Matrix¶
- class probnum.linops.Matrix(A)¶
Bases:
LambdaLinearOperator
A linear operator defined via a matrix.
- Parameters:
A (Union[ArrayLike, scipy.sparse.spmatrix]) – The explicit matrix.
Attributes Summary
Transpose of the linear operator.
Data type of the linear operator.
Whether the
LinearOperator
represents a lower triangular matrix.Whether the
LinearOperator
\(L \in \mathbb{R}^{n \times n}\) is (strictly) positive-definite, i.e. \(x^T L x > 0\) for \(x \in \mathbb{R}^n\).Whether input dimension matches output dimension.
Whether the
LinearOperator
\(L\) is symmetric, i.e. \(L = L^T\).Whether the
LinearOperator
represents an upper triangular matrix.Number of linear operator dimensions.
Shape of the linear operator.
Product of the
shape
entries.Methods Summary
__call__
(x[, axis])Apply the linear operator to an input array along a specified axis.
astype
(dtype[, order, casting, subok, copy])Cast a linear operator to a different
dtype
.broadcast_matmat
([matmat, method])Broadcasting for a (implicitly defined) matrix-matrix product.
broadcast_matvec
([matvec, method])Broadcasting for a (implicitly defined) matrix-vector product.
cholesky
([lower])Computes a Cholesky decomposition of the
LinearOperator
.cond
([p])Compute the condition number of the linear operator.
det
()Determinant of the linear operator.
diagonal
()Diagonal 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.
solve
(b)Solves linear systems
A @ x = b
, whereb
is either a vector or a (stack of) matrices.Compute or approximate the closest symmetric
LinearOperator
in the Frobenius norm.todense
([cache])Dense matrix representation of the linear operator.
trace
()Trace of the linear operator.
transpose
(*axes)Transpose this linear operator.
Attributes Documentation
- T¶
Transpose of the linear operator.
- dtype¶
Data type of the linear operator.
- is_lower_triangular¶
Whether the
LinearOperator
represents a lower triangular matrix.If this is
None
, it is unknown whether the matrix is lower triangular or not.
- is_positive_definite¶
Whether the
LinearOperator
\(L \in \mathbb{R}^{n \times n}\) is (strictly) positive-definite, i.e. \(x^T L x > 0\) for \(x \in \mathbb{R}^n\).If this is
None
, it is unknown whether the matrix is positive-definite or not. Only symmetric operators can be positive-definite.- Raises:
ValueError – When setting
is_positive_definite
toTrue
whileis_symmetric
isFalse
.
- is_square¶
Whether input dimension matches output dimension.
- is_symmetric¶
Whether the
LinearOperator
\(L\) is symmetric, i.e. \(L = L^T\).If this is
None
, it is unknown whether the operator is symmetric or not. Only square operators can be symmetric.- Raises:
ValueError – When setting
is_symmetric
toTrue
on a non-squareLinearOperator
.
- is_upper_triangular¶
Whether the
LinearOperator
represents an upper triangular matrix.If this is
None
, it is unknown whether the matrix is upper triangular or not.
- ndim¶
Number of linear operator dimensions.
Defined analogously to
numpy.ndarray.ndim
.
- shape¶
Shape of the linear operator.
Defined as a tuple of the output and input dimension of operator.
- size¶
Product of the
shape
entries.Defined analogously to
numpy.ndarray.size
.
Methods Documentation
- __call__(x, axis=None)¶
Apply the linear operator to an input array along a specified axis.
- Parameters:
x (np.ndarray) – Input array.
axis (int) – Axis along which to apply the linear operator. Guaranteed to be positive and valid, i.e.
axis
is a valid index into the shape ofx
, andx
has the correct shape alongaxis
.
- Returns:
apply_result – Array resulting in the application of the linear operator to
x
alongaxis
.- Return type:
np.ndarray
- Raises:
ValueError – If the shape of
x
is invalid.numpy.AxisError – If the axis argument is not within the valid range.
- astype(dtype, order='K', casting='unsafe', subok=True, copy=True)¶
Cast a linear operator to a different
dtype
.- Parameters:
dtype (DTypeLike) – Data type to which the linear operator is cast.
order (str) – Memory layout order of the result.
casting (str) – Controls what kind of data casting may occur.
subok (bool) – If True, then sub-classes will be passed-through (default). False is currently not supported for linear operators.
copy (bool) – Whether to return a new linear operator, even if
dtype
is the same.
- Raises:
TypeError – If the linear operator can not be cast to the desired
dtype
according to the givencasting
rule.NotImplementedError – If
subok
is set toTrue
.
- Return type:
- static broadcast_matmat(matmat=None, method=False)¶
Broadcasting for a (implicitly defined) matrix-matrix product.
Convenience function / decorator to broadcast the definition of a matrix-matrix product to stacks of matrices. This can be used to easily construct a new linear operator only from a matrix-matrix product.
- static broadcast_matvec(matvec=None, method=False)¶
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.
- cholesky(lower=True)¶
Computes a Cholesky decomposition of the
LinearOperator
.The Cholesky decomposition of a symmetric positive-definite matrix \(A \in \mathbb{R}^{n \times n}\) is given by \(A = L L^T\), where the unique Cholesky factor \(L \in \mathbb{R}^{n \times n}\) of \(A\) is a lower triangular matrix with a positive diagonal.
As a side-effect, this method will set the value of
is_positive_definite
toTrue
, if the computation of the Cholesky factorization succeeds. Otherwise,is_positive_definite
will be set toFalse
.The result of this computation will be cached. If
cholesky()
is first called withlower=True
first and afterwards withlower=False
or vice-versa, the method simply returns the transpose of the cached value.- Parameters:
lower (bool) – If this is set to
False
, this method computes and returns the upper triangular Cholesky factor \(U = L^T\), for which \(A = U^T U\). By default (True
), the method computes the lower triangular Cholesky factor \(L\).- Returns:
The lower or upper Cholesky factor of the
LinearOperator
, depending on the value of the parameterlower
. The result will have its propertiesis_upper_triangular
/is_lower_triangular
set accordingly.- Return type:
cholesky_factor
- Raises:
numpy.linalg.LinAlgError – If the
LinearOperator
is not symmetric, i.e. ifis_symmetric
is not set toTrue
.numpy.linalg.LinAlgError – If the
LinearOperator
is not positive definite.
- 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
- Raises:
numpy.linalg.LinAlgError – If
cond()
is called on a non-square matrix.
- det()¶
Determinant of the linear operator.
- Returns:
The determinant of the linear operator.
- Return type:
det
- Raises:
numpy.linalg.LinAlgError – If
det()
is called on a non-square matrix.
- eigvals()¶
Eigenvalue spectrum of the linear operator.
- Raises:
numpy.linalg.LinAlgError – If
eigvals()
is called on a non-square operator.- Return type:
- inv()¶
Inverse of the linear operator.
- Returns:
inv – Inverse of this linear operator, which is again a LinearOperator.
- Return type:
- Raises:
numpy.linalg.LinAlgError – If
inv()
is called on a non-square linear operator.
- logabsdet()¶
Log absolute determinant of the linear operator.
- Returns:
The log absolute determinant of the linear operator.
- Return type:
logabsdet
- Raises:
numpy.linalg.LinAlgError – If
logabsdet()
is called on a non-square matrix.
- rank()¶
Rank of the linear operator.
- Return type:
int64
- solve(b)¶
Solves linear systems
A @ x = b
, whereb
is either a vector or a (stack of) matrices.This method broadcasts like
A.inv() @ b
, but it might not produce the exact same result.- Parameters:
b (ArrayLike) – The right-hand side(s) of the linear system(s). This can either be a vector or a (stack of) matrices.
- Returns:
The solution(s) of the linear system(s). Depending on the shape of
b
,x
is either a vector or a (stack of) matrices.- Return type:
x
- Raises:
numpy.linalg.LinAlgError – If the linear operator is non-square or not invertible.
ValueError – If the shape of
b
does not meet the requirements outlined above.
- symmetrize()¶
Compute or approximate the closest symmetric
LinearOperator
in the Frobenius norm.The closest symmetric matrix to a given square matrix \(A\) in the Frobenius norm is given by
\[\operatorname{sym}(A) := \frac{1}{2} (A + A^T).\]However, for efficiency reasons, it is preferrable to approximate this operator in some cases. For example, a Kronecker product \(K = A \otimes B\) is more efficiently symmetrized by means of
\begin{equation} \operatorname{sym}(A) \otimes \operatorname{sym}(B) = \operatorname{sym}(K) + \frac{1}{2} \left( \frac{1}{2} \left( A \otimes B^T + A^T \otimes B \right) - \operatorname{sym}(K) \right). \end{equation}- Returns:
The closest symmetric
LinearOperator
in the Frobenius norm, or an approximation, which makes a reasonable trade-off between accuracy and efficiency (see above). The resultingLinearOperator
will have itsis_symmetric
property set toTrue
.- Return type:
symmetrized_linop
- Raises:
numpy.linalg.LinAlgError – If this method is called on a non-square
LinearOperator
.
- 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.
- 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:
- Raises:
numpy.linalg.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().
- Parameters:
*axes (int | Tuple[int]) – Permutation of the axes of the
LinearOperator
.- Raises:
ValueError – If the given axis indices do not constitute a valid permutation of the axes.
numpy.AxisError – If the axis indices are out of bounds.
- Return type: