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scipy.linalg.

# cholesky_banded

scipy.linalg.cholesky_banded(ab, overwrite_ab=False, lower=False, check_finite=True)

  Cholesky decompose a banded Hermitian positive-definite matrix

  The matrix a is stored in ab either in lower-diagonal or upper- diagonal ordered form:

      ab[u + i - j, j] == a[i,j]        (if upper form; i <= j)
      ab[    i - j, j] == a[i,j]        (if lower form; i >= j)

  Example of ab (shape of a is (6,6), u=2):

      upper form:
      *   *   a02 a13 a24 a35
      *   a01 a12 a23 a34 a45
      a00 a11 a22 a33 a44 a55

      lower form:
      a00 a11 a22 a33 a44 a55
      a10 a21 a32 a43 a54 *
      a20 a31 a42 a53 *   *

  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:
    ab(u + 1, M) array_like

      Banded matrix

    overwrite_ab bool, optional

      Discard data in ab (may enhance performance)

    lower bool, optional

      Is the matrix in the lower form. (Default is upper form)

    check_finite bool, optional

      Whether to check that the input matrix contains 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:
    c(u + 1, M) ndarray

      Cholesky factorization of a, in the same banded format as ab

  See also

  `cho_solve_banded`

    Solve a linear set equations, given the Cholesky factorization of a banded Hermitian.

  Examples

      >>> import numpy as np
      >>> from scipy.linalg import cholesky_banded
      >>> from numpy import allclose, zeros, diag
      >>> Ab = np.array([[0, 0, 1j, 2, 3j], [0, -1, -2, 3, 4], [9, 8, 7, 6, 9]])
      >>> A = np.diag(Ab[0,2:], k=2) + np.diag(Ab[1,1:], k=1)
      >>> A = A + A.conj().T + np.diag(Ab[2, :])
      >>> c = cholesky_banded(Ab)
      >>> C = np.diag(c[0, 2:], k=2) + np.diag(c[1, 1:], k=1) + np.diag(c[2, :])
      >>> np.allclose(C.conj().T @ C - A, np.zeros((5, 5)))
      True