mjo/interpolation.py: use xrange everywhere.

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

diff --git a/mjo/interpolation.py b/mjo/interpolation.py
index fa784dd8c86fc2ab9406592622599f0320ba8941..a5ad78ddb32589137b72658afa4fe6401f17b202 100644 (file)
--- 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):
@@ -18,7 +18,7 @@ def lagrange_denominator(k, xs):
     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])
+    return product([xs[k] - xs[j] for j in xrange(len(xs)) if j != k])
 
 
 def lagrange_coefficient(k, x, 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 ]
@@ -88,8 +96,8 @@ def lagrange_polynomial(x, xs, ys):
         True
 
     """
-    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)) ])
+    ls = [ lagrange_coefficient(k, x, xs) for k in xrange(len(xs)) ]
+    sigma = sum([ ys[k] * ls[k] for k in xrange(len(xs)) ])
     return sigma
 
 
@@ -111,6 +119,10 @@ def lagrange_interpolate(f, x, xs):
 
     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::
@@ -135,7 +147,11 @@ 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]
@@ -145,7 +161,7 @@ def divided_difference_coefficients(xs):
         [1/2/pi^2, -1/pi^2, 1/2/pi^2]
 
     """
-    coeffs = [ QQ(1)/lagrange_denominator(k, xs) for k in range(0, len(xs)) ]
+    coeffs = [ QQ(1)/lagrange_denominator(k, xs) for k in xrange(len(xs)) ]
     return coeffs
 
 
@@ -166,6 +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]
@@ -183,7 +203,7 @@ 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 = SR.var('x1,x2')
@@ -215,7 +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)
@@ -229,7 +253,7 @@ def newton_polynomial(x, xs, ys):
 
     N = SR(0)
 
-    for k in range(0, degree+1):
+    for k in xrange(degree+1):
         term  = divided_difference(xs[:k+1], ys[:k+1])
         term *= lagrange_psi(x, xs[:k])
         N += term
@@ -304,7 +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)
@@ -319,10 +347,10 @@ def hermite_interpolant(x, xs, ys, y_primes):
 
     """
     s1 = sum([ ys[k] * hermite_coefficient(k, x, xs)
-               for k in range(0, len(xs)) ])
+               for k in xrange(len(xs)) ])
 
     s2 = sum([ y_primes[k] * hermite_deriv_coefficient(k, x, xs)
-               for k in range(0, len(xs)) ])
+               for k in xrange(len(xs)) ])
 
     return (s1 + s2)