X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Finterpolation.py;h=cbba68e03a4e2bd3ae5601ec51844aada3cf435d;hb=f352ef9a0ab5fbc784becdef65e20fe10d80e67e;hp=5d65d154e1ce567a7a408c305abcf91596176f4c;hpb=89ea7cefd713fbd44e6838603185a70ace6edf54;p=sage.d.git

diff --git a/mjo/interpolation.py b/mjo/interpolation.py
index 5d65d15..cbba68e 100644
--- a/mjo/interpolation.py
+++ b/mjo/interpolation.py
@@ -1,5 +1,5 @@
 from sage.all import *
-from misc import product
+product = prod
 
 
 def lagrange_denominator(k, xs):
@@ -94,6 +94,42 @@ def lagrange_polynomial(x, xs, ys):
 
 
 
+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
@@ -150,7 +186,7 @@ def divided_difference(xs, ys):
         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)