mars.tensor.linalg.cholesky#

mars.tensor.linalg.cholesky(a, lower=False)[source]#

Cholesky decomposition.

Return the Cholesky decomposition, L * L.H, of the square matrix a, where L is lower-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. Only L is actually returned.

Parameters

a ((..., M, M) array_like) – Hermitian (symmetric if all elements are real), positive-definite input matrix.
lower (bool) – Whether to compute the upper or lower triangular Cholesky factorization. Default is upper-triangular.

Returns

L – Upper or lower-triangular Cholesky factor of a.

Return type

(…, M, M) array_like

Raises

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

Notes

Broadcasting rules apply, see the mt.linalg documentation for details.

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

\[A \mathbf{x} = \mathbf{b}\]

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

First, we solve for \(\mathbf{y}\) in

\[L \mathbf{y} = \mathbf{b},\]

and then for \(\mathbf{x}\) in

\[L.H \mathbf{x} = \mathbf{y}.\]

Examples

>>> import mars.tensor as mt

>>> A = mt.array([[1,-2j],[2j,5]])
>>> A.execute()
array([[ 1.+0.j,  0.-2.j],
       [ 0.+2.j,  5.+0.j]])
>>> L = mt.linalg.cholesky(A, lower=True)
>>> L.execute()
array([[ 1.+0.j,  0.+0.j],
       [ 0.+2.j,  1.+0.j]])
>>> mt.dot(L, L.T.conj()).execute() # 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?
>>> mt.linalg.cholesky(A, lower=True).execute()
array([[ 1.+0.j,  0.+0.j],
       [ 0.+2.j,  1.+0.j]])