Function chebvander

numpy/polynomial/chebyshev.py:1406–1456

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 .. math:: V[..., i] = T_i(x), where ``0 <= i <= deg``. The leading indices of `V` index the elements of `x` and t

(x, deg)

def chebvander(x, deg):
    """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

    .. math:: V[..., i] = T_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 = chebvander(x, n)``, then ``np.dot(V, c)`` and
    ``chebval(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 Chebyshev series of the same degree and sample points.

    Parameters
    ----------
    x : array_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.
    deg : int
        Degree of the resulting matrix.

    Returns
    -------
    vander : ndarray
        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 Chebyshev polynomial.  The dtype will be the same as
        the converted `x`.

    """
    ideg = pu._as_int(deg, "deg")
    if ideg < 0:
        raise ValueError("deg must be non-negative")

    x = np.array(x, copy=None, ndmin=1) + 0.0
    dims = (ideg + 1,) + x.shape
    dtyp = x.dtype
    v = np.empty(dims, dtype=dtyp)
    # Use forward recursion to generate the entries.
    v[0] = x * 0 + 1
    if ideg > 0:
        x2 = 2 * x
        v[1] = x
        for i in range(2, ideg + 1):
            v[i] = v[i - 1] * x2 - v[i - 2]
    return np.moveaxis(v, 0, -1)


def chebvander2d(x, y, deg):