scipy.sparse.linalg.

LinearOperator#

class scipy.sparse.linalg.LinearOperator(*args, **kwargs)[source]#

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:
shapetuple

Matrix dimensions (M, N).

matveccallable f(v)

Returns returns A @ v.

rmatveccallable f(v)

Returns A^H @ v, where A^H is the conjugate transpose of A.

matmatcallable f(V)

Returns A @ V, where V is a dense matrix with dimensions (N, K).

dtypedtype

Data type of the matrix.

rmatmatcallable f(V)

Returns A^H @ V, where V is a dense matrix with dimensions (M, K).

Attributes:
argstuple

For linear operators describing products etc. of other linear operators, the operands of the binary operation.

ndimint

Number of dimensions (this is always 2)

Methods

__call__(x)

Call self as a function.

adjoint()

Hermitian adjoint.

dot(x)

Matrix-matrix or matrix-vector multiplication.

matmat(X)

Matrix-matrix multiplication.

matvec(x)

Matrix-vector multiplication.

rmatmat(X)

Adjoint matrix-matrix multiplication.

rmatvec(x)

Adjoint matrix-vector multiplication.

transpose()

Transpose this linear operator.

__mul__

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.

It is highly recommended to explicitly specify the dtype, otherwise it is determined automatically at the cost of a single matvec application on int8 zero vector using the promoted dtype of the output. Python int could be difficult to automatically cast to numpy integers in the definition of the matvec so the determination may be inaccurate. It is assumed that matmat, rmatvec, and rmatmat would result in the same dtype of the output given an int8 input as matvec.

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.

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=int8>
>>> A.matvec(np.ones(2))
array([ 2.,  3.])
>>> A @ np.ones(2)
array([ 2.,  3.])
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