mjo/interpolation.py

   1 from sage.all import *
   2 load('~/.sage/init.sage')
   3
   4 def lagrange_coefficient(k, x, xs):
   5     """
   6     Returns the coefficient function l_{k}(variable) of y_{k} in the
   7     Lagrange polynomial of f. See,
   8
   9       http://en.wikipedia.org/wiki/Lagrange_polynomial
  10
  11     for more information.
  12
  13     INPUT:
  14
  15       - ``k`` -- the index of the coefficient.
  16
  17       - ``x`` -- the symbolic variable to use for the first argument
  18         of l_{k}.
  19
  20       - ``xs`` -- The list of points at which the function values are
  21         known.
  22
  23     OUTPUT:
  24
  25     A symbolic function of one variable.
  26
  27     TESTS::
  28
  29         sage: xs = [ -pi/2, -pi/6, 0, pi/6, pi/2 ]
  30         sage: lagrange_coefficient(0, x, xs)
  31         1/8*(pi - 6*x)*(pi - 2*x)*(pi + 6*x)*x/pi^4
  32
  33     """
  34     numerator = product([x - xs[j] for j in range(0, len(xs)) if j != k])
  35     denominator = product([xs[k] - xs[j] for j in range(0, len(xs)) if j != k])
  36
  37     return (numerator / denominator)
  38
  39
  40
  41 def lagrange_polynomial(f, x, xs):
  42     """
  43     Return the Lagrange form of the interpolation polynomial in `x` of
  44     `f` at the points `xs`.
  45
  46     INPUT:
  47
  48       - ``f`` - The function to interpolate.
  49
  50       - ``x`` - The independent variable of the resulting polynomial.
  51
  52       - ``xs`` - The list of points at which we interpolate `f`.
  53
  54     OUTPUT:
  55
  56     A symbolic function (polynomial) interpolating `f` at `xs`.
  57
  58     TESTS::
  59
  60         sage: xs = [ -pi/2, -pi/6, 0, pi/6, pi/2 ]
  61         sage: L = lagrange_polynomial(sin, x, xs)
  62         sage: expected  = 27/16*(pi - 6*x)*(pi - 2*x)*(pi + 2*x)*x/pi^4
  63         sage: expected -= 1/8*(pi - 6*x)*(pi - 2*x)*(pi + 6*x)*x/pi^4
  64         sage: expected -= 1/8*(pi - 6*x)*(pi + 2*x)*(pi + 6*x)*x/pi^4
  65         sage: expected += 27/16*(pi - 2*x)*(pi + 2*x)*(pi + 6*x)*x/pi^4
  66         sage: bool(L == expected)
  67         True
  68
  69     """
  70     ys = [ f(xs[k]) for k in range(0, len(xs)) ]
  71     ls = [ lagrange_coefficient(k, x, xs) for k in range(0, len(xs)) ]
  72     sigma = sum([ ys[k] * ls[k] for k in range(0, len(xs)) ])
  73     return sigma