romtools.vector_space.utils.orthogonalizer#

An orthogonalizer is used to orthogonalize a basis at the end of the construction of a vector space. Specifically, given a basis

\[\boldsymbol \Phi \in \mathbb{R}^{N \times K}\]

the orthogonalizer will compute a new, orthogonalized basis \(\boldsymbol \Phi_{\ast}\) where

\[\boldsymbol \Phi_{\ast}^T \mathbf{W} \boldsymbol \Phi_{\ast} = \mathbf{I}.\]

In the above, \(\mathbf{W}\) is a weighting matrix (typically the cell volumes).

Classes

`EuclideanL2Orthogonalizer`([qrFnc])	Orthogonalizes the basis in the standard Euclidean L2 inner product, i.e., the output basis will satisfy
`EuclideanMatrixWeightedL2Orthogonalizer`(...)	Orthogonalizes the basis in an SPD matrix-weighted Euclidean L2 inner product, i.e., the output basis will satisfy
`EuclideanVectorWeightedL2Orthogonalizer`(...)	Orthogonalizes the basis in vector-weighted Euclidean L2 inner product, i.e., the output basis will satisfy
`NoOpOrthogonalizer`()	No op class (doesn't do anything)
`Orthogonalizer`(args, *kwargs)	Interface for the Orthogonalizer class.

class romtools.vector_space.utils.orthogonalizer.EuclideanL2Orthogonalizer(qrFnc=None)[source]#

Bases: object

Orthogonalizes the basis in the standard Euclidean L2 inner product, i.e., the output basis will satisfy

\[\boldsymbol \Phi_{\ast}^T \boldsymbol \Phi_{\ast} = \mathbf{I}.\]

This class conforms to the Orthogonalizer protocol.

orthogonalize(my_array)[source]#

Orthogonalizes the input matrix and returns the orthogonalized matrix.

Parameters:: my_array (np.ndarray) – The input matrix to be orthogonalized.
Returns:: The orthogonalized matrix.
Return type:: np.ndarray

class romtools.vector_space.utils.orthogonalizer.EuclideanMatrixWeightedL2Orthogonalizer(weighting_matrix, qrFnc=None)[source]#

Bases: object

Orthogonalizes the basis in an SPD matrix-weighted Euclidean L2 inner product, i.e., the output basis will satisfy

\[\boldsymbol \Phi_{\ast}^T \mathbf{M}\boldsymbol \Phi_{\ast} = \mathbf{I},\]

where \(\mathbf{M}\) is the SPD weighting matrix. Typically, this inner product is used for orthogonalizing with respect to a mass matrix

This class conforms to the Orthogonalizer protocol.

orthogonalize(my_array)[source]#

Orthogonalizes the input matrix using the specified weighting matrix and returns the orthogonalized matrix.

Parameters:: my_array (np.ndarray) – The input matrix to be orthogonalized.
Returns:: The orthogonalized matrix.
Return type:: np.ndarray

class romtools.vector_space.utils.orthogonalizer.EuclideanVectorWeightedL2Orthogonalizer(weighting_vector, qrFnc=None)[source]#

Bases: object

Orthogonalizes the basis in vector-weighted Euclidean L2 inner product, i.e., the output basis will satisfy

\[\boldsymbol \Phi_{\ast}^T \mathrm{diag}(\mathbf{w})\boldsymbol \Phi_{\ast} = \mathbf{I},\]

where \(\mathbf{w}\) is the weighting vector. Typically, this inner product is used for orthogonalizing with respect to cell volumes

This class conforms to the Orthogonalizer protocol.

orthogonalize(my_array)[source]#

Orthogonalizes the input matrix using the specified weighting vector and returns the orthogonalized matrix.

Parameters:: my_array (np.ndarray) – The input matrix to be orthogonalized.
Returns:: The orthogonalized matrix.
Return type:: np.ndarray

class romtools.vector_space.utils.orthogonalizer.NoOpOrthogonalizer[source]#

Bases: object

No op class (doesn’t do anything)

This class conforms to the Orthogonalizer protocol.

orthogonalize(my_array)[source]#

Returns the input array.

class romtools.vector_space.utils.orthogonalizer.Orthogonalizer(*args, **kwargs)[source]#

Bases: Protocol

Interface for the Orthogonalizer class.