Anasazi::BasicOrthoManager< ScalarType, MV, OP > Class Template Reference

An implementation of the Anasazi::MatOrthoManager that performs orthogonalization using (potentially) multiple steps of classical Gram-Schmidt. More...

#include <AnasaziBasicOrthoManager.hpp>

Inheritance diagram for Anasazi::BasicOrthoManager< ScalarType, MV, OP >:
Anasazi::MatOrthoManager< ScalarType, MV, OP > Anasazi::OrthoManager< ScalarType, MV >

List of all members.

Public Member Functions

Constructor/Destructor

 BasicOrthoManager (Teuchos::RCP< const OP > Op=Teuchos::null, typename Teuchos::ScalarTraits< ScalarType >::magnitudeType kappa=1.41421356, typename Teuchos::ScalarTraits< ScalarType >::magnitudeType eps=0.0, typename Teuchos::ScalarTraits< ScalarType >::magnitudeType tol=0.20)
 Constructor specifying re-orthogonalization tolerance.
 ~BasicOrthoManager ()
 Destructor.
Methods implementing Anasazi::MatOrthoManager

void projectMat (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< MV > MX=Teuchos::null, Teuchos::Array< Teuchos::RCP< const MV > > MQ=Teuchos::tuple(Teuchos::RCP< const MV >(Teuchos::null))) const
 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().
int normalizeMat (MV &X, Teuchos::RCP< Teuchos::SerialDenseMatrix< int, ScalarType > > B=Teuchos::null, Teuchos::RCP< MV > MX=Teuchos::null) const
 This method takes a multivector X and attempts to compute an orthonormal basis for $colspan(X)$, with respect to innerProd().
int projectAndNormalizeMat (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, Teuchos::RCP< MV > MX=Teuchos::null, Teuchos::Array< Teuchos::RCP< const MV > > MQ=Teuchos::tuple(Teuchos::RCP< const MV >(Teuchos::null))) const
 Given a set of bases Q[i] and a multivector X, this method computes an orthonormal basis for $colspan(X) - \sum_i colspan(Q[i])$.
Error methods

Teuchos::ScalarTraits
< ScalarType >::magnitudeType 
orthonormErrorMat (const MV &X, Teuchos::RCP< const MV > MX=Teuchos::null) const
 This method computes the error in orthonormality of a multivector, measured as the Frobenius norm of the difference innerProd(X,Y) - I. The method has the option of exploiting a caller-provided MX.
Teuchos::ScalarTraits
< ScalarType >::magnitudeType 
orthogErrorMat (const MV &X1, const MV &X2, Teuchos::RCP< const MV > MX1, Teuchos::RCP< const MV > MX2) const
 This method computes the error in orthogonality of two multivectors, measured as the Frobenius norm of innerProd(X,Y). The method has the option of exploiting a caller-provided MX.
Accessor routines

void setKappa (typename Teuchos::ScalarTraits< ScalarType >::magnitudeType kappa)
 Set parameter for re-orthogonalization threshold.
Teuchos::ScalarTraits
< ScalarType >::magnitudeType 
getKappa () const
 Return parameter for re-orthogonalization threshold.

Detailed Description

template<class ScalarType, class MV, class OP>
class Anasazi::BasicOrthoManager< ScalarType, MV, OP >

An implementation of the Anasazi::MatOrthoManager that performs orthogonalization using (potentially) multiple steps of classical Gram-Schmidt.

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

Definition at line 51 of file AnasaziBasicOrthoManager.hpp.


Constructor & Destructor Documentation

template<class ScalarType , class MV , class OP >
Anasazi::BasicOrthoManager< ScalarType, MV, OP >::BasicOrthoManager ( Teuchos::RCP< const OP >  Op = Teuchos::null,
typename Teuchos::ScalarTraits< ScalarType >::magnitudeType  kappa = 1.41421356,
typename Teuchos::ScalarTraits< ScalarType >::magnitudeType  eps = 0.0,
typename Teuchos::ScalarTraits< ScalarType >::magnitudeType  tol = 0.20 
) [inline]

Constructor specifying re-orthogonalization tolerance.

Definition at line 295 of file AnasaziBasicOrthoManager.hpp.

template<class ScalarType , class MV , class OP >
Anasazi::BasicOrthoManager< ScalarType, MV, OP >::~BasicOrthoManager (  )  [inline]

Destructor.

Definition at line 68 of file AnasaziBasicOrthoManager.hpp.


Member Function Documentation

template<class ScalarType , class MV , class OP >
void Anasazi::BasicOrthoManager< ScalarType, MV, OP >::projectMat ( MV &  X,
Teuchos::Array< Teuchos::RCP< const MV > >  Q,
Teuchos::Array< Teuchos::RCP< Teuchos::SerialDenseMatrix< int, ScalarType > > >  C = Teuchos::tuple(Teuchos::RCPTeuchos::SerialDenseMatrix<int,ScalarType> >(Teuchos::null)),
Teuchos::RCP< MV >  MX = Teuchos::null,
Teuchos::Array< Teuchos::RCP< const MV > >  MQ = Teuchos::tuple(Teuchos::RCP<const MV>(Teuchos::null)) 
) const [inline, 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

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

MX [in/out] The image of X under the inner product operator Op. If $ MX != 0$: On input, this is expected to be consistent with Op X. On output, this is updated consistent with updates to X. If $ MX == 0$ or $ Op == 0$: MX is not referenced.
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.

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

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

Implements Anasazi::MatOrthoManager< ScalarType, MV, OP >.

Definition at line 350 of file AnasaziBasicOrthoManager.hpp.

template<class ScalarType , class MV , class OP >
int Anasazi::BasicOrthoManager< ScalarType, MV, OP >::normalizeMat ( MV &  X,
Teuchos::RCP< Teuchos::SerialDenseMatrix< int, ScalarType > >  B = Teuchos::null,
Teuchos::RCP< MV >  MX = Teuchos::null 
) const [inline, virtual]

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

The method uses classical Gram-Schmidt with selective reorthogonalization. As a result, the coefficient matrix B is upper triangular.

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.

The method attempts to find a basis with dimension equal to the number of columns in X. It does this by augmenting linearly dependent vectors in X with random directions. A finite number of these attempts will be made; therefore, it is possible that the dimension of the computed basis is less than the number of vectors in X.

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.

MX [in/out] The image of X under the inner product operator Op. If $ MX != 0$: On input, this is expected to be consistent with Op X. On output, this is updated consistent with updates to X. If $ MX == 0$ or $ Op == 0$: MX is not referenced.
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( Xout, Xin, 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. This matrix is not necessarily triangular (as in a QR factorization); see the documentation of specific orthogonalization managers.
The first rows in B corresponding to the valid columns in X will be upper triangular.

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.

Implements Anasazi::MatOrthoManager< ScalarType, MV, OP >.

Definition at line 546 of file AnasaziBasicOrthoManager.hpp.

template<class ScalarType , class MV , class OP >
int Anasazi::BasicOrthoManager< ScalarType, MV, OP >::projectAndNormalizeMat ( MV &  X,
Teuchos::Array< Teuchos::RCP< const MV > >  Q,
Teuchos::Array< Teuchos::RCP< Teuchos::SerialDenseMatrix< int, ScalarType > > >  C = Teuchos::tuple(Teuchos::RCPTeuchos::SerialDenseMatrix<int,ScalarType> >(Teuchos::null)),
Teuchos::RCP< Teuchos::SerialDenseMatrix< int, ScalarType > >  B = Teuchos::null,
Teuchos::RCP< MV >  MX = Teuchos::null,
Teuchos::Array< Teuchos::RCP< const MV > >  MQ = Teuchos::tuple(Teuchos::RCP<const MV>(Teuchos::null)) 
) const [inline, 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 doe 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.

The method attempts to find a basis with dimension the same as the number of columns in X. It does this by augmenting linearly dependent vectors with random directions. A finite number of these attempts will be made; therefore, it is possible that the dimension of the computed basis is less than the number of vectors in X.

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

\[ \langle X[i], X[j] \rangle = \delta_{ij} \quad \textrm{and} \quad \langle X, Q[i] \rangle = 0\ . \]

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.

MX [in/out] The image of X under the inner product operator Op. If $ MX != 0$: On input, this is expected to be consistent with Op X. On output, this is updated consistent with updates to X. If $ MX == 0$ or $ Op == 0$: MX is not referenced.
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( Xout, Xin, 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. This matrix is not necessarily triangular (as in a QR factorization); see the documentation of specific orthogonalization managers.
The first rows in B corresponding to the valid columns in X will be upper triangular.

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

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

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.

Implements Anasazi::MatOrthoManager< ScalarType, MV, OP >.

Definition at line 594 of file AnasaziBasicOrthoManager.hpp.

template<class ScalarType , class MV , class OP >
Teuchos::ScalarTraits< ScalarType >::magnitudeType Anasazi::BasicOrthoManager< ScalarType, MV, OP >::orthonormErrorMat ( const MV &  X,
Teuchos::RCP< const MV >  MX = Teuchos::null 
) const [inline, virtual]

This method computes the error in orthonormality of a multivector, measured as the Frobenius norm of the difference innerProd(X,Y) - I. The method has the option of exploiting a caller-provided MX.

Implements Anasazi::MatOrthoManager< ScalarType, MV, OP >.

Definition at line 321 of file AnasaziBasicOrthoManager.hpp.

template<class ScalarType , class MV , class OP >
Teuchos::ScalarTraits< ScalarType >::magnitudeType Anasazi::BasicOrthoManager< ScalarType, MV, OP >::orthogErrorMat ( const MV &  X1,
const MV &  X2,
Teuchos::RCP< const MV >  MX1,
Teuchos::RCP< const MV >  MX2 
) const [inline, virtual]

This method computes the error in orthogonality of two multivectors, measured as the Frobenius norm of innerProd(X,Y). The method has the option of exploiting a caller-provided MX.

Implements Anasazi::MatOrthoManager< ScalarType, MV, OP >.

Definition at line 338 of file AnasaziBasicOrthoManager.hpp.

template<class ScalarType , class MV , class OP >
void Anasazi::BasicOrthoManager< ScalarType, MV, OP >::setKappa ( typename Teuchos::ScalarTraits< ScalarType >::magnitudeType  kappa  )  [inline]

Set parameter for re-orthogonalization threshold.

Definition at line 263 of file AnasaziBasicOrthoManager.hpp.

template<class ScalarType , class MV , class OP >
Teuchos::ScalarTraits<ScalarType>::magnitudeType Anasazi::BasicOrthoManager< ScalarType, MV, OP >::getKappa (  )  const [inline]

Return parameter for re-orthogonalization threshold.

Definition at line 266 of file AnasaziBasicOrthoManager.hpp.


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