from sage.all import *
from misc import product

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(x, xs, ys):
    """
    Return the Lagrange form of the interpolation polynomial in `x` of
    at the points (xs[k], ys[k]).

    INPUT:

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

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

      - ``ys`` - The function values at `xs`.

    OUTPUT:

    A symbolic function (polynomial) interpolating each (xs[k], ys[k]).

    TESTS::

        sage: xs = [ -pi/2, -pi/6, 0, pi/6, pi/2 ]
        sage: ys = map(sin, xs)
        sage: L = lagrange_polynomial(x, xs, ys)
        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

    """
    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



def divided_difference_coefficients(xs):
    """
    Assuming some function `f`, compute the coefficients of the
    divided difference f[xs[0], ..., xs[n]].

    TESTS:

        sage: divided_difference_coefficients([0])
        [1]
        sage: divided_difference_coefficients([0, pi])
        [-1/pi, 1/pi]
        sage: divided_difference_coefficients([0, pi, 2*pi])
        [1/2/pi^2, -1/pi^2, 1/2/pi^2]

    """
    coeffs = [ product([ (QQ(1) / (xj - xi)) for xi in xs if xi != xj ])
               for xj in xs ]
    return coeffs

def divided_difference(f, xs):
    """
    Return the Newton divided difference of `f` at the points
    `xs`. Reference:

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

    INPUT:

      - ``f`` -- The function whose divided difference we seek.

      - ``xs`` -- The list of points at which to compute `f`.

    OUTPUT:

    The divided difference of `f` at ``xs``.

    TESTS::

        sage: divided_difference(sin, [0])
        0
        sage: divided_difference(sin, [0, pi])
        0
        sage: divided_difference(sin, [0, pi, 2*pi])
        0

    We try something entirely symbolic::

        sage: f = function('f', x)
        sage: divided_difference(f, [x])
        f(x)
        sage: x1,x2 = var('x1,x2')
        sage: divided_difference(f, [x1,x2])
        (f(x1) - f(x2))/(x1 - x2)

    """
    if (len(xs) == 1):
        # Avoid that goddamned DeprecationWarning when we have a named
        # argument but don't know what it is.
        if len(f.variables()) == 0:
            return f(xs[0])
        else:
            v = f.variables()[0]
            return f({ v: xs[0] })

    # Use the recursive definition.
    numerator  = divided_difference(f, xs[1:])
    numerator -= divided_difference(f, xs[:-1])
    return numerator / (xs[-1] - xs[0])


def newton_polynomial(f, x, xs):
    """
    Return the Newton form of the interpolating polynomial of `f` at
    the points `xs` in the variable `x`.

    INPUT:

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

      - ``x`` -- The independent variable to use for the interpolating
        polynomial.

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

    OUTPUT:

    A symbolic function.

    TESTS:

        sage: xs = [ -pi/2, -pi/6, 0, pi/6, pi/2 ]
        sage: ys = map(sin, xs)
        sage: L = lagrange_polynomial(x, xs, ys)
        sage: N = newton_polynomial(sin, x, xs)
        sage: bool(N == L)
        True

    """
    degree = len(xs) - 1

    N = SR(0)

    for k in range(0, degree+1):
        term  = divided_difference(f, xs[:k+1])
        term *= product([ x - xk for xk in xs[:k]])
        N += term

    return N