X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Finterpolation.py;h=6e0ecb3d79c5129a2e6270177b648142085786d1;hb=HEAD;hp=dc061077eda70879682048775c3ca69e423e9450;hpb=442ca575b029c244e7307b502b63d11611dc205b;p=sage.d.git

diff --git a/mjo/interpolation.py b/mjo/interpolation.py
index dc06107..6e0ecb3 100644
--- a/mjo/interpolation.py
+++ b/mjo/interpolation.py
@@ -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 range(len(xs)) if j != k )
 
 
 def lagrange_coefficient(k, x, xs):
@@ -86,7 +86,7 @@ def lagrange_polynomial(x, xs, ys):
     TESTS::
 
         sage: xs = [ -pi/2, -pi/6, 0, pi/6, pi/2 ]
-        sage: ys = map(sin, xs)
+        sage: ys = list(map(sin, xs))
         sage: L = lagrange_polynomial(x, xs, ys)
         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
@@ -96,9 +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)) ])
-    return sigma
+    ls = [ lagrange_coefficient(k, x, xs) for k in range(len(xs)) ]
+    return sum( ys[k] * ls[k] for k in range(len(xs)) )
 
 
 
@@ -161,8 +160,7 @@ 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)) ]
-    return coeffs
+    return [ ~lagrange_denominator(k, xs) for k in range(len(xs)) ]
 
 
 def divided_difference(xs, ys):
@@ -189,15 +187,15 @@ def divided_difference(xs, ys):
     TESTS::
 
         sage: xs = [0]
-        sage: ys = map(sin, xs)
+        sage: ys = list(map(sin, xs))
         sage: divided_difference(xs, ys)
         0
         sage: xs = [0, pi]
-        sage: ys = map(sin, xs)
+        sage: ys = list(map(sin, xs))
         sage: divided_difference(xs, ys)
         0
         sage: xs = [0, pi, 2*pi]
-        sage: ys = map(sin, xs)
+        sage: ys = list(map(sin, xs))
         sage: divided_difference(xs, ys)
         0
 
@@ -242,23 +240,15 @@ def newton_polynomial(x, xs, ys):
     TESTS::
 
         sage: xs = [ -pi/2, -pi/6, 0, pi/6, pi/2 ]
-        sage: ys = map(sin, xs)
+        sage: ys = list(map(sin, xs))
         sage: L = lagrange_polynomial(x, xs, ys)
         sage: N = newton_polynomial(x, xs, ys)
         sage: bool(N == L)
         True
 
     """
-    degree = len(xs) - 1
-
-    N = SR(0)
-
-    for k in range(0, degree+1):
-        term  = divided_difference(xs[:k+1], ys[:k+1])
-        term *= lagrange_psi(x, xs[:k])
-        N += term
-
-    return N
+    return sum( divided_difference(xs[:k+1], ys[:k+1])*lagrange_psi(x, xs[:k])
+                for k in range(len(xs)) )
 
 
 def hermite_coefficient(k, x, xs):
@@ -335,8 +325,8 @@ def hermite_interpolant(x, xs, ys, y_primes):
     TESTS::
 
         sage: xs = [ 0, pi/6, pi/2 ]
-        sage: ys = map(sin, xs)
-        sage: y_primes = map(cos, xs)
+        sage: ys = list(map(sin, xs))
+        sage: y_primes = list(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
@@ -346,11 +336,11 @@ def hermite_interpolant(x, xs, ys, y_primes):
         True
 
     """
-    s1 = sum([ ys[k] * hermite_coefficient(k, x, xs)
-               for k in range(0, len(xs)) ])
+    s1 = sum( ys[k] * hermite_coefficient(k, x, xs)
+               for k in range(len(xs)) )
 
-    s2 = sum([ y_primes[k] * hermite_deriv_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(len(xs)) )
 
     return (s1 + s2)
 
@@ -375,4 +365,4 @@ def lagrange_psi(x, xs):
 
     """
 
-    return product([ (x - xj) for xj in xs ])
+    return product( (x - xj) for xj in xs )