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.

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:
dndarray, shape (ndim,)

The diagonal elements of the array.

endarray, shape (ndim-1,)

The off-diagonal elements of the array.

eigvals_onlybool, optional

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

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 matrix a.

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 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]. Only returned if eigvals_only=False.

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

>>> import numpy as np
>>> 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
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