bundles / numpy 2.4.4 / numpy / linalg / cholesky
_ArrayFunctionDispatcher
numpy.linalg:cholesky
source: /numpy/linalg/_linalg.py :778
Signature
def cholesky ( a , / , upper = False ) Summary
Cholesky decomposition.
Extended Summary
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_likeHermitian (symmetric if all elements are real), positive-definite input matrix.
upper: boolIf
True, the result must be the upper-triangular Cholesky factor. IfFalse, the result must be the lower-triangular Cholesky factor. Default:False.
Returns
L: (..., M, M) array_likeLower or upper-triangular Cholesky factor of
a. Returns a matrix object ifais a matrix object.
Raises
: LinAlgErrorIf the decomposition fails, for example, if
ais not positive-definite.
Notes
Broadcasting rules apply, see the numpy.linalg documentation for details.
The Cholesky decomposition is often used as a fast way of solving
(when A is both Hermitian/symmetric and positive-definite).
First, we solve for in
and then for in
Examples
import numpy as np A = np.array([[1,-2j],[2j,5]]) A L = np.linalg.cholesky(A) L np.dot(L, L.T.conj()) # verify that L * L.H = A A = [[1,-2j],[2j,5]] # what happens if A is only array_like? np.linalg.cholesky(A) # an ndarray object is returned✓
np.linalg.cholesky(np.matrix(A)) np.linalg.cholesky(A, upper=True)✗
See also
- scipy.linalg.cho_factor
Cholesky decomposition of a matrix, to use in
scipy.linalg.cho_solve.- scipy.linalg.cholesky
Similar function in SciPy.
- scipy.linalg.cholesky_banded
Cholesky decompose a banded Hermitian positive-definite matrix.
Aliases
-
numpy.linalg.cholesky