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/propertiesshape
(pair of integers) anddtype
(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 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 thatmatmat
,rmatvec
, andrmatmat
would result in the same dtype of the output given an int8 input asmatvec
.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.])