From d6c9163d4844d519559a1a35cd7094fc97572973 Mon Sep 17 00:00:00 2001
From: Michael Orlitzky <michael@orlitzky.com>
Date: Thu, 25 Oct 2012 20:07:56 -0400
Subject: [PATCH] Add the 'product' function to misc.py. Add interpolation.py
 with two Lagrange polynomial functions.

---
 mjo/all.py           |  1 +
 mjo/interpolation.py | 73 ++++++++++++++++++++++++++++++++++++++++++++
 mjo/misc.py          | 22 +++++++++++++
 3 files changed, 96 insertions(+)
 create mode 100644 mjo/interpolation.py

diff --git a/mjo/all.py b/mjo/all.py
index 58a993d..4169cee 100644
--- a/mjo/all.py
+++ b/mjo/all.py
@@ -3,6 +3,7 @@ Import all of the other code, so that the user doesn't have to do it
 in his script. Instead, he can just `from mjo.all import *`.
 """
 
+from interpolation import *
 from misc import *
 from plot import *
 from symbolic import *
diff --git a/mjo/interpolation.py b/mjo/interpolation.py
new file mode 100644
index 0000000..e3e6933
--- /dev/null
+++ b/mjo/interpolation.py
@@ -0,0 +1,73 @@
+from sage.all import *
+load('~/.sage/init.sage')
+
+def lagrange_coefficient(k, x, xs):
+    """
+    Returns the coefficient function l_{k}(variable) of y_{k} in the
+    Lagrange polynomial of f. See,
+
+      http://en.wikipedia.org/wiki/Lagrange_polynomial
+
+    for more information.
+
+    INPUT:
+
+      - ``k`` -- the index of the coefficient.
+
+      - ``x`` -- the symbolic variable to use for the first argument
+        of l_{k}.
+
+      - ``xs`` -- The list of points at which the function values are
+        known.
+
+    OUTPUT:
+
+    A symbolic function of one variable.
+
+    TESTS::
+
+        sage: xs = [ -pi/2, -pi/6, 0, pi/6, pi/2 ]
+        sage: lagrange_coefficient(0, 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])
+
+    return (numerator / denominator)
+
+
+
+def lagrange_polynomial(f, x, xs):
+    """
+    Return the Lagrange form of the interpolation polynomial in `x` of
+    `f` at the points `xs`.
+
+    INPUT:
+
+      - ``f`` - The function to interpolate.
+
+      - ``x`` - The independent variable of the resulting polynomial.
+
+      - ``xs`` - The list of points at which we interpolate `f`.
+
+    OUTPUT:
+
+    A symbolic function (polynomial) interpolating `f` at `xs`.
+
+    TESTS::
+
+        sage: xs = [ -pi/2, -pi/6, 0, pi/6, pi/2 ]
+        sage: L = lagrange_polynomial(sin, x, xs)
+        sage: expected  = 27/16*(pi - 6*x)*(pi - 2*x)*(pi + 2*x)*x/pi^4
+        sage: expected -= 1/8*(pi - 6*x)*(pi - 2*x)*(pi + 6*x)*x/pi^4
+        sage: expected -= 1/8*(pi - 6*x)*(pi + 2*x)*(pi + 6*x)*x/pi^4
+        sage: expected += 27/16*(pi - 2*x)*(pi + 2*x)*(pi + 6*x)*x/pi^4
+        sage: bool(L == expected)
+        True
+
+    """
+    ys = [ f(xs[k]) for k in range(0, len(xs)) ]
+    ls = [ lagrange_coefficient(k, x, xs) for k in range(0, len(xs)) ]
+    sigma = sum([ ys[k] * ls[k] for k in range(0, len(xs)) ])
+    return sigma
diff --git a/mjo/misc.py b/mjo/misc.py
index 28113b6..f5a7bcf 100644
--- a/mjo/misc.py
+++ b/mjo/misc.py
@@ -3,6 +3,7 @@ Stuff that doesn't fit anywhere else.
 """
 
 from sage.all import *
+from functools import reduce
 
 def legend_latex(obj):
     """
@@ -11,3 +12,24 @@ def legend_latex(obj):
     legend label.
     """
     return '$%s$' % latex(obj)
+
+def product(factors):
+    """
+    Returns the product of the elements in the list `factors`. If the
+    list is empty, we return 1.
+
+    TESTS:
+
+    Normal integer multiplication::
+
+        sage: product([1,2,3])
+        6
+
+    And with symbolic variables::
+
+        sage: x,y,z = var('x,y,z')
+        sage: product([x,y,z])
+        x*y*z
+
+    """
+    return reduce(operator.mul, factors, 1)
-- 
2.43.2