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scipy.linalg.

# ldl

scipy.linalg.ldl(A, lower=True, hermitian=True, overwrite_a=False, check_finite=True)

  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.

  The documentation is written assuming array arguments are of specified “core” shapes. However, array argument(s) of this function may have additional “batch” dimensions prepended to the core shape. In this case, the array is treated as a batch of lower-dimensional slices; see Batched Linear Operations for details. Note that calls with zero-size batches are unsupported and will raise a `ValueError`.

  Parameters:
    A array_like

      Square input array

    lower bool, optional

      This switches between the lower and upper triangular outer factors of the factorization. Lower triangular (`lower=True`) is the default.

    hermitian bool, 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_a bool, optional

      Allow overwriting data in A (may enhance performance). The default is False. See overwrite_* arguments for details.

    check_finite bool, 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:
    lu ndarray

      The (possibly) permuted upper/lower triangular outer factor of the factorization.

    d ndarray

      The block diagonal multiplier of the factorization.

    perm ndarray

      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.

  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, 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.]])