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bundles / scipy 1.17.1 / scipy / linalg / _decomp / eigvalsh_tridiagonal

function

scipy.linalg._decomp:eigvalsh_tridiagonal

source: /scipy/linalg/_decomp.py :1130

Signature

def   eigvalsh_tridiagonal ( d ,  e ,  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 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.

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 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_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. 'sterf' and 'stev' can only be used when select='a'.

Returns

w : (M,) ndarray

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

Raises

: LinAlgError

If eigenvalue computation does not converge.

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))
✓

See also

eigh_tridiagonal

eigenvalues and right eigenvectors for symmetric/Hermitian tridiagonal matrices

Aliases

  • scipy.linalg.eigvalsh_tridiagonal

Referenced by

This package