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 to the power of N - i - 1. Such a matrix with a geometric progression in each row is named Van Der Monde, or Vandermonde matrix, from Alexandre-Theophile Vandermonde.
Parameters: | x : array_like
N : int, optional
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Returns: | out : ndarray
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References
[R276] | Wikipedia, “Vandermonde matrix”, http://en.wikipedia.org/wiki/Vandermonde_matrix |
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