orthax.orthvander

orthax.orthvander(x, deg, rec)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

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

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

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

Parameters:

  • x (jax.Array) – 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.

  • deg (int) – Degree of the resulting matrix.

  • rec (AbstractRecurrenceRelation) – Recurrence relation for the family of orthogonal polynomials.

Returns:

vander (jax.Array) – 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 polynomial. The dtype will be the same as the converted x.