X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Finterpolation.py;h=dc061077eda70879682048775c3ca69e423e9450;hb=2fa2a7a2f6a5d5f1628a4f5cf3301d5e7f670038;hp=5d65d154e1ce567a7a408c305abcf91596176f4c;hpb=89ea7cefd713fbd44e6838603185a70ace6edf54;p=sage.d.git

diff --git a/mjo/interpolation.py b/mjo/interpolation.py
index 5d65d15..dc06107 100644
--- a/mjo/interpolation.py
+++ b/mjo/interpolation.py
@@ -1,5 +1,5 @@
 from sage.all import *
-from misc import product
+product = prod
 
 
 def lagrange_denominator(k, xs):
@@ -44,11 +44,15 @@ def lagrange_coefficient(k, x, xs):
 
     A symbolic expression of one variable.
 
+    SETUP::
+
+        sage: from mjo.interpolation import lagrange_coefficient
+
     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
+        1/8*(pi + 6*x)*(pi - 2*x)*(pi - 6*x)*x/pi^4
 
     """
     numerator = lagrange_psi(x, xs)/(x - xs[k])
@@ -60,7 +64,7 @@ def lagrange_coefficient(k, x, xs):
 
 def lagrange_polynomial(x, xs, ys):
     """
-    Return the Lagrange form of the interpolation polynomial in `x` of
+    Return the Lagrange form of the interpolating polynomial in `x`
     at the points (xs[k], ys[k]).
 
     INPUT:
@@ -75,6 +79,10 @@ def lagrange_polynomial(x, xs, ys):
 
     A symbolic expression (polynomial) interpolating each (xs[k], ys[k]).
 
+    SETUP::
+
+        sage: from mjo.interpolation import lagrange_polynomial
+
     TESTS::
 
         sage: xs = [ -pi/2, -pi/6, 0, pi/6, pi/2 ]
@@ -94,12 +102,56 @@ def lagrange_polynomial(x, xs, ys):
 
 
 
+def lagrange_interpolate(f, x, xs):
+    """
+    Interpolate the function ``f`` at the points ``xs`` using the
+    Lagrange form of the interpolating polynomial.
+
+    INPUT:
+
+      - ``f`` -- The function to interpolate.
+
+      - ``x`` -- The independent variable of the resulting polynomial.
+
+      - ``xs`` -- A list of points at which to interpolate ``f``.
+
+    OUTPUT:
+
+    A polynomial in ``x`` which interpolates ``f`` at ``xs``.
+
+    SETUP::
+
+        sage: from mjo.interpolation import lagrange_interpolate
+
+    EXAMPLES:
+
+    We're exact on polynomials of degree `n` if we use `n+1` points::
+
+        sage: t = SR.symbol('t', domain='real')
+        sage: lagrange_interpolate(x^2, t, [-1,0,1]).simplify_rational()
+        t^2
+
+    """
+    # f should be a function of one variable.
+    z = f.variables()[0]
+    # We're really just doing map(f, xs) here; the additional
+    # gymnastics are to avoid a warning when calling `f` with an
+    # unnamed argument.
+    ys = [ f({z: xk}) for xk in xs ]
+    return lagrange_polynomial(x, xs, ys)
+
+
+
 def divided_difference_coefficients(xs):
     """
     Assuming some function `f`, compute the coefficients of the
     divided difference f[xs[0], ..., xs[n]].
 
-    TESTS:
+    SETUP::
+
+        sage: from mjo.interpolation import divided_difference_coefficients
+
+    TESTS::
 
         sage: divided_difference_coefficients([0])
         [1]
@@ -130,6 +182,10 @@ def divided_difference(xs, ys):
 
     The (possibly symbolic) divided difference function.
 
+    SETUP::
+
+        sage: from mjo.interpolation import divided_difference
+
     TESTS::
 
         sage: xs = [0]
@@ -147,10 +203,10 @@ def divided_difference(xs, ys):
 
     We try something entirely symbolic::
 
-        sage: f = function('f', x)
+        sage: f = function('f')(x)
         sage: divided_difference([x], [f(x=x)])
         f(x)
-        sage: x1,x2 = var('x1,x2')
+        sage: x1,x2 = SR.var('x1,x2')
         sage: divided_difference([x1,x2], [f(x=x1),f(x=x2)])
         f(x1)/(x1 - x2) - f(x2)/(x1 - x2)
 
@@ -179,7 +235,11 @@ def newton_polynomial(x, xs, ys):
 
     A symbolic expression.
 
-    TESTS:
+    SETUP::
+
+        sage: from mjo.interpolation import lagrange_polynomial, newton_polynomial
+
+    TESTS::
 
         sage: xs = [ -pi/2, -pi/6, 0, pi/6, pi/2 ]
         sage: ys = map(sin, xs)
@@ -268,7 +328,11 @@ def hermite_interpolant(x, xs, ys, y_primes):
 
     A symbolic expression.
 
-    TESTS:
+    SETUP::
+
+        sage: from mjo.interpolation import hermite_interpolant
+
+    TESTS::
 
         sage: xs = [ 0, pi/6, pi/2 ]
         sage: ys = map(sin, xs)