From f6e1d203ea962a391bcb71ffbf8dd380f09fa833 Mon Sep 17 00:00:00 2001
From: Michael Orlitzky <michael@orlitzky.com>
Date: Fri, 26 Oct 2012 14:22:56 -0400
Subject: [PATCH] Make the divided_difference() function rely on
 divided_difference_coefficients(). Make divided_difference() and
 newton_polynomial() take a list of function values instead of the function
 itself.

---
 mjo/interpolation.py | 67 +++++++++++++++++++++-----------------------
 1 file changed, 32 insertions(+), 35 deletions(-)

diff --git a/mjo/interpolation.py b/mjo/interpolation.py
index d5011ea..1d594c3 100644
--- a/mjo/interpolation.py
+++ b/mjo/interpolation.py
@@ -93,70 +93,67 @@ def divided_difference_coefficients(xs):
                for xj in xs ]
     return coeffs
 
-def divided_difference(f, xs):
+def divided_difference(xs, ys):
     """
-    Return the Newton divided difference of `f` at the points
-    `xs`. Reference:
+    Return the Newton divided difference of the points (xs[k],
+    ys[k]). Reference:
 
       http://en.wikipedia.org/wiki/Divided_differences
 
     INPUT:
 
-      - ``f`` -- The function whose divided difference we seek.
+      - ``xs`` -- The list of x-values.
 
-      - ``xs`` -- The list of points at which to compute `f`.
+      - ``ys`` -- The function values at `xs`.
 
     OUTPUT:
 
-    The divided difference of `f` at ``xs``.
+    The (possibly symbolic) divided difference function.
 
     TESTS::
 
-        sage: divided_difference(sin, [0])
+        sage: xs = [0]
+        sage: ys = map(sin, xs)
+        sage: divided_difference(xs, ys)
         0
-        sage: divided_difference(sin, [0, pi])
+        sage: xs = [0, pi]
+        sage: ys = map(sin, xs)
+        sage: divided_difference(xs, ys)
         0
-        sage: divided_difference(sin, [0, pi, 2*pi])
+        sage: xs = [0, pi, 2*pi]
+        sage: ys = map(sin, xs)
+        sage: divided_difference(xs, ys)
         0
 
     We try something entirely symbolic::
 
         sage: f = function('f', x)
-        sage: divided_difference(f, [x])
+        sage: divided_difference([x], [f(x=x)])
         f(x)
         sage: x1,x2 = var('x1,x2')
-        sage: divided_difference(f, [x1,x2])
-        (f(x1) - f(x2))/(x1 - x2)
+        sage: divided_difference([x1,x2], [f(x=x1),f(x=x2)])
+        f(x1)/(x1 - x2) - 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):
+    coeffs = divided_difference_coefficients(xs)
+    v_cs = vector(coeffs)
+    v_ys = vector(ys)
+    return v_cs.dot_product(v_ys)
+
+
+def newton_polynomial(x, xs, ys):
     """
-    Return the Newton form of the interpolating polynomial of `f` at
-    the points `xs` in the variable `x`.
+    Return the Newton form of the interpolating polynomial of the
+    points (xs[k], ys[k]) 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`.
+      - ``xs`` -- The list of x-values.
+
+      - ``ys`` -- The function values at `xs`.
 
     OUTPUT:
 
@@ -167,7 +164,7 @@ def newton_polynomial(f, x, xs):
         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: N = newton_polynomial(x, xs, ys)
         sage: bool(N == L)
         True
 
@@ -177,7 +174,7 @@ def newton_polynomial(f, x, xs):
     N = SR(0)
 
     for k in range(0, degree+1):
-        term  = divided_difference(f, xs[:k+1])
+        term  = divided_difference(xs[:k+1], ys[:k+1])
         term *= product([ x - xk for xk in xs[:k]])
         N += term
 
-- 
2.44.2