bundles / scipy 1.17.1 / scipy / linalg / _decomp / eig_banded

function

`scipy.linalg._decomp:eig_banded`

source: /scipy/linalg/_decomp.py :661

Signature

   def     eig_banded (    a_band    ,    lower    =  False   ,    eigvals_only    =  False   ,    overwrite_a_band    =  False   ,    select    =  a   ,    select_range    =  None   ,    max_ev    =  0   ,    check_finite    =  True     )

Summary

Solve real symmetric or complex Hermitian band matrix eigenvalue problem.

Extended Summary

Find eigenvalues w and optionally right eigenvectors v of a

a v[:,i] = w[i] v[:,i]
v.H v    = identity

The matrix a is stored in a_band either in lower diagonal or upper diagonal ordered form:

a_band[u + i - j, j] == a[i,j] (if upper form; i <= j) a_band[ i - j, j] == a[i,j] (if lower form; i >= j)

where u is the number of bands above the diagonal.

Example of a_band (shape of a is (6,6), u=2)

upper form:
*   *   a02 a13 a24 a35
*   a01 a12 a23 a34 a45
a00 a11 a22 a33 a44 a55

lower form:
a00 a11 a22 a33 a44 a55
a10 a21 a32 a43 a54 *
a20 a31 a42 a53 *   *

Cells marked with * are not used.

The documentation is written assuming array arguments are of specified "core" shapes. However, array argument(s) of this function may have additional "batch" dimensions prepended to the core shape. In this case, the array is treated as a batch of lower-dimensional slices; see linalg_batch for details. Note that calls with zero-size batches are unsupported and will raise a ValueError.

Parameters

a_band : (u+1, M) array_like

The bands of the M by M matrix a.

lower : bool, optional

Is the matrix in the lower form. (Default is upper form)

eigvals_only : bool, optional

Compute only the eigenvalues and no eigenvectors. (Default: calculate also eigenvectors)

overwrite_a_band : bool, optional

Discard data in a_band (may enhance performance)

select : {'a', 'v', 'i'}, optional

Which eigenvalues to calculate

======  ========================================
select  calculated
======  ========================================
'a'     All eigenvalues
'v'     Eigenvalues in the interval (min, max]
'i'     Eigenvalues with indices min <= i <= max
======  ========================================

select_range : (min, max), optional

Range of selected eigenvalues

max_ev : int, optional

For select=='v', maximum number of eigenvalues expected. For other values of select, has no meaning.

In doubt, leave this parameter untouched.

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.

Returns

w : (M,) ndarray: The eigenvalues, in ascending order, each repeated according to its multiplicity.
v : (M, M) float or complex ndarray: The normalized eigenvector corresponding to the eigenvalue w[i] is the column v[:,i]. Only returned if eigvals_only=False.

Raises

: LinAlgError: If eigenvalue computation does not converge.

Examples

import numpy as np
from scipy.linalg import eig_banded
A = np.array([[1, 5, 2, 0], [5, 2, 5, 2], [2, 5, 3, 5], [0, 2, 5, 4]])
Ab = np.array([[1, 2, 3, 4], [5, 5, 5, 0], [2, 2, 0, 0]])
w, v = eig_banded(Ab, lower=True)
np.allclose(A @ v - v @ np.diag(w), np.zeros((4, 4)))
w = eig_banded(Ab, lower=True, eigvals_only=True)

✓

✗

Request only the eigenvalues between ``[-3, 4]``

w, v = eig_banded(Ab, lower=True, select='v', select_range=[-3, 4])

✓

✗

Aliases

scipy.linalg.eig_banded