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# scipy.linalg.solve_banded

scipy.linalg.solve_banded(l_and_u, ab, b, overwrite_ab=False, overwrite_b=False, check_finite=True)

  Solve the equation a x = b for x, assuming a is banded matrix.

  The matrix a is stored in ab using the matrix diagonal ordered form:

      ab[u + i - j, j] == a[i,j]

  Example of ab (shape of a is (6,6), u =1, l =2):

      *    a01  a12  a23  a34  a45
      a00  a11  a22  a33  a44  a55
      a10  a21  a32  a43  a54   *
      a20  a31  a42  a53   *    *

  Parameters:
    (l, u)(integer, integer)

      Number of non-zero lower and upper diagonals

    ab(l + 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)

    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. Returned shape depends on the shape of b.

  Examples

  Solve the banded system a x = b, where:

          [5  2 -1  0  0]       [0]
          [1  4  2 -1  0]       [1]
      a = [0  1  3  2 -1]   b = [2]
          [0  0  1  2  2]       [2]
          [0  0  0  1  1]       [3]

  There is one nonzero diagonal below the main diagonal (l = 1), and two above (u = 2). The diagonal banded form of the matrix is:

           [*  * -1 -1 -1]
      ab = [*  2  2  2  2]
           [5  4  3  2  1]
           [1  1  1  1  *]

      >>> import numpy as np
      >>> from scipy.linalg import solve_banded
      >>> ab = np.array([[0,  0, -1, -1, -1],
      ...                [0,  2,  2,  2,  2],
      ...                [5,  4,  3,  2,  1],
      ...                [1,  1,  1,  1,  0]])
      >>> b = np.array([0, 1, 2, 2, 3])
      >>> x = solve_banded((1, 2), ab, b)
      >>> x
      array([-2.37288136,  3.93220339, -4.        ,  4.3559322 , -1.3559322 ])