`scipy.linalg._decomp:eigh_tridiagonal`

source: /scipy/linalg/_decomp.py :1212

Signature

   def     eigh_tridiagonal (    d    ,    e    ,    eigvals_only    =  False   ,    select    =  a   ,    select_range    =  None   ,    check_finite    =  True   ,    tol    =  0.0   ,    lapack_driver    =  auto     )

Summary

Solve eigenvalue problem for a real symmetric tridiagonal matrix.

Extended Summary

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 linalg_batch for details. Note that calls with zero-size batches are unsupported and will raise a ValueError.

Parameters

d : ndarray, shape (ndim,)

The diagonal elements of the array.

e : ndarray, shape (ndim-1,)

The off-diagonal elements of the array.

eigvals_only : bool, 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_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', 'stev', or 'stevd'. When 'auto' (default), it will use 'stevd' 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.

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)))

`scipy.linalg._decomp:eigh_tridiagonal`

Signature

Summary

Extended Summary

Parameters

Returns

Raises

Notes

Examples

See also

Aliases

Referenced by

This package