From: Michael Orlitzky <michael@orlitzky.com>
Date: Fri, 26 Oct 2012 19:20:25 +0000 (-0400)
Subject: Add the Hermite functions to interpolation.py.
X-Git-Url: http://gitweb.michael.orlitzky.com/?a=commitdiff_plain;h=c2eb40221f6eb4bea022fcb885f25268c1dbd1e3;p=sage.d.git

Add the Hermite functions to interpolation.py.
---

diff --git a/mjo/interpolation.py b/mjo/interpolation.py
index 1d594c3..70c35c1 100644
--- a/mjo/interpolation.py
+++ b/mjo/interpolation.py
@@ -22,7 +22,7 @@ def lagrange_coefficient(k, x, xs):
 
     OUTPUT:
 
-    A symbolic function of one variable.
+    A symbolic expression of one variable.
 
     TESTS::
 
@@ -53,7 +53,7 @@ def lagrange_polynomial(x, xs, ys):
 
     OUTPUT:
 
-    A symbolic function (polynomial) interpolating each (xs[k], ys[k]).
+    A symbolic expression (polynomial) interpolating each (xs[k], ys[k]).
 
     TESTS::
 
@@ -157,7 +157,7 @@ def newton_polynomial(x, xs, ys):
 
     OUTPUT:
 
-    A symbolic function.
+    A symbolic expression.
 
     TESTS:
 
@@ -179,3 +179,93 @@ def newton_polynomial(x, xs, ys):
         N += term
 
     return N
+
+
+def hermite_coefficient(k, x, xs):
+    """
+    Return the Hermite coefficient h_{k}(x). See Atkinson, p. 160.
+
+    INPUT:
+
+      - ``k`` -- The index of the coefficient.
+
+      - ``x`` -- The symbolic variable to use as the argument of h_{k}.
+
+      - ``xs`` -- The list of points at which the function values are
+        known.
+
+    OUTPUT:
+
+    A symbolic expression.
+
+    """
+    lk = lagrange_coefficient(k, x, xs)
+    return (1 - 2*lk.diff(x)(x=xs[k])*(x - xs[k]))*(lk**2)
+
+
+def hermite_deriv_coefficient(k, x, xs):
+    """
+    Return the Hermite derivative coefficient, \tilde{h}_{k}(x). See
+    Atkinson, p. 160.
+
+    INPUT:
+
+      - ``k`` -- The index of the coefficient.
+
+      - ``x`` -- The symbolic variable to use as the argument of h_{k}.
+
+      - ``xs`` -- The list of points at which the function values are
+        known.
+
+    OUTPUT:
+
+    A symbolic expression.
+
+    """
+    lk = lagrange_coefficient(k, x, xs)
+    return (x - xs[k])*(lk**2)
+
+
+def hermite_interpolant(x, xs, ys, y_primes):
+    """
+    Return the Hermite interpolant `H(x)` such that H(xs[k]) = ys[k]
+    and H'(xs[k]) = y_primes[k] for each k.
+
+    Reference: Atkinson, p. 160.
+
+    INPUT:
+
+      - ``x`` -- The symbolic variable to use as the argument of H(x).
+
+      - ``xs`` -- The list of points at which the function values are
+        known.
+
+      - ``ys`` -- The function values at the `xs`.
+
+      - ``y_primes`` -- The derivatives at the `xs`.
+
+    OUTPUT:
+
+    A symbolic expression.
+
+    TESTS:
+
+        sage: xs = [ 0, pi/6, pi/2 ]
+        sage: ys = map(sin, xs)
+        sage: y_primes = map(cos, xs)
+        sage: H = hermite_interpolant(x, xs, ys, y_primes)
+        sage: expected  = -27/4*(pi - 6*x)*(pi - 2*x)^2*sqrt(3)*x^2/pi^4
+        sage: expected += (5*(pi - 2*x)/pi + 1)*(pi - 6*x)^2*x^2/pi^4
+        sage: expected += 81/2*((pi - 6*x)/pi + 1)*(pi - 2*x)^2*x^2/pi^4
+        sage: expected += (pi - 6*x)^2*(pi - 2*x)^2*x/pi^4
+        sage: bool(H == expected)
+        True
+
+    """
+    s1 = sum([ ys[k] * hermite_coefficient(k, x, xs)
+               for k in range(0, len(xs)) ])
+
+    s2 = sum([ y_primes[k] * hermite_deriv_coefficient(k, x, xs)
+               for k in range(0, len(xs)) ])
+
+    return (s1 + s2)