Anasazi's templated virtual class for providing routines for orthogonalization and orthonormalization of multivectors. More...

#include <AnasaziOrthoManager.hpp>

Inheritance diagram for Anasazi::OrthoManager< ScalarType, MV >:

Public Member Functions
Constructor/Destructor
	OrthoManager ()
	Default constructor. More...

virtual	~OrthoManager ()
	Destructor. More...

Orthogonalization methods
virtual void	innerProd (const MV &X, const MV &Y, Teuchos::SerialDenseMatrix< int, ScalarType > &Z) const =0
	Provides the inner product defining the orthogonality concepts. More...

virtual void	norm (const MV &X, std::vector< typename Teuchos::ScalarTraits< ScalarType >::magnitudeType > &normvec) const =0
	Provides the norm induced by innerProd(). More...

virtual void	project (MV &X, Teuchos::Array< Teuchos::RCP< const MV > > Q, Teuchos::Array< Teuchos::RCP< Teuchos::SerialDenseMatrix< int, ScalarType > > > C=Teuchos::tuple(Teuchos::RCP< Teuchos::SerialDenseMatrix< int, ScalarType > >(Teuchos::null))) const =0
	Given a list of mutually orthogonal and internally orthonormal bases `Q`, this method projects a multivector `X` onto the space orthogonal to the individual `Q[i]`, optionally returning the coefficients of `X` for the individual `Q[i]`. All of this is done with respect to the inner product innerProd(). More...

virtual int	normalize (MV &X, Teuchos::RCP< Teuchos::SerialDenseMatrix< int, ScalarType > > B=Teuchos::null) const =0
	This method takes a multivector `X` and attempts to compute a basis for . This basis is orthonormal with respect to innerProd(). More...

virtual int	projectAndNormalize (MV &X, Teuchos::Array< Teuchos::RCP< const MV > > Q, Teuchos::Array< Teuchos::RCP< Teuchos::SerialDenseMatrix< int, ScalarType > > > C=Teuchos::tuple(Teuchos::RCP< Teuchos::SerialDenseMatrix< int, ScalarType > >(Teuchos::null)), Teuchos::RCP< Teuchos::SerialDenseMatrix< int, ScalarType > > B=Teuchos::null) const =0
	Given a set of bases `Q[i]` and a multivector `X`, this method computes an orthonormal basis for $colspan(X) - \sum_i colspan(Q[i])$ . More...

Error methods
virtual Teuchos::ScalarTraits < ScalarType >::magnitudeType	orthonormError (const MV &X) const =0
	This method computes the error in orthonormality of a multivector. More...

virtual Teuchos::ScalarTraits < ScalarType >::magnitudeType	orthogError (const MV &X1, const MV &X2) const =0
	This method computes the error in orthogonality of two multivectors. More...

Detailed Description

template<class ScalarType, class MV>
class Anasazi::OrthoManager< ScalarType, MV >

Anasazi's templated virtual class for providing routines for orthogonalization and orthonormalization of multivectors.

This class defines concepts of orthogonality through the definition of an inner product. It also provides computational routines for orthogonalization.

A concrete implementation of this class is necessary. The user can create their own implementation if those supplied are not suitable for their needs.

Author: Chris Baker, Ulrich Hetmaniuk, Rich Lehoucq, and Heidi Thornquist

Definition at line 87 of file AnasaziOrthoManager.hpp.

Constructor & Destructor Documentation

template<class ScalarType, class MV>

Anasazi::OrthoManager< ScalarType, MV >::OrthoManager ( )

inline

Default constructor.

Definition at line 92 of file AnasaziOrthoManager.hpp.

template<class ScalarType, class MV>

virtual Anasazi::OrthoManager< ScalarType, MV >::~OrthoManager ( )

inlinevirtual

Destructor.

Definition at line 95 of file AnasaziOrthoManager.hpp.

Member Function Documentation

template<class ScalarType, class MV>

virtual void Anasazi::OrthoManager< ScalarType, MV >::innerProd	(	const MV &	X,
		const MV &	Y,
		Teuchos::SerialDenseMatrix< int, ScalarType > &	Z
	)		const

pure virtual

Provides the inner product defining the orthogonality concepts.

All concepts of orthogonality discussed in this class are defined with respect to this inner product.

Note: This is potentially different from MultiVecTraits::MvTransMv(). For example, it is customary in many eigensolvers to exploit a mass matrix M for the inner product: .

Parameters

Z	[out] `Z(i,j)` contains the inner product of `X[i]` and `Y[i]`: $Z(i,j) = \langle X[i], Y[i] \rangle$

Implemented in Anasazi::MatOrthoManager< ScalarType, MV, OP >, and Anasazi::TsqrOrthoManager< Scalar, MV >.

template<class ScalarType, class MV>

virtual void Anasazi::OrthoManager< ScalarType, MV >::norm	(	const MV &	X,
		std::vector< typename Teuchos::ScalarTraits< ScalarType >::magnitudeType > &	normvec
	)		const

pure virtual

Provides the norm induced by innerProd().

This computes the norm for each column of a multivector. This is the norm induced by innerProd():

$\|x\| = \sqrt{\langle x, x \rangle}$

Parameters

normvec [out] Vector of norms, whose i-th entry corresponds to the i-th column of X

Precondition

normvec.size() == GetNumberVecs(X)

Implemented in Anasazi::MatOrthoManager< ScalarType, MV, OP >.

template<class ScalarType, class MV>

virtual void Anasazi::OrthoManager< ScalarType, MV >::project	(	MV &	X,
		Teuchos::Array< Teuchos::RCP< const MV > >	Q,
		Teuchos::Array< Teuchos::RCP< Teuchos::SerialDenseMatrix< int, ScalarType > > >	C = `Teuchos::tuple(Teuchos::RCP< Teuchos::SerialDenseMatrix< int, ScalarType > >(Teuchos::null))`
	)		const

pure virtual

Given a list of mutually orthogonal and internally orthonormal bases Q, this method projects a multivector X onto the space orthogonal to the individual Q[i], optionally returning the coefficients of X for the individual Q[i]. All of this is done with respect to the inner product innerProd().

After calling this routine, X will be orthogonal to each of the Q[i].

Parameters

X

[in/out] The multivector to be modified.
On output, the columns of X will be orthogonal to each Q[i], satisfying

$\langle Q[i], X_{out} \rangle = 0$

Also,

$X_{out} = X_{in} - \sum_i Q[i] \langle Q[i], X_{in} \rangle$

Q

[in] A list of multivector bases specifying the subspaces to be orthogonalized against, satisfying

$\langle Q[i], Q[j] \rangle = I \quad\textrm{if}\quad i=j$

and

$\langle Q[i], Q[j] \rangle = 0 \quad\textrm{if}\quad i \neq j\ .$

C

[out] The coefficients of X in the bases Q[i]. If C[i] is a non-null pointer and C[i] matches the dimensions of X and Q[i], then the coefficients computed during the orthogonalization routine will be stored in the matrix C[i], similar to calling

innerProd( Q[i], X, C[i] );

If C[i] points to a Teuchos::SerialDenseMatrix with size inconsistent with X and Q[i], then a std::invalid_argument exception will be thrown.
Otherwise, if C.size() < i or C[i] is a null pointer, the caller will not have access to the computed coefficients.

Implemented in Anasazi::MatOrthoManager< ScalarType, MV, OP >, and Anasazi::TsqrOrthoManager< Scalar, MV >.

template<class ScalarType, class MV>

virtual int Anasazi::OrthoManager< ScalarType, MV >::normalize	(	MV &	X,
		Teuchos::RCP< Teuchos::SerialDenseMatrix< int, ScalarType > >	B = `Teuchos::null`
	)		const

pure virtual

This method takes a multivector X and attempts to compute a basis for $ colspan(X)$ . This basis is orthonormal with respect to innerProd().

This routine returns an integer rank stating the rank of the computed basis. If X does not have full rank and the normalize() routine does not attempt to augment the subspace, then rank may be smaller than the number of columns in X. In this case, only the first rank columns of output X and first rank rows of B will be valid.

Parameters

X

[in/out] The multivector to be modified.
On output, the first rank columns of X satisfy

$\langle X[i], X[j] \rangle = \delta_{ij}\ .$

Also,

$X_{in}(1:m,1:n) = X_{out}(1:m,1:rank) B(1:rank,1:n)\ ,$

where m is the number of rows in X and n is the number of columns in X.

B

[out] The coefficients of the original X with respect to the computed basis. If B is a non-null pointer and B matches the dimensions of B, then the coefficients computed during the orthogonalization routine will be stored in B, similar to calling

innerProd( X_{out}, X_{in}, B );

If B points to a Teuchos::SerialDenseMatrix with size inconsistent with X, then a std::invalid_argument exception will be thrown.
Otherwise, if B is null, the caller will not have access to the computed coefficients.

Note: This matrix is not necessarily triangular (as in a QR factorization); see the documentation of specific orthogonalization managers.

Returns: Rank of the basis computed by this method, less than or equal to the number of columns in X. This specifies how many columns in the returned X and rows in the returned B are valid.

Implemented in Anasazi::MatOrthoManager< ScalarType, MV, OP >.

template<class ScalarType, class MV>

virtual int Anasazi::OrthoManager< ScalarType, MV >::projectAndNormalize	(	MV &	X,
		Teuchos::Array< Teuchos::RCP< const MV > >	Q,
		Teuchos::Array< Teuchos::RCP< Teuchos::SerialDenseMatrix< int, ScalarType > > >	C = `Teuchos::tuple(Teuchos::RCP< Teuchos::SerialDenseMatrix< int, ScalarType > >(Teuchos::null))`,
		Teuchos::RCP< Teuchos::SerialDenseMatrix< int, ScalarType > >	B = `Teuchos::null`
	)		const

pure virtual

Given a set of bases Q[i] and a multivector X, this method computes an orthonormal basis for $colspan(X) - \sum_i colspan(Q[i])$ .

This routine returns an integer rank stating the rank of the computed basis. If the subspace $colspan(X) - \sum_i colspan(Q[i])$ does not have dimension as large as the number of columns of X and the orthogonalization manager does not attempt to augment the subspace, then rank may be smaller than the number of columns of X. In this case, only the first rank columns of output X and first rank rows of B will be valid.

Note: This routine guarantees both the orthogonality of the returned basis against the Q[i] as well as the orthonormality of the returned basis. Therefore, this method is not necessarily equivalent to calling project() followed by a call to normalize(); see the documentation for specific orthogonalization managers.

Parameters

X	[in/out] On output, the first `rank` columns of `X` satisfy $\langle X[i], X[j] \rangle = \delta_{ij} \quad \textrm{and} \quad \langle X, Q[i] \rangle = 0\ .$ Also, $X_{in}(1:m,1:n) = X_{out}(1:m,1:rank) B(1:rank,1:n) + \sum_i Q[i] C[i]$ where `m` is the number of rows in `X` and `n` is the number of columns in `X`.
Q	[in] A list of multivector bases specifying the subspaces to be orthogonalized against, satisfying $\langle Q[i], Q[j] \rangle = I \quad\textrm{if}\quad i=j$ and $\langle Q[i], Q[j] \rangle = 0 \quad\textrm{if}\quad i \neq j\ .$
C	[out] The coefficients of `X` in the `Q[i]`. If `C[i]` is a non-null pointer and `C[i]` matches the dimensions of `X` and `Q[i]`, then the coefficients computed during the orthogonalization routine will be stored in the matrix `C[i]`, similar to calling innerProd( Q[i], X, C[i] ); If `C[i]` points to a Teuchos::SerialDenseMatrix with size inconsistent with `X` and `Q[i]`, then a std::invalid_argument exception will be thrown. Otherwise, if `C.size() < i` or `C[i]` is a null pointer, the caller will not have access to the computed coefficients.
B	[out] The coefficients of the original `X` with respect to the computed basis. If `B` is a non-null pointer and `B` matches the dimensions of `B`, then the coefficients computed during the orthogonalization routine will be stored in `B`, similar to calling innerProd( Sout, Sin, B ); If `B` points to a Teuchos::SerialDenseMatrix with size inconsistent with `X`, then a std::invalid_argument exception will be thrown. Otherwise, if `B` is null, the caller will not have access to the computed coefficients.

Note: This matrix is not necessarily triangular (as in a QR factorization); see the documentation of specific orthogonalization managers.

Returns: Rank of the basis computed by this method, less than or equal to the number of columns in X. This specifies how many columns in the returned X and rows in the returned B are valid.

Implemented in Anasazi::MatOrthoManager< ScalarType, MV, OP >, and Anasazi::TsqrOrthoManager< Scalar, MV >.

template<class ScalarType, class MV>

virtual Teuchos::ScalarTraits< ScalarType >::magnitudeType Anasazi::OrthoManager< ScalarType, MV >::orthonormError ( const MV & X ) const

pure virtual

This method computes the error in orthonormality of a multivector.

This method return some measure of $\| \langle X, X \rangle - I \|$ .
See the documentation of specific orthogonalization managers.

Implemented in Anasazi::TsqrOrthoManager< Scalar, MV >, Anasazi::MatOrthoManager< ScalarType, MV, OP >, and Anasazi::MatOrthoManager< Scalar, MV, OP >.

template<class ScalarType, class MV>

virtual Teuchos::ScalarTraits<ScalarType>::magnitudeType Anasazi::OrthoManager< ScalarType, MV >::orthogError	(	const MV &	X1,
		const MV &	X2
	)		const

pure virtual

This method computes the error in orthogonality of two multivectors.

This method return some measure of $\| \langle X1, X2 \rangle \|$ .
See the documentation of specific orthogonalization managers.

Implemented in Anasazi::MatOrthoManager< ScalarType, MV, OP >, Anasazi::MatOrthoManager< Scalar, MV, OP >, and Anasazi::TsqrOrthoManager< Scalar, MV >.

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

AnasaziOrthoManager.hpp

Public Member Functions

Detailed Description

template<class ScalarType, class MV> class Anasazi::OrthoManager< ScalarType, MV >

Constructor & Destructor Documentation

Member Function Documentation

template<class ScalarType, class MV>
class Anasazi::OrthoManager< ScalarType, MV >