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Module « scipy.linalg »

Fonction eig_banded - module scipy.linalg

Signature de la fonction eig_banded

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) 

Description

help(scipy.linalg.eig_banded)

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.
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.

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
--------
>>> 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)))
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])

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