scipy.linalg.eig_banded

scipy.linalg.eig_banded(a_band, lower=False, eigvals_only=False, overwrite_a_band=False, select='a', select_range=None, max_ev=0, check_finite=True)

Solve real symmetric or complex Hermitian band matrix eigenvalue problem.

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.

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)

eigvals_onlybool, optional

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

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

max_evint, optional

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

In doubt, leave this parameter untouched.

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.

v(M, M) float or complex ndarray

The normalized eigenvector corresponding to the eigenvalue w[i] is the column v[:,i].

Raises

LinAlgError

If eigenvalue computation does not converge.

See also

eigvals_banded

eigenvalues for symmetric/Hermitian band matrices

eig

eigenvalues and right eigenvectors of general arrays.

eigh

eigenvalues and right eigenvectors for symmetric/Hermitian arrays

eigh_tridiagonal

eigenvalues and right eigenvectors for symmetric/Hermitian tridiagonal matrices

Examples

>>> 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)))
True
>>> w = eig_banded(Ab, lower=True, eigvals_only=True)
>>> w
array([-4.26200532, -2.22987175,  3.95222349, 12.53965359])

Request only the eigenvalues between [-3, 4]

>>> w, v = eig_banded(Ab, lower=True, select='v', select_range=[-3, 4])
>>> w
array([-2.22987175,  3.95222349])