of file Ifpack_Polynomial.h.

bool Ifpack_Polynomial::ZeroStartingSolution_
private

If true, the starting solution is always the zero vector.

Definition at line 437 of file Ifpack_Polynomial.h.


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