Pseudo-Vandermonde matrix of given degrees.
Returns the pseudo-Vandermonde matrix of degrees deg and sample
points (x, y). The pseudo-Vandermonde matrix is defined by
System Message: WARNING/2 (V[..., deg[1]*i + j] = L_i(x) * L_j(y),
)
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where 0 <= i <= deg[0] and 0 <= j <= deg[1]. The leading indices of
V index the points (x, y) and the last index encodes the degrees of
the Legendre polynomials.
If V = legvander2d(x, y, [xdeg, ydeg]), then the columns of V
correspond to the elements of a 2-D coefficient array c of shape
(xdeg + 1, ydeg + 1) in the order
System Message: WARNING/2 (c_{00}, c_{01}, c_{02} ... , c_{10}, c_{11}, c_{12} ...
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and np.dot(V, c.flat) and legval2d(x, y, 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 2-D Legendre
series of the same degrees and sample points.
Parameters: | x, y : 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].
|
Returns: | vander2d : ndarray
The shape of the returned matrix is x.shape + (order,), where
System Message: WARNING/2 (order = (deg[0]+1)*(deg([1]+1))
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. The dtype will be the same
as the converted x and y.
|
Notes