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')

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:

d : ndarray, shape (ndim,)

The diagonal elements of the array.

e : ndarray, 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_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.

tol : float

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 matrix a.

lapack_driver : str

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 and eigvals_only=False, then a second LAPACK call (to ?STEIN) is used to find the corresponding eigenvectors. ‘sterf’ can only be used when eigvals_only=True and select='a'. ‘stev’ can only be used when select='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 column v[:,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