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

An implementation of the Anasazi::MatOrthoManager that performs orthogonalization using the SVQB iterative orthogonalization technique described by Stathapoulos and Wu. This orthogonalization routine, while not returning the upper triangular factors of the popular Gram-Schmidt method, has a communication cost (measured in number of communication calls) that is independent of the number of columns in the basis. More...

#include <AnasaziSVQBOrthoManager.hpp>

Inheritance diagram for Anasazi::SVQBOrthoManager< ScalarType, MV, OP >:

Anasazi::MatOrthoManager< ScalarType, MV, OP > Anasazi::OrthoManager< ScalarType, MV > List of all members.

Public Member Functions

Constructor/Destructor
 SVQBOrthoManager (Teuchos::RCP< const OP > Op=Teuchos::null, bool debug=false)
 Constructor specifying re-orthogonalization tolerance.
 ~SVQBOrthoManager ()
 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 &X, const MV &Y, Teuchos::RCP< const MV > MX=Teuchos::null, Teuchos::RCP< const MV > MY=Teuchos::null) 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.

Detailed Description

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

An implementation of the Anasazi::MatOrthoManager that performs orthogonalization using the SVQB iterative orthogonalization technique described by Stathapoulos and Wu. This orthogonalization routine, while not returning the upper triangular factors of the popular Gram-Schmidt method, has a communication cost (measured in number of communication calls) that is independent of the number of columns in the basis.

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

Definition at line 56 of file AnasaziSVQBOrthoManager.hpp.


Constructor & Destructor Documentation

template<class ScalarType, class MV, class OP>
Anasazi::SVQBOrthoManager< ScalarType, MV, OP >::SVQBOrthoManager ( Teuchos::RCP< const OP >  Op = Teuchos::null,
bool  debug = false 
)

Constructor specifying re-orthogonalization tolerance.

Definition at line 290 of file AnasaziSVQBOrthoManager.hpp.

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

Destructor.

Definition at line 76 of file AnasaziSVQBOrthoManager.hpp.


Member Function Documentation

template<class ScalarType, class MV, class OP>
void Anasazi::SVQBOrthoManager< 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 [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 337 of file AnasaziSVQBOrthoManager.hpp.

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

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

This method does not compute an upper triangular coefficient matrix B.

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.
In general, B has no non-zero structure.
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 352 of file AnasaziSVQBOrthoManager.hpp.

template<class ScalarType, class MV, class OP>
int Anasazi::SVQBOrthoManager< 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 [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.
In general, B has no non-zero structure.
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 366 of file AnasaziSVQBOrthoManager.hpp.

template<class ScalarType, class MV, class OP>
Teuchos::ScalarTraits< ScalarType >::magnitudeType Anasazi::SVQBOrthoManager< ScalarType, MV, OP >::orthonormErrorMat ( const MV &  X,
Teuchos::RCP< const MV >  MX = Teuchos::null 
) const [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 305 of file AnasaziSVQBOrthoManager.hpp.

template<class ScalarType, class MV, class OP>
Teuchos::ScalarTraits< ScalarType >::magnitudeType Anasazi::SVQBOrthoManager< ScalarType, MV, OP >::orthogErrorMat ( const MV &  X,
const MV &  Y,
Teuchos::RCP< const MV >  MX = Teuchos::null,
Teuchos::RCP< const MV >  MY = Teuchos::null 
) const [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 320 of file AnasaziSVQBOrthoManager.hpp.


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