Add a note about the nonstandard name "Motzkin decomposition."

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

diff --git a/mjo/interpolation.py b/mjo/interpolation.py
index 70c35c1a3ac9e812865185c71b1f4d2cc88be1ed..4be95e2463b7d4b1592a5f4e6cf86ba54e36da7f 100644 (file)
--- a/mjo/interpolation.py
+++ b/mjo/interpolation.py
@@ -1,5 +1,25 @@
 from sage.all import *
-from misc import product
+product = prod
+
+
+def lagrange_denominator(k, xs):
+    """
+    Return the denominator of the kth Lagrange coefficient.
+
+    INPUT:
+
+      - ``k`` -- The index of the coefficient.
+
+      - ``xs`` -- The list of points at which the function values are
+        known.
+
+    OUTPUT:
+
+    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])
+
 
 def lagrange_coefficient(k, x, xs):
     """
@@ -12,9 +32,9 @@ def lagrange_coefficient(k, x, xs):
 
     INPUT:
 
-      - ``k`` -- the index of the coefficient.
+      - ``k`` -- The index of the coefficient.
 
-      - ``x`` -- the symbolic variable to use for the first argument
+      - ``x`` -- The symbolic variable to use for the first argument
         of l_{k}.
 
       - ``xs`` -- The list of points at which the function values are
@@ -28,11 +48,11 @@ def lagrange_coefficient(k, x, xs):
 
         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 = 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])
+    numerator = lagrange_psi(x, xs)/(x - xs[k])
+    denominator = lagrange_denominator(k, xs)
 
     return (numerator / denominator)
 
@@ -40,7 +60,7 @@ 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:
@@ -74,6 +94,42 @@ 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``.
+
+    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
@@ -89,10 +145,10 @@ def divided_difference_coefficients(xs):
         [1/2/pi^2, -1/pi^2, 1/2/pi^2]
 
     """
-    coeffs = [ product([ (QQ(1) / (xj - xi)) for xi in xs if xi != xj ])
-               for xj in xs ]
+    coeffs = [ QQ(1)/lagrange_denominator(k, xs) for k in range(0, len(xs)) ]
     return coeffs
 
+
 def divided_difference(xs, ys):
     """
     Return the Newton divided difference of the points (xs[k],
@@ -127,10 +183,10 @@ 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)
 
@@ -175,7 +231,7 @@ def newton_polynomial(x, xs, ys):
 
     for k in range(0, degree+1):
         term  = divided_difference(xs[:k+1], ys[:k+1])
-        term *= product([ x - xk for xk in xs[:k]])
+        term *= lagrange_psi(x, xs[:k])
         N += term
 
     return N
@@ -269,3 +325,26 @@ def hermite_interpolant(x, xs, ys, y_primes):
                for k in range(0, len(xs)) ])
 
     return (s1 + s2)
+
+
+def lagrange_psi(x, xs):
+    """
+    The function,
+
+      Psi(x) = (x - xs[0])*(x - xs[1])* ... *(x - xs[-1])
+
+    used in Lagrange and Hermite interpolation.
+
+    INPUT:
+
+      - ``x`` -- The independent variable of the resulting expression.
+
+      - ``xs`` -- A list of points.
+
+    OUTPUT:
+
+    A symbolic expression in one variable, `x`.
+
+    """
+
+    return product([ (x - xj) for xj in xs ])