numpy.polynomial.legendre.legvander

polynomial.legendre.legvander(x, deg)[source]

Pseudo-Vandermonde matrix of given degree.

Returns the pseudo-Vandermonde matrix of degree deg and sample points x. The pseudo-Vandermonde matrix is defined by

\[V[..., i] = L_i(x)\]

where 0 <= i <= deg. The leading indices of V index the elements of x and the last index is the degree of the Legendre polynomial.

If c is a 1-D array of coefficients of length n + 1 and V is the array V = legvander(x, n), then np.dot(V, c) and legval(x, c) are the same up to roundoff. This equivalence is useful both for least squares fitting and for the evaluation of a large number of Legendre series of the same degree and sample points.

Parameters:

xarray_like

Array of points. The dtype is converted to float64 or complex128 depending on whether any of the elements are complex. If x is scalar it is converted to a 1-D array.

degint

Degree of the resulting matrix.

Returns:

vanderndarray

The pseudo-Vandermonde matrix. The shape of the returned matrix is x.shape + (deg + 1,), where The last index is the degree of the corresponding Legendre polynomial. The dtype will be the same as the converted x.