from sage.all import *
-load('~/.sage/init.sage')
+from misc import product
def lagrange_coefficient(k, x, xs):
"""
-def lagrange_polynomial(f, x, xs):
+def lagrange_polynomial(x, xs, ys):
"""
Return the Lagrange form of the interpolation polynomial in `x` of
- `f` at the points `xs`.
+ at the points (xs[k], ys[k]).
INPUT:
- - ``f`` - The function to interpolate.
-
- ``x`` - The independent variable of the resulting polynomial.
- ``xs`` - The list of points at which we interpolate `f`.
+ - ``ys`` - The function values at `xs`.
+
OUTPUT:
- A symbolic function (polynomial) interpolating `f` at `xs`.
+ A symbolic function (polynomial) interpolating each (xs[k], ys[k]).
TESTS::
sage: xs = [ -pi/2, -pi/6, 0, pi/6, pi/2 ]
- sage: L = lagrange_polynomial(sin, x, xs)
+ sage: ys = 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
sage: expected -= 1/8*(pi - 6*x)*(pi + 2*x)*(pi + 6*x)*x/pi^4
True
"""
- ys = [ f(xs[k]) for k in range(0, len(xs)) ]
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
-def divided_difference(f, xs):
+
+def divided_difference_coefficients(xs):
+ """
+ Assuming some function `f`, compute the coefficients of the
+ divided difference f[xs[0], ..., xs[n]].
+
+ TESTS:
+
+ sage: divided_difference_coefficients([0])
+ [1]
+ sage: divided_difference_coefficients([0, pi])
+ [-1/pi, 1/pi]
+ sage: divided_difference_coefficients([0, pi, 2*pi])
+ [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 ]
+ return coeffs
+
+def divided_difference(xs, ys):
"""
- Return the Newton divided difference of `f` at the points
- `xs`. Reference:
+ Return the Newton divided difference of the points (xs[k],
+ ys[k]). Reference:
http://en.wikipedia.org/wiki/Divided_differences
INPUT:
- - ``f`` -- The function whose divided difference we seek.
+ - ``xs`` -- The list of x-values.
- - ``xs`` -- The list of points at which to compute `f`.
+ - ``ys`` -- The function values at `xs`.
OUTPUT:
- The divided difference of `f` at ``xs``.
+ The (possibly symbolic) divided difference function.
TESTS::
- sage: divided_difference(sin, [0])
+ sage: xs = [0]
+ sage: ys = map(sin, xs)
+ sage: divided_difference(xs, ys)
0
- sage: divided_difference(sin, [0, pi])
+ sage: xs = [0, pi]
+ sage: ys = map(sin, xs)
+ sage: divided_difference(xs, ys)
0
- sage: divided_difference(sin, [0, pi, 2*pi])
+ sage: xs = [0, pi, 2*pi]
+ sage: ys = map(sin, xs)
+ sage: divided_difference(xs, ys)
0
We try something entirely symbolic::
sage: f = function('f', x)
- sage: divided_difference(f, [x])
+ sage: divided_difference([x], [f(x=x)])
f(x)
sage: x1,x2 = var('x1,x2')
- sage: divided_difference(f, [x1,x2])
- (f(x1) - f(x2))/(x1 - x2)
+ sage: divided_difference([x1,x2], [f(x=x1),f(x=x2)])
+ f(x1)/(x1 - x2) - f(x2)/(x1 - x2)
"""
- if (len(xs) == 1):
- # Avoid that goddamned DeprecationWarning when we have a named
- # argument but don't know what it is.
- if len(f.variables()) == 0:
- return f(xs[0])
- else:
- v = f.variables()[0]
- return f({ v: xs[0] })
-
- # Use the recursive definition.
- numerator = divided_difference(f, xs[1:])
- numerator -= divided_difference(f, xs[:-1])
- return numerator / (xs[-1] - xs[0])
-
-
-def newton_polynomial(f, x, xs):
+ coeffs = divided_difference_coefficients(xs)
+ v_cs = vector(coeffs)
+ v_ys = vector(ys)
+ return v_cs.dot_product(v_ys)
+
+
+def newton_polynomial(x, xs, ys):
"""
- Return the Newton form of the interpolating polynomial of `f` at
- the points `xs` in the variable `x`.
+ Return the Newton form of the interpolating polynomial of the
+ points (xs[k], ys[k]) in the variable `x`.
INPUT:
- - ``f`` -- The function to interpolate.
-
- ``x`` -- The independent variable to use for the interpolating
polynomial.
- - ``xs`` -- The list of points at which to interpolate `f`.
+ - ``xs`` -- The list of x-values.
+
+ - ``ys`` -- The function values at `xs`.
OUTPUT:
TESTS:
sage: xs = [ -pi/2, -pi/6, 0, pi/6, pi/2 ]
- sage: L = lagrange_polynomial(sin, x, xs)
- sage: N = newton_polynomial(sin, x, xs)
+ sage: ys = map(sin, xs)
+ sage: L = lagrange_polynomial(x, xs, ys)
+ sage: N = newton_polynomial(x, xs, ys)
sage: bool(N == L)
True
N = SR(0)
for k in range(0, degree+1):
- term = divided_difference(f, xs[:k+1])
+ term = divided_difference(xs[:k+1], ys[:k+1])
term *= product([ x - xk for xk in xs[:k]])
N += term