LinearOperator

class probnum.linops.LinearOperator(dtype, shape)

Bases: scipy.sparse.linalg.interface.LinearOperator

Finite dimensional linear operators.

This class provides a way to define finite dimensional linear operators without explicitly constructing a matrix representation. Instead it suffices to define a matrix-vector product and a shape attribute. This avoids unnecessary memory usage and can often be more convenient to derive.

LinearOperator instances can be multiplied, added and exponentiated. This happens lazily: the result of these operations is a new, composite LinearOperator, that defers linear operations to the original operators and combines the results.

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 \(A v\).
• rmatvec (callable f(v)) – Returns \(A^H v\), where \(A^H\) is the conjugate transpose of \(A\).
• matmat (callable f(V)) – Returns \(AV\), where \(V\) is a dense matrix with dimensions (N, K).
• dtype (dtype) – Data type of the operator.
• rmatmat (callable f(V)) – Returns \(A^H V\), where \(V\) is a dense matrix with dimensions (M, K).

See also

aslinop

Transform into a LinearOperator.

Examples

>>> import numpy as np
>>> from probnum.linops import LinearOperator
>>> def mv(v):
...     return np.array([2 * v[0] - v[1], 3 * v[1]])
...
>>> A = LinearOperator(shape=(2, 2), matvec=mv)
>>> A
<2x2 _CustomLinearOperator with dtype=float64>
>>> A.matvec(np.array([1., 2.]))
array([0., 6.])
>>> A @ np.ones(2)
array([1., 3.])

Attributes Summary

H	Hermitian adjoint.
T	Transpose this linear operator.
ndim

Methods Summary

__call__(x)	Call self as a function.
adjoint()	Hermitian adjoint.
cond([p])	Compute the condition number of the linear operator.
det()	Determinant of the linear operator.
dot(x)	Matrix-matrix or matrix-vector multiplication.
eigvals()	Eigenvalue spectrum of the linear operator.
inv()	Inverse of the linear operator.
logabsdet()	Log absolute determinant of the linear operator.
matmat(X)	Matrix-matrix multiplication.
matvec(x)	Matrix-vector multiplication.
rank()	Rank of the linear operator.
rmatmat(X)	Adjoint matrix-matrix multiplication.
rmatvec(x)	Adjoint matrix-vector multiplication.
todense()	Dense matrix representation of the linear operator.
trace()	Trace of the linear operator.
transpose()	Transpose this linear operator.

Attributes Documentation

H

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.

Can be abbreviated self.H instead of self.adjoint().

Returns:A_H – Hermitian adjoint of self. Return type:LinearOperator

T

Transpose this linear operator.

Can be abbreviated self.T instead of self.transpose().

ndim = 2

Methods Documentation

__call__(x)

Call self as a function.

adjoint()

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.

Can be abbreviated self.H instead of self.adjoint().

Returns:A_H – Hermitian adjoint of self. Return type:LinearOperator

cond(p=None)

Compute the condition number of the linear operator.

The condition number of the linear operator with respect to the p norm. It measures how much the solution \(x\) of the linear system \(Ax=b\) changes with respect to small changes in \(b\).

Parameters:p ({None, 1, , 2, , inf, 'fro'}, optional) –

Order of the norm:

p	norm for matrices
None	2-norm, computed directly via singular value decomposition
’fro’	Frobenius norm
np.inf	max(sum(abs(x), axis=1))
1	max(sum(abs(x), axis=0))
2	2-norm (largest sing. value)

Returns:cond – The condition number of the linear operator. May be infinite. Return type:{float, inf}

det()

Determinant of the linear operator.

dot(x)

Matrix-matrix or matrix-vector multiplication.

Parameters:x (array_like) – 1-d or 2-d array, representing a vector or matrix. Returns:Ax – 1-d or 2-d array (depending on the shape of x) that represents the result of applying this linear operator on x. Return type:array

eigvals()

Eigenvalue spectrum of the linear operator.

inv()

Inverse of the linear operator.

logabsdet()

Log absolute determinant of the linear operator.

matmat(X)

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 – A matrix or ndarray with shape (M,K) depending on the type of the X argument. Return type:{matrix, ndarray}

Notes

This matmat wraps any user-specified matmat routine or overridden _matmat method to ensure that y has the correct type.

matvec(x)

Matrix-vector multiplication. Performs the operation y=A*x where A is an MxN linear operator and x is a 1-d array or random variable.

Parameters:x ({matrix, ndarray, RandomVariable}) – An array or RandomVariable with shape (N,) or (N,1). Returns:y – A matrix or ndarray or RandomVariable with shape (M,) or (M,1) depending on the type and shape of the x argument. Return type:{matrix, ndarray}

Notes

This matvec wraps the user-specified matvec routine or overridden _matvec method to ensure that y has the correct shape and type.

rank()

Rank of the linear operator.

rmatmat(X)

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 – A matrix or 2D array depending on the type of the input. Return type:{matrix, ndarray}

Notes

This rmatmat wraps the user-specified rmatmat routine.

rmatvec(x)

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 – A matrix or ndarray with shape (N,) or (N,1) depending on the type and shape of the x argument. Return type:{matrix, ndarray}

Notes

This rmatvec wraps the user-specified rmatvec routine or overridden _rmatvec method to ensure that y has the correct shape and type.

todense()

Dense matrix representation of the linear operator.

This method can be computationally very costly depending on the shape of the linear operator. Use with caution.

Returns:matrix – Matrix representation of the linear operator. Return type:np.ndarray

trace()

Trace of the linear operator.

Computes the trace of a square linear operator \(\text{tr}(A) = \sum_{i-1}^n A_ii\).

Returns:trace – Trace of the linear operator. Return type:float Raises:ValueError : If trace() is called on a non-square matrix.

transpose()

Transpose this linear operator.

Can be abbreviated self.T instead of self.transpose().