ldl#
- scipy.linalg.ldl(A, lower=True, hermitian=True, overwrite_a=False, check_finite=True)[source]#
Computes the LDLt or Bunch-Kaufman factorization of a symmetric/ hermitian matrix.
This function returns a block diagonal matrix D consisting blocks of size at most 2x2 and also a possibly permuted unit lower triangular matrix
L
such that the factorizationA = L D L^H
orA = L D L^T
holds. If lower is False then (again possibly permuted) upper triangular matrices are returned as outer factors.The permutation array can be used to triangularize the outer factors simply by a row shuffle, i.e.,
lu[perm, :]
is an upper/lower triangular matrix. This is also equivalent to multiplication with a permutation matrixP.dot(lu)
, whereP
is a column-permuted identity matrixI[:, perm]
.Depending on the value of the boolean lower, only upper or lower triangular part of the input array is referenced. Hence, a triangular matrix on entry would give the same result as if the full matrix is supplied.
- Parameters:
- Aarray_like
Square input array
- lowerbool, optional
This switches between the lower and upper triangular outer factors of the factorization. Lower triangular (
lower=True
) is the default.- hermitianbool, optional
For complex-valued arrays, this defines whether
A = A.conj().T
orA = A.T
is assumed. For real-valued arrays, this switch has no effect.- overwrite_abool, optional
Allow overwriting data in A (may enhance performance). The default is False.
- check_finitebool, 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.
- Returns:
- lundarray
The (possibly) permuted upper/lower triangular outer factor of the factorization.
- dndarray
The block diagonal multiplier of the factorization.
- permndarray
The row-permutation index array that brings lu into triangular form.
- Raises:
- ValueError
If input array is not square.
- ComplexWarning
If a complex-valued array with nonzero imaginary parts on the diagonal is given and hermitian is set to True.
Notes
This function uses
?SYTRF
routines for symmetric matrices and?HETRF
routines for Hermitian matrices from LAPACK. See [1] for the algorithm details.Depending on the lower keyword value, only lower or upper triangular part of the input array is referenced. Moreover, this keyword also defines the structure of the outer factors of the factorization.
Added in version 1.1.0.
References
[1]J.R. Bunch, L. Kaufman, Some stable methods for calculating inertia and solving symmetric linear systems, Math. Comput. Vol.31, 1977. DOI:10.2307/2005787
Examples
Given an upper triangular array
a
that represents the full symmetric array with its entries, obtainl
, ‘d’ and the permutation vector perm:>>> import numpy as np >>> from scipy.linalg import ldl >>> a = np.array([[2, -1, 3], [0, 2, 0], [0, 0, 1]]) >>> lu, d, perm = ldl(a, lower=0) # Use the upper part >>> lu array([[ 0. , 0. , 1. ], [ 0. , 1. , -0.5], [ 1. , 1. , 1.5]]) >>> d array([[-5. , 0. , 0. ], [ 0. , 1.5, 0. ], [ 0. , 0. , 2. ]]) >>> perm array([2, 1, 0]) >>> lu[perm, :] array([[ 1. , 1. , 1.5], [ 0. , 1. , -0.5], [ 0. , 0. , 1. ]]) >>> lu.dot(d).dot(lu.T) array([[ 2., -1., 3.], [-1., 2., 0.], [ 3., 0., 1.]])