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# scipy.linalg.solve_toeplitz

scipy.linalg.solve_toeplitz(c_or_cr, b, check_finite=True)

  Solve a Toeplitz system using Levinson Recursion

  The Toeplitz matrix has constant diagonals, with c as its first column and r as its first row. If r is not given, `r == conjugate(c)` is assumed.

  Parameters:
    c_or_cr array_like or tuple of (array_like, array_like)

      The vector `c`, or a tuple of arrays (`c`, `r`). Whatever the actual shape of `c`, it will be converted to a 1-D array. If not supplied, `r = conjugate(c)` is assumed; in this case, if c[0] is real, the Toeplitz matrix is Hermitian. r[0] is ignored; the first row of the Toeplitz matrix is `[c[0], r[1:]]`. Whatever the actual shape of `r`, it will be converted to a 1-D array.

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

      Right-hand side in `T x = b`.

    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 (result entirely NaNs) if the inputs do contain infinities or NaNs.

  Returns:
    x(M,) or (M, K) ndarray

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

  See also

  `toeplitz`

    Toeplitz matrix

  Notes

  The solution is computed using Levinson-Durbin recursion, which is faster than generic least-squares methods, but can be less numerically stable.

  Examples

  Solve the Toeplitz system T x = b, where:

          [ 1 -1 -2 -3]       [1]
      T = [ 3  1 -1 -2]   b = [2]
          [ 6  3  1 -1]       [2]
          [10  6  3  1]       [5]

  To specify the Toeplitz matrix, only the first column and the first row are needed.

      >>> import numpy as np
      >>> c = np.array([1, 3, 6, 10])    # First column of T
      >>> r = np.array([1, -1, -2, -3])  # First row of T
      >>> b = np.array([1, 2, 2, 5])

      >>> from scipy.linalg import solve_toeplitz, toeplitz
      >>> x = solve_toeplitz((c, r), b)
      >>> x
      array([ 1.66666667, -1.        , -2.66666667,  2.33333333])

  Check the result by creating the full Toeplitz matrix and multiplying it by x. We should get b.

      >>> T = toeplitz(c, r)
      >>> T.dot(x)
      array([ 1.,  2.,  2.,  5.])