eja: add another non-element subalgebra example.

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

diff --git a/mjo/interpolation.py b/mjo/interpolation.py
index 5c8fa288de43c0ffa8b580cece5a0566945f88b9..d3b406a56d0b9c50fbb1ffd48b4ff454ee5b6a03 100644 (file)
--- 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 xrange(len(xs)) if j != k )
 
 
 def lagrange_coefficient(k, x, xs):
@@ -44,6 +44,10 @@ 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 ]
@@ -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 ]
@@ -88,9 +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 xrange(len(xs)) ]
+    return sum( ys[k] * ls[k] for k in xrange(len(xs)) )
 
 
 
@@ -111,6 +118,10 @@ 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 +146,11 @@ 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]
@@ -145,8 +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 xrange(len(xs)) ]
 
 
 def divided_difference(xs, ys):
@@ -166,6 +180,10 @@ 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 +201,7 @@ 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 +233,11 @@ 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)
@@ -225,16 +247,8 @@ def newton_polynomial(x, xs, ys):
         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 xrange(len(xs)) )
 
 
 def hermite_coefficient(k, x, xs):
@@ -304,7 +318,11 @@ 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)
@@ -318,11 +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 xrange(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 xrange(len(xs)) )
 
     return (s1 + s2)
 
@@ -347,4 +365,4 @@ def lagrange_psi(x, xs):
 
     """
 
-    return product([ (x - xj) for xj in xs ])
+    return product( (x - xj) for xj in xs )