from sage.all import *
load('~/.sage/init.sage')

def lagrange_coefficient(k, x, xs):
    """
    Returns the coefficient function l_{k}(variable) of y_{k} in the
    Lagrange polynomial of f. See,

      http://en.wikipedia.org/wiki/Lagrange_polynomial

    for more information.

    INPUT:

      - ``k`` -- the index of the coefficient.

      - ``x`` -- the symbolic variable to use for the first argument
        of l_{k}.

      - ``xs`` -- The list of points at which the function values are
        known.

    OUTPUT:

    A symbolic function of one variable.

    TESTS::

        sage: xs = [ -pi/2, -pi/6, 0, pi/6, pi/2 ]
        sage: lagrange_coefficient(0, x, xs)
        1/8*(pi - 6*x)*(pi - 2*x)*(pi + 6*x)*x/pi^4

    """
    numerator = product([x - xs[j] for j in range(0, len(xs)) if j != k])
    denominator = product([xs[k] - xs[j] for j in range(0, len(xs)) if j != k])

    return (numerator / denominator)



def lagrange_polynomial(f, x, xs):
    """
    Return the Lagrange form of the interpolation polynomial in `x` of
    `f` at the points `xs`.

    INPUT:

      - ``f`` - The function to interpolate.

      - ``x`` - The independent variable of the resulting polynomial.

      - ``xs`` - The list of points at which we interpolate `f`.

    OUTPUT:

    A symbolic function (polynomial) interpolating `f` at `xs`.

    TESTS::

        sage: xs = [ -pi/2, -pi/6, 0, pi/6, pi/2 ]
        sage: L = lagrange_polynomial(sin, x, xs)
        sage: expected  = 27/16*(pi - 6*x)*(pi - 2*x)*(pi + 2*x)*x/pi^4
        sage: expected -= 1/8*(pi - 6*x)*(pi - 2*x)*(pi + 6*x)*x/pi^4
        sage: expected -= 1/8*(pi - 6*x)*(pi + 2*x)*(pi + 6*x)*x/pi^4
        sage: expected += 27/16*(pi - 2*x)*(pi + 2*x)*(pi + 6*x)*x/pi^4
        sage: bool(L == expected)
        True

    """
    ys = [ f(xs[k]) for k in range(0, len(xs)) ]
    ls = [ lagrange_coefficient(k, x, xs) for k in range(0, len(xs)) ]
    sigma = sum([ ys[k] * ls[k] for k in range(0, len(xs)) ])
    return sigma