flopscope.

flopscope.numpy.vander

fnp.vander(x, N=None, increasing=False)[flopscope source][numpy source]

Generate a Vandermonde matrix.

Adapted from NumPy docs np.vander

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Typecustom
NumPy Refnp.vander
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Vandermonde matrix; cost = len(x)*(N-1).

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 flopscope.numpy as fnp
>>> x = flops.array([1, 2, 3, 5])
>>> N = 3
>>> flops.vander(x, N)
array([[ 1,  1,  1],
       [ 4,  2,  1],
       [ 9,  3,  1],
       [25,  5,  1]])
>>> flops.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 = flops.array([1, 2, 3, 5])
>>> flops.vander(x)
array([[  1,   1,   1,   1],
       [  8,   4,   2,   1],
       [ 27,   9,   3,   1],
       [125,  25,   5,   1]])
>>> flops.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:

>>> flops.linalg.det(flops.vander(x))
48.000000000000043 # may vary
>>> (5-3)*(5-2)*(5-1)*(3-2)*(3-1)*(2-1)
48