bundles / scipy 1.17.1 / scipy / linalg / _decomp / eigvalsh

function

`scipy.linalg._decomp:eigvalsh`

source: /scipy/linalg/_decomp.py :917

Signature

   def     eigvalsh (    a    ,    b    =  None   ,  * ,     lower    =  True   ,    overwrite_a    =  False   ,    overwrite_b    =  False   ,    type    =  1   ,    check_finite    =  True   ,    subset_by_index    =  None   ,    subset_by_value    =  None   ,    driver    =  None     )

Summary

Solves a standard or generalized eigenvalue problem for a complex Hermitian or real symmetric matrix.

Extended Summary

Find eigenvalues array w of array a, where b is positive definite such that for every eigenvalue λ (i-th entry of w) and its eigenvector vi (i-th column of v) satisfies

              a @ vi = λ * b @ vi
vi.conj().T @ a @ vi = λ
vi.conj().T @ b @ vi = 1

In the standard problem, b is assumed to be the identity matrix.

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

a : (M, M) array_like

A complex Hermitian or real symmetric matrix whose eigenvalues will be computed.

b : (M, M) array_like, optional

A complex Hermitian or real symmetric definite positive matrix in. If omitted, identity matrix is assumed.

lower : bool, optional

Whether the pertinent array data is taken from the lower or upper triangle of a and, if applicable, b. (Default: lower)

overwrite_a : bool, optional

Whether to overwrite data in a (may improve performance). Default is False.

overwrite_b : bool, optional

Whether to overwrite data in b (may improve performance). Default is False.

type : int, optional

For the generalized problems, this keyword specifies the problem type to be solved for w and v (only takes 1, 2, 3 as possible inputs)

1 =>     a @ v = w @ b @ v
2 => a @ b @ v = w @ v
3 => b @ a @ v = w @ v

This keyword is ignored for standard problems.

check_finite : bool, optional

Whether to check that the input matrices contain 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.

subset_by_index : iterable, optional

If provided, this two-element iterable defines the start and the end indices of the desired eigenvalues (ascending order and 0-indexed). To return only the second smallest to fifth smallest eigenvalues, [1, 4] is used. [n-3, n-1] returns the largest three. Only available with "evr", "evx", and "gvx" drivers. The entries are directly converted to integers via int().

subset_by_value : iterable, optional

If provided, this two-element iterable defines the half-open interval (a, b] that, if any, only the eigenvalues between these values are returned. Only available with "evr", "evx", and "gvx" drivers. Use np.inf for the unconstrained ends.

driver : str, optional

Defines which LAPACK driver should be used. Valid options are "ev", "evd", "evr", "evx" for standard problems and "gv", "gvd", "gvx" for generalized (where b is not None) problems. See the Notes section of scipy.linalg.eigh.

Returns

w : (N,) ndarray: The N (N<=M) selected eigenvalues, in ascending order, each repeated according to its multiplicity.

Raises

: LinAlgError: If eigenvalue computation does not converge, an error occurred, or b matrix is not definite positive. Note that if input matrices are not symmetric or Hermitian, no error will be reported but results will be wrong.

Notes

This function does not check the input array for being Hermitian/symmetric in order to allow for representing arrays with only their upper/lower triangular parts.

This function serves as a one-liner shorthand for scipy.linalg.eigh with the option eigvals_only=True to get the eigenvalues and not the eigenvectors. Here it is kept as a legacy convenience. It might be beneficial to use the main function to have full control and to be a bit more pythonic.

Examples

For more examples see `scipy.linalg.eigh`.

import numpy as np
from scipy.linalg import eigvalsh
A = np.array([[6, 3, 1, 5], [3, 0, 5, 1], [1, 5, 6, 2], [5, 1, 2, 2]])
w = eigvalsh(A)

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Aliases

scipy.linalg.eigvalsh