mars.tensor.linalg.cholesky#
- mars.tensor.linalg.cholesky(a, lower=False)[源代码]#
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.
- 参数
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.
- 返回
L – Upper or lower-triangular Cholesky factor of a.
- 返回类型
(…, M, M) array_like
- 引发
LinAlgError – If the decomposition fails, for example, if a is not positive-definite.
提示
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
\[L.H \mathbf{x} = \mathbf{y}.\]实际案例
>>> 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]])