Linear Operators¶

Module: pycsou.core.linop

Abstract classes for multidimensional linear maps.

LinearOperator(shape[, dtype, is_explicit, …])

Base class for linear operators.

class LinearOperator(shape: Tuple[int, int], dtype: Optional[type] = None, is_explicit: bool = False, is_dense: bool = False, is_sparse: bool = False, is_dask: bool = False, is_symmetric: bool = False, lipschitz_cst: float = inf)[source]¶

Bases: pycsou.core.map.DifferentiableMap

Base class for linear operators.

Any instance/subclass of this class must at least implement the abstract methods __call__ and adjoint.

Notes

This class supports the following arithmetic operators +, -, *, @, ** and /, implemented with the class methods __add__/__radd__, __sub__/__neg__, __mul__/__rmul__, __matmul__, __pow__, __truediv__. Such arithmetic operators can be used to add, substract, scale, compose, exponentiate or evaluate LinearOperator instances.

__init__(shape: Tuple[int, int], dtype: Optional[type] = None, is_explicit: bool = False, is_dense: bool = False, is_sparse: bool = False, is_dask: bool = False, is_symmetric: bool = False, lipschitz_cst: float = inf)[source]¶

Parameters

shape (Tuple[int, int]) – Shape of the linear operator.
dtype (Optional[type]) – Data type of the linear operator.
is_explicit (bool) – If True, the linear operator is specified explicitly in terms of a Numpy/Scipy/Dask array.
is_dense (bool) – If True, the linear operator is specified explicitly in terms of a Numpy array.
is_sparse (bool) – If True, the linear operator is specified explicitly in terms of a Scipy sparse matrix.
is_dask (bool) – If True, the linear operator is specified explicitly in terms of a Dask array.
is_symmetric (bool) – Whether the linear operator is symmetric or not.
lipschitz_cst (float) – Lipschitz constant (maximal singular value) of the linear operator if known. Default to \(+\infty\).

matvec(x: Union[numbers.Number, numpy.ndarray]) → Union[numbers.Number, numpy.ndarray][source]¶: Alias for self.__call__ to comply with Scipy’s interface.

abstract adjoint(y: Union[numbers.Number, numpy.ndarray]) → Union[numbers.Number, numpy.ndarray][source]¶

Evaluates the adjoint of the operator at a point.

Parameters: y (Union[Number, np.ndarray]) – Point at which the adjoint should be evaluated.
Returns: Evaluation of the adjoint at y.
Return type: Union[Number, np.ndarray]

transpose(y: Union[numbers.Number, numpy.ndarray]) → Union[numbers.Number, numpy.ndarray][source]¶

Evaluates the tranpose of the operator at a point.

Parameters: y (Union[Number, np.ndarray]) – Point at which the transpose should be evaluated.
Returns: Evaluation of the transpose at y.
Return type: Union[Number, np.ndarray]

Notes

For real-valued operators, the adjoint and the transpose coincide. For complex-valued operators, we can define the transpose as \(\mathbf{A}^T \mathbf{y}=\overline{\mathbf{A}^\ast \overline{\mathbf{y}}}\).

jacobianT(arg: Union[numbers.Number, numpy.ndarray, None] = None) → pycsou.core.linop.LinearOperator[source]¶

Transpose of the Jacobian matrix of the differentiable map evaluated at arg.

Parameters: arg (Union[Number, np.ndarray]) – Point at which the transposed Jacobian matrix is evaluated.
Returns: Linear operator associated to the transposed Jacobian matrix.
Return type: LinearOperator

get_adjointOp() → Union[pycsou.core.linop.LinearOperator, pycsou.core.linop.AdjointLinearOperator][source]¶

Return the adjoint operator as a LinearOperator instance.

Returns: The adjoint operator.
Return type: Union[‘LinearOperator’, ‘AdjointLinearOperator’]

property H¶: Alias for self.get_adjointOp.

get_transposeOp() → pycsou.core.linop.TransposeLinearOperator[source]¶

Return the transpose operator as a LinearOperator instance.

Returns: The transpose operator.
Return type: TransposeLinearOperator

Notes

For real-valued operators, the adjoint and the transpose coincide. For complex-valued operators, we can define the transpose as \(\mathbf{A}^T \mathbf{y}=\overline{\mathbf{A}^\ast \overline{\mathbf{y}}}\).

property T¶: Alias for self.get_adjointOp.

property RangeGram¶

Range-Gram operator, obtained by the composition self * self.H (adjoint followed by operator).

Returns: The Range-Gram operator.
Return type: SymmetricLinearOperator

property DomainGram¶

Domain-Gram operator, obtained by the composition self.H * self (operator followed by adjoint).

Returns: The Domain-Gram operator.
Return type: SymmetricLinearOperator

eigenvals(k: int, which='LM', **kwargs) → numpy.ndarray[source]¶

Find k eigenvalues of a square operator.

Parameters

k (int) – The number of eigenvalues and eigenvectors desired. k must be strictly smaller than the dimension of the operator. It is not possible to compute all eigenvectors of a matrix.
which (str, [‘LM’ | ‘SM’ | ‘LR’ | ‘SR’ | ‘LI’ | ‘SI’]) –
Which k eigenvalues to find:
- ‘LM’ : largest magnitude
- ‘SM’ : smallest magnitude
- ‘LR’ : largest real part
- ‘SR’ : smallest real part
- ‘LI’ : largest imaginary part
- ‘SI’ : smallest imaginary part
kwargs (dict) – A dict of additional keyword arguments values accepted by Scipy’s functions scipy.sparse.linalg.eigs() and scipy.sparse.linalg.eigsh().

Returns

Array containing the k requested eigenvalues.

Return type

np.ndarray

Raises

NotImplementedError – If the linear operator is not square.

Notes

This function calls one of the two Scipy’s functions: scipy.sparse.linalg.eigs() and scipy.sparse.linalg.eigsh(). See the documentation of these two functions for more information on their behaviour and the underlying ARPACK functions they rely on.

See also

svds()

singularvals(k: int, which='LM', **kwargs) → numpy.ndarray[source]¶

Compute the largest or smallest k singular values of an operator. The order of the singular values is not guaranteed.

Parameters

k (int) – Number of singular values to compute. Must be 1 <= k < min(self.shape).
which (str, [‘LM’ | ‘SM’]) –
Which k singular values to find:
- ‘LM’ : largest magnitude
- ‘SM’ : smallest magnitude
kwargs (dict) – A dict of additional keyword arguments values accepted by Scipy’s function scipy.sparse.linalg.svds().

Returns

Array containing the k requested singular values.

Return type

np.ndarray

Examples

>>> sig = np.repeat([0., 1., 0.], 10)
>>> filter = signal.hann(5); filter[filter.size//2:] = 0
>>> ConvOp = Convolve1D(size=sig.size, filter=filter)
>>> np.round((ConvOp.singularvals(k=3, which='LM', tol=1e-3)), 2)
array([0.5, 0.5, 0.5])

Notes

This function calls the Scipy’s function: scipy.sparse.linalg.svds(). See the documentation of this function for more information on its behaviour and the underlying ARPACK/LOBPCG functions it relies on.

See also

eigenvals()

compute_lipschitz_cst(**kwargs)[source]¶

Compute the Lipschitz constant of the operator.

Parameters: kwargs (dict) – A dict of additional keyword arguments values accepted by Scipy’s functions scipy.sparse.linalg.eigs(), scipy.sparse.linalg.eigsh(), scipy.sparse.linalg.svds().
Returns: Nothing: The Lipschitz constant is stored in the attribute self.lipschitz_cst.
Return type: None

Examples

>>> sig = np.repeat([0., 1., 0.], 10)
>>> filter = signal.hann(5); filter[filter.size//2:] = 0
>>> ConvOp = Convolve1D(size=sig.size, filter=filter)
>>> ConvOp.compute_lipschitz_cst(tol=1e-2); np.round(ConvOp.lipschitz_cst,1)
0.5

Notes

The Lipschtiz constant of a linear operator is its largest singular value. This function therefore calls the methods self.singularvals with k=1 and which='LM' to perform this computation.

Warning

For high-dimensional linear operators this method can be very time-consuming. Reducing the computation accuracy with the optional argument tol: float may help reduce the computational burden. See Scipy’s functions scipy.sparse.linalg.eigs(), scipy.sparse.linalg.eigsh(), scipy.sparse.linalg.svds() for more on this parameter.

todense() → DenseLinearOperator[source]¶

Convert the operator to a DenseLinearOperator.

Returns: The dense linear operator representation.
Return type: DenseLinearOperator

tosparse() → SparseLinearOperator[source]¶

Convert the operator to a SparseLinearOperator.

Returns: The sparse linear operator representation.
Return type: SparseLinearOperator

tosciop() → scipy.sparse.linalg.interface.LinearOperator[source]¶

Convert the operator to a Scipy scipy.sparse.linalg.LinearOperator.

Returns: The Scipy linear operator representation.
Return type: scipy.sparse.linalg.LinearOperator

property SciOp¶: Alias for method self.tosciop.

topylop() → pylops.LinearOperator.LinearOperator[source]¶

Convert the operator to a Pylops pylops.LinearOperator.

Returns: The Pylops linear operator representation.
Return type: pylops.LinearOperator

property PyLop¶: Alias for method self.topylop.

cond(**kwargs) → float[source]¶

Compute the condition number of the operator.

Parameters: kwargs (dict) – A dict of additional keyword arguments values accepted by Pylops’ method pylops.LinearOperator.cond().
Returns: Condition number.
Return type: float

pinv(y: Union[numbers.Number, numpy.ndarray], eps: numbers.Number = 0, **kwargs) → Union[numbers.Number, numpy.ndarray][source]¶

Evaluate the pseudo-inverse of the operator at y.

Parameters

y (Union[Number, np.ndarray]) – Point at which the pseudo-inverse is evaluated.
eps (Number) – Tikhonov damping.
kwargs – Arbitrary keyword arguments accepted by the function: pylops.optimization.leastsquares.NormalEquationsInversion().

Returns

Evaluation of the pseudo-inverse of the operator at y.

Return type

numpy.ndarray

Notes

This is a wrapper around the function pylops.optimization.leastsquares.NormalEquationsInversion(). Additional information can be found in the help of this function.

property PinvOp¶: Return the pseudo-inverse of the operator as a LinearOperator.

property dagger¶: Alias for self.PinvOp.

property RowProjector¶: Orthogonal projection operator onto the rows of self. It is given by self.dagger * self.

property ColProjector¶: Orthogonal projection operator onto the columns of self. It is given by self * self.dagger.

class AdjointLinearOperator(LinOp: pycsou.core.linop.LinearOperator)[source]¶

Bases: pycsou.core.linop.LinearOperator

__init__(LinOp: pycsou.core.linop.LinearOperator)[source]¶

Parameters

shape (Tuple[int, int]) – Shape of the linear operator.
dtype (Optional[type]) – Data type of the linear operator.
is_explicit (bool) – If True, the linear operator is specified explicitly in terms of a Numpy/Scipy/Dask array.
is_dense (bool) – If True, the linear operator is specified explicitly in terms of a Numpy array.
is_sparse (bool) – If True, the linear operator is specified explicitly in terms of a Scipy sparse matrix.
is_dask (bool) – If True, the linear operator is specified explicitly in terms of a Dask array.
is_symmetric (bool) – Whether the linear operator is symmetric or not.
lipschitz_cst (float) – Lipschitz constant (maximal singular value) of the linear operator if known. Default to \(+\infty\).

adjoint(x: Union[numbers.Number, numpy.ndarray]) → Union[numbers.Number, numpy.ndarray][source]¶

Evaluates the adjoint of the operator at a point.

Parameters: y (Union[Number, np.ndarray]) – Point at which the adjoint should be evaluated.
Returns: Evaluation of the adjoint at y.
Return type: Union[Number, np.ndarray]

compute_lipschitz_cst(**kwargs)[source]¶

Compute the Lipschitz constant of the operator.

Parameters: kwargs (dict) – A dict of additional keyword arguments values accepted by Scipy’s functions scipy.sparse.linalg.eigs(), scipy.sparse.linalg.eigsh(), scipy.sparse.linalg.svds().
Returns: Nothing: The Lipschitz constant is stored in the attribute self.lipschitz_cst.
Return type: None

Examples

>>> sig = np.repeat([0., 1., 0.], 10)
>>> filter = signal.hann(5); filter[filter.size//2:] = 0
>>> ConvOp = Convolve1D(size=sig.size, filter=filter)
>>> ConvOp.compute_lipschitz_cst(tol=1e-2); np.round(ConvOp.lipschitz_cst,1)
0.5

Notes

The Lipschtiz constant of a linear operator is its largest singular value. This function therefore calls the methods self.singularvals with k=1 and which='LM' to perform this computation.

Warning

For high-dimensional linear operators this method can be very time-consuming. Reducing the computation accuracy with the optional argument tol: float may help reduce the computational burden. See Scipy’s functions scipy.sparse.linalg.eigs(), scipy.sparse.linalg.eigsh(), scipy.sparse.linalg.svds() for more on this parameter.

class TransposeLinearOperator(LinOp: pycsou.core.linop.LinearOperator)[source]¶

Bases: pycsou.core.linop.LinearOperator

__init__(LinOp: pycsou.core.linop.LinearOperator)[source]¶

Parameters

shape (Tuple[int, int]) – Shape of the linear operator.
dtype (Optional[type]) – Data type of the linear operator.
is_explicit (bool) – If True, the linear operator is specified explicitly in terms of a Numpy/Scipy/Dask array.
is_dense (bool) – If True, the linear operator is specified explicitly in terms of a Numpy array.
is_sparse (bool) – If True, the linear operator is specified explicitly in terms of a Scipy sparse matrix.
is_dask (bool) – If True, the linear operator is specified explicitly in terms of a Dask array.
is_symmetric (bool) – Whether the linear operator is symmetric or not.
lipschitz_cst (float) – Lipschitz constant (maximal singular value) of the linear operator if known. Default to \(+\infty\).

adjoint(x: Union[numbers.Number, numpy.ndarray]) → Union[numbers.Number, numpy.ndarray][source]¶

Evaluates the adjoint of the operator at a point.

Parameters: y (Union[Number, np.ndarray]) – Point at which the adjoint should be evaluated.
Returns: Evaluation of the adjoint at y.
Return type: Union[Number, np.ndarray]

class LinOpSum(LinOp1: pycsou.core.linop.LinearOperator, LinOp2: pycsou.core.linop.LinearOperator, dtype: Optional[type] = None)[source]¶

Bases: pycsou.core.linop.LinearOperator, pycsou.core.map.DiffMapSum

__init__(LinOp1: pycsou.core.linop.LinearOperator, LinOp2: pycsou.core.linop.LinearOperator, dtype: Optional[type] = None)[source]¶

Parameters

shape (Tuple[int, int]) – Shape of the linear operator.
dtype (Optional[type]) – Data type of the linear operator.
is_explicit (bool) – If True, the linear operator is specified explicitly in terms of a Numpy/Scipy/Dask array.
is_dense (bool) – If True, the linear operator is specified explicitly in terms of a Numpy array.
is_sparse (bool) – If True, the linear operator is specified explicitly in terms of a Scipy sparse matrix.
is_dask (bool) – If True, the linear operator is specified explicitly in terms of a Dask array.
is_symmetric (bool) – Whether the linear operator is symmetric or not.
lipschitz_cst (float) – Lipschitz constant (maximal singular value) of the linear operator if known. Default to \(+\infty\).

adjoint(y: Union[numbers.Number, numpy.ndarray]) → Union[numbers.Number, numpy.ndarray][source]¶

Evaluates the adjoint of the operator at a point.

Parameters: y (Union[Number, np.ndarray]) – Point at which the adjoint should be evaluated.
Returns: Evaluation of the adjoint at y.
Return type: Union[Number, np.ndarray]

class LinOpComp(LinOp1: pycsou.core.linop.LinearOperator, LinOp2: pycsou.core.linop.LinearOperator, dtype: Optional[type] = None)[source]¶

Bases: pycsou.core.linop.LinearOperator, pycsou.core.map.DiffMapComp

__init__(LinOp1: pycsou.core.linop.LinearOperator, LinOp2: pycsou.core.linop.LinearOperator, dtype: Optional[type] = None)[source]¶

Parameters

shape (Tuple[int, int]) – Shape of the linear operator.
dtype (Optional[type]) – Data type of the linear operator.
is_explicit (bool) – If True, the linear operator is specified explicitly in terms of a Numpy/Scipy/Dask array.
is_dense (bool) – If True, the linear operator is specified explicitly in terms of a Numpy array.
is_sparse (bool) – If True, the linear operator is specified explicitly in terms of a Scipy sparse matrix.
is_dask (bool) – If True, the linear operator is specified explicitly in terms of a Dask array.
is_symmetric (bool) – Whether the linear operator is symmetric or not.
lipschitz_cst (float) – Lipschitz constant (maximal singular value) of the linear operator if known. Default to \(+\infty\).

adjoint(y: Union[numbers.Number, numpy.ndarray]) → Union[numbers.Number, numpy.ndarray][source]¶

Evaluates the adjoint of the operator at a point.

Parameters: y (Union[Number, np.ndarray]) – Point at which the adjoint should be evaluated.
Returns: Evaluation of the adjoint at y.
Return type: Union[Number, np.ndarray]

class SymmetricLinearOperator(LinOp: pycsou.core.linop.LinearOperator)[source]¶

Bases: pycsou.core.linop.LinearOperator

__init__(LinOp: pycsou.core.linop.LinearOperator)[source]¶

Parameters

shape (Tuple[int, int]) – Shape of the linear operator.
dtype (Optional[type]) – Data type of the linear operator.
is_explicit (bool) – If True, the linear operator is specified explicitly in terms of a Numpy/Scipy/Dask array.
is_dense (bool) – If True, the linear operator is specified explicitly in terms of a Numpy array.
is_sparse (bool) – If True, the linear operator is specified explicitly in terms of a Scipy sparse matrix.
is_dask (bool) – If True, the linear operator is specified explicitly in terms of a Dask array.
is_symmetric (bool) – Whether the linear operator is symmetric or not.
lipschitz_cst (float) – Lipschitz constant (maximal singular value) of the linear operator if known. Default to \(+\infty\).

adjoint(y: Union[numbers.Number, numpy.ndarray]) → Union[numbers.Number, numpy.ndarray][source]¶

Evaluates the adjoint of the operator at a point.

Parameters: y (Union[Number, np.ndarray]) – Point at which the adjoint should be evaluated.
Returns: Evaluation of the adjoint at y.
Return type: Union[Number, np.ndarray]

class UnitaryOperator(size: int, dtype: Optional[type] = None, is_explicit: bool = False, is_dense: bool = False, is_sparse: bool = False, is_dask: bool = False, is_symmetric: bool = False)[source]¶

Bases: pycsou.core.linop.LinearOperator

__init__(size: int, dtype: Optional[type] = None, is_explicit: bool = False, is_dense: bool = False, is_sparse: bool = False, is_dask: bool = False, is_symmetric: bool = False)[source]¶

Parameters

shape (Tuple[int, int]) – Shape of the linear operator.
dtype (Optional[type]) – Data type of the linear operator.
is_explicit (bool) – If True, the linear operator is specified explicitly in terms of a Numpy/Scipy/Dask array.
is_dense (bool) – If True, the linear operator is specified explicitly in terms of a Numpy array.
is_sparse (bool) – If True, the linear operator is specified explicitly in terms of a Scipy sparse matrix.
is_dask (bool) – If True, the linear operator is specified explicitly in terms of a Dask array.
is_symmetric (bool) – Whether the linear operator is symmetric or not.
lipschitz_cst (float) – Lipschitz constant (maximal singular value) of the linear operator if known. Default to \(+\infty\).

property RangeGram¶

Range-Gram operator, obtained by the composition self * self.H (adjoint followed by operator).

Returns: The Range-Gram operator.
Return type: SymmetricLinearOperator

property DomainGram¶

Domain-Gram operator, obtained by the composition self.H * self (operator followed by adjoint).

Returns: The Domain-Gram operator.
Return type: SymmetricLinearOperator

eigenvals(k: int, which='LM', **kwargs) → numpy.ndarray[source]¶

Find k eigenvalues of a square operator.

Parameters

k (int) – The number of eigenvalues and eigenvectors desired. k must be strictly smaller than the dimension of the operator. It is not possible to compute all eigenvectors of a matrix.
which (str, [‘LM’ | ‘SM’ | ‘LR’ | ‘SR’ | ‘LI’ | ‘SI’]) –
Which k eigenvalues to find:
- ‘LM’ : largest magnitude
- ‘SM’ : smallest magnitude
- ‘LR’ : largest real part
- ‘SR’ : smallest real part
- ‘LI’ : largest imaginary part
- ‘SI’ : smallest imaginary part
kwargs (dict) – A dict of additional keyword arguments values accepted by Scipy’s functions scipy.sparse.linalg.eigs() and scipy.sparse.linalg.eigsh().

Returns

Array containing the k requested eigenvalues.

Return type

np.ndarray

Raises

NotImplementedError – If the linear operator is not square.

Notes

This function calls one of the two Scipy’s functions: scipy.sparse.linalg.eigs() and scipy.sparse.linalg.eigsh(). See the documentation of these two functions for more information on their behaviour and the underlying ARPACK functions they rely on.

See also

svds()

singularvals(k: int, which='LM', **kwargs) → numpy.ndarray[source]¶

Compute the largest or smallest k singular values of an operator. The order of the singular values is not guaranteed.

Parameters

k (int) – Number of singular values to compute. Must be 1 <= k < min(self.shape).
which (str, [‘LM’ | ‘SM’]) –
Which k singular values to find:
- ‘LM’ : largest magnitude
- ‘SM’ : smallest magnitude
kwargs (dict) – A dict of additional keyword arguments values accepted by Scipy’s function scipy.sparse.linalg.svds().

Returns

Array containing the k requested singular values.

Return type

np.ndarray

Examples

>>> sig = np.repeat([0., 1., 0.], 10)
>>> filter = signal.hann(5); filter[filter.size//2:] = 0
>>> ConvOp = Convolve1D(size=sig.size, filter=filter)
>>> np.round((ConvOp.singularvals(k=3, which='LM', tol=1e-3)), 2)
array([0.5, 0.5, 0.5])

Notes

This function calls the Scipy’s function: scipy.sparse.linalg.svds(). See the documentation of this function for more information on its behaviour and the underlying ARPACK/LOBPCG functions it relies on.

See also

eigenvals()

compute_lipschitz_cst(**kwargs)[source]¶

Compute the Lipschitz constant of the operator.

Parameters: kwargs (dict) – A dict of additional keyword arguments values accepted by Scipy’s functions scipy.sparse.linalg.eigs(), scipy.sparse.linalg.eigsh(), scipy.sparse.linalg.svds().
Returns: Nothing: The Lipschitz constant is stored in the attribute self.lipschitz_cst.
Return type: None

Examples

>>> sig = np.repeat([0., 1., 0.], 10)
>>> filter = signal.hann(5); filter[filter.size//2:] = 0
>>> ConvOp = Convolve1D(size=sig.size, filter=filter)
>>> ConvOp.compute_lipschitz_cst(tol=1e-2); np.round(ConvOp.lipschitz_cst,1)
0.5

Notes

The Lipschtiz constant of a linear operator is its largest singular value. This function therefore calls the methods self.singularvals with k=1 and which='LM' to perform this computation.

Warning

For high-dimensional linear operators this method can be very time-consuming. Reducing the computation accuracy with the optional argument tol: float may help reduce the computational burden. See Scipy’s functions scipy.sparse.linalg.eigs(), scipy.sparse.linalg.eigsh(), scipy.sparse.linalg.svds() for more on this parameter.

pinv(y: Union[numbers.Number, numpy.ndarray], eps: numbers.Number = 0, **kwargs) → Union[numbers.Number, numpy.ndarray][source]¶

Evaluate the pseudo-inverse of the operator at y.

Parameters

y (Union[Number, np.ndarray]) – Point at which the pseudo-inverse is evaluated.
eps (Number) – Tikhonov damping.
kwargs – Arbitrary keyword arguments accepted by the function: pylops.optimization.leastsquares.NormalEquationsInversion().

Returns

Evaluation of the pseudo-inverse of the operator at y.

Return type

numpy.ndarray

Notes

This is a wrapper around the function pylops.optimization.leastsquares.NormalEquationsInversion(). Additional information can be found in the help of this function.

property PinvOp¶: Return the pseudo-inverse of the operator as a LinearOperator.

cond(**kwargs)[source]¶

Compute the condition number of the operator.

Parameters: kwargs (dict) – A dict of additional keyword arguments values accepted by Pylops’ method pylops.LinearOperator.cond().
Returns: Condition number.
Return type: float

class LinOpPinv(LinOp: pycsou.core.linop.LinearOperator, eps: numbers.Number = 0)[source]¶

Bases: pycsou.core.linop.LinearOperator

__init__(LinOp: pycsou.core.linop.LinearOperator, eps: numbers.Number = 0)[source]¶

Parameters

shape (Tuple[int, int]) – Shape of the linear operator.
dtype (Optional[type]) – Data type of the linear operator.
is_explicit (bool) – If True, the linear operator is specified explicitly in terms of a Numpy/Scipy/Dask array.
is_dense (bool) – If True, the linear operator is specified explicitly in terms of a Numpy array.
is_sparse (bool) – If True, the linear operator is specified explicitly in terms of a Scipy sparse matrix.
is_dask (bool) – If True, the linear operator is specified explicitly in terms of a Dask array.
is_symmetric (bool) – Whether the linear operator is symmetric or not.
lipschitz_cst (float) – Lipschitz constant (maximal singular value) of the linear operator if known. Default to \(+\infty\).

adjoint(y: Union[numbers.Number, numpy.ndarray]) → Union[numbers.Number, numpy.ndarray][source]¶

Evaluates the adjoint of the operator at a point.

Parameters: y (Union[Number, np.ndarray]) – Point at which the adjoint should be evaluated.
Returns: Evaluation of the adjoint at y.
Return type: Union[Number, np.ndarray]