scipy.linalg.

eigvals_banded#

scipy.linalg.eigvals_banded(a_band, lower=False, overwrite_a_band=False, select='a', select_range=None, check_finite=True)[source]#

Solve real symmetric or complex Hermitian band matrix eigenvalue problem.

Find eigenvalues w 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 Batched Linear Operations for details.

Parameters:

a_band(u+1, M) array_like

The bands of the M by M matrix a.

lowerbool, optional

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

overwrite_a_bandbool, 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

check_finitebool, 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.

Raises:

LinAlgError: If eigenvalue computation does not converge.

See also

eig_banded: eigenvalues and right eigenvectors for symmetric/Hermitian band matrices
eigvalsh_tridiagonal: eigenvalues of symmetric/Hermitian tridiagonal matrices
eigvals: eigenvalues of general arrays
eigh: eigenvalues and right eigenvectors for symmetric/Hermitian arrays
eig: eigenvalues and right eigenvectors for non-symmetric arrays

Examples

>>> import numpy as np
>>> from scipy.linalg import eigvals_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 = eigvals_banded(Ab, lower=True)
>>> w
array([-4.26200532, -2.22987175,  3.95222349, 12.53965359])