scipy.linalg.eigvalsh_tridiagonal(d, e, 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 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.

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




All eigenvalues


Eigenvalues in the interval (min, max]


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.


The absolute tolerance to which each eigenvalue is required (only used when lapack_driver='stebz'). 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 function to use, can be ‘auto’, ‘stemr’, ‘stebz’, ‘sterf’, or ‘stev’. When ‘auto’ (default), it will use ‘stemr’ if select='a' and ‘stebz’ otherwise. ‘sterf’ and ‘stev’ can only be used when select='a'.

w(M,) ndarray

The eigenvalues, in ascending order, each repeated according to its multiplicity.


If eigenvalue computation does not converge.

See also


eigenvalues and right eiegenvectors for symmetric/Hermitian tridiagonal matrices


>>> import numpy as np
>>> from scipy.linalg import eigvalsh_tridiagonal, eigvalsh
>>> d = 3*np.ones(4)
>>> e = -1*np.ones(3)
>>> w = eigvalsh_tridiagonal(d, e)
>>> A = np.diag(d) + np.diag(e, k=1) + np.diag(e, k=-1)
>>> w2 = eigvalsh(A)  # Verify with other eigenvalue routines
>>> np.allclose(w - w2, np.zeros(4))