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This module defines a class hierarchy that implements a kind of "lazy"
matrix representation, called the ``LinearOperator``. It can be used to do
linear algebra with extremely large sparse or structured matrices, without
representing those explicitly in memory. Such matrices can be added,
multiplied, transposed, etc.
As a motivating example, suppose you want have a matrix where almost all of
the elements have the value one. The standard sparse matrix representation
skips the storage of zeros, but not ones. By contrast, a LinearOperator is
able to represent such matrices efficiently. First, we need a compact way to
represent an all-ones matrix::
>>> import numpy as np
>>> class Ones(LinearOperator):
... def __init__(self, shape):
... super().__init__(dtype=None, shape=shape)
... def _matvec(self, x):
... return np.repeat(x.sum(), self.shape[0])
Instances of this class emulate ``np.ones(shape)``, but using a constant
amount of storage, independent of ``shape``. The ``_matvec`` method specifies
how this linear operator multiplies with (operates on) a vector. We can now
add this operator to a sparse matrix that stores only offsets from one::
>>> from scipy.sparse import csr_matrix
>>> offsets = csr_matrix([[1, 0, 2], [0, -1, 0], [0, 0, 3]])
>>> A = aslinearoperator(offsets) + Ones(offsets.shape)
>>> A.dot([1, 2, 3])
array([13, 4, 15])
The result is the same as that given by its dense, explicitly-stored
counterpart::
>>> (np.ones(A.shape, A.dtype) + offsets.toarray()).dot([1, 2, 3])
array([13, 4, 15])
Several algorithms in the ``scipy.sparse`` library are able to operate on
``LinearOperator`` instances.
� N)�
isspmatrix)�isshape� isintlike�asmatrix�is_pydata_spmatrix�LinearOperator�aslinearoperatorc s� e Zd ZdZdZ� fdd�Zdd� Zdd� Zd d
� Zd d � Z d
d� Z
dd� Z dd� Z dd� Z
dd� Zdd� Zdd� Zdd � Zd d � Zd d � Zd!d"� Zd#d$� Zd%d&� Zd'd(� Zd)d*� Zd+d,� Zd-d.� Zd/d0� Ze e�Z d1d2� Z e e �Z d3d4� Z d5d6� Z!� Z"S )7r aj Common interface for performing matrix vector products
Many iterative methods (e.g. cg, gmres) do not need to know the
individual entries of a matrix to solve a linear system A*x=b.
Such solvers only require the computation of matrix vector
products, A*v where v is a dense vector. This class serves as
an abstract interface between iterative solvers and matrix-like
objects.
To construct a concrete LinearOperator, either pass appropriate
callables to the constructor of this class, or subclass it.
A subclass must implement either one of the methods ``_matvec``
and ``_matmat``, and the attributes/properties ``shape`` (pair of
integers) and ``dtype`` (may be None). It may call the ``__init__``
on this class to have these attributes validated. Implementing
``_matvec`` automatically implements ``_matmat`` (using a naive
algorithm) and vice-versa.
Optionally, a subclass may implement ``_rmatvec`` or ``_adjoint``
to implement the Hermitian adjoint (conjugate transpose). As with
``_matvec`` and ``_matmat``, implementing either ``_rmatvec`` or
``_adjoint`` implements the other automatically. Implementing
``_adjoint`` is preferable; ``_rmatvec`` is mostly there for
backwards compatibility.
Parameters
----------
shape : tuple
Matrix dimensions (M, N).
matvec : callable f(v)
Returns returns A * v.
rmatvec : callable f(v)
Returns A^H * v, where A^H is the conjugate transpose of A.
matmat : callable f(V)
Returns A * V, where V is a dense matrix with dimensions (N, K).
dtype : dtype
Data type of the matrix.
rmatmat : callable f(V)
Returns A^H * V, where V is a dense matrix with dimensions (M, K).
Attributes
----------
args : tuple
For linear operators describing products etc. of other linear
operators, the operands of the binary operation.
ndim : int
Number of dimensions (this is always 2)
See Also
--------
aslinearoperator : Construct LinearOperators
Notes
-----
The user-defined matvec() function must properly handle the case
where v has shape (N,) as well as the (N,1) case. The shape of
the return type is handled internally by LinearOperator.
LinearOperator instances can also be multiplied, added with each
other and exponentiated, all lazily: the result of these operations
is always a new, composite LinearOperator, that defers linear
operations to the original operators and combines the results.
More details regarding how to subclass a LinearOperator and several
examples of concrete LinearOperator instances can be found in the
external project `PyLops <https://pylops.readthedocs.io>`_.
Examples
--------
>>> import numpy as np
>>> from scipy.sparse.linalg import LinearOperator
>>> def mv(v):
... return np.array([2*v[0], 3*v[1]])
...
>>> A = LinearOperator((2,2), matvec=mv)
>>> A
<2x2 _CustomLinearOperator with dtype=float64>
>>> A.matvec(np.ones(2))
array([ 2., 3.])
>>> A * np.ones(2)
array([ 2., 3.])
� c s\ | t u r tt | ��t�S tt | ��| �}t|�jt jkr,t|�jt jkr,tjdt dd� |S )NzMLinearOperator subclass should implement at least one of _matvec and _matmat.r )�category�
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� s �zLinearOperator.__new__c C s@ |dur t �|�}t|�}t|�std|f ��|| _|| _dS )z�Initialize this LinearOperator.
To be called by subclasses. ``dtype`` may be None; ``shape`` should
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ValueError�shape)�selfr r! r r r �__init__� s
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�zLinearOperator._init_dtypec s t �� fdd�|jD ��S )z�Default matrix-matrix multiplication handler.
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define matrix multiplication (though in a very suboptimal way).
c � g | ] }� � |�d d���qS �r$ � )r' �reshape��.0�col�r" r r �
<listcomp>� � z*LinearOperator._matmat.<locals>.<listcomp>)r �hstack�T�r" �Xr r1 r r � s zLinearOperator._matmatc C s | � |�dd��S )ay Default matrix-vector multiplication handler.
If self is a linear operator of shape (M, N), then this method will
be called on a shape (N,) or (N, 1) ndarray, and should return a
shape (M,) or (M, 1) ndarray.
This default implementation falls back on _matmat, so defining that
will define matrix-vector multiplication as well.
r$ r, )�matmatr- �r" �xr r r r � s
zLinearOperator._matvecc C s� t �|�}| j\}}|j|fkr|j|dfkrtd��| �|�}t|t j�r+t|�}nt �|�}|j dkr<|�
|�}|S |j dkrI|�
|d�}|S td��)ax Matrix-vector multiplication.
Performs the operation y=A*x where A is an MxN linear
operator and x is a column vector or 1-d array.
Parameters
----------
x : {matrix, ndarray}
An array with shape (N,) or (N,1).
Returns
-------
y : {matrix, ndarray}
A matrix or ndarray with shape (M,) or (M,1) depending
on the type and shape of the x argument.
Notes
-----
This matvec wraps the user-specified matvec routine or overridden
_matvec method to ensure that y has the correct shape and type.
r, �dimension mismatchr z/invalid shape returned by user-defined matvec()) r �
asanyarrayr! r r �
isinstance�matrixr r& �ndimr- �r" r: �M�N�yr r r r' � �
� �zLinearOperator.matvecc C s� t �|�}| j\}}|j|fkr|j|dfkrtd��| �|�}t|t j�r+t|�}nt �|�}|j dkr<|�
|�}|S |j dkrI|�
|d�}|S td��)a� Adjoint matrix-vector multiplication.
Performs the operation y = A^H * x where A is an MxN linear
operator and x is a column vector or 1-d array.
Parameters
----------
x : {matrix, ndarray}
An array with shape (M,) or (M,1).
Returns
-------
y : {matrix, ndarray}
A matrix or ndarray with shape (N,) or (N,1) depending
on the type and shape of the x argument.
Notes
-----
This rmatvec wraps the user-specified rmatvec routine or overridden
_rmatvec method to ensure that y has the correct shape and type.
r, r; r z0invalid shape returned by user-defined rmatvec()) r r< r! r �_rmatvecr= r> r r& r? r- r@ r r r �rmatvec� rD zLinearOperator.rmatvecc C s t | �jtjkr
t�| j�|�S )z6Default implementation of _rmatvec; defers to adjoint.)r �_adjointr �NotImplementedError�Hr' r9 r r r rE ' s zLinearOperator._rmatvecc C sl t �|�}|jdkrtd|j ��|jd | jd kr%td| j|jf ��| �|�}t|t j�r4t|�}|S )aP Matrix-matrix multiplication.
Performs the operation y=A*X where A is an MxN linear
operator and X dense N*K matrix or ndarray.
Parameters
----------
X : {matrix, ndarray}
An array with shape (N,K).
Returns
-------
Y : {matrix, ndarray}
A matrix or ndarray with shape (M,K) depending on
the type of the X argument.
Notes
-----
This matmat wraps any user-specified matmat routine or overridden
_matmat method to ensure that y has the correct type.
r �(expected 2-d ndarray or matrix, not %d-dr r, �dimension mismatch: %r, %r) r r< r? r r! r r= r> r �r" r7 �Yr r r r8 / s
�
�
zLinearOperator.matmatc C sl t �|�}|jdkrtd|j ��|jd | jd kr%td| j|jf ��| �|�}t|t j�r4t|�}|S )a; Adjoint matrix-matrix multiplication.
Performs the operation y = A^H * x where A is an MxN linear
operator and x is a column vector or 1-d array, or 2-d array.
The default implementation defers to the adjoint.
Parameters
----------
X : {matrix, ndarray}
A matrix or 2D array.
Returns
-------
Y : {matrix, ndarray}
A matrix or 2D array depending on the type of the input.
Notes
-----
This rmatmat wraps the user-specified rmatmat routine.
r rJ r rK ) r r< r? r r! �_rmatmatr= r> r rL r r r �rmatmatX s
�
�
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| � |�S rP )�dotr9 r r r �__mul__� �
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t| |�S t�|�rt| |�S t�|�}|jdks*|jdkr/|jd dkr/| � |�S |jdkr9| �
|�S t d| ��)ar Matrix-matrix or matrix-vector multiplication.
Parameters
----------
x : array_like
1-d or 2-d array, representing a vector or matrix.
Returns
-------
Ax : array
1-d or 2-d array (depending on the shape of x) that represents
the result of applying this linear operator on x.
r, r z+expected 1-d or 2-d array or matrix, got %r) r= r �_ProductLinearOperatorr �isscalar�_ScaledLinearOperatorr& r? r! r' r8 r r9 r r r rS � s
"
�zLinearOperator.dotc C � t �|�r td��| �|�S �Nz0Scalar operands are not allowed, use '*' instead)r rW r rT �r" �otherr r r �
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zLinearOperator.__matmul__c C rY rZ )r rW r �__rmul__r[ r r r � __rmatmul__� r^ zLinearOperator.__rmatmul__c C � t �|�r
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zLinearOperator.__repr__c C � | � � S )a� Hermitian adjoint.
Returns the Hermitian adjoint of self, aka the Hermitian
conjugate or Hermitian transpose. For a complex matrix, the
Hermitian adjoint is equal to the conjugate transpose.
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Returns
-------
A_H : LinearOperator
Hermitian adjoint of self.
)rG r1 r r r �adjoint� s zLinearOperator.adjointc C rp )z�Transpose this linear operator.
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_transposer1 r r r � transpose� s zLinearOperator.transposec C � t | �S )z6Default implementation of _adjoint; defers to rmatvec.)�_AdjointLinearOperatorr1 r r r rG � � zLinearOperator._adjointc C rt )z? Default implementation of _transpose; defers to rmatvec + conj)�_TransposedLinearOperatorr1 r r r rr � rv zLinearOperator._transpose)#rm �
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z _CustomLinearOperator._matmatc C rR rP )r} r9 r r r r rU z _CustomLinearOperator._matvecc C s | j }|d u r td��| � |�S )Nzrmatvec is not defined)r~ rH )r" r: �funcr r r rE s
z _CustomLinearOperator._rmatvecc r� rP )r r rN r6 r r r rN r� z _CustomLinearOperator._rmatmatc C s. t | jd | jd f| j| j| j| j| jd�S )Nr, r )r! r' rF r8 rO r )r r! r~ r} r r� r r1 r r r rG s �z _CustomLinearOperator._adjoint)NNNN) rm rx ry rz r# r r rE rN rG r| r r r r r � s �
r c �@ e Zd ZdZ� fdd�Zdd� Zdd� Zdd � Zd
d � Z� Z S ) ru z$Adjoint of arbitrary Linear Operatorc �8 |j d |j d f}t� j|j|d� || _|f| _d S �Nr, r )r r! �r! r r# r �Ar �r" r� r! r r r r# + � z _AdjointLinearOperator.__init__c C � | j �|�S rP )r� rE r9 r r r r 1 rk z _AdjointLinearOperator._matvecc C r� rP )r� r r9 r r r rE 4 rk z _AdjointLinearOperator._rmatvecc C r� rP )r� rN r9 r r r r 7 rk z _AdjointLinearOperator._matmatc C r� rP )r� r r9 r r r rN : rk z _AdjointLinearOperator._rmatmat�
rm rx ry rz r# r rE r rN r| r r r r ru ) s ru c r� ) rw z*Transposition of arbitrary Linear Operatorc r� r� r� r� r r r r# ? r� z"_TransposedLinearOperator.__init__c C � t �| j�t �|���S rP )r �conjr� rE r9 r r r r E � z!_TransposedLinearOperator._matvecc C r� rP )r r� r� r r9 r r r rE I � z"_TransposedLinearOperator._rmatvecc C r� rP )r r� r� rN r9 r r r r L r� z!_TransposedLinearOperator._matmatc C r� rP )r r� r� r r9 r r r rN P r� z"_TransposedLinearOperator._rmatmatr� r r r r rw = s rw c C s@ |d u rg }| D ]}|d urt |d�r|�|j� qt�|g �S )Nr )�hasattr�appendr r �find_common_type)� operators�dtypesr r r r �
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z_SumLinearOperator.__init__c C � | j d �|�| j d �|� S �Nr r, �r r' r9 r r r r g � z_SumLinearOperator._matvecc C r� r� �r rF r9 r r r rE j r� z_SumLinearOperator._rmatvecc C r� r� �r rO r9 r r r rN m r� z_SumLinearOperator._rmatmatc C r� r� �r r8 r9 r r r r p r� z_SumLinearOperator._matmatc C s | j \}}|j|j S rP �r rI r� r r r rG s �
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���z _ProductLinearOperator.__init__c C � | j d �| j d �|��S r� r� r9 r r r r � � z _ProductLinearOperator._matvecc C r� �Nr, r r� r9 r r r rE � r� z _ProductLinearOperator._rmatvecc C r� r� r� r9 r r r rN � r� z _ProductLinearOperator._rmatmatc C r� r� r� r9 r r r r � r� z _ProductLinearOperator._matmatc C s | j \}}|j|j S rP r� r� r r r rG � r� z _ProductLinearOperator._adjointr� r r r r rV x s rV c r� )
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z _ScaledLinearOperator.__init__c C � | j d | j d �|� S r� r� r9 r r r r � � z _ScaledLinearOperator._matvecc C � t �| jd �| jd �|� S r� )r r� r rF r9 r r r rE � r� z _ScaledLinearOperator._rmatvecc C r� r� )r r� r rO r9 r r r rN � r� z _ScaledLinearOperator._rmatmatc C r� r� r� r9 r r r r � r� z _ScaledLinearOperator._matmatc C s | j \}}|jt�|� S rP )r rI r r� )r" r� r� r r r rG � s
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z _PowerLinearOperator._adjoint) rm rx ry r# r� r rE rN r rG r| r r r r rd � s rd c s, e Zd Z� fdd�Zdd� Zdd� Z� ZS )�MatrixLinearOperatorc s* t � �|j|j� || _d | _|f| _d S rP )r r# r r! r� �_MatrixLinearOperator__adjr )r" r� r r r r# � s z MatrixLinearOperator.__init__c C r� rP )r� rS r6 r r r r � rk z MatrixLinearOperator._matmatc C s | j d u r
t| �| _ | j S rP )r� �_AdjointMatrixOperatorr1 r r r rG � s
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� Z� Z S )�IdentityOperatorNc s t � �||� d S rP )r r# )r" r! r r r r r# � s zIdentityOperator.__init__c C � |S rP r r9 r r r r � � zIdentityOperator._matvecc C r� rP r r9 r r r rE � r� zIdentityOperator._rmatvecc C r� rP r r9 r r r rN � r� zIdentityOperator._rmatmatc C r� rP r r9 r r r r � r� zIdentityOperator._matmatc C s | S rP r r1 r r r rG r� zIdentityOperator._adjointrP r� r r r r r� � s r� c C s� t | t�r| S t | tj�st | tj�r(| jdkr td��t�t�| ��} t | �S t
| �s0t | �r4t | �S t | d�rgt | d�rgd}d}d}t | d�rL| j
}t | d�rT| j}t | d�r\| j}t| j| j|||d �S td
��) a� Return A as a LinearOperator.
'A' may be any of the following types:
- ndarray
- matrix
- sparse matrix (e.g. csr_matrix, lil_matrix, etc.)
- LinearOperator
- An object with .shape and .matvec attributes
See the LinearOperator documentation for additional information.
Notes
-----
If 'A' has no .dtype attribute, the data type is determined by calling
:func:`LinearOperator.matvec()` - set the .dtype attribute to prevent this
call upon the linear operator creation.
Examples
--------
>>> from scipy.sparse.linalg import aslinearoperator
>>> M = np.array([[1,2,3],[4,5,6]], dtype=np.int32)
>>> aslinearoperator(M)
<2x3 MatrixLinearOperator with dtype=int32>
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atleast_2dr& r� r r r� rF rO r r! r' � TypeError)r� rF rO r r r r r s.
�rP )rz r �numpyr � scipy.sparser �scipy.sparse._sputilsr r r r �__all__r r ru rw r� rg rV rX rd r� r� r� r r r r r �<module> s, * I.
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