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)[source]#
- 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. - 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) 
- 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]. 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])