from sage.all import *
-from misc import product
+product = prod
def lagrange_denominator(k, xs):
+def lagrange_interpolate(f, x, xs):
+ """
+ Interpolate the function ``f`` at the points ``xs`` using the
+ Lagrange form of the interpolating polynomial.
+
+ INPUT:
+
+ - ``f`` -- The function to interpolate.
+
+ - ``x`` -- The independent variable of the resulting polynomial.
+
+ - ``xs`` -- A list of points at which to interpolate ``f``.
+
+ OUTPUT:
+
+ A polynomial in ``x`` which interpolates ``f`` at ``xs``.
+
+ EXAMPLES:
+
+ We're exact on polynomials of degree `n` if we use `n+1` points::
+
+ sage: t = SR.symbol('t', domain='real')
+ sage: lagrange_interpolate(x^2, t, [-1,0,1]).simplify_rational()
+ t^2
+
+ """
+ # f should be a function of one variable.
+ z = f.variables()[0]
+ # We're really just doing map(f, xs) here; the additional
+ # gymnastics are to avoid a warning when calling `f` with an
+ # unnamed argument.
+ ys = [ f({z: xk}) for xk in xs ]
+ return lagrange_polynomial(x, xs, ys)
+
+
+
def divided_difference_coefficients(xs):
"""
Assuming some function `f`, compute the coefficients of the
sage: f = function('f', x)
sage: divided_difference([x], [f(x=x)])
f(x)
- sage: x1,x2 = var('x1,x2')
+ sage: x1,x2 = SR.var('x1,x2')
sage: divided_difference([x1,x2], [f(x=x1),f(x=x2)])
f(x1)/(x1 - x2) - f(x2)/(x1 - x2)