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Module « scipy.linalg »

Fonction eigh - module scipy.linalg

Signature de la fonction eigh

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

Description

help(scipy.linalg.eigh)

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

Find eigenvalues array ``w`` and optionally eigenvectors array ``v`` 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.

Parameters
----------
a : (M, M) array_like
    A complex Hermitian or real symmetric matrix whose eigenvalues and
    eigenvectors 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)
eigvals_only : bool, optional
    Whether to calculate only eigenvalues and no eigenvectors.
    (Default: both are calculated)
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.
    The default for standard problems is "evr". For generalized problems,
    "gvd" is used for full set, and "gvx" for subset requested cases.
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.
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.
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.

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

See Also
--------
eigvalsh : eigenvalues of symmetric or Hermitian arrays
eig : eigenvalues and right eigenvectors for non-symmetric arrays
eigh_tridiagonal : eigenvalues and right eiegenvectors for
    symmetric/Hermitian tridiagonal matrices

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. Also, note that even though not taken into account,
finiteness check applies to the whole array and unaffected by "lower"
keyword.

This function uses LAPACK drivers for computations in all possible keyword
combinations, prefixed with ``sy`` if arrays are real and ``he`` if
complex, e.g., a float array with "evr" driver is solved via
"syevr", complex arrays with "gvx" driver problem is solved via "hegvx"
etc.

As a brief summary, the slowest and the most robust driver is the
classical ``<sy/he>ev`` which uses symmetric QR. ``<sy/he>evr`` is seen as
the optimal choice for the most general cases. However, there are certain
occasions that ``<sy/he>evd`` computes faster at the expense of more
memory usage. ``<sy/he>evx``, while still being faster than ``<sy/he>ev``,
often performs worse than the rest except when very few eigenvalues are
requested for large arrays though there is still no performance guarantee.

Note that the underlying LAPACK algorithms are different depending on whether
`eigvals_only` is True or False --- thus the eigenvalues may differ
depending on whether eigenvectors are requested or not. The difference is
generally of the order of machine epsilon times the largest eigenvalue,
so is likely only visible for zero or nearly zero eigenvalues.

For the generalized problem, normalization with respect to the given
type argument::

        type 1 and 3 :      v.conj().T @ a @ v = w
        type 2       : inv(v).conj().T @ a @ inv(v) = w

        type 1 or 2  :      v.conj().T @ b @ v  = I
        type 3       : v.conj().T @ inv(b) @ v  = I


Examples
--------
>>> import numpy as np
>>> from scipy.linalg import eigh
>>> A = np.array([[6, 3, 1, 5], [3, 0, 5, 1], [1, 5, 6, 2], [5, 1, 2, 2]])
>>> w, v = eigh(A)
>>> np.allclose(A @ v - v @ np.diag(w), np.zeros((4, 4)))
True

Request only the eigenvalues

>>> w = eigh(A, eigvals_only=True)

Request eigenvalues that are less than 10.

>>> A = np.array([[34, -4, -10, -7, 2],
...               [-4, 7, 2, 12, 0],
...               [-10, 2, 44, 2, -19],
...               [-7, 12, 2, 79, -34],
...               [2, 0, -19, -34, 29]])
>>> eigh(A, eigvals_only=True, subset_by_value=[-np.inf, 10])
array([6.69199443e-07, 9.11938152e+00])

Request the second smallest eigenvalue and its eigenvector

>>> w, v = eigh(A, subset_by_index=[1, 1])
>>> w
array([9.11938152])
>>> v.shape  # only a single column is returned
(5, 1)

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