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

The columns of the output matrix are powers of the input vector. The order of the powers is determined by the :None:None:`increasing` boolean argument. Specifically, when :None:None:`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.

versionadded

Returns

out : ndarray

Vandermonde matrix. If :None:None:`increasing` is False, the first column is x^(N-1) , the second x^(N-2) and so forth. If :None:None:`increasing` is True, the columns are x^0, x^1, ..., x^(N-1) .

Generate a Vandermonde matrix.

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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