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¶

The matrix L of the decomposition.

Type:	`numpy.matrix` or `scipy.sparse.spmatrix`

P¶

The permutation matrix. P @ A @ P.T is the matrix A permuted by the permutation of the decomposition

Type:	`scipy.sparse.dok_matrix`

append_block_decomposition(dec)[source]¶

Makes a new decomposition where this decompsition and the passed one are appended.

Parameters:	dec (DecompositionBase) – The decomposition that should be appended to this decomposition.
Returns:	dec – A new decomposition where this decompsition and the passed one are appended. This desomposition represents the first block and the passed one the second block.
Return type:	DecompositionBase

as_LDL_Decomposition()[source]¶

as_any_type(*type_strs, copy=False)¶

Converts this 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_same_type(dec, copy=False)¶

Converts the passed decompositions to the same type as this decomposition.

Parameters:	doc (DecompositionBase) – The decomposition that should be converted. copy (bool) – Whether the data of the decomposition that sould be converted should always be copied or only if needed.
Returns:	If the type of the passed decomposition is not same type as this decomposition, a decomposition of this type is returned which represents the same decomposed matrix as the passed 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]¶

Converts this 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¶

The composed matrix represented by this decomposition.

Type:	`numpy.matrix` or `scipy.sparse.spmatrix`

copy()¶

Copies this decomposition.

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

inverse_L_right_side_multiplication(x, dtype=None)[source]¶

inverse_matrix_both_sides_multiplication(x, y=None, dtype=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. dtype (numpy.dtype) – Type to use in computation. optional, default: Determined by the method.
Returns:	The result of y.H @ A @ x.
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, dtype=None)[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]. dtype (numpy.dtype) – Type to use in computation. optional, default: Determined by the method.
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¶

Whether this is a decompositon with permutation.

Type:	`bool`

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 decomposition is 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. `DecompositionInvalidDecompositionTypeFile` – If the files contains another decomposition type.

matrix_both_sides_multiplication(x, y=None, dtype=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. dtype (numpy.dtype) – Type to use in computation. optional, default: Determined by the method.
Returns:	The result of y.H @ A @ x.
Return type:	numpy.ndarray or scipy.sparse.spmatrix

matrix_right_side_multiplication(x, dtype=None)[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]. dtype (numpy.dtype) – Type to use in computation. optional, default: Determined by the method.
Returns:	The result of A @ x.
Return type:	numpy.ndarray or scipy.sparse.spmatrix

n¶

The dimension of the squared decomposed matrix.

Type:	`int`

p¶

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

Type:	`numpy.ndarray`

p_inverse¶

The permutation vector that undoes the permutation.

Type:	`numpy.ndarray`

permute_matrix(A)¶

Permutes 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, dtype=None)¶

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]. dtype (numpy.dtype) – Type to use in computation. optional, default: Determined by the method.
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'¶

The type of this decomposition represented as string.

Type:	`str`

unpermute_matrix(A)¶

Unpermutes 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¶

The permutation matrix.

Type:	`scipy.sparse.dia_matrix`

L¶

The matrix L of the decomposition.

Type:	`numpy.matrix` or `scipy.sparse.spmatrix`

LD¶

A matrix whose diagonal values are the diagonal values of D and whose off-diagonal values are those of L.

Type:	`numpy.matrix` or `scipy.sparse.spmatrix`

P¶

The permutation matrix. P @ A @ P.T is the matrix A permuted by the permutation of the decomposition

Type:	`scipy.sparse.dok_matrix`

append_block_decomposition(dec)[source]¶

Makes a new decomposition where this decompsition and the passed one are appended.

Parameters:	dec (DecompositionBase) – The decomposition that should be appended to this decomposition.
Returns:	dec – A new decomposition where this decompsition and the passed one are appended. This desomposition represents the first block and the passed one the second block.
Return type:	DecompositionBase

as_LDL_DecompositionCompressed()[source]¶

as_LL_Decomposition()[source]¶

as_any_type(*type_strs, copy=False)¶

Converts this 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_same_type(dec, copy=False)¶

Converts the passed decompositions to the same type as this decomposition.

Parameters:	doc (DecompositionBase) – The decomposition that should be converted. copy (bool) – Whether the data of the decomposition that sould be converted should always be copied or only if needed.
Returns:	If the type of the passed decomposition is not same type as this decomposition, a decomposition of this type is returned which represents the same decomposed matrix as the passed 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]¶

Converts this 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¶

The composed matrix represented by this decomposition.

Type:	`numpy.matrix` or `scipy.sparse.spmatrix`

copy()¶

Copies this decomposition.

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

d¶

The diagonal vector of the matrix D of the decomposition.

Type:	`numpy.ndarray`

inverse_matrix_both_sides_multiplication(x, y=None, dtype=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. dtype (numpy.dtype) – Type to use in computation. optional, default: Determined by the method.
Returns:	The result of y.H @ A @ x.
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, dtype=None)[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]. dtype (numpy.dtype) – Type to use in computation. optional, default: Determined by the method.
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¶

Whether this is a decompositon with permutation.

Type:	`bool`

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 decomposition is 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. `DecompositionInvalidDecompositionTypeFile` – If the files contains another decomposition type.

matrix_both_sides_multiplication(x, y=None, dtype=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. dtype (numpy.dtype) – Type to use in computation. optional, default: Determined by the method.
Returns:	The result of y.H @ A @ x.
Return type:	numpy.ndarray or scipy.sparse.spmatrix

matrix_right_side_multiplication(x, dtype=None)[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]. dtype (numpy.dtype) – Type to use in computation. optional, default: Determined by the method.
Returns:	The result of A @ x.
Return type:	numpy.ndarray or scipy.sparse.spmatrix

n¶

The dimension of the squared decomposed matrix.

Type:	`int`

p¶

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

Type:	`numpy.ndarray`

p_inverse¶

The permutation vector that undoes the permutation.

Type:	`numpy.ndarray`

permute_matrix(A)¶

Permutes 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, dtype=None)¶

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]. dtype (numpy.dtype) – Type to use in computation. optional, default: Determined by the method.
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'¶

The type of this decomposition represented as string.

Type:	`str`

unpermute_matrix(A)¶

Unpermutes 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¶

The permutation matrix.

Type:	`scipy.sparse.dia_matrix`

L¶

The matrix L of the decomposition.

Type:	`numpy.matrix` or `scipy.sparse.spmatrix`

LD¶

A matrix whose diagonal values are the diagonal values of D and whose off-diagonal values are those of L.

Type:	`numpy.matrix` or `scipy.sparse.spmatrix`

P¶

The permutation matrix. P @ A @ P.T is the matrix A permuted by the permutation of the decomposition

Type:	`scipy.sparse.dok_matrix`

append_block_decomposition(dec)[source]¶

Makes a new decomposition where this decompsition and the passed one are appended.

Parameters:	dec (DecompositionBase) – The decomposition that should be appended to this decomposition.
Returns:	dec – A new decomposition where this decompsition and the passed one are appended. This desomposition represents the first block and the passed one the second block.
Return type:	DecompositionBase

as_LDL_Decomposition()[source]¶

as_any_type(*type_strs, copy=False)¶

Converts this 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_same_type(dec, copy=False)¶

Converts the passed decompositions to the same type as this decomposition.

Parameters:	doc (DecompositionBase) – The decomposition that should be converted. copy (bool) – Whether the data of the decomposition that sould be converted should always be copied or only if needed.
Returns:	If the type of the passed decomposition is not same type as this decomposition, a decomposition of this type is returned which represents the same decomposed matrix as the passed 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]¶

Converts this 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¶

The composed matrix represented by this decomposition.

Type:	`numpy.matrix` or `scipy.sparse.spmatrix`

copy()¶

Copies this decomposition.

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

d¶

The diagonal vector of the matrix D of the decomposition.

Type:	`numpy.ndarray`

inverse_matrix_both_sides_multiplication(x, y=None, dtype=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. dtype (numpy.dtype) – Type to use in computation. optional, default: Determined by the method.
Returns:	The result of y.H @ A @ x.
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, dtype=None)[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]. dtype (numpy.dtype) – Type to use in computation. optional, default: Determined by the method.
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¶

Whether this is a decompositon with permutation.

Type:	`bool`

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 decomposition is 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. `DecompositionInvalidDecompositionTypeFile` – If the files contains another decomposition type.

matrix_both_sides_multiplication(x, y=None, dtype=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. dtype (numpy.dtype) – Type to use in computation. optional, default: Determined by the method.
Returns:	The result of y.H @ A @ x.
Return type:	numpy.ndarray or scipy.sparse.spmatrix

matrix_right_side_multiplication(x, dtype=None)[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]. dtype (numpy.dtype) – Type to use in computation. optional, default: Determined by the method.
Returns:	The result of A @ x.
Return type:	numpy.ndarray or scipy.sparse.spmatrix

n¶

The dimension of the squared decomposed matrix.

Type:	`int`

p¶

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

Type:	`numpy.ndarray`

p_inverse¶

The permutation vector that undoes the permutation.

Type:	`numpy.ndarray`

permute_matrix(A)¶

Permutes 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, dtype=None)¶

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]. dtype (numpy.dtype) – Type to use in computation. optional, default: Determined by the method.
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'¶

The type of this decomposition represented as string.

Type:	`str`

unpermute_matrix(A)¶

Unpermutes 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 all other 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¶

The permutation matrix. P @ A @ P.T is the matrix A permuted by the permutation of the decomposition

Type:	`scipy.sparse.dok_matrix`

append_block_decomposition(dec)[source]¶

Makes a new decomposition where this decompsition and the passed one are appended.

Parameters:	dec (DecompositionBase) – The decomposition that should be appended to this decomposition.
Returns:	dec – A new decomposition where this decompsition and the passed one are appended. This desomposition represents the first block and the passed one the second block.
Return type:	DecompositionBase

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

Converts this 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_same_type(dec, copy=False)[source]¶

Converts the passed decompositions to the same type as this decomposition.

Parameters:	doc (DecompositionBase) – The decomposition that should be converted. copy (bool) – Whether the data of the decomposition that sould be converted should always be copied or only if needed.
Returns:	If the type of the passed decomposition is not same type as this decomposition, a decomposition of this type is returned which represents the same decomposed matrix as the passed 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]¶

Converts this 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¶

The composed matrix represented by this decomposition.

Type:	`numpy.matrix` or `scipy.sparse.spmatrix`

copy()[source]¶

Copies this decomposition.

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

inverse_matrix_both_sides_multiplication(x, y=None, dtype=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. dtype (numpy.dtype) – Type to use in computation. optional, default: Determined by the method.
Returns:	The result of y.H @ A @ x.
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, dtype=None)[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]. dtype (numpy.dtype) – Type to use in computation. optional, default: Determined by the method.
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¶

Whether this is a decompositon with permutation.

Type:	`bool`

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 decomposition is 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. `DecompositionInvalidDecompositionTypeFile` – If the files contains another decomposition type.

matrix_both_sides_multiplication(x, y=None, dtype=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. dtype (numpy.dtype) – Type to use in computation. optional, default: Determined by the method.
Returns:	The result of y.H @ A @ x.
Return type:	numpy.ndarray or scipy.sparse.spmatrix

matrix_right_side_multiplication(x, dtype=None)[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]. dtype (numpy.dtype) – Type to use in computation. optional, default: Determined by the method.
Returns:	The result of A @ x.
Return type:	numpy.ndarray or scipy.sparse.spmatrix

n¶

The dimension of the squared decomposed matrix.

Type:	`int`

p¶

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

Type:	`numpy.ndarray`

p_inverse¶

The permutation vector that undoes the permutation.

Type:	`numpy.ndarray`

permute_matrix(A)[source]¶

Permutes 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, dtype=None)[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]. dtype (numpy.dtype) – Type to use in computation. optional, default: Determined by the method.
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'¶

The type of this decomposition represented as string.

Type:	`str`

unpermute_matrix(A)[source]¶

Unpermutes 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

Matrix decompositions¶

LL decomposition¶

LDL decomposition¶

LDL decomposition compressed¶

Base decomposition¶

matrix-decomposition

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