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Module « numpy.matlib »

Fonction vander - module numpy.matlib

Signature de la fonction vander

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

Description

help(numpy.matlib.vander)

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.

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
--------
>>> import numpy as np
>>> 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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