NumPy: numpy.vander() function

numpy.vander() function

The numpy.vander() function returns a Vandermonde matrix of a given input array. The Vandermonde matrix is a matrix with the elements of an input vector raised to different powers, where the power is determined by the position of the element in the vector.

Syntax:

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

Parameters:

Return value:

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).

Example: Generating a Vandermonde matrix using NumPy

>>> import numpy as np
>>> a = np.array ([1,2,4,6])
>>> Y=4
>>> np.vander(a, Y)
array([[  1,   1,   1,   1],
       [  8,   4,   2,   1],
       [ 64,  16,   4,   1],
       [216,  36,   6,   1]])
>>> np.column_stack([a**(Y-1-i) for i in range(Y)])
array([[  1,   1,   1,   1],
       [  8,   4,   2,   1],
       [ 64,  16,   4,   1],   
       [216,  36,   6,   1]])

In the above code, the np.vander function is used to create the Vandermonde matrix for the input array a and power Y. The resulting matrix has Y columns, with the first column consisting of ones, and subsequent columns consisting of powers of the input array a, raised to decreasing powers of Y-1.
The np.column_stack function is used to stack the individual arrays created using the list comprehension. This creates the same Vandermonde matrix as the one generated using np.vander.

Pictorial Presentation:

Example: Vandermonde Matrix Generation with Numpy

>>> import numpy as np
>>> a = np.array([1,2,4,5])
>>> np.vander(a)
array([[  1,   1,   1,   1],
       [  8,   4,   2,   1],
       [ 64,  16,   4,   1],in
       [125,  25,   5,   1]])
>>> np.vander(a, increasing=True)
array([[  1,   1,   1,   1],
       [  1,   2,   4,   8],
       [  1,   4,  16,  64],
       [  1,   5,  25, 125]])

In the above code, the first example generates a Vandermonde matrix of order n (where n is the length of a) with decreasing powers of elements. The resulting matrix has dimensions (n, n), with the i-th row consisting of the elements a**(n-1-i), where ** denotes exponentiation.
The second example generates a Vandermonde matrix with increasing powers of elements by passing the increasing=True parameter to np.vander(). The resulting matrix has the same dimensions as the first example, but the i-th row consists of the elements a**i.
Both examples demonstrate the versatility of np.vander() for generating Vandermonde matrices with varying order and power configurations.

Pictorial Presentation:

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

>>> import numpy as np
>>> a = np.array([1,2,4,5])
>>> np.linalg.det(np.vander(a))
72.000000000000071
>>> (5-4)*(5-2)*(5-1)*(4-2)*(4-1)*(2-1)
72

Python - NumPy Code Editor:

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