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Module « scipy.linalg »

Fonction eigvalsh - module scipy.linalg

Signature de la fonction eigvalsh

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) 

Description

help(scipy.linalg.eigvalsh)

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

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.

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.

See Also
--------
eigh : eigenvalues and right eigenvectors for symmetric/Hermitian arrays
eigvals : eigenvalues of general arrays
eigvals_banded : eigenvalues for symmetric/Hermitian band matrices
eigvalsh_tridiagonal : eigenvalues of 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.

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)
>>> w
array([-3.74637491, -0.76263923,  6.08502336, 12.42399079])

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