eigvalsh_tridiagonal#
- 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.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.
- 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
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 valueeps*|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. ‘sterf’ and ‘stev’ can only be used whenselect='a'
.
- Returns:
- w(M,) ndarray
The eigenvalues, in ascending order, each repeated according to its multiplicity.
- Raises:
- LinAlgError
If eigenvalue computation does not converge.
See also
eigh_tridiagonal
eigenvalues and right eiegenvectors for symmetric/Hermitian tridiagonal matrices
Examples
>>> 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)) True