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

#include <Teuchos_SerialQRDenseSolver.hpp>

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

Public Member Functions
Constructor/Destructor Methods
	SerialQRDenseSolver ()
	Default constructor; matrix should be set using setMatrix(), LHS and RHS set with setVectors(). More...

virtual	~SerialQRDenseSolver ()
	SerialQRDenseSolver destructor. More...

Set Methods
int	setMatrix (const RCP< SerialDenseMatrix< OrdinalType, ScalarType > > &A)
	Sets the pointers for coefficient matrix. More...

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). More...

Strategy Modifying Methods
void	factorWithEquilibration (bool flag)
	Causes equilibration to be called just before the matrix factorization as part of the call to `factor`. More...

void	solveWithTranspose (bool flag)
	If `flag` is true, causes all subsequent function calls to work with the adjoint of this matrix, otherwise not. More...

void	solveWithTransposeFlag (Teuchos::ETransp trans)
	All subsequent function calls will work with the transpose-type set by this method (`Teuchos::NO_TRANS` or Teuchos::CONJ_TRANS). More...

Factor/Solve/Invert Methods
int	factor ()
	Computes the in-place QR factorization of the matrix using the LAPACK routine _GETRF or the Eigen class HouseholderQR. More...

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

int	computeEquilibrateScaling ()
	Determines if this matrix should be scaled. More...

int	equilibrateMatrix ()
	Equilibrates the this matrix. More...

int	equilibrateRHS ()
	Equilibrates the current RHS. More...

int	unequilibrateLHS ()
	Unscales the solution vectors if equilibration was used to solve the system. More...

int	formQ ()
	Explicitly forms the unitary matrix Q. More...

int	formR ()
	Explicitly forms the upper triangular matrix R. More...

int	multiplyQ (ETransp transq, SerialDenseMatrix< OrdinalType, ScalarType > &C)
	Left multiply the input matrix by the unitary matrix Q or its adjoint. More...

int	solveR (ETransp transr, SerialDenseMatrix< OrdinalType, ScalarType > &C)
	Solve input matrix on the left with the upper triangular matrix R or its adjoint. More...

Query methods
bool	transpose ()
	Returns true if adjoint of this matrix has and will be used. More...

bool	factored ()
	Returns true if matrix is factored (factor available via getFactoredMatrix()). More...

bool	equilibratedA ()
	Returns true if factor is equilibrated (factor available via getFactoredMatrix()). More...

bool	equilibratedB ()
	Returns true if RHS is equilibrated (RHS available via getRHS()). More...

bool	shouldEquilibrate ()
	Returns true if the LAPACK general rules for equilibration suggest you should equilibrate the system. More...

bool	solved ()
	Returns true if the current set of vectors has been solved. More...

bool	formedQ ()
	Returns true if Q has been formed explicitly. More...

bool	formedR ()
	Returns true if R has been formed explicitly. More...

Data Accessor methods
RCP< SerialDenseMatrix < OrdinalType, ScalarType > >	getMatrix () const
	Returns pointer to current matrix. More...

RCP< SerialDenseMatrix < OrdinalType, ScalarType > >	getFactoredMatrix () const
	Returns pointer to factored matrix (assuming factorization has been performed). More...

RCP< SerialDenseMatrix < OrdinalType, ScalarType > >	getQ () const
	Returns pointer to Q (assuming factorization has been performed). More...

RCP< SerialDenseMatrix < OrdinalType, ScalarType > >	getR () const
	Returns pointer to R (assuming factorization has been performed). More...

RCP< SerialDenseMatrix < OrdinalType, ScalarType > >	getLHS () const
	Returns pointer to current LHS. More...

RCP< SerialDenseMatrix < OrdinalType, ScalarType > >	getRHS () const
	Returns pointer to current RHS. More...

OrdinalType	numRows () const
	Returns row dimension of system. More...

OrdinalType	numCols () const
	Returns column dimension of system. More...

std::vector< ScalarType >	tau () const
	Returns pointer to pivot vector (if factorization has been computed), zero otherwise. More...

MagnitudeType	ANORM () const
	Returns the absolute value of the largest element of this matrix (returns -1 if not yet computed). More...

I/O methods
void	Print (std::ostream &os) const
	Print service methods; defines behavior of ostream << operator. More...

Public Member Functions inherited from Teuchos::CompObject
	CompObject ()
	Default constructor. More...

	CompObject (const CompObject &source)
	Copy Constructor. More...

virtual	~CompObject ()
	Destructor. More...

void	setFlopCounter (const Flops &FlopCounter)
	Set the internal Teuchos::Flops() pointer. More...

void	setFlopCounter (const CompObject &compObject)
	Set the internal Teuchos::Flops() pointer to the flop counter of another Teuchos::CompObject. More...

void	unsetFlopCounter ()
	Set the internal Teuchos::Flops() pointer to 0 (no flops counted). More...

Flops *	getFlopCounter () const
	Get the pointer to the Teuchos::Flops() object associated with this object, returns 0 if none. More...

void	resetFlops () const
	Resets the number of floating point operations to zero for this multi-std::vector. More...

double	getFlops () const
	Returns the number of floating point operations with this multi-std::vector. More...

void	updateFlops (int addflops) const
	Increment Flop count for this object. More...

void	updateFlops (long int addflops) const
	Increment Flop count for this object. More...

void	updateFlops (double addflops) const
	Increment Flop count for this object. More...

void	updateFlops (float addflops) const
	Increment Flop count for this object. More...

Public Member Functions inherited from Teuchos::Object
	Object (int tracebackModeIn=-1)
	Default Constructor. More...

	Object (const char *label, int tracebackModeIn=-1)
	Labeling Constructor. More...

	Object (const std::string &label, int tracebackModeIn=-1)
	Create an Object with the given label, and optionally, with the given traceback mode. More...

virtual	~Object ()
	Destructor (virtual, for safety of derived classes). More...

virtual void	print (std::ostream &os) const
	Print the object to the given output stream. More...

virtual int	reportError (const std::string message, int errorCode) const
	Report an error with this Object. More...

virtual void	setLabel (const char *theLabel)

virtual const char *	label () const
	Access the object's label (LEGACY; return std::string instead). More...

Public Member Functions inherited from Teuchos::BLAS< OrdinalType, ScalarType >
	BLAS (void)
	Default constructor. More...

	BLAS (const BLAS< OrdinalType, ScalarType > &)
	Copy constructor. More...

virtual	~BLAS (void)
	Destructor. More...

Public Member Functions inherited from Teuchos::DefaultBLASImpl< OrdinalType, ScalarType >
	DefaultBLASImpl (void)
	Default constructor. More...

	DefaultBLASImpl (const DefaultBLASImpl< OrdinalType, ScalarType > &)
	Copy constructor. More...

virtual	~DefaultBLASImpl (void)
	Destructor. More...

template<typename alpha_type , typename A_type , typename x_type , typename beta_type >
void	GEMV (ETransp trans, const OrdinalType &m, const OrdinalType &n, const alpha_type alpha, const A_type A, const OrdinalType &lda, const x_type x, const OrdinalType &incx, const beta_type beta, ScalarType *y, const OrdinalType &incy) const
	Performs the matrix-vector operation: `y` `<-` `alphaAx+betay` or `y` `<-` `alphaA'x+betay` where `A` is a general `m` by `n` matrix. More...

template<typename A_type >
void	TRMV (EUplo uplo, ETransp trans, EDiag diag, const OrdinalType &n, const A_type A, const OrdinalType &lda, ScalarType x, const OrdinalType &incx) const
	Performs the matrix-vector operation: `x` `<-` `Ax` or `x` `<-` `A'x` where `A` is a unit/non-unit `n` by `n` upper/lower triangular matrix. More...

template<typename alpha_type , typename x_type , typename y_type >
void	GER (const OrdinalType &m, const OrdinalType &n, const alpha_type alpha, const x_type x, const OrdinalType &incx, const y_type y, const OrdinalType &incy, ScalarType *A, const OrdinalType &lda) const
	Performs the rank 1 operation: `A` `<-` `alphaxy'+A`. More...

template<typename alpha_type , typename A_type , typename B_type , typename beta_type >
void	GEMM (ETransp transa, ETransp transb, const OrdinalType &m, const OrdinalType &n, const OrdinalType &k, const alpha_type alpha, const A_type A, const OrdinalType &lda, const B_type B, const OrdinalType &ldb, const beta_type beta, ScalarType *C, const OrdinalType &ldc) const
	General matrix-matrix multiply. More...

void	SWAP (const OrdinalType &n, ScalarType const x, const OrdinalType &incx, ScalarType const y, const OrdinalType &incy) const
	Swap the entries of x and y. More...

template<typename alpha_type , typename A_type , typename B_type , typename beta_type >
void	SYMM (ESide side, EUplo uplo, const OrdinalType &m, const OrdinalType &n, const alpha_type alpha, const A_type A, const OrdinalType &lda, const B_type B, const OrdinalType &ldb, const beta_type beta, ScalarType *C, const OrdinalType &ldc) const
	Performs the matrix-matrix operation: `C` `<-` `alphaAB+betaC` or `C` `<-` `alphaBA+betaC` where `A` is an `m` by `m` or `n` by `n` symmetric matrix and `B` is a general matrix. More...

template<typename alpha_type , typename A_type , typename beta_type >
void	SYRK (EUplo uplo, ETransp trans, const OrdinalType &n, const OrdinalType &k, const alpha_type alpha, const A_type A, const OrdinalType &lda, const beta_type beta, ScalarType C, const OrdinalType &ldc) const
	Performs the symmetric rank k operation: `C` <- `alphaAA'+betaC` or `C` <- `alphaA'A+betaC`, where `alpha` and `beta` are scalars, `C` is an `n` by `n` symmetric matrix and `A` is an `n` by `k` matrix in the first case or `k` by `n` matrix in the second case. More...

template<typename alpha_type , typename A_type >
void	TRMM (ESide side, EUplo uplo, ETransp transa, EDiag diag, const OrdinalType &m, const OrdinalType &n, const alpha_type alpha, const A_type A, const OrdinalType &lda, ScalarType B, const OrdinalType &ldb) const
	Performs the matrix-matrix operation: `B` `<-` `alphaop`(A)B or `B` `<-` `alphaBop`(A) where `op(A)` is an unit/non-unit, upper/lower triangular matrix and `B` is a general matrix. More...

template<typename alpha_type , typename A_type >
void	TRSM (ESide side, EUplo uplo, ETransp transa, EDiag diag, const OrdinalType &m, const OrdinalType &n, const alpha_type alpha, const A_type A, const OrdinalType &lda, ScalarType B, const OrdinalType &ldb) const
	Solves the matrix equations: `op(A)X=alphaB` or `Xop`(A)=alphaB where `X` and `B` are `m` by `n` matrices, `A` is a unit/non-unit, upper/lower triangular matrix and `op(A)` is `A` or `A'`. The matrix `X` is overwritten on `B`. More...

void	ROTG (ScalarType da, ScalarType db, rotg_c_type c, ScalarType s) const
	Computes a Givens plane rotation. More...

void	ROT (const OrdinalType &n, ScalarType dx, const OrdinalType &incx, ScalarType dy, const OrdinalType &incy, MagnitudeType c, ScalarType s) const
	Applies a Givens plane rotation. More...

void	SCAL (const OrdinalType &n, const ScalarType &alpha, ScalarType *x, const OrdinalType &incx) const
	Scale the vector `x` by the constant `alpha`. More...

void	COPY (const OrdinalType &n, const ScalarType x, const OrdinalType &incx, ScalarType y, const OrdinalType &incy) const
	Copy the vector `x` to the vector `y`. More...

template<typename alpha_type , typename x_type >
void	AXPY (const OrdinalType &n, const alpha_type alpha, const x_type x, const OrdinalType &incx, ScalarType y, const OrdinalType &incy) const
	Perform the operation: `y` `<-` `y+alpha*x`. More...

ScalarTraits< ScalarType > ::magnitudeType	ASUM (const OrdinalType &n, const ScalarType *x, const OrdinalType &incx) const
	Sum the absolute values of the entries of `x`. More...

template<typename x_type , typename y_type >
ScalarType	DOT (const OrdinalType &n, const x_type x, const OrdinalType &incx, const y_type y, const OrdinalType &incy) const
	Form the dot product of the vectors `x` and `y`. More...

ScalarTraits< ScalarType > ::magnitudeType	NRM2 (const OrdinalType &n, const ScalarType *x, const OrdinalType &incx) const
	Compute the 2-norm of the vector `x`. More...

OrdinalType	IAMAX (const OrdinalType &n, const ScalarType *x, const OrdinalType &incx) const
	Return the index of the element of `x` with the maximum magnitude. More...

Public Member Functions inherited from Teuchos::LAPACK< OrdinalType, ScalarType >
	LAPACK (void)
	Default Constructor. More...

	LAPACK (const LAPACK< OrdinalType, ScalarType > &lapack)
	Copy Constructor. More...

virtual	~LAPACK (void)
	Destructor. More...

void	PTTRF (const OrdinalType &n, ScalarType d, ScalarType e, OrdinalType *info) const
	Computes the `LDL'` factorization of a Hermitian/symmetric positive definite tridiagonal matrix `A`. More...

void	PTTRS (const OrdinalType &n, const OrdinalType &nrhs, const ScalarType d, const ScalarType e, ScalarType B, const OrdinalType &ldb, OrdinalType info) const
	Solves a tridiagonal system `AX=B` using the D*L' factorization of `A` computed by PTTRF. More...

void	POTRF (const char &UPLO, const OrdinalType &n, ScalarType A, const OrdinalType &lda, OrdinalType info) const
	Computes Cholesky factorization of a real symmetric positive definite matrix `A`. More...

void	POTRS (const char &UPLO, const OrdinalType &n, const OrdinalType &nrhs, const ScalarType A, const OrdinalType &lda, ScalarType B, const OrdinalType &ldb, OrdinalType *info) const
	Solves a system of linear equations `A*X=B`, where `A` is a symmetric positive definite matrix factored by POTRF and the `nrhs` solutions are returned in `B`. More...

void	POTRI (const char &UPLO, const OrdinalType &n, ScalarType A, const OrdinalType &lda, OrdinalType info) const
	Computes the inverse of a real symmetric positive definite matrix `A` using the Cholesky factorization `A` from POTRF. More...

void	POCON (const char &UPLO, const OrdinalType &n, const ScalarType A, const OrdinalType &lda, const ScalarType &anorm, ScalarType rcond, ScalarType WORK, OrdinalType IWORK, OrdinalType *info) const
	Estimates the reciprocal of the condition number (1-norm) of a real symmetric positive definite matrix `A` using the Cholesky factorization from POTRF. More...

void	POSV (const char &UPLO, const OrdinalType &n, const OrdinalType &nrhs, ScalarType A, const OrdinalType &lda, ScalarType B, const OrdinalType &ldb, OrdinalType *info) const
	Computes the solution to a real system of linear equations `A*X=B`, where `A` is a symmetric positive definite matrix and the `nrhs` solutions are returned in `B`. More...

void	POEQU (const OrdinalType &n, const ScalarType A, const OrdinalType &lda, MagnitudeType S, MagnitudeType scond, MagnitudeType amax, OrdinalType *info) const
	Computes row and column scalings intended to equilibrate a symmetric positive definite matrix `A` and reduce its condition number (w.r.t. 2-norm). More...

void	PORFS (const char &UPLO, const OrdinalType &n, const OrdinalType &nrhs, const ScalarType A, const OrdinalType &lda, const ScalarType AF, const OrdinalType &ldaf, const ScalarType B, const OrdinalType &ldb, ScalarType X, const OrdinalType &ldx, ScalarType FERR, ScalarType BERR, ScalarType WORK, OrdinalType IWORK, OrdinalType *info) const
	Improves the computed solution to a system of linear equations when the coefficient matrix is symmetric positive definite, and provides error bounds and backward error estimates for the solution. More...

TEUCHOS_DEPRECATED void	POSVX (const char &FACT, const char &UPLO, const OrdinalType &n, const OrdinalType &nrhs, ScalarType A, const OrdinalType &lda, ScalarType AF, const OrdinalType &ldaf, const char &EQUED, ScalarType S, ScalarType B, const OrdinalType &ldb, ScalarType X, const OrdinalType &ldx, ScalarType rcond, ScalarType FERR, ScalarType BERR, ScalarType WORK, OrdinalType IWORK, OrdinalType *info) const
	Uses the Cholesky factorization to compute the solution to a real system of linear equations `A*X=B`, where `A` is symmetric positive definite. System can be equilibrated by POEQU and iteratively refined by PORFS, if requested. More...

void	POSVX (const char &FACT, const char &UPLO, const OrdinalType &n, const OrdinalType &nrhs, ScalarType A, const OrdinalType &lda, ScalarType AF, const OrdinalType &ldaf, char EQUED, ScalarType S, ScalarType B, const OrdinalType &ldb, ScalarType X, const OrdinalType &ldx, ScalarType rcond, ScalarType FERR, ScalarType BERR, ScalarType WORK, OrdinalType IWORK, OrdinalType info) const

void	GELS (const char &TRANS, const OrdinalType &m, const OrdinalType &n, const OrdinalType &nrhs, ScalarType A, const OrdinalType &lda, ScalarType B, const OrdinalType &ldb, ScalarType WORK, const OrdinalType &lwork, OrdinalType info) const
	Solves an over/underdetermined real `m` by `n` linear system `A` using QR or LQ factorization of A. More...

void	GELSS (const OrdinalType &m, const OrdinalType &n, const OrdinalType &nrhs, ScalarType A, const OrdinalType &lda, ScalarType B, const OrdinalType &ldb, MagnitudeType S, const MagnitudeType rcond, OrdinalType rank, ScalarType WORK, const OrdinalType &lwork, MagnitudeType RWORK, OrdinalType *info) const
	Use the SVD to solve a possibly rank-deficient linear least-squares problem. More...

void	GELSS (const OrdinalType &m, const OrdinalType &n, const OrdinalType &nrhs, ScalarType A, const OrdinalType &lda, ScalarType B, const OrdinalType &ldb, ScalarType S, const ScalarType &rcond, OrdinalType rank, ScalarType WORK, const OrdinalType &lwork, OrdinalType info) const
	Legacy GELSS interface for real-valued ScalarType. More...

void	GGLSE (const OrdinalType &m, const OrdinalType &n, const OrdinalType &p, ScalarType A, const OrdinalType &lda, ScalarType B, const OrdinalType &ldb, ScalarType C, ScalarType D, ScalarType X, ScalarType WORK, const OrdinalType &lwork, OrdinalType *info) const
	Solves the linear equality-constrained least squares (LSE) problem where `A` is an `m` by `n` matrix,`B` is a `p` by `n` matrix `C` is a given `m-vector`, and D is a given `p-vector`. More...

void	GEQRF (const OrdinalType &m, const OrdinalType &n, ScalarType A, const OrdinalType &lda, ScalarType TAU, ScalarType WORK, const OrdinalType &lwork, OrdinalType info) const
	Computes a QR factorization of a general `m` by `n` matrix `A`. More...

void	GEQR2 (const OrdinalType &m, const OrdinalType &n, ScalarType A[], const OrdinalType &lda, ScalarType TAU[], ScalarType WORK[], OrdinalType *const info) const
	BLAS 2 version of GEQRF, with known workspace size. More...

void	GETRF (const OrdinalType &m, const OrdinalType &n, ScalarType A, const OrdinalType &lda, OrdinalType IPIV, OrdinalType *info) const
	Computes an LU factorization of a general `m` by `n` matrix `A` using partial pivoting with row interchanges. More...

void	GETRS (const char &TRANS, const OrdinalType &n, const OrdinalType &nrhs, const ScalarType A, const OrdinalType &lda, const OrdinalType IPIV, ScalarType B, const OrdinalType &ldb, OrdinalType info) const
	Solves a system of linear equations `AX=B` or `A'X=B` with a general `n` by `n` matrix `A` using the LU factorization computed by GETRF. More...

void	LASCL (const char &TYPE, const OrdinalType &kl, const OrdinalType &ku, const MagnitudeType cfrom, const MagnitudeType cto, const OrdinalType &m, const OrdinalType &n, ScalarType A, const OrdinalType &lda, OrdinalType info) const
	Multiplies the `m` by `n` matrix `A` by the real scalar `cto/cfrom`. More...

void	GEQP3 (const OrdinalType &m, const OrdinalType &n, ScalarType A, const OrdinalType &lda, OrdinalType jpvt, ScalarType TAU, ScalarType WORK, const OrdinalType &lwork, MagnitudeType RWORK, OrdinalType info) const
	Computes a QR factorization with column pivoting of a matrix `A:` AP = QR using Level 3 BLAS. More...

void	LASWP (const OrdinalType &N, ScalarType A[], const OrdinalType &LDA, const OrdinalType &K1, const OrdinalType &K2, const OrdinalType IPIV[], const OrdinalType &INCX) const
	Apply a series of row interchanges to the matrix A. More...

void	GBTRF (const OrdinalType &m, const OrdinalType &n, const OrdinalType &kl, const OrdinalType &ku, ScalarType A, const OrdinalType &lda, OrdinalType IPIV, OrdinalType *info) const
	Computes an LU factorization of a general banded `m` by `n` matrix `A` using partial pivoting with row interchanges. More...

void	GBTRS (const char &TRANS, const OrdinalType &n, const OrdinalType &kl, const OrdinalType &ku, const OrdinalType &nrhs, const ScalarType A, const OrdinalType &lda, const OrdinalType IPIV, ScalarType B, const OrdinalType &ldb, OrdinalType info) const
	Solves a system of linear equations `AX=B` or `A'X=B` with a general banded `n` by `n` matrix `A` using the LU factorization computed by GBTRF. More...

void	GTTRF (const OrdinalType &n, ScalarType dl, ScalarType d, ScalarType du, ScalarType du2, OrdinalType IPIV, OrdinalType info) const
	Computes an LU factorization of a `n` by `n` tridiagonal matrix `A` using partial pivoting with row interchanges. More...

void	GTTRS (const char &TRANS, const OrdinalType &n, const OrdinalType &nrhs, const ScalarType dl, const ScalarType d, const ScalarType du, const ScalarType du2, const OrdinalType IPIV, ScalarType B, const OrdinalType &ldb, OrdinalType *info) const
	Solves a system of linear equations `AX=B` or `A'X=B` or `A^H*X=B` with a tridiagonal matrix `A` using the LU factorization computed by GTTRF. More...

void	GETRI (const OrdinalType &n, ScalarType A, const OrdinalType &lda, const OrdinalType IPIV, ScalarType WORK, const OrdinalType &lwork, OrdinalType info) const
	Computes the inverse of a matrix `A` using the LU factorization computed by GETRF. More...

void	LATRS (const char &UPLO, const char &TRANS, const char &DIAG, const char &NORMIN, const OrdinalType &N, ScalarType A, const OrdinalType &LDA, ScalarType X, MagnitudeType SCALE, MagnitudeType CNORM, OrdinalType *INFO) const
	Robustly solve a possibly singular triangular linear system. More...

void	GECON (const char &NORM, const OrdinalType &n, const ScalarType A, const OrdinalType &lda, const ScalarType &anorm, ScalarType rcond, ScalarType WORK, OrdinalType IWORK, OrdinalType *info) const
	Estimates the reciprocal of the condition number of a general real matrix `A`, in either the 1-norm or the infinity-norm, using the LU factorization computed by GETRF. More...

void	GBCON (const char &NORM, const OrdinalType &n, const OrdinalType &kl, const OrdinalType &ku, const ScalarType A, const OrdinalType &lda, OrdinalType IPIV, const ScalarType &anorm, ScalarType rcond, ScalarType WORK, OrdinalType IWORK, OrdinalType info) const
	Estimates the reciprocal of the condition number of a general banded real matrix `A`, in either the 1-norm or the infinity-norm, using the LU factorization computed by GETRF. More...

ScalarTraits< ScalarType > ::magnitudeType	LANGB (const char &NORM, const OrdinalType &n, const OrdinalType &kl, const OrdinalType &ku, const ScalarType A, const OrdinalType &lda, MagnitudeType WORK) const
	Returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of an `n` by `n` band matrix `A`, with `kl` sub-diagonals and `ku` super-diagonals. More...

void	GESV (const OrdinalType &n, const OrdinalType &nrhs, ScalarType A, const OrdinalType &lda, OrdinalType IPIV, ScalarType B, const OrdinalType &ldb, OrdinalType info) const
	Computes the solution to a real system of linear equations `A*X=B`, where `A` is factored through GETRF and the `nrhs` solutions are computed through GETRS. More...

void	GEEQU (const OrdinalType &m, const OrdinalType &n, const ScalarType A, const OrdinalType &lda, ScalarType R, ScalarType C, ScalarType rowcond, ScalarType colcond, ScalarType amax, OrdinalType *info) const
	Computes row and column scalings intended to equilibrate an `m` by `n` matrix `A` and reduce its condition number. More...

void	GERFS (const char &TRANS, const OrdinalType &n, const OrdinalType &nrhs, const ScalarType A, const OrdinalType &lda, const ScalarType AF, const OrdinalType &ldaf, const OrdinalType IPIV, const ScalarType B, const OrdinalType &ldb, ScalarType X, const OrdinalType &ldx, ScalarType FERR, ScalarType BERR, ScalarType WORK, OrdinalType IWORK, OrdinalType info) const
	Improves the computed solution to a system of linear equations and provides error bounds and backward error estimates for the solution. Use after GETRF/GETRS. More...

void	GBEQU (const OrdinalType &m, const OrdinalType &n, const OrdinalType &kl, const OrdinalType &ku, const ScalarType A, const OrdinalType &lda, MagnitudeType R, MagnitudeType C, MagnitudeType rowcond, MagnitudeType colcond, MagnitudeType amax, OrdinalType *info) const
	Computes row and column scalings intended to equilibrate an `m` by `n` banded matrix `A` and reduce its condition number. More...

void	GBRFS (const char &TRANS, const OrdinalType &n, const OrdinalType &kl, const OrdinalType &ku, const OrdinalType &nrhs, const ScalarType A, const OrdinalType &lda, const ScalarType AF, const OrdinalType &ldaf, const OrdinalType IPIV, const ScalarType B, const OrdinalType &ldb, ScalarType X, const OrdinalType &ldx, ScalarType FERR, ScalarType BERR, ScalarType WORK, OrdinalType IWORK, OrdinalType info) const
	Improves the computed solution to a banded system of linear equations and provides error bounds and backward error estimates for the solution. Use after GBTRF/GBTRS. More...

TEUCHOS_DEPRECATED void	GESVX (const char &FACT, const char &TRANS, const OrdinalType &n, const OrdinalType &nrhs, ScalarType A, const OrdinalType &lda, ScalarType AF, const OrdinalType &ldaf, OrdinalType IPIV, const char &EQUED, ScalarType R, ScalarType C, ScalarType B, const OrdinalType &ldb, ScalarType X, const OrdinalType &ldx, ScalarType rcond, ScalarType FERR, ScalarType BERR, ScalarType WORK, OrdinalType IWORK, OrdinalType *info) const
	Uses the LU factorization to compute the solution to a real system of linear equations `A*X=B`, returning error bounds on the solution and a condition estimate. More...

void	GESVX (const char &FACT, const char &TRANS, const OrdinalType &n, const OrdinalType &nrhs, ScalarType A, const OrdinalType &lda, ScalarType AF, const OrdinalType &ldaf, OrdinalType IPIV, char EQUED, ScalarType R, ScalarType C, ScalarType B, const OrdinalType &ldb, ScalarType X, const OrdinalType &ldx, ScalarType rcond, ScalarType FERR, ScalarType BERR, ScalarType WORK, OrdinalType IWORK, OrdinalType info) const

void	SYTRD (const char &UPLO, const OrdinalType &n, ScalarType A, const OrdinalType &lda, ScalarType D, ScalarType E, ScalarType TAU, ScalarType WORK, const OrdinalType &lwork, OrdinalType info) const
	Reduces a real symmetric matrix `A` to tridiagonal form by orthogonal similarity transformations. More...

void	GEHRD (const OrdinalType &n, const OrdinalType &ilo, const OrdinalType &ihi, ScalarType A, const OrdinalType &lda, ScalarType TAU, ScalarType WORK, const OrdinalType &lwork, OrdinalType info) const
	Reduces a real general matrix `A` to upper Hessenberg form by orthogonal similarity transformations. More...

void	TRTRS (const char &UPLO, const char &TRANS, const char &DIAG, const OrdinalType &n, const OrdinalType &nrhs, const ScalarType A, const OrdinalType &lda, ScalarType B, const OrdinalType &ldb, OrdinalType *info) const
	Solves a triangular linear system of the form `AX=B` or `ATX=B`, where `A` is a triangular matrix. More...

void	TRTRI (const char &UPLO, const char &DIAG, const OrdinalType &n, const ScalarType A, const OrdinalType &lda, OrdinalType info) const
	Computes the inverse of an upper or lower triangular matrix `A`. More...

void	SPEV (const char &JOBZ, const char &UPLO, const OrdinalType &n, ScalarType AP, ScalarType W, ScalarType Z, const OrdinalType &ldz, ScalarType WORK, OrdinalType *info) const
	Computes the eigenvalues and, optionally, eigenvectors of a symmetric `n` by `n` matrix `A` in packed storage. More...

void	SYEV (const char &JOBZ, const char &UPLO, const OrdinalType &n, ScalarType A, const OrdinalType &lda, ScalarType W, ScalarType WORK, const OrdinalType &lwork, OrdinalType info) const
	Computes all the eigenvalues and, optionally, eigenvectors of a symmetric `n` by `n` matrix A. More...

void	SYGV (const OrdinalType &itype, const char &JOBZ, const char &UPLO, const OrdinalType &n, ScalarType A, const OrdinalType &lda, ScalarType B, const OrdinalType &ldb, ScalarType W, ScalarType WORK, const OrdinalType &lwork, OrdinalType *info) const
	Computes all the eigenvalues and, optionally, eigenvectors of a symmetric `n` by `n` matrix pencil `{A`,B}, where `A` is symmetric and `B` is symmetric positive-definite. More...

void	HEEV (const char &JOBZ, const char &UPLO, const OrdinalType &n, ScalarType A, const OrdinalType &lda, MagnitudeType W, ScalarType WORK, const OrdinalType &lwork, MagnitudeType RWORK, OrdinalType *info) const
	Computes all the eigenvalues and, optionally, eigenvectors of a Hermitian `n` by `n` matrix A. More...

void	HEGV (const OrdinalType &itype, const char &JOBZ, const char &UPLO, const OrdinalType &n, ScalarType A, const OrdinalType &lda, ScalarType B, const OrdinalType &ldb, MagnitudeType W, ScalarType WORK, const OrdinalType &lwork, MagnitudeType RWORK, OrdinalType info) const
	Computes all the eigenvalues and, optionally, eigenvectors of a generalized Hermitian-definite `n` by `n` matrix pencil `{A`,B}, where `A` is Hermitian and `B` is Hermitian positive-definite. More...

void	STEQR (const char &COMPZ, const OrdinalType &n, MagnitudeType D, MagnitudeType E, ScalarType Z, const OrdinalType &ldz, MagnitudeType WORK, OrdinalType *info) const
	Computes the eigenvalues and, optionally, eigenvectors of a symmetric tridiagonal `n` by `n` matrix `A` using implicit QL/QR. The eigenvectors can only be computed if `A` was reduced to tridiagonal form by SYTRD. More...

void	PTEQR (const char &COMPZ, const OrdinalType &n, MagnitudeType D, MagnitudeType E, ScalarType Z, const OrdinalType &ldz, MagnitudeType WORK, OrdinalType *info) const
	Computes the eigenvalues and, optionally, eigenvectors of a symmetric positive-definite tridiagonal `n` by `n` matrix `A` using BDSQR, after factoring the matrix with PTTRF. More...

void	HSEQR (const char &JOB, const char &COMPZ, const OrdinalType &n, const OrdinalType &ilo, const OrdinalType &ihi, ScalarType H, const OrdinalType &ldh, ScalarType WR, ScalarType WI, ScalarType Z, const OrdinalType &ldz, ScalarType WORK, const OrdinalType &lwork, OrdinalType info) const
	Computes the eigenvalues of a real upper Hessenberg matrix `H` and, optionally, the matrices `T` and `Z` from the Schur decomposition, where T is an upper quasi-triangular matrix and Z contains the Schur vectors. More...

void	GEES (const char &JOBVS, const char &SORT, OrdinalType &(ptr2func)(ScalarType , ScalarType ), const OrdinalType &n, ScalarType A, const OrdinalType &lda, OrdinalType sdim, ScalarType WR, ScalarType WI, ScalarType VS, const OrdinalType &ldvs, ScalarType WORK, const OrdinalType &lwork, OrdinalType BWORK, OrdinalType *info) const

void	GEES (const char &JOBVS, const char &SORT, OrdinalType &(ptr2func)(ScalarType ), const OrdinalType &n, ScalarType A, const OrdinalType &lda, OrdinalType sdim, ScalarType W, ScalarType VS, const OrdinalType &ldvs, ScalarType WORK, const OrdinalType &lwork, MagnitudeType RWORK, OrdinalType BWORK, OrdinalType info) const

void	GEES (const char &JOBVS, const OrdinalType &n, ScalarType A, const OrdinalType &lda, OrdinalType sdim, MagnitudeType WR, MagnitudeType WI, ScalarType VS, const OrdinalType &ldvs, ScalarType WORK, const OrdinalType &lwork, MagnitudeType RWORK, OrdinalType BWORK, OrdinalType *info) const

void	GEEV (const char &JOBVL, const char &JOBVR, const OrdinalType &n, ScalarType A, const OrdinalType &lda, MagnitudeType WR, MagnitudeType WI, ScalarType VL, const OrdinalType &ldvl, ScalarType VR, const OrdinalType &ldvr, ScalarType WORK, const OrdinalType &lwork, MagnitudeType RWORK, OrdinalType info) const
	Computes for an `n` by `n` real nonsymmetric matrix `A`, the eigenvalues and, optionally, the left and/or right eigenvectors. More...

void	GEEVX (const char &BALANC, const char &JOBVL, const char &JOBVR, const char &SENSE, const OrdinalType &n, ScalarType A, const OrdinalType &lda, ScalarType WR, ScalarType WI, ScalarType VL, const OrdinalType &ldvl, ScalarType VR, const OrdinalType &ldvr, OrdinalType ilo, OrdinalType ihi, MagnitudeType SCALE, MagnitudeType abnrm, MagnitudeType RCONDE, MagnitudeType RCONDV, ScalarType WORK, const OrdinalType &lwork, OrdinalType IWORK, OrdinalType info) const

void	GGEVX (const char &BALANC, const char &JOBVL, const char &JOBVR, const char &SENSE, const OrdinalType &n, ScalarType A, const OrdinalType &lda, ScalarType B, const OrdinalType &ldb, MagnitudeType ALPHAR, MagnitudeType ALPHAI, ScalarType BETA, ScalarType VL, const OrdinalType &ldvl, ScalarType VR, const OrdinalType &ldvr, OrdinalType ilo, OrdinalType ihi, MagnitudeType lscale, MagnitudeType rscale, MagnitudeType abnrm, MagnitudeType bbnrm, MagnitudeType RCONDE, MagnitudeType RCONDV, ScalarType WORK, const OrdinalType &lwork, OrdinalType IWORK, OrdinalType BWORK, OrdinalType *info) const

void	GGEV (const char &JOBVL, const char &JOBVR, const OrdinalType &n, ScalarType A, const OrdinalType &lda, ScalarType B, const OrdinalType &ldb, MagnitudeType ALPHAR, MagnitudeType ALPHAI, ScalarType BETA, ScalarType VL, const OrdinalType &ldvl, ScalarType VR, const OrdinalType &ldvr, ScalarType WORK, const OrdinalType &lwork, OrdinalType *info) const

void	TRSEN (const char &JOB, const char &COMPQ, const OrdinalType SELECT, const OrdinalType &n, ScalarType T, const OrdinalType &ldt, ScalarType Q, const OrdinalType &ldq, MagnitudeType WR, MagnitudeType WI, OrdinalType M, ScalarType S, MagnitudeType SEP, ScalarType WORK, const OrdinalType &lwork, OrdinalType IWORK, const OrdinalType &liwork, OrdinalType *info) const

void	TGSEN (const OrdinalType &ijob, const OrdinalType &wantq, const OrdinalType &wantz, const OrdinalType SELECT, const OrdinalType &n, ScalarType A, const OrdinalType &lda, ScalarType B, const OrdinalType &ldb, MagnitudeType ALPHAR, MagnitudeType ALPHAI, MagnitudeType BETA, ScalarType Q, const OrdinalType &ldq, ScalarType Z, const OrdinalType &ldz, OrdinalType M, MagnitudeType PL, MagnitudeType PR, MagnitudeType DIF, ScalarType WORK, const OrdinalType &lwork, OrdinalType IWORK, const OrdinalType &liwork, OrdinalType *info) const

void	GGES (const char &JOBVL, const char &JOBVR, const char &SORT, OrdinalType &(ptr2func)(ScalarType , ScalarType , ScalarType ), const OrdinalType &n, ScalarType A, const OrdinalType &lda, ScalarType B, const OrdinalType &ldb, OrdinalType sdim, MagnitudeType ALPHAR, MagnitudeType ALPHAI, MagnitudeType BETA, ScalarType VL, const OrdinalType &ldvl, ScalarType VR, const OrdinalType &ldvr, ScalarType WORK, const OrdinalType &lwork, OrdinalType BWORK, OrdinalType *info) const

void	GESVD (const char &JOBU, const char &JOBVT, const OrdinalType &m, const OrdinalType &n, ScalarType A, const OrdinalType &lda, MagnitudeType S, ScalarType U, const OrdinalType &ldu, ScalarType V, const OrdinalType &ldv, ScalarType WORK, const OrdinalType &lwork, MagnitudeType RWORK, OrdinalType *info) const
	Computes the singular values (and optionally, vectors) of a real matrix `A`. More...

void	ORMQR (const char &SIDE, const char &TRANS, const OrdinalType &m, const OrdinalType &n, const OrdinalType &k, ScalarType A, const OrdinalType &lda, const ScalarType TAU, ScalarType C, const OrdinalType &ldc, ScalarType WORK, const OrdinalType &lwork, OrdinalType *info) const

void	ORM2R (const char &SIDE, const char &TRANS, const OrdinalType &m, const OrdinalType &n, const OrdinalType &k, const ScalarType A[], const OrdinalType &lda, const ScalarType TAU[], ScalarType C[], const OrdinalType &ldc, ScalarType WORK[], OrdinalType *const info) const
	BLAS 2 version of ORMQR; known workspace size. More...

void	UNMQR (const char &SIDE, const char &TRANS, const OrdinalType &m, const OrdinalType &n, const OrdinalType &k, ScalarType A, const OrdinalType &lda, const ScalarType TAU, ScalarType C, const OrdinalType &ldc, ScalarType WORK, const OrdinalType &lwork, OrdinalType *info) const
	Apply Householder reflectors (complex case). More...

void	UNM2R (const char &SIDE, const char &TRANS, const OrdinalType &M, const OrdinalType &N, const OrdinalType &K, const ScalarType A[], const OrdinalType &LDA, const ScalarType TAU[], ScalarType C[], const OrdinalType &LDC, ScalarType WORK[], OrdinalType *const INFO) const
	BLAS 2 version of UNMQR; known workspace size. More...

void	ORGQR (const OrdinalType &m, const OrdinalType &n, const OrdinalType &k, ScalarType A, const OrdinalType &lda, const ScalarType TAU, ScalarType WORK, const OrdinalType &lwork, OrdinalType info) const
	Compute explicit Q factor from QR factorization (GEQRF) (real case). More...

void	UNGQR (const OrdinalType &m, const OrdinalType &n, const OrdinalType &k, ScalarType A, const OrdinalType &lda, const ScalarType TAU, ScalarType WORK, const OrdinalType &lwork, OrdinalType info) const
	Compute explicit QR factor from QR factorization (GEQRF) (complex case). More...

void	ORGHR (const OrdinalType &n, const OrdinalType &ilo, const OrdinalType &ihi, ScalarType A, const OrdinalType &lda, const ScalarType TAU, ScalarType WORK, const OrdinalType &lwork, OrdinalType info) const
	Generates a real orthogonal matrix `Q` which is the product of `ihi-ilo` elementary reflectors of order `n`, as returned by GEHRD. On return `Q` is stored in `A`. More...

void	ORMHR (const char &SIDE, const char &TRANS, const OrdinalType &m, const OrdinalType &n, const OrdinalType &ilo, const OrdinalType &ihi, const ScalarType A, const OrdinalType &lda, const ScalarType TAU, ScalarType C, const OrdinalType &ldc, ScalarType WORK, const OrdinalType &lwork, OrdinalType *info) const
	Overwrites the general real `m` by `n` matrix `C` with the product of `C` and `Q`, which is a product of `ihi-ilo` elementary reflectors, as returned by GEHRD. More...

void	TREVC (const char &SIDE, const char &HOWMNY, OrdinalType select, const OrdinalType &n, const ScalarType T, const OrdinalType &ldt, ScalarType VL, const OrdinalType &ldvl, ScalarType VR, const OrdinalType &ldvr, const OrdinalType &mm, OrdinalType m, ScalarType WORK, OrdinalType *info) const

void	TREVC (const char &SIDE, const OrdinalType &n, const ScalarType T, const OrdinalType &ldt, ScalarType VL, const OrdinalType &ldvl, ScalarType VR, const OrdinalType &ldvr, const OrdinalType &mm, OrdinalType m, ScalarType WORK, MagnitudeType RWORK, OrdinalType *info) const

TEUCHOS_DEPRECATED void	TREXC (const char &COMPQ, const OrdinalType &n, ScalarType T, const OrdinalType &ldt, ScalarType Q, const OrdinalType &ldq, const OrdinalType &ifst, const OrdinalType &ilst, ScalarType WORK, OrdinalType info) const

void	TREXC (const char &COMPQ, const OrdinalType &n, ScalarType T, const OrdinalType &ldt, ScalarType Q, const OrdinalType &ldq, OrdinalType ifst, OrdinalType ilst, ScalarType WORK, OrdinalType info) const

void	TGEVC (const char &SIDE, const char &HOWMNY, const OrdinalType SELECT, const OrdinalType &n, ScalarType S, const OrdinalType &lds, ScalarType P, const OrdinalType &ldp, ScalarType VL, const OrdinalType &ldvl, ScalarType VR, const OrdinalType &ldvr, const OrdinalType &mm, OrdinalType M, ScalarType WORK, OrdinalType info) const

void	LARTG (const ScalarType &f, const ScalarType &g, MagnitudeType c, ScalarType s, ScalarType *r) const
	Gnerates a plane rotation that zeros out the second component of the input vector. More...

void	LARFG (const OrdinalType &n, ScalarType alpha, ScalarType x, const OrdinalType &incx, ScalarType *tau) const
	Generates an elementary reflector of order `n` that zeros out the last `n-1` components of the input vector. More...

TEUCHOS_DEPRECATED void	GEBAL (const char &JOBZ, const OrdinalType &n, ScalarType A, const OrdinalType &lda, const OrdinalType &ilo, const OrdinalType &ihi, MagnitudeType scale, OrdinalType *info) const
	Balances a general matrix A, through similarity transformations to make the rows and columns as close in norm as possible. More...

void	GEBAL (const char &JOBZ, const OrdinalType &n, ScalarType A, const OrdinalType &lda, OrdinalType ilo, OrdinalType ihi, MagnitudeType scale, OrdinalType *info) const

void	GEBAK (const char &JOBZ, const char &SIDE, const OrdinalType &n, const OrdinalType &ilo, const OrdinalType &ihi, const MagnitudeType scale, const OrdinalType &m, ScalarType V, const OrdinalType &ldv, OrdinalType *info) const
	Forms the left or right eigenvectors of a general matrix that has been balanced by GEBAL by backward transformation of the computed eigenvectors `V`. More...

ScalarType	LARND (const OrdinalType &idist, OrdinalType *seed) const
	Returns a random number from a uniform or normal distribution. More...

void	LARNV (const OrdinalType &idist, OrdinalType seed, const OrdinalType &n, ScalarType v) const
	Returns a vector of random numbers from a chosen distribution. More...

ScalarType	LAMCH (const char &CMACH) const
	Determines machine parameters for floating point characteristics. More...

OrdinalType	ILAENV (const OrdinalType &ispec, const std::string &NAME, const std::string &OPTS, const OrdinalType &N1=-1, const OrdinalType &N2=-1, const OrdinalType &N3=-1, const OrdinalType &N4=-1) const
	Chooses problem-dependent parameters for the local environment. More...

ScalarType	LAPY2 (const ScalarType &x, const ScalarType &y) const
	Computes x^2 + y^2 safely, to avoid overflow. More...

Additional Inherited Members
Public Types inherited from Teuchos::DefaultBLASImpl< OrdinalType, ScalarType >
typedef details::GivensRotator < ScalarType >::c_type	rotg_c_type
	The type used for c in ROTG. More...

Static Public Member Functions inherited from Teuchos::Object
static void	setTracebackMode (int tracebackModeValue)
	Set the value of the Object error traceback report mode. More...

static int	getTracebackMode ()
	Get the value of the Object error traceback report mode. More...

Related Functions inherited from Teuchos::Object
std::ostream &	operator<< (std::ostream &os, const Teuchos::Object &obj)
	Print the given Object to the given output stream. More...

Detailed Description

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

A class for solving dense linear problems.

The Teuchos::SerialQRDenseSolver 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::SerialQRDenseSolver 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::SerialQRDenseSolver vs. Teuchos::LAPACK

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

When you should use Teuchos::LAPACK: If you are simply looking for a convenient wrapper around the Fortran LAPACK routines and you have a well-conditioned problem, you should probably use Teuchos::LAPACK directly.
When you should use Teuchos::SerialQRDenseSolver: If you want to (or potentially want to) solve ill-conditioned problems or want to work with a more object-oriented interface, you should probably use Teuchos::SerialQRDenseSolver.

Constructing Teuchos::SerialQRDenseSolver Objects

There is a single Teuchos::SerialQRDenseSolver 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:

setMatrix() - Sets the matrix.
setVectors() - Sets the left and right hand side vector(s).

Vector and Utility Functions

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

Factorizations.
Solves.
Equilibration.
Norms.

Strategies for Solving Linear Systems In many cases, linear least squares systems can be accurately solved by simply computing the QR 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::SerialQRDenseSolver 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 to determine if equilibration might be useful.

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

Definition at line 132 of file Teuchos_SerialQRDenseSolver.hpp.