# numpy.vander

numpy.vander(x, N=None, increasing=False)

  Generate a Vandermonde matrix.

  The columns of the output matrix are powers of the input vector. The order of the powers is determined by the increasing boolean argument. Specifically, when increasing is False, the i-th output column is the input vector raised element-wise to the power of `N - i - 1`. Such a matrix with a geometric progression in each row is named for Alexandre- Theophile Vandermonde.

  Parameters
    x array_like

      1-D input array.

    N int, optional

      Number of columns in the output. If N is not specified, a square array is returned (`N = len(x)`).

    increasing bool, optional

      Order of the powers of the columns. If True, the powers increase from left to right, if False (the default) they are reversed.

      New in version 1.9.0.

  Returns
    out ndarray

      Vandermonde matrix. If increasing is False, the first column is `x^(N-1)`, the second `x^(N-2)` and so forth. If increasing is True, the columns are `x^0, x^1, ..., x^(N-1)`.

  See also

  `polynomial.polynomial.polyvander`

  Examples

      >>> x = np.array([1, 2, 3, 5])
      >>> N = 3
      >>> np.vander(x, N)
      array([[ 1,  1,  1],
             [ 4,  2,  1],
             [ 9,  3,  1],
             [25,  5,  1]])

      >>> np.column_stack([x**(N-1-i) for i in range(N)])
      array([[ 1,  1,  1],
             [ 4,  2,  1],
             [ 9,  3,  1],
             [25,  5,  1]])

      >>> x = np.array([1, 2, 3, 5])
      >>> np.vander(x)
      array([[  1,   1,   1,   1],
             [  8,   4,   2,   1],
             [ 27,   9,   3,   1],
             [125,  25,   5,   1]])
      >>> np.vander(x, increasing=True)
      array([[  1,   1,   1,   1],
             [  1,   2,   4,   8],
             [  1,   3,   9,  27],
             [  1,   5,  25, 125]])

  The determinant of a square Vandermonde matrix is the product of the differences between the values of the input vector:

      >>> np.linalg.det(np.vander(x))
      48.000000000000043 # may vary
      >>> (5-3)*(5-2)*(5-1)*(3-2)*(3-1)*(2-1)
      48

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