X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Finterpolation.py;h=5d65d154e1ce567a7a408c305abcf91596176f4c;hb=ff5b1d1514a419bbf8140658778d1e69e89b2664;hp=70c35c1a3ac9e812865185c71b1f4d2cc88be1ed;hpb=c2eb40221f6eb4bea022fcb885f25268c1dbd1e3;p=sage.d.git

diff --git a/mjo/interpolation.py b/mjo/interpolation.py
index 70c35c1..5d65d15 100644
--- a/mjo/interpolation.py
+++ b/mjo/interpolation.py
@@ -1,6 +1,26 @@
 from sage.all import *
 from misc import product
 
+
+def lagrange_denominator(k, xs):
+    """
+    Return the denominator of the kth Lagrange coefficient.
+
+    INPUT:
+
+      - ``k`` -- The index of the coefficient.
+
+      - ``xs`` -- The list of points at which the function values are
+        known.
+
+    OUTPUT:
+
+    The product of all xs[j] with j != k.
+
+    """
+    return product([xs[k] - xs[j] for j in range(0, len(xs)) if j != k])
+
+
 def lagrange_coefficient(k, x, xs):
     """
     Returns the coefficient function l_{k}(variable) of y_{k} in the
@@ -12,9 +32,9 @@ def lagrange_coefficient(k, x, xs):
 
     INPUT:
 
-      - ``k`` -- the index of the coefficient.
+      - ``k`` -- The index of the coefficient.
 
-      - ``x`` -- the symbolic variable to use for the first argument
+      - ``x`` -- The symbolic variable to use for the first argument
         of l_{k}.
 
       - ``xs`` -- The list of points at which the function values are
@@ -31,8 +51,8 @@ def lagrange_coefficient(k, 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])
+    numerator = lagrange_psi(x, xs)/(x - xs[k])
+    denominator = lagrange_denominator(k, xs)
 
     return (numerator / denominator)
 
@@ -89,10 +109,10 @@ def divided_difference_coefficients(xs):
         [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 ]
+    coeffs = [ QQ(1)/lagrange_denominator(k, xs) for k in range(0, len(xs)) ]
     return coeffs
 
+
 def divided_difference(xs, ys):
     """
     Return the Newton divided difference of the points (xs[k],
@@ -175,7 +195,7 @@ def newton_polynomial(x, xs, ys):
 
     for k in range(0, degree+1):
         term  = divided_difference(xs[:k+1], ys[:k+1])
-        term *= product([ x - xk for xk in xs[:k]])
+        term *= lagrange_psi(x, xs[:k])
         N += term
 
     return N
@@ -269,3 +289,26 @@ def hermite_interpolant(x, xs, ys, y_primes):
                for k in range(0, len(xs)) ])
 
     return (s1 + s2)
+
+
+def lagrange_psi(x, xs):
+    """
+    The function,
+
+      Psi(x) = (x - xs[0])*(x - xs[1])* ... *(x - xs[-1])
+
+    used in Lagrange and Hermite interpolation.
+
+    INPUT:
+
+      - ``x`` -- The independent variable of the resulting expression.
+
+      - ``xs`` -- A list of points.
+
+    OUTPUT:
+
+    A symbolic expression in one variable, `x`.
+
+    """
+
+    return product([ (x - xj) for xj in xs ])