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
  74
  75
  76 def divided_difference(f, xs):
  77     """
  78     Return the Newton divided difference of `f` at the points
  79     `xs`. Reference:
  80
  81       http://en.wikipedia.org/wiki/Divided_differences
  82
  83     INPUT:
  84
  85       - ``f`` -- The function whose divided difference we seek.
  86
  87       - ``xs`` -- The list of points at which to compute `f`.
  88
  89     OUTPUT:
  90
  91     The divided difference of `f` at ``xs``.
  92
  93     TESTS::
  94
  95         sage: divided_difference(sin, [0])
  96         0
  97         sage: divided_difference(sin, [0, pi])
  98         0
  99         sage: divided_difference(sin, [0, pi, 2*pi])
 100         0
 101
 102     We try something entirely symbolic::
 103
 104         sage: f = function('f', x)
 105         sage: divided_difference(f, [x])
 106         f(x)
 107         sage: x1,x2 = var('x1,x2')
 108         sage: divided_difference(f, [x1,x2])
 109         (f(x1) - f(x2))/(x1 - x2)
 110
 111     """
 112     if (len(xs) == 1):
 113         # Avoid that goddamned DeprecationWarning when we have a named
 114         # argument but don't know what it is.
 115         if len(f.variables()) == 0:
 116             return f(xs[0])
 117         else:
 118             v = f.variables()[0]
 119             return f({ v: xs[0] })
 120
 121     # Use the recursive definition.
 122     numerator  = divided_difference(f, xs[1:])
 123     numerator -= divided_difference(f, xs[:-1])
 124     return numerator / (xs[-1] - xs[0])
 125
 126
 127 def newton_polynomial(f, x, xs):
 128     """
 129     Return the Newton form of the interpolating polynomial of `f` at
 130     the points `xs` in the variable `x`.
 131
 132     INPUT:
 133
 134       - ``f`` -- The function to interpolate.
 135
 136       - ``x`` -- The independent variable to use for the interpolating
 137         polynomial.
 138
 139       - ``xs`` -- The list of points at which to interpolate `f`.
 140
 141     OUTPUT:
 142
 143     A symbolic function.
 144
 145     TESTS:
 146
 147         sage: xs = [ -pi/2, -pi/6, 0, pi/6, pi/2 ]
 148         sage: L = lagrange_polynomial(sin, x, xs)
 149         sage: N = newton_polynomial(sin, x, xs)
 150         sage: bool(N == L)
 151         True
 152
 153     """
 154     degree = len(xs) - 1
 155
 156     N = SR(0)
 157
 158     for k in range(0, degree+1):
 159         term  = divided_difference(f, xs[:k+1])
 160         term *= product([ x - xk for xk in xs[:k]])
 161         N += term
 162
 163     return N