# 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 *   *

  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