scipy.linalg.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 factorization A = L D L^H or A = 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 matrix P.dot(lu), where P is a column-permuted identity matrix I[:, 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 or A = 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.

See also

cholesky, lu

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.

New 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, obtain l, ‘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.]])