Skip to main content
`Ctrl`+`K`
scipy.linalg.

# solveh_banded

scipy.linalg.solveh_banded(ab, b, overwrite_ab=False, overwrite_b=False, lower=False, check_finite=True)

  Solve the equation `a @ x = b` for `x`, where `a` is the Hermitian positive-definite banded matrix defined by ab.

  Uses Thomas’ Algorithm, which is more efficient than standard LU factorization, but should only be used for Hermitian positive-definite matrices.

  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), number of upper diagonals, `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 *   *

  Cells marked with * are not used.

  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.

  Parameters:
    ab(`u` + 1, M) array_like

      Banded matrix

    b(M,) or (M, K) array_like

      Right-hand side

    overwrite_ab bool, optional

      Discard data in ab (may enhance performance)

    overwrite_b bool, optional

      Discard data in b (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 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:
    x(M,) or (M, K) ndarray

      The solution to the system `a x = b`. Shape of return matches shape of b.

  Notes

  In the case of a non-positive definite matrix `a`, the solver `solve_banded` may be used.

  Examples

  Solve the banded system `A x = b`, where:

          [ 4  2 -1  0  0  0]       [1]
          [ 2  5  2 -1  0  0]       [2]
      A = [-1  2  6  2 -1  0]   b = [2]
          [ 0 -1  2  7  2 -1]       [3]
          [ 0  0 -1  2  8  2]       [3]
          [ 0  0  0 -1  2  9]       [3]

      >>> import numpy as np
      >>> from scipy.linalg import solveh_banded

  `ab` contains the main diagonal and the nonzero diagonals below the main diagonal. That is, we use the lower form:

      >>> ab = np.array([[ 4,  5,  6,  7, 8, 9],
      ...                [ 2,  2,  2,  2, 2, 0],
      ...                [-1, -1, -1, -1, 0, 0]])
      >>> b = np.array([1, 2, 2, 3, 3, 3])
      >>> x = solveh_banded(ab, b, lower=True)
      >>> x
      array([ 0.03431373,  0.45938375,  0.05602241,  0.47759104,  0.17577031,
              0.34733894])

  Solve the Hermitian banded system `H x = b`, where:

          [ 8   2-1j   0     0  ]        [ 1  ]
      H = [2+1j  5     1j    0  ]    b = [1+1j]
          [ 0   -1j    9   -2-1j]        [1-2j]
          [ 0    0   -2+1j   6  ]        [ 0  ]

  In this example, we put the upper diagonals in the array `hb`:

      >>> hb = np.array([[0, 2-1j, 1j, -2-1j],
      ...                [8,  5,    9,   6  ]])
      >>> b = np.array([1, 1+1j, 1-2j, 0])
      >>> x = solveh_banded(hb, b)
      >>> x
      array([ 0.07318536-0.02939412j,  0.11877624+0.17696461j,
              0.10077984-0.23035393j, -0.00479904-0.09358128j])