numpy.vander¶

numpy.
vander
(x, N=None, increasing=False)[source]¶ 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 ith output column is the input vector raised elementwise 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
1D 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^(N1)
, the secondx^(N2)
and so forth. If increasing is True, the columns arex^0, x^1, ..., x^(N1)
.See also
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**(N1i) 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 >>> (53)*(52)*(51)*(32)*(31)*(21) 48