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lagvander3d(x, y, z, deg)

Returns the pseudo-Vandermonde matrix of degrees :None:None:`deg` and sample points :None:None:`(x, y, z)`. If :None:None:`l, m, n` are the given degrees in :None:None:`x, y, z`, then The pseudo-Vandermonde matrix is defined by

$$V[..., (m+1)(n+1)i + (n+1)j + k] = L_i(x)*L_j(y)*L_k(z),$$

where :None:None:`0 <= i <= l`, :None:None:`0 <= j <= m`, and :None:None:`0 <= j <= n`. The leading indices of :None:None:`V` index the points :None:None:`(x, y, z)` and the last index encodes the degrees of the Laguerre polynomials.

If V = lagvander3d(x, y, z, [xdeg, ydeg, zdeg]) , then the columns of :None:None:`V` correspond to the elements of a 3-D coefficient array :None:None:`c` of shape (xdeg + 1, ydeg + 1, zdeg + 1) in the order

$$c_{000}, c_{001}, c_{002},... , c_{010}, c_{011}, c_{012},...$$

and np.dot(V, c.flat) and lagval3d(x, y, z, c) will be the same up to roundoff. This equivalence is useful both for least squares fitting and for the evaluation of a large number of 3-D Laguerre series of the same degrees and sample points.

Notes

versionadded

Parameters

x, y, z : 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.
deg : list of ints: List of maximum degrees of the form [x_deg, y_deg, z_deg].

Returns

vander3d : ndarray: The shape of the returned matrix is x.shape + (order,) , where $order = (deg[0]+1)*(deg[1]+1)*(deg[2]+1)$ . The dtype will be the same as the converted x, y, and z.

Pseudo-Vandermonde matrix of given degrees.

Examples

See :

Back References

The following pages refer to to this document either explicitly or contain code examples using this.

numpy.polynomial.laguerre.lagvander2d numpy.polynomial.laguerre.lagvander3d

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