Teuchos::SerialDenseSolver< OrdinalType, ScalarType > Class Template Reference

A class for solving dense linear problems. More...

#include <Teuchos_SerialDenseSolver.hpp>

Inheritance diagram for Teuchos::SerialDenseSolver< OrdinalType, ScalarType >:

Teuchos::CompObject Teuchos::Object Teuchos::BLAS< OrdinalType, ScalarType > Teuchos::LAPACK< OrdinalType, ScalarType > List of all members.

Public Types

typedef Teuchos::ScalarTraits<
ScalarType >::magnitudeType 
MagnitudeType

Public Member Functions

Constructor/Destructor Methods
 SerialDenseSolver ()
 Default constructor; matrix should be set using setMatrix(), LHS and RHS set with setVectors().
virtual ~SerialDenseSolver ()
 SerialDenseSolver destructor.
Set Methods
int setMatrix (const RCP< SerialDenseMatrix< OrdinalType, ScalarType > > &A)
 Sets the pointers for coefficient matrix.
int setVectors (const RCP< SerialDenseMatrix< OrdinalType, ScalarType > > &X, const RCP< SerialDenseMatrix< OrdinalType, ScalarType > > &B)
 Sets the pointers for left and right hand side vector(s).
Strategy Modifying Methods
void factorWithEquilibration (bool flag)
 Causes equilibration to be called just before the matrix factorization as part of the call to factor.
void solveWithTranspose (bool flag)
 If flag is true, causes all subsequent function calls to work with the transpose of this matrix, otherwise not.
void solveToRefinedSolution (bool flag)
 Causes all solves to compute solution to best ability using iterative refinement.
void estimateSolutionErrors (bool flag)
 Causes all solves to estimate the forward and backward solution error.
Factor/Solve/Invert Methods
int factor ()
 Computes the in-place LU factorization of the matrix using the LAPACK routine _GETRF.
int solve ()
 Computes the solution X to AX = B for the this matrix and the B provided to SetVectors()..
int invert ()
 Inverts the this matrix.
int computeEquilibrateScaling ()
 Computes the scaling vector S(i) = 1/sqrt(A(i,i)) of the this matrix.
int equilibrateMatrix ()
 Equilibrates the this matrix.
int equilibrateRHS ()
 Equilibrates the current RHS.
int applyRefinement ()
 Apply Iterative Refinement.
int unequilibrateLHS ()
 Unscales the solution vectors if equilibration was used to solve the system.
int reciprocalConditionEstimate (MagnitudeType &Value)
 Returns the reciprocal of the 1-norm condition number of the this matrix.
Query methods
bool transpose ()
 Returns true if transpose of this matrix has and will be used.
bool factored ()
 Returns true if matrix is factored (factor available via AF() and LDAF()).
bool equilibratedA ()
 Returns true if factor is equilibrated (factor available via AF() and LDAF()).
bool equilibratedB ()
 Returns true if RHS is equilibrated (RHS available via B() and LDB()).
bool shouldEquilibrate ()
 Returns true if the LAPACK general rules for equilibration suggest you should equilibrate the system.
bool solutionErrorsEstimated ()
 Returns true if forward and backward error estimated have been computed (available via FERR() and BERR()).
bool inverted ()
 Returns true if matrix inverse has been computed (inverse available via AF() and LDAF()).
bool reciprocalConditionEstimated ()
 Returns true if the condition number of the this matrix has been computed (value available via ReciprocalConditionEstimate()).
bool solved ()
 Returns true if the current set of vectors has been solved.
bool solutionRefined ()
 Returns true if the current set of vectors has been refined.
Data Accessor methods
RCP< SerialDenseMatrix< OrdinalType,
ScalarType > > 
getMatrix () const
 Returns pointer to current matrix.
RCP< SerialDenseMatrix< OrdinalType,
ScalarType > > 
getFactoredMatrix () const
 Returns pointer to factored matrix (assuming factorization has been performed).
RCP< SerialDenseMatrix< OrdinalType,
ScalarType > > 
getLHS () const
 Returns pointer to current LHS.
RCP< SerialDenseMatrix< OrdinalType,
ScalarType > > 
getRHS () const
 Returns pointer to current RHS.
OrdinalType numRows () const
 Returns row dimension of system.
OrdinalType numCols () const
 Returns column dimension of system.
std::vector< OrdinalType > IPIV () const
 Returns pointer to pivot vector (if factorization has been computed), zero otherwise.
MagnitudeType ANORM () const
 Returns the 1-Norm of the this matrix (returns -1 if not yet computed).
MagnitudeType RCOND () const
 Returns the reciprocal of the condition number of the this matrix (returns -1 if not yet computed).
MagnitudeType ROWCND () const
 Ratio of smallest to largest row scale factors for the this matrix (returns -1 if not yet computed).
MagnitudeType COLCND () const
 Ratio of smallest to largest column scale factors for the this matrix (returns -1 if not yet computed).
MagnitudeType AMAX () const
 Returns the absolute value of the largest entry of the this matrix (returns -1 if not yet computed).
std::vector< MagnitudeTypeFERR () const
 Returns a pointer to the forward error estimates computed by LAPACK.
std::vector< MagnitudeTypeBERR () const
 Returns a pointer to the backward error estimates computed by LAPACK.
std::vector< MagnitudeTypeR () const
 Returns a pointer to the row scaling vector used for equilibration.
std::vector< MagnitudeTypeC () const
 Returns a pointer to the column scale vector used for equilibration.
I/O methods
void Print (std::ostream &os) const
 Print service methods; defines behavior of ostream << operator.

Protected Member Functions

void allocateWORK ()
void allocateIWORK ()
void resetMatrix ()
void resetVectors ()

Protected Attributes

bool equilibrate_
bool shouldEquilibrate_
bool equilibratedA_
bool equilibratedB_
bool transpose_
bool factored_
bool estimateSolutionErrors_
bool solutionErrorsEstimated_
bool solved_
bool inverted_
bool reciprocalConditionEstimated_
bool refineSolution_
bool solutionRefined_
Teuchos::ETransp TRANS_
OrdinalType M_
OrdinalType N_
OrdinalType Min_MN_
OrdinalType LDA_
OrdinalType LDAF_
OrdinalType INFO_
OrdinalType LWORK_
std::vector< OrdinalType > IPIV_
std::vector< int > IWORK_
MagnitudeType ANORM_
MagnitudeType RCOND_
MagnitudeType ROWCND_
MagnitudeType COLCND_
MagnitudeType AMAX_
RCP< SerialDenseMatrix< OrdinalType,
ScalarType > > 
Matrix_
RCP< SerialDenseMatrix< OrdinalType,
ScalarType > > 
LHS_
RCP< SerialDenseMatrix< OrdinalType,
ScalarType > > 
RHS_
RCP< SerialDenseMatrix< OrdinalType,
ScalarType > > 
Factor_
ScalarType * A_
ScalarType * AF_
std::vector< MagnitudeTypeFERR_
std::vector< MagnitudeTypeBERR_
std::vector< ScalarType > WORK_
std::vector< MagnitudeTypeR_
std::vector< MagnitudeTypeC_

Detailed Description

template<typename OrdinalType, typename ScalarType>
class Teuchos::SerialDenseSolver< OrdinalType, ScalarType >

A class for solving dense linear problems.

The Teuchos::SerialDenseSolver class enables the definition, in terms of Teuchos::SerialDenseMatrix and Teuchos::SerialDenseVector objects, of a dense linear problem, followed by the solution of that problem via the most sophisticated techniques available in LAPACK.

The Teuchos::SerialDenseSolver class is intended to provide full-featured support for solving linear problems for general dense rectangular (or square) matrices. It is written on top of BLAS and LAPACK and thus has excellent performance and numerical capabilities. Using this class, one can either perform simple factorizations and solves or apply all the tricks available in LAPACK to get the best possible solution for very ill-conditioned problems.

Teuchos::SerialDenseSolver vs. Teuchos::LAPACK

The Teuchos::LAPACK class provides access to most of the same functionality as Teuchos::SerialDenseSolver. The primary difference is that Teuchos::LAPACK is a "thin" layer on top of LAPACK and Teuchos::SerialDenseSolver attempts to provide easy access to the more sophisticated aspects of solving dense linear and eigensystems.

Constructing Teuchos::SerialDenseSolver Objects

There is a single Teuchos::SerialDenseSolver constructor. However, the matrix, right hand side and solution vectors must be set prior to executing most methods in this class.

Setting vectors used for linear solves

The matrix A, the left hand side X and the right hand side B (when solving AX = B, for X), can be set by appropriate set methods. Each of these three objects must be an Teuchos::SerialDenseMatrix or and Teuchos::SerialDenseVector object. The set methods are as follows:

Vector and Utility Functions

Once a Teuchos::SerialDenseSolver is constructed, several mathematical functions can be applied to the object. Specifically:

Strategies for Solving Linear Systems In many cases, linear systems can be accurately solved by simply computing the LU factorization of the matrix and then performing a forward back solve with a given set of right hand side vectors. However, in some instances, the factorization may be very poorly conditioned and this simple approach may not work. In these situations, equilibration and iterative refinement may improve the accuracy, or prevent a breakdown in the factorization.

Teuchos::SerialDenseSolver will use equilibration with the factorization if, once the object is constructed and before it is factored, you call the function factorWithEquilibration(true) to force equilibration to be used. If you are uncertain if equilibration should be used, you may call the function shouldEquilibrate() which will return true if equilibration could possibly help. shouldEquilibrate() uses guidelines specified in the LAPACK User Guide, namely if SCOND < 0.1 and AMAX < Underflow or AMAX > Overflow, to determine if equilibration might be useful.

Teuchos::SerialDenseSolver will use iterative refinement after a forward/back solve if you call solveToRefinedSolution(true). It will also compute forward and backward error estimates if you call estimateSolutionErrors(true). Access to the forward (back) error estimates is available via FERR() (BERR()).

Examples using Teuchos::SerialDenseSolver can be found in the Teuchos test directories.

Examples:

DenseMatrix/cxx_main.cpp.

Definition at line 119 of file Teuchos_SerialDenseSolver.hpp.


Constructor & Destructor Documentation

template<typename OrdinalType, typename ScalarType>
Teuchos::SerialDenseSolver< OrdinalType, ScalarType >::SerialDenseSolver (  ) 

Default constructor; matrix should be set using setMatrix(), LHS and RHS set with setVectors().

Definition at line 392 of file Teuchos_SerialDenseSolver.hpp.

template<typename OrdinalType, typename ScalarType>
Teuchos::SerialDenseSolver< OrdinalType, ScalarType >::~SerialDenseSolver (  )  [virtual]

SerialDenseSolver destructor.

Definition at line 429 of file Teuchos_SerialDenseSolver.hpp.


Member Function Documentation

template<typename OrdinalType, typename ScalarType>
int Teuchos::SerialDenseSolver< OrdinalType, ScalarType >::setMatrix ( const RCP< SerialDenseMatrix< OrdinalType, ScalarType > > &  A  ) 

Sets the pointers for coefficient matrix.

Definition at line 474 of file Teuchos_SerialDenseSolver.hpp.

template<typename OrdinalType, typename ScalarType>
int Teuchos::SerialDenseSolver< OrdinalType, ScalarType >::setVectors ( const RCP< SerialDenseMatrix< OrdinalType, ScalarType > > &  X,
const RCP< SerialDenseMatrix< OrdinalType, ScalarType > > &  B 
)

Sets the pointers for left and right hand side vector(s).

Row dimension of X must match column dimension of matrix A, row dimension of B must match row dimension of A. X and B must have the same dimensions.

Definition at line 491 of file Teuchos_SerialDenseSolver.hpp.

template<typename OrdinalType, typename ScalarType>
void Teuchos::SerialDenseSolver< OrdinalType, ScalarType >::factorWithEquilibration ( bool  flag  )  [inline]

Causes equilibration to be called just before the matrix factorization as part of the call to factor.

This function must be called before the factorization is performed.

Definition at line 156 of file Teuchos_SerialDenseSolver.hpp.

template<typename OrdinalType, typename ScalarType>
void Teuchos::SerialDenseSolver< OrdinalType, ScalarType >::solveWithTranspose ( bool  flag  )  [inline]

If flag is true, causes all subsequent function calls to work with the transpose of this matrix, otherwise not.

Definition at line 159 of file Teuchos_SerialDenseSolver.hpp.

template<typename OrdinalType, typename ScalarType>
void Teuchos::SerialDenseSolver< OrdinalType, ScalarType >::solveToRefinedSolution ( bool  flag  )  [inline]

Causes all solves to compute solution to best ability using iterative refinement.

Definition at line 162 of file Teuchos_SerialDenseSolver.hpp.

template<typename OrdinalType, typename ScalarType>
void Teuchos::SerialDenseSolver< OrdinalType, ScalarType >::estimateSolutionErrors ( bool  flag  ) 

Causes all solves to estimate the forward and backward solution error.

Error estimates will be in the arrays FERR and BERR, resp, after the solve step is complete. These arrays are accessible via the FERR() and BERR() access functions.

Definition at line 514 of file Teuchos_SerialDenseSolver.hpp.

template<typename OrdinalType, typename ScalarType>
int Teuchos::SerialDenseSolver< OrdinalType, ScalarType >::factor (  ) 

Computes the in-place LU factorization of the matrix using the LAPACK routine _GETRF.

Returns:
Integer error code, set to 0 if successful.

Definition at line 524 of file Teuchos_SerialDenseSolver.hpp.

template<typename OrdinalType, typename ScalarType>
int Teuchos::SerialDenseSolver< OrdinalType, ScalarType >::solve (  ) 

Computes the solution X to AX = B for the this matrix and the B provided to SetVectors()..

Returns:
Integer error code, set to 0 if successful.

Definition at line 561 of file Teuchos_SerialDenseSolver.hpp.

template<typename OrdinalType, typename ScalarType>
int Teuchos::SerialDenseSolver< OrdinalType, ScalarType >::invert (  ) 

Inverts the this matrix.

Returns:
Integer error code, set to 0 if successful. Otherwise returns the LAPACK error code INFO.

Definition at line 776 of file Teuchos_SerialDenseSolver.hpp.

template<typename OrdinalType, typename ScalarType>
int Teuchos::SerialDenseSolver< OrdinalType, ScalarType >::computeEquilibrateScaling (  ) 

Computes the scaling vector S(i) = 1/sqrt(A(i,i)) of the this matrix.

Returns:
Integer error code, set to 0 if successful. Otherwise returns the LAPACK error code INFO.

Definition at line 655 of file Teuchos_SerialDenseSolver.hpp.

template<typename OrdinalType, typename ScalarType>
int Teuchos::SerialDenseSolver< OrdinalType, ScalarType >::equilibrateMatrix (  ) 

Equilibrates the this matrix.

Returns:
Integer error code, set to 0 if successful. Otherwise returns the LAPACK error code INFO.

Definition at line 676 of file Teuchos_SerialDenseSolver.hpp.

template<typename OrdinalType, typename ScalarType>
int Teuchos::SerialDenseSolver< OrdinalType, ScalarType >::equilibrateRHS (  ) 

Equilibrates the current RHS.

Returns:
Integer error code, set to 0 if successful. Otherwise returns the LAPACK error code INFO.

Definition at line 719 of file Teuchos_SerialDenseSolver.hpp.

template<typename OrdinalType, typename ScalarType>
int Teuchos::SerialDenseSolver< OrdinalType, ScalarType >::applyRefinement (  ) 

Apply Iterative Refinement.

Returns:
Integer error code, set to 0 if successful. Otherwise returns the LAPACK error code INFO.

Definition at line 626 of file Teuchos_SerialDenseSolver.hpp.

template<typename OrdinalType, typename ScalarType>
int Teuchos::SerialDenseSolver< OrdinalType, ScalarType >::unequilibrateLHS (  ) 

Unscales the solution vectors if equilibration was used to solve the system.

Returns:
Integer error code, set to 0 if successful. Otherwise returns the LAPACK error code INFO.

Definition at line 750 of file Teuchos_SerialDenseSolver.hpp.

template<typename OrdinalType, typename ScalarType>
int Teuchos::SerialDenseSolver< OrdinalType, ScalarType >::reciprocalConditionEstimate ( MagnitudeType Value  ) 

Returns the reciprocal of the 1-norm condition number of the this matrix.

Parameters:
Value Out On return contains the reciprocal of the 1-norm condition number of the this matrix.
Returns:
Integer error code, set to 0 if successful. Otherwise returns the LAPACK error code INFO.

Definition at line 806 of file Teuchos_SerialDenseSolver.hpp.

template<typename OrdinalType, typename ScalarType>
bool Teuchos::SerialDenseSolver< OrdinalType, ScalarType >::transpose (  )  [inline]

Returns true if transpose of this matrix has and will be used.

Definition at line 236 of file Teuchos_SerialDenseSolver.hpp.

template<typename OrdinalType, typename ScalarType>
bool Teuchos::SerialDenseSolver< OrdinalType, ScalarType >::factored (  )  [inline]

Returns true if matrix is factored (factor available via AF() and LDAF()).

Definition at line 239 of file Teuchos_SerialDenseSolver.hpp.

template<typename OrdinalType, typename ScalarType>
bool Teuchos::SerialDenseSolver< OrdinalType, ScalarType >::equilibratedA (  )  [inline]

Returns true if factor is equilibrated (factor available via AF() and LDAF()).

Definition at line 242 of file Teuchos_SerialDenseSolver.hpp.

template<typename OrdinalType, typename ScalarType>
bool Teuchos::SerialDenseSolver< OrdinalType, ScalarType >::equilibratedB (  )  [inline]

Returns true if RHS is equilibrated (RHS available via B() and LDB()).

Definition at line 245 of file Teuchos_SerialDenseSolver.hpp.

template<typename OrdinalType, typename ScalarType>
bool Teuchos::SerialDenseSolver< OrdinalType, ScalarType >::shouldEquilibrate (  )  [inline]

Returns true if the LAPACK general rules for equilibration suggest you should equilibrate the system.

Definition at line 248 of file Teuchos_SerialDenseSolver.hpp.

template<typename OrdinalType, typename ScalarType>
bool Teuchos::SerialDenseSolver< OrdinalType, ScalarType >::solutionErrorsEstimated (  )  [inline]

Returns true if forward and backward error estimated have been computed (available via FERR() and BERR()).

Definition at line 251 of file Teuchos_SerialDenseSolver.hpp.

template<typename OrdinalType, typename ScalarType>
bool Teuchos::SerialDenseSolver< OrdinalType, ScalarType >::inverted (  )  [inline]

Returns true if matrix inverse has been computed (inverse available via AF() and LDAF()).

Definition at line 254 of file Teuchos_SerialDenseSolver.hpp.

template<typename OrdinalType, typename ScalarType>
bool Teuchos::SerialDenseSolver< OrdinalType, ScalarType >::reciprocalConditionEstimated (  )  [inline]

Returns true if the condition number of the this matrix has been computed (value available via ReciprocalConditionEstimate()).

Definition at line 257 of file Teuchos_SerialDenseSolver.hpp.

template<typename OrdinalType, typename ScalarType>
bool Teuchos::SerialDenseSolver< OrdinalType, ScalarType >::solved (  )  [inline]

Returns true if the current set of vectors has been solved.

Definition at line 260 of file Teuchos_SerialDenseSolver.hpp.

template<typename OrdinalType, typename ScalarType>
bool Teuchos::SerialDenseSolver< OrdinalType, ScalarType >::solutionRefined (  )  [inline]

Returns true if the current set of vectors has been refined.

Definition at line 263 of file Teuchos_SerialDenseSolver.hpp.

template<typename OrdinalType, typename ScalarType>
RCP<SerialDenseMatrix<OrdinalType, ScalarType> > Teuchos::SerialDenseSolver< OrdinalType, ScalarType >::getMatrix (  )  const [inline]

Returns pointer to current matrix.

Definition at line 270 of file Teuchos_SerialDenseSolver.hpp.

template<typename OrdinalType, typename ScalarType>
RCP<SerialDenseMatrix<OrdinalType, ScalarType> > Teuchos::SerialDenseSolver< OrdinalType, ScalarType >::getFactoredMatrix (  )  const [inline]

Returns pointer to factored matrix (assuming factorization has been performed).

Definition at line 273 of file Teuchos_SerialDenseSolver.hpp.

template<typename OrdinalType, typename ScalarType>
RCP<SerialDenseMatrix<OrdinalType, ScalarType> > Teuchos::SerialDenseSolver< OrdinalType, ScalarType >::getLHS (  )  const [inline]

Returns pointer to current LHS.

Definition at line 276 of file Teuchos_SerialDenseSolver.hpp.

template<typename OrdinalType, typename ScalarType>
RCP<SerialDenseMatrix<OrdinalType, ScalarType> > Teuchos::SerialDenseSolver< OrdinalType, ScalarType >::getRHS (  )  const [inline]

Returns pointer to current RHS.

Definition at line 279 of file Teuchos_SerialDenseSolver.hpp.

template<typename OrdinalType, typename ScalarType>
OrdinalType Teuchos::SerialDenseSolver< OrdinalType, ScalarType >::numRows (  )  const [inline]

Returns row dimension of system.

Definition at line 282 of file Teuchos_SerialDenseSolver.hpp.

template<typename OrdinalType, typename ScalarType>
OrdinalType Teuchos::SerialDenseSolver< OrdinalType, ScalarType >::numCols (  )  const [inline]

Returns column dimension of system.

Definition at line 285 of file Teuchos_SerialDenseSolver.hpp.

template<typename OrdinalType, typename ScalarType>
std::vector<OrdinalType> Teuchos::SerialDenseSolver< OrdinalType, ScalarType >::IPIV (  )  const [inline]

Returns pointer to pivot vector (if factorization has been computed), zero otherwise.

Definition at line 288 of file Teuchos_SerialDenseSolver.hpp.

template<typename OrdinalType, typename ScalarType>
MagnitudeType Teuchos::SerialDenseSolver< OrdinalType, ScalarType >::ANORM (  )  const [inline]

Returns the 1-Norm of the this matrix (returns -1 if not yet computed).

Definition at line 291 of file Teuchos_SerialDenseSolver.hpp.

template<typename OrdinalType, typename ScalarType>
MagnitudeType Teuchos::SerialDenseSolver< OrdinalType, ScalarType >::RCOND (  )  const [inline]

Returns the reciprocal of the condition number of the this matrix (returns -1 if not yet computed).

Definition at line 294 of file Teuchos_SerialDenseSolver.hpp.

template<typename OrdinalType, typename ScalarType>
MagnitudeType Teuchos::SerialDenseSolver< OrdinalType, ScalarType >::ROWCND (  )  const [inline]

Ratio of smallest to largest row scale factors for the this matrix (returns -1 if not yet computed).

If ROWCND() is >= 0.1 and AMAX() is not close to overflow or underflow, then equilibration is not needed.

Definition at line 299 of file Teuchos_SerialDenseSolver.hpp.

template<typename OrdinalType, typename ScalarType>
MagnitudeType Teuchos::SerialDenseSolver< OrdinalType, ScalarType >::COLCND (  )  const [inline]

Ratio of smallest to largest column scale factors for the this matrix (returns -1 if not yet computed).

If COLCND() is >= 0.1 then equilibration is not needed.

Definition at line 304 of file Teuchos_SerialDenseSolver.hpp.

template<typename OrdinalType, typename ScalarType>
MagnitudeType Teuchos::SerialDenseSolver< OrdinalType, ScalarType >::AMAX (  )  const [inline]

Returns the absolute value of the largest entry of the this matrix (returns -1 if not yet computed).

Definition at line 307 of file Teuchos_SerialDenseSolver.hpp.

template<typename OrdinalType, typename ScalarType>
std::vector<MagnitudeType> Teuchos::SerialDenseSolver< OrdinalType, ScalarType >::FERR (  )  const [inline]

Returns a pointer to the forward error estimates computed by LAPACK.

Definition at line 310 of file Teuchos_SerialDenseSolver.hpp.

template<typename OrdinalType, typename ScalarType>
std::vector<MagnitudeType> Teuchos::SerialDenseSolver< OrdinalType, ScalarType >::BERR (  )  const [inline]

Returns a pointer to the backward error estimates computed by LAPACK.

Definition at line 313 of file Teuchos_SerialDenseSolver.hpp.

template<typename OrdinalType, typename ScalarType>
std::vector<MagnitudeType> Teuchos::SerialDenseSolver< OrdinalType, ScalarType >::R (  )  const [inline]

Returns a pointer to the row scaling vector used for equilibration.

Definition at line 316 of file Teuchos_SerialDenseSolver.hpp.

template<typename OrdinalType, typename ScalarType>
std::vector<MagnitudeType> Teuchos::SerialDenseSolver< OrdinalType, ScalarType >::C (  )  const [inline]

Returns a pointer to the column scale vector used for equilibration.

Definition at line 319 of file Teuchos_SerialDenseSolver.hpp.

template<typename OrdinalType, typename ScalarType>
void Teuchos::SerialDenseSolver< OrdinalType, ScalarType >::Print ( std::ostream &  os  )  const

Print service methods; defines behavior of ostream << operator.

Definition at line 833 of file Teuchos_SerialDenseSolver.hpp.


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