Add the 'product' function to misc.py.

[sage.d.git] / mjo / interpolation.py

diff --git a/mjo/interpolation.py b/mjo/interpolation.py
new file mode 100644 (file)
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