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Module « numpy.linalg »

Fonction cholesky - module numpy.linalg

Signature de la fonction cholesky

def cholesky(a, /, *, upper=False) 

Description

help(numpy.linalg.cholesky)

Cholesky decomposition.

Return the lower or upper Cholesky decomposition, ``L * L.H`` or
``U.H * U``, of the square matrix ``a``, where ``L`` is lower-triangular,
``U`` is upper-triangular, and ``.H`` is the conjugate transpose operator
(which is the ordinary transpose if ``a`` is real-valued). ``a`` must be
Hermitian (symmetric if real-valued) and positive-definite. No checking is
performed to verify whether ``a`` is Hermitian or not. In addition, only
the lower or upper-triangular and diagonal elements of ``a`` are used.
Only ``L`` or ``U`` is actually returned.

Parameters
----------
a : (..., M, M) array_like
    Hermitian (symmetric if all elements are real), positive-definite
    input matrix.
upper : bool
    If ``True``, the result must be the upper-triangular Cholesky factor.
    If ``False``, the result must be the lower-triangular Cholesky factor.
    Default: ``False``.

Returns
-------
L : (..., M, M) array_like
    Lower or upper-triangular Cholesky factor of `a`. Returns a matrix
    object if `a` is a matrix object.

Raises
------
LinAlgError
   If the decomposition fails, for example, if `a` is not
   positive-definite.

See Also
--------
scipy.linalg.cholesky : Similar function in SciPy.
scipy.linalg.cholesky_banded : Cholesky decompose a banded Hermitian
                               positive-definite matrix.
scipy.linalg.cho_factor : Cholesky decomposition of a matrix, to use in
                          `scipy.linalg.cho_solve`.

Notes
-----
Broadcasting rules apply, see the `numpy.linalg` documentation for
details.

The Cholesky decomposition is often used as a fast way of solving

.. math:: A \mathbf{x} = \mathbf{b}

(when `A` is both Hermitian/symmetric and positive-definite).

First, we solve for :math:`\mathbf{y}` in

.. math:: L \mathbf{y} = \mathbf{b},

and then for :math:`\mathbf{x}` in

.. math:: L^{H} \mathbf{x} = \mathbf{y}.

Examples
--------
>>> import numpy as np
>>> A = np.array([[1,-2j],[2j,5]])
>>> A
array([[ 1.+0.j, -0.-2.j],
       [ 0.+2.j,  5.+0.j]])
>>> L = np.linalg.cholesky(A)
>>> L
array([[1.+0.j, 0.+0.j],
       [0.+2.j, 1.+0.j]])
>>> np.dot(L, L.T.conj()) # verify that L * L.H = A
array([[1.+0.j, 0.-2.j],
       [0.+2.j, 5.+0.j]])
>>> A = [[1,-2j],[2j,5]] # what happens if A is only array_like?
>>> np.linalg.cholesky(A) # an ndarray object is returned
array([[1.+0.j, 0.+0.j],
       [0.+2.j, 1.+0.j]])
>>> # But a matrix object is returned if A is a matrix object
>>> np.linalg.cholesky(np.matrix(A))
matrix([[ 1.+0.j,  0.+0.j],
        [ 0.+2.j,  1.+0.j]])
>>> # The upper-triangular Cholesky factor can also be obtained.
>>> np.linalg.cholesky(A, upper=True)
array([[1.-0.j, 0.-2.j],
       [0.-0.j, 1.-0.j]])

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