Function chebinterpolate

numpy/polynomial/chebyshev.py:1779–1840

Interpolate a function at the Chebyshev points of the first kind. Returns the Chebyshev series that interpolates `func` at the Chebyshev points of the first kind in the interval [-1, 1]. The interpolating series tends to a minmax approximation to `func` with increasing `deg` if the

(func, deg, args=())

def chebinterpolate(func, deg, args=()):
    """Interpolate a function at the Chebyshev points of the first kind.

    Returns the Chebyshev series that interpolates `func` at the Chebyshev
    points of the first kind in the interval [-1, 1]. The interpolating
    series tends to a minmax approximation to `func` with increasing `deg`
    if the function is continuous in the interval.

    Parameters
    ----------
    func : function
        The function to be approximated. It must be a function of a single
        variable of the form ``f(x, a, b, c...)``, where ``a, b, c...`` are
        extra arguments passed in the `args` parameter.
    deg : int
        Degree of the interpolating polynomial
    args : tuple, optional
        Extra arguments to be used in the function call. Default is no extra
        arguments.

    Returns
    -------
    coef : ndarray, shape (deg + 1,)
        Chebyshev coefficients of the interpolating series ordered from low to
        high.

    Examples
    --------
    >>> import numpy.polynomial.chebyshev as C
    >>> C.chebinterpolate(lambda x: np.tanh(x) + 0.5, 8)
    array([  5.00000000e-01,   8.11675684e-01,  -9.86864911e-17,
            -5.42457905e-02,  -2.71387850e-16,   4.51658839e-03,
             2.46716228e-17,  -3.79694221e-04,  -3.26899002e-16])

    Notes
    -----
    The Chebyshev polynomials used in the interpolation are orthogonal when
    sampled at the Chebyshev points of the first kind. If it is desired to
    constrain some of the coefficients they can simply be set to the desired
    value after the interpolation, no new interpolation or fit is needed. This
    is especially useful if it is known apriori that some of coefficients are
    zero. For instance, if the function is even then the coefficients of the
    terms of odd degree in the result can be set to zero.

    """
    deg = np.asarray(deg)

    # check arguments.
    if deg.ndim > 0 or deg.dtype.kind not in 'iu' or deg.size == 0:
        raise TypeError("deg must be an int")
    if deg < 0:
        raise ValueError("expected deg >= 0")

    order = deg + 1
    xcheb = chebpts1(order)
    yfunc = func(xcheb, *args)
    m = chebvander(xcheb, deg)
    c = np.dot(m.T, yfunc)