scipy 1.8.0 Pypi GitHub Homepage
Other Docs
ParametersRaisesReturnsBackRef
schur(a, output='real', lwork=None, overwrite_a=False, sort=None, check_finite=True)

The Schur decomposition is:

A = Z T Z^H

where Z is unitary and T is either upper-triangular, or for real Schur decomposition (output='real'), quasi-upper triangular. In the quasi-triangular form, 2x2 blocks describing complex-valued eigenvalue pairs may extrude from the diagonal.

Parameters

a : (M, M) array_like

Matrix to decompose

output : {'real', 'complex'}, optional

Construct the real or complex Schur decomposition (for real matrices).

lwork : int, optional

Work array size. If None or -1, it is automatically computed.

overwrite_a : bool, optional

Whether to overwrite data in a (may improve performance).

sort : {None, callable, 'lhp', 'rhp', 'iuc', 'ouc'}, optional

Specifies whether the upper eigenvalues should be sorted. A callable may be passed that, given a eigenvalue, returns a boolean denoting whether the eigenvalue should be sorted to the top-left (True). Alternatively, string parameters may be used:

'lhp'   Left-hand plane (x.real < 0.0)
'rhp'   Right-hand plane (x.real > 0.0)
'iuc'   Inside the unit circle (x*x.conjugate() <= 1.0)
'ouc'   Outside the unit circle (x*x.conjugate() > 1.0)

Defaults to None (no sorting).

check_finite : bool, optional

Whether to check that the input matrix contains only finite numbers. Disabling may give a performance gain, but may result in problems (crashes, non-termination) if the inputs do contain infinities or NaNs.

Raises

LinAlgError

Error raised under three conditions:

  1. The algorithm failed due to a failure of the QR algorithm to compute all eigenvalues.

  2. If eigenvalue sorting was requested, the eigenvalues could not be reordered due to a failure to separate eigenvalues, usually because of poor conditioning.

  3. If eigenvalue sorting was requested, roundoff errors caused the leading eigenvalues to no longer satisfy the sorting condition.

Returns

T : (M, M) ndarray

Schur form of A. It is real-valued for the real Schur decomposition.

Z : (M, M) ndarray

An unitary Schur transformation matrix for A. It is real-valued for the real Schur decomposition.

sdim : int

If and only if sorting was requested, a third return value will contain the number of eigenvalues satisfying the sort condition.

Compute Schur decomposition of a matrix.

See Also

rsf2csf

Convert real Schur form to complex Schur form

Examples

>>> from scipy.linalg import schur, eigvals
... A = np.array([[0, 2, 2], [0, 1, 2], [1, 0, 1]])
... T, Z = schur(A)
... T array([[ 2.65896708, 1.42440458, -1.92933439], [ 0. , -0.32948354, -0.49063704], [ 0. , 1.31178921, -0.32948354]])
>>> Z
array([[0.72711591, -0.60156188, 0.33079564],
       [0.52839428, 0.79801892, 0.28976765],
       [0.43829436, 0.03590414, -0.89811411]])
>>> T2, Z2 = schur(A, output='complex')
... T2 array([[ 2.65896708, -1.22839825+1.32378589j, 0.42590089+1.51937378j], [ 0. , -0.32948354+0.80225456j, -0.59877807+0.56192146j], [ 0. , 0. , -0.32948354-0.80225456j]])
>>> eigvals(T2)
array([2.65896708, -0.32948354+0.80225456j, -0.32948354-0.80225456j])

An arbitrary custom eig-sorting condition, having positive imaginary part, which is satisfied by only one eigenvalue

>>> T3, Z3, sdim = schur(A, output='complex', sort=lambda x: x.imag > 0)
... sdim 1
See :

Back References

The following pages refer to to this document either explicitly or contain code examples using this.

scipy.linalg._decomp_schur.rsf2csf scipy.linalg._decomp_schur.schur numpy.linalg.eig

Local connectivity graph

Hover to see nodes names; edges to Self not shown, Caped at 50 nodes.

Using a canvas is more power efficient and can get hundred of nodes ; but does not allow hyperlinks; , arrows or text (beyond on hover)

SVG is more flexible but power hungry; and does not scale well to 50 + nodes.

All aboves nodes referred to, (or are referred from) current nodes; Edges from Self to other have been omitted (or all nodes would be connected to the central node "self" which is not useful). Nodes are colored by the library they belong to, and scaled with the number of references pointing them


GitHub : /scipy/linalg/_decomp_schur.py#17
type: <class 'function'>
Commit: