`numpy.polynomial.polyutils:_vander_nd`

source: /numpy/polynomial/polyutils.py :364

Signature

def _vander_nd ( vander_fs , points , degrees )

Summary

A generalization of the Vandermonde matrix for N dimensions

Extended Summary

The result is built by combining the results of 1d Vandermonde matrices,

W [i_{0}, \dots, i_{M}, j_{0}, \dots, j_{N}] = k = 0 \prod N V_{k} (x_{k}) [i_{0}, \dots, i_{M}, j_{k}]

where

N M V_{k} x_{k} 0 \leq j_{k} = len(points) = len(degrees) = len(vander_fs) = points[k].ndim = vander_fs[k] = points[k] \leq degrees[k]

Expanding the one-dimensional $V_{k}$ functions gives:

W [i_{0}, \dots, i_{M}, j_{0}, \dots, j_{N}] = k = 0 \prod N B_{k, j_{k}} (x_{k} [i_{0}, \dots, i_{M}])

where $B_{k, m}$ is the m'th basis of the polynomial construction used along dimension $k$ . For a regular polynomial, $B_{k, m} (x) = P_{m} (x) = x^{m}$ .

Parameters

vander_fs : Sequence[function(array_like, int) -> ndarray]: The 1d vander function to use for each axis, such as polyvander
points : Sequence[array_like]: Arrays of point coordinates, all of the same shape. The dtypes will be converted to either float64 or complex128 depending on whether any of the elements are complex. Scalars are converted to 1-D arrays. This must be the same length as vander_fs.
degrees : Sequence[int]: The maximum degree (inclusive) to use for each axis. This must be the same length as vander_fs.

Returns

vander_nd : ndarray: An array of shape points[0].shape + tuple(d + 1 for d in degrees).

Aliases

numpy.polynomial.polyutils._vander_nd