Kronecker¶

class probnum.linops.Kronecker(A, B)

Kronecker product of two linear operators.

The Kronecker product 1 $$A \otimes B$$ of two linear operators $$A$$ and $$B$$ is given by

$\begin{split}A \otimes B = \begin{bmatrix} A_{11} B & \dots & A_{1 m_1} B \\ \vdots & \ddots & \vdots \\ A_{n_11} B & \dots & A_{n_1 m_1} B \end{bmatrix}\end{split}$

where $$A_{ij}v=A(v_j e_i)$$, where $$e_i$$ is the $$i^{\text{th}}$$ unit vector. The result is a new linear operator mapping from $$\mathbb{R}^{n_1n_2}$$ to $$\mathbb{R}^{m_1m_2}$$. By recognizing that $$(A \otimes B)\operatorname{vec}(X) = AXB^{\top}$$, the Kronecker product can be understood as “translation” between matrix multiplication and (row-wise) vectorization.

Parameters
• A (np.ndarray or LinearOperator) – The first operator.

• B (np.ndarray or LinearOperator) – The second operator.

• dtype (dtype) – Data type of the operator.

References

1

Van Loan, C. F., The ubiquitous Kronecker product, Journal of Computational and Applied Mathematics, 2000, 123, 85-100

SymmetricKronecker

The symmetric Kronecker product of two linear operators.

Attributes Summary

 T rtype LinearOperator dtype Data type of the linear operator. is_square Whether input dimension matches output dimension. ndim Number of linear operator dimensions. shape Shape of the linear operator. size rtype int

Methods Summary

 __call__(x[, axis]) Call self as a function. astype(dtype[, order, casting, subok, copy]) Cast a linear operator to a different dtype. broadcast_matmat(matmat) Broadcasting for a (implicitly defined) matrix-matrix product. broadcast_matvec(matvec) Broadcasting for a (implicitly defined) matrix-vector product. broadcast_rmatmat(rmatmat) rtype broadcast_rmatvec(rmatvec) rtype cond([p]) Compute the condition number of the linear operator. det() Determinant of the linear operator. eigvals() Eigenvalue spectrum of the linear operator. inv() Inverse of the linear operator. Log absolute determinant of the linear operator. rank() Rank of the linear operator. todense([cache]) Dense matrix representation of the linear operator. trace() Trace of the linear operator. transpose(*axes) Transpose this linear operator.

Attributes Documentation

T
Return type

LinearOperator

dtype

Data type of the linear operator.

Return type

dtype

is_square

Whether input dimension matches output dimension.

Return type

bool

ndim

Number of linear operator dimensions.

Defined analogously to numpy.ndarray.ndim.

Return type

int

shape

Shape of the linear operator.

Defined as a tuple of the output and input dimension of operator.

Return type

Tuple[int, int]

size
Return type

int

Methods Documentation

__call__(x, axis=None)

Call self as a function.

Return type

ndarray

astype(dtype, order='K', casting='unsafe', subok=True, copy=True)

Cast a linear operator to a different dtype.

Parameters
Return type

LinearOperator

Broadcasting for a (implicitly defined) matrix-matrix product.

Convenience function / decorator to broadcast the definition of a matrix-matrix product to vectors. This can be used to easily construct a new linear operator only from a matrix-matrix product.

Return type

Broadcasting for a (implicitly defined) matrix-vector product.

Convenience function / decorator to broadcast the definition of a matrix-vector product. This can be used to easily construct a new linear operator only from a matrix-vector product.

Return type
Return type
Return type
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

The condition number of the linear operator. May be infinite.

Return type

cond

det()

Determinant of the linear operator.

Return type

inexact

eigvals()

Eigenvalue spectrum of the linear operator.

Return type

ndarray

inv()

Inverse of the linear operator.

Return type

LinearOperator

logabsdet()

Log absolute determinant of the linear operator.

Return type

inexact

rank()

Rank of the linear operator.

Return type

int64

todense(cache=True)

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

LinAlgError : – If trace() is called on a non-square matrix.

transpose(*axes)

Transpose this linear operator.

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

Return type

LinearOperator