scipy.linalg.eigh_tridiagonal¶
-
scipy.linalg.
eigh_tridiagonal
(d, e, eigvals_only=False, select='a', select_range=None, check_finite=True, tol=0.0, lapack_driver='auto')[source]¶ Solve eigenvalue problem for a real symmetric tridiagonal matrix.
Find eigenvalues w and optionally right eigenvectors v of
a
:a v[:,i] = w[i] v[:,i] v.H v = identity
For a real symmetric matrix
a
with diagonal elements d and off-diagonal elements e.- Parameters
- dndarray, shape (ndim,)
The diagonal elements of the array.
- endarray, shape (ndim-1,)
The off-diagonal elements of the array.
- 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.
- tolfloat
The absolute tolerance to which each eigenvalue is required (only used when ‘stebz’ is the lapack_driver). An eigenvalue (or cluster) is considered to have converged if it lies in an interval of this width. If <= 0. (default), the value
eps*|a|
is used where eps is the machine precision, and|a|
is the 1-norm of the matrixa
.- lapack_driverstr
LAPACK function to use, can be ‘auto’, ‘stemr’, ‘stebz’, ‘sterf’, or ‘stev’. When ‘auto’ (default), it will use ‘stemr’ if
select='a'
and ‘stebz’ otherwise. When ‘stebz’ is used to find the eigenvalues andeigvals_only=False
, then a second LAPACK call (to?STEIN
) is used to find the corresponding eigenvectors. ‘sterf’ can only be used wheneigvals_only=True
andselect='a'
. ‘stev’ can only be used whenselect='a'
.
- Returns
- w(M,) ndarray
The eigenvalues, in ascending order, each repeated according to its multiplicity.
- v(M, M) ndarray
The normalized eigenvector corresponding to the eigenvalue
w[i]
is the columnv[:,i]
.
- Raises
- LinAlgError
If eigenvalue computation does not converge.
See also
eigvalsh_tridiagonal
eigenvalues of symmetric/Hermitian tridiagonal matrices
eig
eigenvalues and right eigenvectors for non-symmetric arrays
eigh
eigenvalues and right eigenvectors for symmetric/Hermitian arrays
eig_banded
eigenvalues and right eigenvectors for symmetric/Hermitian band matrices
Notes
This function makes use of LAPACK
S/DSTEMR
routines.Examples
>>> from scipy.linalg import eigh_tridiagonal >>> d = 3*np.ones(4) >>> e = -1*np.ones(3) >>> w, v = eigh_tridiagonal(d, e) >>> A = np.diag(d) + np.diag(e, k=1) + np.diag(e, k=-1) >>> np.allclose(A @ v - v @ np.diag(w), np.zeros((4, 4))) True