MATRIX-DECOMPOSITION¶

This is matrix-decomposition, a library for decompose (factorize) dense and sparse matrices in Python.

Release info¶

There are several ways to obtain and install this package.

Conda¶

To install this package with conda run:

conda install -c jore matrix-decomposition

https://anaconda.org/jore/matrix-decomposition

pip¶

To install this package with pip run:

pip install 'matrix-decomposition'

https://pypi.python.org/pypi/matrix-decomposition

GitHub¶

To clone this package with git run:

git clone https://github.com/jor-/matrix-decomposition.git

To install this package after that with python run:

cd matrix-decomposition; python setup.py install

https://github.com/jor-/matrix-decomposition

Documentation¶

https://matrix-decomposition.readthedocs.io

Test status¶

Contents¶

Functions¶

Several functions are included in this package. The most important are summarized here.

decompose a matrix¶

matrix.decompose(A, permutation=None, return_type=None, check_finite=True, overwrite_A=False)[source]¶

Computes a decomposition of a matrix.

Parameters:	A (numpy.ndarray or scipy.sparse.spmatrix) – Matrix to be decomposed. It is assumed, that A is Hermitian. The matrix must be a squared matrix. permutation (str or numpy.ndarray) – The symmetric permutation method that is applied to the matrix before it is decomposed. It has to be a value in `matrix.UNIVERSAL_PERMUTATION_METHODS`. If A is sparse, it can also be a value in `matrix.SPARSE_ONLY_PERMUTATION_METHODS`. It is also possible to directly pass a permutation vector. optional, default: no permutation return_type (str) – The type of the decomposition that should be calculated. It has to be a value in `matrix.DECOMPOSITION_TYPES`. If return_type is None the type of the returned decomposition is chosen by the function itself. optional, default: the type of the decomposition is chosen by the function itself check_finite (bool) – Whether to check that the input matrix contains only finite numbers. Disabling may result in problems (crashes, non-termination) if the inputs do contain infinities or NaNs. Disabling gives a performance gain. optional, default: True overwrite_A (bool) – Whether it is allowed to overwrite A. Enabling may result in performance gain. optional, default: False
Returns:	A decomposition of A of type return_type.
Return type:	matrix.decompositions.DecompositionBase
Raises:	`matrix.errors.NoDecompositionPossibleError` – If the decomposition of A is not possible. `matrix.errors.MatrixNotSquareError` – If A is not a square matrix. `matrix.errors.MatrixNotFiniteError` – If A is not a finite matrix and check_finite is True.

matrix.UNIVERSAL_PERMUTATION_METHODS = ('none', 'decreasing_diagonal_values', 'increasing_diagonal_values', 'decreasing_absolute_diagonal_values', 'increasing_absolute_diagonal_values')¶: Supported permutation methods for decompose dense and sparse matrices.

matrix.SPARSE_ONLY_PERMUTATION_METHODS = ()¶: Supported permutation methods only for sparse matrices.

matrix.DECOMPOSITION_TYPES = ('LDL', 'LDL_compressed', 'LL')¶: Supported types of decompositions.

approximate a matrix¶

matrix.approximate.decomposition(A, min_diag_B=None, max_diag_B=None, min_diag_D=None, max_diag_D=None, min_abs_value_D=None, permutation=None, overwrite_A=False, return_type=None)[source]¶

Computes an approximative decomposition of a matrix with the specified properties.

Returns a decomposition of A if has such a decomposition and otherwise a decomposition of an approximation of A.

Parameters:	A (numpy.ndarray or scipy.sparse.spmatrix) – The matrix that should be approximated by a decomposition. It is assumed, that A is Hermitian. The matrix must be a squared matrix. min_diag_B (numpy.ndarray or float) – Each component of the diagonal of the composed matrix B of an approximated LDL decomposition is forced to be greater or equal to min_diag_B. optional, default : No minimal value is forced. max_diag_B (numpy.ndarray or float) – Each component of the diagonal of the composed matrix B of an approximated LDL decomposition is forced to be lower or equal to max_diag_B. optional, default : No maximal value is forced. min_diag_D (float) – Each component of the diagonal of the matrix D in an approximated LDL decomposition is forced to be greater or equal to min_diag_D. min_diag_D must be greater or equal to 0. optional, default : 0 max_diag_D (float) – Each component of the diagonal of the matrix D in an approximated LDL decomposition is forced to be lower or equal to max_diag_D. optional, default : No maximal value is forced. min_abs_value_D (float) – Absolute values below min_abs_value_D are considered as zero in the matrix D of an approximated LDL decomposition. optional, default : _matrixThe square root of the resolution of the underlying data type. permutation (str or numpy.ndarray) – The symmetric permutation method that is applied to the matrix before it is decomposed. It has to be a value in `matrix.APPROXIMATION_PERMUTATION_METHODS`. If A is sparse, it can also be a value in `matrix.SPARSE_ONLY_PERMUTATION_METHODS`. It is also possible to directly pass a permutation vector. optional, default: The permutation is chosen by the algorithm. overwrite_A (bool) – Whether it is allowed to overwrite A. Enabling may result in performance gain. optional, default: False return_type (str) – The type of the decomposition that should be returned. It has to be a value in `matrix.DECOMPOSITION_TYPES`. optional, default : The type of the decomposition is chosen by the function itself.
Returns:	An (approximative) decomposition of A of type return_type.
Return type:	matrix.decompositions.DecompositionBase
Raises:	`matrix.errors.MatrixNotSquareError` – If A is not a square matrix. `matrix.errors.MatrixComplexDiagonalValueError` – If A has complex diagonal values.

matrix.approximate.positive_semidefinite_matrix(A, min_diag_B=None, max_diag_B=None, min_diag_D=None, max_diag_D=None, min_abs_value_D=None, permutation=None, overwrite_A=False)[source]¶

Computes a positive semidefinite approximation of A.

Returns A if A is positive semidefinite and otherwise an approximation of A.

Parameters:	A (numpy.ndarray or scipy.sparse.spmatrix) – The matrix that should be approximated. It is assumed, that A is Hermitian. The matrix must be a squared matrix. min_diag_B (numpy.ndarray or float) – Each component of the diagonal of the returned matrix is forced to be greater or equal to min_diag_B. optional, default : No minimal value is forced. max_diag_B (numpy.ndarray or float) – Each component of the diagonal of the returned matrix is forced to be lower or equal to max_diag_B. optional, default : No maximal value is forced. min_diag_D (float) – Each component of the diagonal of the matrix D in a LDL decomposition of the returned matrix is forced to be greater or equal to min_diag_D. min_diag_D must be greater or equal to 0. optional, default : 0 max_diag_D (float) – Each component of the diagonal of the matrix D in a LDL decomposition of the returned matrix is forced to be lower or equal to max_diag_D. optional, default : No maximal value is forced. min_abs_value_D (float) – Absolute values below min_abs_value_D are considered as zero in the matrix D in a LDL decomposition of the returned matrix. optional, default : _matrixThe square root of the resolution of the underlying data type. permutation (str or numpy.ndarray) – The symmetric permutation method that is applied to the matrix before it is decomposed. It has to be a value in `matrix.APPROXIMATION_PERMUTATION_METHODS`. If A is sparse, it can also be a value in `matrix.SPARSE_ONLY_PERMUTATION_METHODS`. It is also possible to directly pass a permutation vector. optional, default: The permutation is chosen by the algorithm. overwrite_A (bool) – Whether it is allowed to overwrite A. Enabling may result in performance gain. optional, default: False
Returns:	B – An approximation of A which has a LDL decomposition.
Return type:	numpy.ndarray or scipy.sparse.spmatrix (same type as A)
Raises:	`matrix.errors.MatrixNotSquareError` – If A is not a square matrix. `matrix.errors.MatrixComplexDiagonalValueError` – If A has complex diagonal values.

matrix.approximate.positive_definite_matrix(A, min_diag_B=None, max_diag_B=None, min_diag_D=None, max_diag_D=None, min_abs_value_D=None, permutation=None, overwrite_A=False)[source]¶

Computes a positive definite approximation of A.

Returns A if A is positive definite and otherwise an approximation of A.

Parameters:	A (numpy.ndarray or scipy.sparse.spmatrix) – The matrix that should be approximated. It is assumed, that A is Hermitian. The matrix must be a squared matrix. min_diag_B (numpy.ndarray or float) – Each component of the diagonal of the returned matrix is forced to be greater or equal to min_diag_B. optional, default : No minimal value is forced. max_diag_B (numpy.ndarray or float) – Each component of the diagonal of the returned matrix is forced to be lower or equal to max_diag_B. optional, default : No maximal value is forced. min_diag_D (float) – Each component of the diagonal of the matrix D in a LDL decomposition of the returned matrix is forced to be greater or equal to min_diag_D. min_diag_D must be greater than 0. optional, default : The square root of the resolution of the underlying data type. max_diag_D (float) – Each component of the diagonal of the matrix D in a LDL decomposition of the returned matrix is forced to be lower or equal to max_diag_D. optional, default : No maximal value is forced. min_abs_value_D (float) – Absolute values below min_abs_value_D are considered as zero in the matrix D in a LDL decomposition of the returned matrix. optional, default : _matrixThe square root of the resolution of the underlying data type. permutation (str or numpy.ndarray) – The symmetric permutation method that is applied to the matrix before it is decomposed. It has to be a value in `matrix.APPROXIMATION_PERMUTATION_METHODS`. If A is sparse, it can also be a value in `matrix.SPARSE_ONLY_PERMUTATION_METHODS`. It is also possible to directly pass a permutation vector. optional, default: The permutation is chosen by the algorithm. overwrite_A (bool) – Whether it is allowed to overwrite A. Enabling may result in performance gain. optional, default: False
Returns:	B – An approximation of A which has a LDL decomposition.
Return type:	numpy.ndarray or scipy.sparse.spmatrix (same type as A)
Raises:	`matrix.errors.MatrixNotSquareError` – If A is not a square matrix. `matrix.errors.MatrixComplexDiagonalValueError` – If A has complex diagonal values.

matrix.APPROXIMATION_PERMUTATION_METHODS = ('none', 'decreasing_diagonal_values', 'increasing_diagonal_values', 'decreasing_absolute_diagonal_values', 'increasing_absolute_diagonal_values', 'minimal_difference')¶: Supported permutation methods for approximate dense and sparse matrices.

examine a matrix¶

matrix.is_invertible(A, check_finite=True)[source]¶

Returns whether the passed matrix is an invertible matrix.

Parameters:	A (numpy.ndarray or scipy.sparse.spmatrix) – The matrix that should be checked. It is assumed, that A is Hermitian. The matrix must be a squared matrix. check_finite (bool) – Whether to check that A contain only finite numbers. Disabling may result in problems (crashes, non-termination) if they contain infinities or NaNs. Disabling gives a performance gain. optional, default: True
Returns:	Whether A is invertible.
Return type:	bool
Raises:	`matrix.errors.MatrixNotFiniteError` – If A is not a finite matrix and check_finite is True.

matrix.is_positive_semidefinite(A, check_finite=True)[source]¶

Returns whether the passed matrix is positive semi-definite.

Parameters:	A (numpy.ndarray or scipy.sparse.spmatrix) – The matrix that should be checked. It is assumed, that A is Hermitian. The matrix must be a squared matrix. check_finite (bool) – Whether to check that A contain only finite numbers. Disabling may result in problems (crashes, non-termination) if they contain infinities or NaNs. Disabling gives a performance gain. optional, default: True
Returns:	Whether A is positive semi-definite.
Return type:	bool
Raises:	`matrix.errors.MatrixNotFiniteError` – If A is not a finite matrix and check_finite is True.

matrix.is_positive_definite(A, check_finite=True)[source]¶

Returns whether the passed matrix is positive definite.

Parameters:	A (numpy.ndarray or scipy.sparse.spmatrix) – The matrix that should be checked. It is assumed, that A is Hermitian. The matrix must be a squared matrix. check_finite (bool) – Whether to check that A contain only finite numbers. Disabling may result in problems (crashes, non-termination) if they contain infinities or NaNs. Disabling gives a performance gain. optional, default: True
Returns:	Whether A is positive definite.
Return type:	bool
Raises:	`matrix.errors.MatrixNotFiniteError` – If A is not a finite matrix and check_finite is True.

solve system of linear equations¶

matrix.solve(A, b, overwrite_b=False, check_finite=True)[source]¶

Solves the equation A x = b regarding x.

Parameters:	A (numpy.ndarray or scipy.sparse.spmatrix) – The matrix that should be checked. It is assumed, that A is Hermitian. The matrix must be a squared matrix. b (numpy.ndarray) – Right-hand side vector or matrix in equation A x = b. Ii must hold b.shape[0] == A.shape[0]. overwrite_b (bool) – Allow overwriting data in b. Enabling gives a performance gain. optional, default: False check_finite (bool) – Whether to check that A and b` contain only finite numbers. Disabling may result in problems (crashes, non-termination) if they contain infinities or NaNs. Disabling gives a performance gain. optional, default: True
Returns:	An x so that A x = b. The shape of x matches the shape of b.
Return type:	numpy.ndarray
Raises:	`matrix.errors.MatrixNotSquareError` – If A is not a square matrix. `matrix.errors.MatrixNotFiniteError` – If A is not a finite matrix and check_finite is True. `matrix.errors.MatrixSingularError` – If A is singular.

Matrix decompositions¶

Several matrix decompositions are supported. They are available in matrix.decompositions:

LL decomposition¶

class matrix.decompositions.LL_Decomposition(L=None, p=None)[source]¶

Bases: matrix.decompositions.DecompositionBase

A matrix decomposition where \(LL^H\) is the decomposed (permuted) matrix.

L is a lower triangle matrix with ones on the diagonal. This decomposition is also called Cholesky decomposition.

Parameters:	L (numpy.ndarray or scipy.sparse.spmatrix) – The matrix L of the decomposition. optional, If it is not set yet, it must be set later. p (numpy.ndarray) – The permutation vector used for the decomposition. This decomposition is of A[p[:, np.newaxis], p[np.newaxis, :]] where A is a matrix. optional, default: no permutation

L¶: numpy.matrix or scipy.sparse.spmatrix – The matrix L of the decomposition.

P¶: scipy.sparse.dok_matrix – The permutation matrix. P @ A @ P.T is the matrix A permuted by the permutation of the decomposition

as_LDL_Decomposition()[source]¶

as_any_type(*type_strs, copy=False)¶

Convert decomposition to any of the passed types.

Parameters:	type_strs (str) – The decomposition types to any of them this this decomposition is converted. copy* (bool) – Whether the data of this decomposition should always be copied or only if needed.
Returns:	If the type of this decomposition is not in type_strs, a decomposition of type type_str[0] is returned which represents the same decomposed matrix as this decomposition. Otherwise this decomposition or a copy of it is returned, depending on copy.
Return type:	matrix.decompositions.DecompositionBase

as_type(type_str, copy=False)[source]¶

Convert decomposition to passed type.

Parameters:	type_str (str) – The decomposition type to which this decomposition is converted. copy (bool) – Whether the data of this decomposition should always be copied or only if needed.
Returns:	If the type of this decomposition is not type_str, a decomposition of type type_str is returned which represents the same decomposed matrix as this decomposition. Otherwise this decomposition or a copy of it is returned, depending on copy.
Return type:	matrix.decompositions.DecompositionBase

check_finite(check_finite=True)¶

Check if this is a decomposition representing a finite matrix.

Parameters:	check_finite (bool) – Whether to perform this check. default: True
Raises:	`matrix.errors.DecompositionNotFiniteError` – If this is a decomposition representing a non-finite matrix.

check_invertible()¶

Check if this is a decomposition representing an invertible matrix.

Raises:	`matrix.errors.DecompositionSingularError` – If this is a decomposition representing a singular matrix.

composed_matrix¶: numpy.matrix or scipy.sparse.spmatrix – The composed matrix represented by this decomposition.

copy()¶

Copy this decomposition.

Returns:	A copy of this decomposition.
Return type:	matrix.decompositions.DecompositionBase

inverse_matrix_both_sides_multiplication(x, y=None)[source]¶

Calculates the both sides (matrix-matrix or matrix-vector) product y.H @ B @ x, where B is the mattrix inverse of the composed matrix represented by this decomposition.

Parameters:	x (numpy.ndarray or scipy.sparse.spmatrix) – Vector or matrix in the product y.H @ B @ x. It must hold self.n == x.shape[0]. y (numpy.ndarray or scipy.sparse.spmatrix) – Vector or matrix in the product y.H @ B @ x. It must hold self.n == y.shape[0]. optional, default: If y is not passed, x is used as y.
Returns:	The result of x.H @ A @ y.
Return type:	numpy.ndarray or scipy.sparse.spmatrix
Raises:	`matrix.errors.DecompositionSingularError` – If this is a decomposition representing a singular matrix.

inverse_matrix_right_side_multiplication(x)[source]¶

Calculates the right side (matrix-matrix or matrix-vector) product B @ x, where B is the matrix inverse of the composed matrix represented by this decomposition.

Parameters:	x (numpy.ndarray or scipy.sparse.spmatrix) – Vector or matrix in the product in the matrix-matrix or matrix-vector B @ x. It must hold self.n == x.shape[0].
Returns:	The result of B @ x.
Return type:	numpy.ndarray or scipy.sparse.spmatrix
Raises:	`matrix.errors.DecompositionSingularError` – If this is a decomposition representing a singular matrix.

is_almost_equal(other, rtol=0.0001, atol=1e-06)[source]¶

Whether this decomposition is close to passed decomposition.

Parameters:	other (str) – The decomposition which to compare to this decomposition. rtol (float) – The relative tolerance parameter. atol (float) – The absolute tolerance parameter.
Returns:	Whether this decomposition is close to passed decomposition.
Return type:	bool

is_equal(other)[source]¶

Whether this decomposition is equal to passed decomposition.

Parameters:	other (str) – The decomposition which to compare to this decomposition.
Returns:	Whether this decomposition is equal to passed decomposition.
Return type:	bool

is_finite()[source]¶

Returns whether this is a decomposition representing a finite matrix.

Returns:	Whether this is a decomposition representing a finite matrix.
Return type:	bool

is_invertible()¶

Returns whether this is a decomposition representing an invertible matrix.

Returns:	Whether this is a decomposition representing an invertible matrix.
Return type:	bool

is_permuted¶: bool – Whether this is a decompositon with permutation.

is_positive_definite()[source]¶

Returns whether this is a decomposition of a positive definite matrix.

Returns:	Whether this is a decomposition of a positive definite matrix.
Return type:	bool

is_positive_semidefinite()[source]¶

Returns whether this is a decomposition of a positive semi-definite matrix.

Returns:	Whether this is a decomposition of a positive semi-definite matrix.
Return type:	bool

is_singular()[source]¶

Returns whether this is a decomposition representing a singular matrix.

Returns:	Whether this is a decomposition representing a singular matrix.
Return type:	bool

is_sparse()[source]¶

Returns whether this is a decomposition of a sparse matrix.

Returns:	Whether this is a decomposition of a sparse matrix.
Return type:	bool

is_type(type_str)¶

Whether this is a decomposition of the passed type.

Parameters:	type_str (str) – The decomposition type according to which is checked.
Returns:	Whether this is a decomposition of the passed type.
Return type:	bool

load(filename)¶

Loads a decomposition of this type.

Parameters:	filename (str) – Where the decomposition is saved.
Raises:	`FileNotFoundError` – If the files are not found in the passed directory.

matrix_both_sides_multiplication(x, y=None)[source]¶

Calculates the both sides (matrix-matrix or matrix-vector) product y.H @ A @ x, where A is the composed matrix represented by this decomposition.

Parameters:	x (numpy.ndarray or scipy.sparse.spmatrix) – Vector or matrix in the product y.H @ A @ x. It must hold self.n == x.shape[0]. y (numpy.ndarray or scipy.sparse.spmatrix) – Vector or matrix in the product y.H @ A @ x. It must hold self.n == y.shape[0]. optional, default: If y is not passed, x is used as y.
Returns:	The result of x.H @ A @ y.
Return type:	numpy.ndarray or scipy.sparse.spmatrix

matrix_right_side_multiplication(x)[source]¶

Calculates the right side (matrix-matrix or matrix-vector) product A @ x, where A is the composed matrix represented by this decomposition.

Parameters:	x (numpy.ndarray or scipy.sparse.spmatrix) – Vector or matrix in the product in the matrix-matrix or matrix-vector A @ x. It must hold self.n == x.shape[0].
Returns:	The result of A @ x.
Return type:	numpy.ndarray or scipy.sparse.spmatrix

n¶: int – The dimension of the squared decomposed matrix.

p¶: numpy.ndarray – The permutation vector. A[p[:, np.newaxis], p[np.newaxis, :]] is the matrix A permuted by the permutation of the decomposition

p_inverse¶: numpy.ndarray – The permutation vector that undos the permutation.

permute_matrix(A)¶

Permute a matrix by the permutation of the decomposition.

Parameters:	A (numpy.ndarray or scipy.sparse.spmatrix) – The matrix that should be permuted.
Returns:	The matrix A permuted by the permutation of the decomposition.
Return type:	numpy.ndarray or scipy.sparse.spmatrix

save(filename)¶

Saves this decomposition.

Parameters:	filename (str) – Where this decomposition should be saved.

solve(b)¶

Solves the equation A x = b regarding x, where A is the composed matrix represented by this decomposition.

Parameters:	b (numpy.ndarray or scipy.sparse.spmatrix) – Right-hand side vector or matrix in equation A x = b. It must hold self.n == b.shape[0].
Returns:	An x so that A x = b. The shape of x matches the shape of b.
Return type:	numpy.ndarray or scipy.sparse.spmatrix
Raises:	`matrix.errors.DecompositionSingularError` – If this is a decomposition representing a singular matrix.

type_str = 'LL'¶: str – The type of this decomposition represented as string.

unpermute_matrix(A)¶

Unpermute a matrix permuted by the permutation of the decomposition.

Parameters:	A (numpy.ndarray or scipy.sparse.spmatrix) – The matrix that should be unpermuted.
Returns:	The matrix A unpermuted by the permutation of the decomposition.
Return type:	numpy.ndarray or scipy.sparse.spmatrix

LDL decomposition¶

class matrix.decompositions.LDL_Decomposition(L=None, d=None, p=None)[source]¶

Bases: matrix.decompositions.DecompositionBase

A matrix decomposition where \(LDL^H\) is the decomposed (permuted) matrix.

L is a lower triangle matrix with ones on the diagonal. D is a diagonal matrix. Only the diagonal values of D are stored.

Parameters:

L (numpy.ndarray or scipy.sparse.spmatrix) – The matrix L of the decomposition. optional, If it is not set yet, it must be set later.
d (numpy.ndarray) – The vector of the diagonal components of D of the decompositon. optional, If it is not set yet, it must be set later.
p (numpy.ndarray) – The permutation vector used for the decomposition. This decomposition is of A[p[:, np.newaxis], p[np.newaxis, :]] where A is a matrix. optional, default: no permutation

D¶: scipy.sparse.dia_matrix – The permutation matrix.

L¶: numpy.matrix or scipy.sparse.spmatrix – The matrix L of the decomposition.

LD¶: numpy.matrix or scipy.sparse.spmatrix – A matrix whose diagonal values are the diagonal values of D and whose off-diagonal values are those of L.

P¶: scipy.sparse.dok_matrix – The permutation matrix. P @ A @ P.T is the matrix A permuted by the permutation of the decomposition

as_LDL_DecompositionCompressed()[source]¶

as_LL_Decomposition()[source]¶

as_any_type(*type_strs, copy=False)¶

Convert decomposition to any of the passed types.

Parameters:	type_strs (str) – The decomposition types to any of them this this decomposition is converted. copy* (bool) – Whether the data of this decomposition should always be copied or only if needed.
Returns:	If the type of this decomposition is not in type_strs, a decomposition of type type_str[0] is returned which represents the same decomposed matrix as this decomposition. Otherwise this decomposition or a copy of it is returned, depending on copy.
Return type:	matrix.decompositions.DecompositionBase

as_type(type_str, copy=False)[source]¶

Convert decomposition to passed type.

Parameters:	type_str (str) – The decomposition type to which this decomposition is converted. copy (bool) – Whether the data of this decomposition should always be copied or only if needed.
Returns:	If the type of this decomposition is not type_str, a decomposition of type type_str is returned which represents the same decomposed matrix as this decomposition. Otherwise this decomposition or a copy of it is returned, depending on copy.
Return type:	matrix.decompositions.DecompositionBase

check_finite(check_finite=True)¶

Check if this is a decomposition representing a finite matrix.

Parameters:	check_finite (bool) – Whether to perform this check. default: True
Raises:	`matrix.errors.DecompositionNotFiniteError` – If this is a decomposition representing a non-finite matrix.

check_invertible()¶

Check if this is a decomposition representing an invertible matrix.

Raises:	`matrix.errors.DecompositionSingularError` – If this is a decomposition representing a singular matrix.

composed_matrix¶: numpy.matrix or scipy.sparse.spmatrix – The composed matrix represented by this decomposition.

copy()¶

Copy this decomposition.

Returns:	A copy of this decomposition.
Return type:	matrix.decompositions.DecompositionBase

d¶: numpy.ndarray – The diagonal vector of the matrix D of the decomposition.

inverse_matrix_both_sides_multiplication(x, y=None)[source]¶

Calculates the both sides (matrix-matrix or matrix-vector) product y.H @ B @ x, where B is the mattrix inverse of the composed matrix represented by this decomposition.

Parameters:	x (numpy.ndarray or scipy.sparse.spmatrix) – Vector or matrix in the product y.H @ B @ x. It must hold self.n == x.shape[0]. y (numpy.ndarray or scipy.sparse.spmatrix) – Vector or matrix in the product y.H @ B @ x. It must hold self.n == y.shape[0]. optional, default: If y is not passed, x is used as y.
Returns:	The result of x.H @ A @ y.
Return type:	numpy.ndarray or scipy.sparse.spmatrix
Raises:	`matrix.errors.DecompositionSingularError` – If this is a decomposition representing a singular matrix.

inverse_matrix_right_side_multiplication(x)[source]¶

Calculates the right side (matrix-matrix or matrix-vector) product B @ x, where B is the matrix inverse of the composed matrix represented by this decomposition.

Parameters:	x (numpy.ndarray or scipy.sparse.spmatrix) – Vector or matrix in the product in the matrix-matrix or matrix-vector B @ x. It must hold self.n == x.shape[0].
Returns:	The result of B @ x.
Return type:	numpy.ndarray or scipy.sparse.spmatrix
Raises:	`matrix.errors.DecompositionSingularError` – If this is a decomposition representing a singular matrix.

is_almost_equal(other, rtol=0.0001, atol=1e-06)[source]¶

Whether this decomposition is close to passed decomposition.

Parameters:	other (str) – The decomposition which to compare to this decomposition. rtol (float) – The relative tolerance parameter. atol (float) – The absolute tolerance parameter.
Returns:	Whether this decomposition is close to passed decomposition.
Return type:	bool

is_equal(other)[source]¶

Whether this decomposition is equal to passed decomposition.

Parameters:	other (str) – The decomposition which to compare to this decomposition.
Returns:	Whether this decomposition is equal to passed decomposition.
Return type:	bool

is_finite()[source]¶

Returns whether this is a decomposition representing a finite matrix.

Returns:	Whether this is a decomposition representing a finite matrix.
Return type:	bool

is_invertible()¶

Returns whether this is a decomposition representing an invertible matrix.

Returns:	Whether this is a decomposition representing an invertible matrix.
Return type:	bool

is_permuted¶: bool – Whether this is a decompositon with permutation.

is_positive_definite()[source]¶

Returns whether this is a decomposition of a positive definite matrix.

Returns:	Whether this is a decomposition of a positive definite matrix.
Return type:	bool

is_positive_semidefinite()[source]¶

Returns whether this is a decomposition of a positive semi-definite matrix.

Returns:	Whether this is a decomposition of a positive semi-definite matrix.
Return type:	bool

is_singular()[source]¶

Returns whether this is a decomposition representing a singular matrix.

Returns:	Whether this is a decomposition representing a singular matrix.
Return type:	bool

is_sparse()[source]¶

Returns whether this is a decomposition of a sparse matrix.

Returns:	Whether this is a decomposition of a sparse matrix.
Return type:	bool

is_type(type_str)¶

Whether this is a decomposition of the passed type.

Parameters:	type_str (str) – The decomposition type according to which is checked.
Returns:	Whether this is a decomposition of the passed type.
Return type:	bool

load(filename)¶

Loads a decomposition of this type.

Parameters:	filename (str) – Where the decomposition is saved.
Raises:	`FileNotFoundError` – If the files are not found in the passed directory.

matrix_both_sides_multiplication(x, y=None)[source]¶

Calculates the both sides (matrix-matrix or matrix-vector) product y.H @ A @ x, where A is the composed matrix represented by this decomposition.

Parameters:	x (numpy.ndarray or scipy.sparse.spmatrix) – Vector or matrix in the product y.H @ A @ x. It must hold self.n == x.shape[0]. y (numpy.ndarray or scipy.sparse.spmatrix) – Vector or matrix in the product y.H @ A @ x. It must hold self.n == y.shape[0]. optional, default: If y is not passed, x is used as y.
Returns:	The result of x.H @ A @ y.
Return type:	numpy.ndarray or scipy.sparse.spmatrix

matrix_right_side_multiplication(x)[source]¶

Calculates the right side (matrix-matrix or matrix-vector) product A @ x, where A is the composed matrix represented by this decomposition.

Parameters:	x (numpy.ndarray or scipy.sparse.spmatrix) – Vector or matrix in the product in the matrix-matrix or matrix-vector A @ x. It must hold self.n == x.shape[0].
Returns:	The result of A @ x.
Return type:	numpy.ndarray or scipy.sparse.spmatrix

n¶: int – The dimension of the squared decomposed matrix.

p¶: numpy.ndarray – The permutation vector. A[p[:, np.newaxis], p[np.newaxis, :]] is the matrix A permuted by the permutation of the decomposition

p_inverse¶: numpy.ndarray – The permutation vector that undos the permutation.

permute_matrix(A)¶

Permute a matrix by the permutation of the decomposition.

Parameters:	A (numpy.ndarray or scipy.sparse.spmatrix) – The matrix that should be permuted.
Returns:	The matrix A permuted by the permutation of the decomposition.
Return type:	numpy.ndarray or scipy.sparse.spmatrix

save(filename)¶

Saves this decomposition.

Parameters:	filename (str) – Where this decomposition should be saved.

solve(b)¶

Solves the equation A x = b regarding x, where A is the composed matrix represented by this decomposition.

Parameters:	b (numpy.ndarray or scipy.sparse.spmatrix) – Right-hand side vector or matrix in equation A x = b. It must hold self.n == b.shape[0].
Returns:	An x so that A x = b. The shape of x matches the shape of b.
Return type:	numpy.ndarray or scipy.sparse.spmatrix
Raises:	`matrix.errors.DecompositionSingularError` – If this is a decomposition representing a singular matrix.

type_str = 'LDL'¶: str – The type of this decomposition represented as string.

unpermute_matrix(A)¶

Unpermute a matrix permuted by the permutation of the decomposition.

Parameters:	A (numpy.ndarray or scipy.sparse.spmatrix) – The matrix that should be unpermuted.
Returns:	The matrix A unpermuted by the permutation of the decomposition.
Return type:	numpy.ndarray or scipy.sparse.spmatrix

LDL decomposition compressed¶

class matrix.decompositions.LDL_DecompositionCompressed(LD=None, p=None)[source]¶

Bases: matrix.decompositions.DecompositionBase

A matrix decomposition where \(LDL^H\) is the decomposed (permuted) matrix.

L is a lower triangle matrix with ones on the diagonal. D is a diagonal matrix. L and D are stored in one matrix whose diagonal values are the diagonal values of D and whose off-diagonal values are those of L.

Parameters:

LD (numpy.ndarray or scipy.sparse.spmatrix) – A matrix whose diagonal values are the diagonal values of D and whose off-diagonal values are those of L. optional, If it is not set yet, it must be set later.
p (numpy.ndarray) – The permutation vector used for the decomposition. This decomposition is of A[p[:, np.newaxis], p[np.newaxis, :]] where A is a matrix. optional, default: no permutation

D¶: scipy.sparse.dia_matrix – The permutation matrix.

L¶: numpy.matrix or scipy.sparse.spmatrix – The matrix L of the decomposition.

LD¶: numpy.matrix or scipy.sparse.spmatrix – A matrix whose diagonal values are the diagonal values of D and whose off-diagonal values are those of L.

P¶: scipy.sparse.dok_matrix – The permutation matrix. P @ A @ P.T is the matrix A permuted by the permutation of the decomposition

as_LDL_Decomposition()[source]¶

as_any_type(*type_strs, copy=False)¶

Convert decomposition to any of the passed types.

Parameters:	type_strs (str) – The decomposition types to any of them this this decomposition is converted. copy* (bool) – Whether the data of this decomposition should always be copied or only if needed.
Returns:	If the type of this decomposition is not in type_strs, a decomposition of type type_str[0] is returned which represents the same decomposed matrix as this decomposition. Otherwise this decomposition or a copy of it is returned, depending on copy.
Return type:	matrix.decompositions.DecompositionBase

as_type(type_str, copy=False)[source]¶

Convert decomposition to passed type.

Parameters:	type_str (str) – The decomposition type to which this decomposition is converted. copy (bool) – Whether the data of this decomposition should always be copied or only if needed.
Returns:	If the type of this decomposition is not type_str, a decomposition of type type_str is returned which represents the same decomposed matrix as this decomposition. Otherwise this decomposition or a copy of it is returned, depending on copy.
Return type:	matrix.decompositions.DecompositionBase

check_finite(check_finite=True)¶

Check if this is a decomposition representing a finite matrix.

Parameters:	check_finite (bool) – Whether to perform this check. default: True
Raises:	`matrix.errors.DecompositionNotFiniteError` – If this is a decomposition representing a non-finite matrix.

check_invertible()¶

Check if this is a decomposition representing an invertible matrix.

Raises:	`matrix.errors.DecompositionSingularError` – If this is a decomposition representing a singular matrix.

composed_matrix¶: numpy.matrix or scipy.sparse.spmatrix – The composed matrix represented by this decomposition.

copy()¶

Copy this decomposition.

Returns:	A copy of this decomposition.
Return type:	matrix.decompositions.DecompositionBase

d¶: numpy.ndarray – The diagonal vector of the matrix D of the decomposition.

inverse_matrix_both_sides_multiplication(x, y=None)[source]¶

Calculates the both sides (matrix-matrix or matrix-vector) product y.H @ B @ x, where B is the mattrix inverse of the composed matrix represented by this decomposition.

Parameters:	x (numpy.ndarray or scipy.sparse.spmatrix) – Vector or matrix in the product y.H @ B @ x. It must hold self.n == x.shape[0]. y (numpy.ndarray or scipy.sparse.spmatrix) – Vector or matrix in the product y.H @ B @ x. It must hold self.n == y.shape[0]. optional, default: If y is not passed, x is used as y.
Returns:	The result of x.H @ A @ y.
Return type:	numpy.ndarray or scipy.sparse.spmatrix
Raises:	`matrix.errors.DecompositionSingularError` – If this is a decomposition representing a singular matrix.

inverse_matrix_right_side_multiplication(x)[source]¶

Calculates the right side (matrix-matrix or matrix-vector) product B @ x, where B is the matrix inverse of the composed matrix represented by this decomposition.

Parameters:	x (numpy.ndarray or scipy.sparse.spmatrix) – Vector or matrix in the product in the matrix-matrix or matrix-vector B @ x. It must hold self.n == x.shape[0].
Returns:	The result of B @ x.
Return type:	numpy.ndarray or scipy.sparse.spmatrix
Raises:	`matrix.errors.DecompositionSingularError` – If this is a decomposition representing a singular matrix.

is_almost_equal(other, rtol=0.0001, atol=1e-06)[source]¶

Whether this decomposition is close to passed decomposition.

Parameters:	other (str) – The decomposition which to compare to this decomposition. rtol (float) – The relative tolerance parameter. atol (float) – The absolute tolerance parameter.
Returns:	Whether this decomposition is close to passed decomposition.
Return type:	bool

is_equal(other)[source]¶

Whether this decomposition is equal to passed decomposition.

Parameters:	other (str) – The decomposition which to compare to this decomposition.
Returns:	Whether this decomposition is equal to passed decomposition.
Return type:	bool

is_finite()[source]¶

Returns whether this is a decomposition representing a finite matrix.

Returns:	Whether this is a decomposition representing a finite matrix.
Return type:	bool

is_invertible()¶

Returns whether this is a decomposition representing an invertible matrix.

Returns:	Whether this is a decomposition representing an invertible matrix.
Return type:	bool

is_permuted¶: bool – Whether this is a decompositon with permutation.

is_positive_definite()[source]¶

Returns whether this is a decomposition of a positive definite matrix.

Returns:	Whether this is a decomposition of a positive definite matrix.
Return type:	bool

is_positive_semidefinite()[source]¶

Returns whether this is a decomposition of a positive semi-definite matrix.

Returns:	Whether this is a decomposition of a positive semi-definite matrix.
Return type:	bool

is_singular()[source]¶

Returns whether this is a decomposition representing a singular matrix.

Returns:	Whether this is a decomposition representing a singular matrix.
Return type:	bool

is_sparse()[source]¶

Returns whether this is a decomposition of a sparse matrix.

Returns:	Whether this is a decomposition of a sparse matrix.
Return type:	bool

is_type(type_str)¶

Whether this is a decomposition of the passed type.

Parameters:	type_str (str) – The decomposition type according to which is checked.
Returns:	Whether this is a decomposition of the passed type.
Return type:	bool

load(filename)¶

Loads a decomposition of this type.

Parameters:	filename (str) – Where the decomposition is saved.
Raises:	`FileNotFoundError` – If the files are not found in the passed directory.

matrix_both_sides_multiplication(x, y=None)[source]¶

Calculates the both sides (matrix-matrix or matrix-vector) product y.H @ A @ x, where A is the composed matrix represented by this decomposition.

Parameters:	x (numpy.ndarray or scipy.sparse.spmatrix) – Vector or matrix in the product y.H @ A @ x. It must hold self.n == x.shape[0]. y (numpy.ndarray or scipy.sparse.spmatrix) – Vector or matrix in the product y.H @ A @ x. It must hold self.n == y.shape[0]. optional, default: If y is not passed, x is used as y.
Returns:	The result of x.H @ A @ y.
Return type:	numpy.ndarray or scipy.sparse.spmatrix

matrix_right_side_multiplication(x)[source]¶

Calculates the right side (matrix-matrix or matrix-vector) product A @ x, where A is the composed matrix represented by this decomposition.

Parameters:	x (numpy.ndarray or scipy.sparse.spmatrix) – Vector or matrix in the product in the matrix-matrix or matrix-vector A @ x. It must hold self.n == x.shape[0].
Returns:	The result of A @ x.
Return type:	numpy.ndarray or scipy.sparse.spmatrix

n¶: int – The dimension of the squared decomposed matrix.

p¶: numpy.ndarray – The permutation vector. A[p[:, np.newaxis], p[np.newaxis, :]] is the matrix A permuted by the permutation of the decomposition

p_inverse¶: numpy.ndarray – The permutation vector that undos the permutation.

permute_matrix(A)¶

Permute a matrix by the permutation of the decomposition.

Parameters:	A (numpy.ndarray or scipy.sparse.spmatrix) – The matrix that should be permuted.
Returns:	The matrix A permuted by the permutation of the decomposition.
Return type:	numpy.ndarray or scipy.sparse.spmatrix

save(filename)¶

Saves this decomposition.

Parameters:	filename (str) – Where this decomposition should be saved.

solve(b)¶

Solves the equation A x = b regarding x, where A is the composed matrix represented by this decomposition.

Parameters:	b (numpy.ndarray or scipy.sparse.spmatrix) – Right-hand side vector or matrix in equation A x = b. It must hold self.n == b.shape[0].
Returns:	An x so that A x = b. The shape of x matches the shape of b.
Return type:	numpy.ndarray or scipy.sparse.spmatrix
Raises:	`matrix.errors.DecompositionSingularError` – If this is a decomposition representing a singular matrix.

type_str = 'LDL_compressed'¶: str – The type of this decomposition represented as string.

unpermute_matrix(A)¶

Unpermute a matrix permuted by the permutation of the decomposition.

Parameters:	A (numpy.ndarray or scipy.sparse.spmatrix) – The matrix that should be unpermuted.
Returns:	The matrix A unpermuted by the permutation of the decomposition.
Return type:	numpy.ndarray or scipy.sparse.spmatrix

base decomposition¶

class matrix.decompositions.DecompositionBase(p=None)[source]¶

Bases: object

A matrix decomposition.

This class is a base class for matrix decompositions.

Parameters:	p (numpy.ndarray) – The permutation vector used for the decomposition. This decomposition is of A[p[:, np.newaxis], p[np.newaxis, :]] where A is a matrix. optional, default: no permutation

P¶: scipy.sparse.dok_matrix – The permutation matrix. P @ A @ P.T is the matrix A permuted by the permutation of the decomposition

as_any_type(*type_strs, copy=False)[source]¶

Convert decomposition to any of the passed types.

Parameters:	type_strs (str) – The decomposition types to any of them this this decomposition is converted. copy* (bool) – Whether the data of this decomposition should always be copied or only if needed.
Returns:	If the type of this decomposition is not in type_strs, a decomposition of type type_str[0] is returned which represents the same decomposed matrix as this decomposition. Otherwise this decomposition or a copy of it is returned, depending on copy.
Return type:	matrix.decompositions.DecompositionBase

as_type(type_str, copy=False)[source]¶

Convert decomposition to passed type.

Parameters:	type_str (str) – The decomposition type to which this decomposition is converted. copy (bool) – Whether the data of this decomposition should always be copied or only if needed.
Returns:	If the type of this decomposition is not type_str, a decomposition of type type_str is returned which represents the same decomposed matrix as this decomposition. Otherwise this decomposition or a copy of it is returned, depending on copy.
Return type:	matrix.decompositions.DecompositionBase

check_finite(check_finite=True)[source]¶

Check if this is a decomposition representing a finite matrix.

Parameters:	check_finite (bool) – Whether to perform this check. default: True
Raises:	`matrix.errors.DecompositionNotFiniteError` – If this is a decomposition representing a non-finite matrix.

check_invertible()[source]¶

Check if this is a decomposition representing an invertible matrix.

Raises:	`matrix.errors.DecompositionSingularError` – If this is a decomposition representing a singular matrix.

composed_matrix¶: numpy.matrix or scipy.sparse.spmatrix – The composed matrix represented by this decomposition.

copy()[source]¶

Copy this decomposition.

Returns:	A copy of this decomposition.
Return type:	matrix.decompositions.DecompositionBase

inverse_matrix_both_sides_multiplication(x, y=None)[source]¶

Calculates the both sides (matrix-matrix or matrix-vector) product y.H @ B @ x, where B is the mattrix inverse of the composed matrix represented by this decomposition.

Parameters:	x (numpy.ndarray or scipy.sparse.spmatrix) – Vector or matrix in the product y.H @ B @ x. It must hold self.n == x.shape[0]. y (numpy.ndarray or scipy.sparse.spmatrix) – Vector or matrix in the product y.H @ B @ x. It must hold self.n == y.shape[0]. optional, default: If y is not passed, x is used as y.
Returns:	The result of x.H @ A @ y.
Return type:	numpy.ndarray or scipy.sparse.spmatrix
Raises:	`matrix.errors.DecompositionSingularError` – If this is a decomposition representing a singular matrix.

inverse_matrix_right_side_multiplication(x)[source]¶

Calculates the right side (matrix-matrix or matrix-vector) product B @ x, where B is the matrix inverse of the composed matrix represented by this decomposition.

Parameters:	x (numpy.ndarray or scipy.sparse.spmatrix) – Vector or matrix in the product in the matrix-matrix or matrix-vector B @ x. It must hold self.n == x.shape[0].
Returns:	The result of B @ x.
Return type:	numpy.ndarray or scipy.sparse.spmatrix
Raises:	`matrix.errors.DecompositionSingularError` – If this is a decomposition representing a singular matrix.

is_almost_equal(other, rtol=0.0001, atol=1e-06)[source]¶

Whether this decomposition is close to passed decomposition.

Parameters:	other (str) – The decomposition which to compare to this decomposition. rtol (float) – The relative tolerance parameter. atol (float) – The absolute tolerance parameter.
Returns:	Whether this decomposition is close to passed decomposition.
Return type:	bool

is_equal(other)[source]¶

Whether this decomposition is equal to passed decomposition.

Parameters:	other (str) – The decomposition which to compare to this decomposition.
Returns:	Whether this decomposition is equal to passed decomposition.
Return type:	bool

is_finite()[source]¶

Returns whether this is a decomposition representing a finite matrix.

Returns:	Whether this is a decomposition representing a finite matrix.
Return type:	bool

is_invertible()[source]¶

Returns whether this is a decomposition representing an invertible matrix.

Returns:	Whether this is a decomposition representing an invertible matrix.
Return type:	bool

is_permuted¶: bool – Whether this is a decompositon with permutation.

is_positive_definite()[source]¶

Returns whether this is a decomposition of a positive definite matrix.

Returns:	Whether this is a decomposition of a positive definite matrix.
Return type:	bool

is_positive_semidefinite()[source]¶

Returns whether this is a decomposition of a positive semi-definite matrix.

Returns:	Whether this is a decomposition of a positive semi-definite matrix.
Return type:	bool

is_singular()[source]¶

Returns whether this is a decomposition representing a singular matrix.

Returns:	Whether this is a decomposition representing a singular matrix.
Return type:	bool

is_sparse()[source]¶

Returns whether this is a decomposition of a sparse matrix.

Returns:	Whether this is a decomposition of a sparse matrix.
Return type:	bool

is_type(type_str)[source]¶

Whether this is a decomposition of the passed type.

Parameters:	type_str (str) – The decomposition type according to which is checked.
Returns:	Whether this is a decomposition of the passed type.
Return type:	bool

load(filename)[source]¶

Loads a decomposition of this type.

Parameters:	filename (str) – Where the decomposition is saved.
Raises:	`FileNotFoundError` – If the files are not found in the passed directory.

matrix_both_sides_multiplication(x, y=None)[source]¶

Calculates the both sides (matrix-matrix or matrix-vector) product y.H @ A @ x, where A is the composed matrix represented by this decomposition.

Parameters:	x (numpy.ndarray or scipy.sparse.spmatrix) – Vector or matrix in the product y.H @ A @ x. It must hold self.n == x.shape[0]. y (numpy.ndarray or scipy.sparse.spmatrix) – Vector or matrix in the product y.H @ A @ x. It must hold self.n == y.shape[0]. optional, default: If y is not passed, x is used as y.
Returns:	The result of x.H @ A @ y.
Return type:	numpy.ndarray or scipy.sparse.spmatrix

matrix_right_side_multiplication(x)[source]¶

Calculates the right side (matrix-matrix or matrix-vector) product A @ x, where A is the composed matrix represented by this decomposition.

Parameters:	x (numpy.ndarray or scipy.sparse.spmatrix) – Vector or matrix in the product in the matrix-matrix or matrix-vector A @ x. It must hold self.n == x.shape[0].
Returns:	The result of A @ x.
Return type:	numpy.ndarray or scipy.sparse.spmatrix

n¶: int – The dimension of the squared decomposed matrix.

p¶: numpy.ndarray – The permutation vector. A[p[:, np.newaxis], p[np.newaxis, :]] is the matrix A permuted by the permutation of the decomposition

p_inverse¶: numpy.ndarray – The permutation vector that undos the permutation.

permute_matrix(A)[source]¶

Permute a matrix by the permutation of the decomposition.

Parameters:	A (numpy.ndarray or scipy.sparse.spmatrix) – The matrix that should be permuted.
Returns:	The matrix A permuted by the permutation of the decomposition.
Return type:	numpy.ndarray or scipy.sparse.spmatrix

save(filename)[source]¶

Saves this decomposition.

Parameters:	filename (str) – Where this decomposition should be saved.

solve(b)[source]¶

Solves the equation A x = b regarding x, where A is the composed matrix represented by this decomposition.

Parameters:	b (numpy.ndarray or scipy.sparse.spmatrix) – Right-hand side vector or matrix in equation A x = b. It must hold self.n == b.shape[0].
Returns:	An x so that A x = b. The shape of x matches the shape of b.
Return type:	numpy.ndarray or scipy.sparse.spmatrix
Raises:	`matrix.errors.DecompositionSingularError` – If this is a decomposition representing a singular matrix.

type_str = 'base'¶: str – The type of this decomposition represented as string.

unpermute_matrix(A)[source]¶

Unpermute a matrix permuted by the permutation of the decomposition.

Parameters:	A (numpy.ndarray or scipy.sparse.spmatrix) – The matrix that should be unpermuted.
Returns:	The matrix A unpermuted by the permutation of the decomposition.
Return type:	numpy.ndarray or scipy.sparse.spmatrix

Errors¶

This is an overview about the exceptions that could arise in this package. They are available in matrix.errors:

If a matrix should be decomposed with matrix.decompose and the desired decomposition is not computable, the following exceptions can be raised:

NoDecompositionPossibleError¶

class matrix.errors.NoDecompositionPossibleError(base, desired_type)[source]¶

Bases: matrix.errors.BaseError

It is to possible to calculate a desired matrix decomposition.

NoDecompositionPossibleWithProblematicSubdecompositionError¶

class matrix.errors.NoDecompositionPossibleWithProblematicSubdecompositionError(base, desired_type, problematic_leading_principal_submatrix_index, subdecomposition=None)[source]¶

Bases: matrix.errors.NoDecompositionPossibleError

It is not possible to calculate a desired matrix decomposition. Only a subdecompostion could be calculated

NoDecompositionPossibleTooManyEntriesError¶

class matrix.errors.NoDecompositionPossibleTooManyEntriesError(matrix, desired_type)[source]¶

Bases: matrix.errors.NoDecompositionPossibleError

The matrix decomposition is not possible for this matrix because it would have too many entries.

NoDecompositionConversionImplementedError¶

class matrix.errors.NoDecompositionConversionImplementedError(decomposition, desired_type)[source]¶

Bases: matrix.errors.NoDecompositionPossibleError

A decomposition conversion is not implemented for this type.

If a matrix has an invalid characteristic, the following exceptions can occur:

MatrixNotSquareError¶

class matrix.errors.MatrixNotSquareError(matrix)[source]¶

Bases: matrix.errors.MatrixError

A matrix is not a square matrix although this is required.

MatrixNotFiniteError¶

class matrix.errors.MatrixNotFiniteError(matrix)[source]¶

Bases: matrix.errors.MatrixError

A matrix has non-finite entries although a finite matrix is required.

MatrixSingularError¶

class matrix.errors.MatrixSingularError(matrix)[source]¶

Bases: matrix.errors.MatrixError

A matrix is singular although an invertible matrix is required.

MatrixNotHermitianError¶

class matrix.errors.MatrixNotHermitianError(matrix, i=None, j=None)[source]¶

Bases: matrix.errors.MatrixError

A matrix is not Hermitian although a Hermitian matrix is required.

MatrixComplexDiagonalValueError¶

class matrix.errors.MatrixComplexDiagonalValueError(matrix, i=None)[source]¶

Bases: matrix.errors.MatrixNotHermitianError

A matrix has complex diagonal values although real diagonal values are required.

All these exceptions are based on the following exception:

MatrixError¶

class matrix.errors.MatrixError(matrix, message=None)[source]¶

Bases: matrix.errors.BaseError

An exception related to a matrix.

If the matrix represented by a decomposition has an invalid characteristic, the following exceptions can occur:

DecompositionNotFiniteError¶

class matrix.errors.DecompositionNotFiniteError(decomposition)[source]¶

Bases: matrix.errors.DecompositionError

A decomposition of a matrix has non-finite entries although a finite matrix is required.

DecompositionSingularError¶

class matrix.errors.DecompositionSingularError(decomposition)[source]¶

Bases: matrix.errors.DecompositionError

A decomposition represents a singular matrix although a non-singular matrix is required.

If a decomposition should be loaded from a file which is not a valid decomposition file, the following exception is raised:

DecompositionInvalidFile¶

class matrix.errors.DecompositionInvalidFile(filename)[source]¶

Bases: matrix.errors.DecompositionError, OSError

A decomposition indicated that a decomposition should be loaded from an invalid file.

If a decomposition should be loaded from a file which caontains a type which does not fit to the type of the decomposition where it should be loaded into, the following exception is raised:

DecompositionInvalidDecompositionTypeFile¶

class matrix.errors.DecompositionInvalidDecompositionTypeFile(filename, type_file, type_needed)[source]¶

Bases: matrix.errors.DecompositionInvalidFile

A decomposition indicated that a decomposition should be loaded from an file in which another decomposition type is stored.

All these exceptions are based on the following exception:

DecompositionError¶

class matrix.errors.DecompositionError(decomposition, message=None)[source]¶

Bases: matrix.errors.BaseError

An exception related to a decomposition.

The following exception is the base exception from which all other exceptions in this package are derived:

BaseError¶

class matrix.errors.BaseError(message)[source]¶

Bases: Exception

This is the base exception for all exceptions in this package.

Changelog¶

0.8¶

Approximation functions are replaced by more sophisticated approximation functions.

Explicit function for approximating a matrix by a positive (semi)definite matrix is added.

Universal save and load functions are added.

Decompositions obtain is_equal and is_almost_equal methods.

Functions to multiply the matrix represented by a decomposition or its inverse with a matrix or a vector are added.

Allow to directly pass a permutation vector to approximate and decompose methods.

0.7¶

Lineare systems associated to matrices or decompositions can now be solved.

Invertibility of matrices and decompositions can now be examined.

Decompositions can now be examined to see if they contain only finite values.

0.6¶

Decompositions are now saveable and loadable.

0.5¶

Matrices can now be approximated by decompositions.

0.4¶

Positive definiteness and positive semi-definiteness of matrices and decompositions can now be examined.

0.3¶

Dense and sparse matrices are now decomposable into several types (LL, LDL, LDL compressed).

0.2¶

Decompositons are now convertable to other decompositon types.

Decompositions are now comparable.

0.1¶

Several decompositions types are added (LL, LDL, LDL compressed).

Several permutation capabilities added.

Indices and tables¶

Copyright¶

This program is free software: you can redistribute it and/or modify it under the terms of the GNU Affero General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version.

This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Affero General Public License for more details.

You should have received a copy of the GNU Affero General Public License along with this program. If not, see <http://www.gnu.org/licenses/>.