Class xlifepp::OrthoManager#

template<class ScalarType, class MV> class OrthoManager#

Inheritance diagram for xlifepp::OrthoManager:

Collaboration diagram for xlifepp::OrthoManager:

xlifepp’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.

Subclassed by xlifepp::MatOrthoManager< ScalarType, MV, OP >

Constructor/Destructor

inline OrthoManager()#: Default constructor.

inline virtual ~OrthoManager()#: Destructor.

Orthogonalization methods

virtual void innerProd(const MV &X, const MV &Y, MatrixEigenDense<ScalarType> &Z) const = 0#

Provides the inner product defining the orthogonality concepts.

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

Parameters:

X, Y – multi vectors
Z – [out] Z(i,j) contains the inner product of X[i] and Y[i]:

$Z (i, j) = ⟨ X [i], Y [i] ⟩$

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: $x^{H} M x$ .

virtual void norm(const MV &X, std::vector<typename NumTraits<ScalarType>::RealScalar> &normvec) const = 0#

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{⟨ x, x ⟩}

Parameters:

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

Pre:

normvec.size() == GetNumberVecs(X)

virtual void project(MV &X, std::vector<SmartPtr<const MV>> Q, std::vector<SmartPtr<MatrixEigenDense<ScalarType>>> C) 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().

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
$⟨ Q [i], X_{o u t} ⟩ = 0$
Also,
$X_{o u t} = X_{i n} - \sum_{i} Q [i] ⟨ Q [i], X_{i n} ⟩$
Q – [in] A list of multivector bases specifying the subspaces to be orthogonalized against, satisfying

$⟨ Q [i], Q [j] ⟩ = I if i = j$
and
$⟨ Q [i], Q [j] ⟩ = 0 if i \neq j .$
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 [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.

virtual int normalize(MV &X, SmartPtr<MatrixEigenDense<ScalarType>> B = _smPtrNull) const = 0#

This method takes a multivector X and attempts to compute a basis for $c o l s p a n (X)$ .

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
$⟨ X [i], X [j] ⟩ = δ_{i j} .$
Also,
$X_{i n} (1 : m, 1 : n) = X_{o u t} (1 : m, 1 : r a n k) B (1 : r a n k, 1 : n),$
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.

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.

Note

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

virtual int projectAndNormalize(MV &X, std::vector<SmartPtr<const MV>> Q, std::vector<SmartPtr<MatrixEigenDense<ScalarType>>> C = std::vector<SmartPtr<MatrixEigenDense<ScalarType>>>(1, _smPtrNull), SmartPtr<MatrixEigenDense<ScalarType>> B = _smPtrNull) const = 0#

Given a set of bases Q[i] and a multivector X, this method computes an orthonormal basis for $c o l s p a n (X) - \sum_{i} c o l s p a n (Q [i])$ .

This routine returns an integer rank stating the rank of the computed basis. If the subspace $c o l s p a n (X) - \sum_{i} c o l s p a n (Q [i])$ 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.

Parameters:

X – [in/out] On output, the first rank columns of X satisfy

$⟨ X [i], X [j] ⟩ = δ_{i j} and ⟨ X, Q [i] ⟩ = 0 .$
Also,
$X_{i n} (1 : m, 1 : n) = X_{o u t} (1 : m, 1 : r a n k) B (1 : r a n k, 1 : n) + \sum_{i} Q [i] C [i]$
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

$⟨ Q [i], Q [j] ⟩ = I if i = j$
and
$⟨ Q [i], Q [j] ⟩ = 0 if i \neq j .$
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 [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.

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.

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.

Note

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

Error methods

virtual NumTraits<ScalarType>::RealScalar orthonormError(const MV &X) const = 0#

This method computes the error in orthonormality of a multivector.

This method return some measure of $‖ ⟨ X, X ⟩ - I ‖$

.

See the documentation of specific orthogonalization managers.

virtual NumTraits<ScalarType>::RealScalar orthogError(const MV &X1, const MV &X2) const = 0#

This method computes the error in orthogonality of two multivectors.

This method return some measure of $‖ ⟨ X 1, X 2 ⟩ ‖$

.

See the documentation of specific orthogonalization managers.