numpy.vander

numpy.vander(x, N=None)

Generate a Van der Monde matrix.

The columns of the output matrix are decreasing powers of the input vector. Specifically, 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

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

Returns :

out : ndarray

Van der Monde matrix of order N. The first column is x^(N-1), the second x^(N-2) and so forth.

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]])

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
>>> (5-3)*(5-2)*(5-1)*(3-2)*(3-1)*(2-1)
48