mjo/interpolation.py

   1 from sage.all import *
   2 from misc import product
   3
   4
   5 def lagrange_denominator(k, xs):
   6     """
   7     Return the denominator of the kth Lagrange coefficient.
   8
   9     INPUT:
  10
  11       - ``k`` -- The index of the coefficient.
  12
  13       - ``xs`` -- The list of points at which the function values are
  14         known.
  15
  16     OUTPUT:
  17
  18     The product of all xs[j] with j != k.
  19
  20     """
  21     return product([xs[k] - xs[j] for j in range(0, len(xs)) if j != k])
  22
  23
  24 def lagrange_coefficient(k, x, xs):
  25     """
  26     Returns the coefficient function l_{k}(variable) of y_{k} in the
  27     Lagrange polynomial of f. See,
  28
  29       http://en.wikipedia.org/wiki/Lagrange_polynomial
  30
  31     for more information.
  32
  33     INPUT:
  34
  35       - ``k`` -- The index of the coefficient.
  36
  37       - ``x`` -- The symbolic variable to use for the first argument
  38         of l_{k}.
  39
  40       - ``xs`` -- The list of points at which the function values are
  41         known.
  42
  43     OUTPUT:
  44
  45     A symbolic expression of one variable.
  46
  47     TESTS::
  48
  49         sage: xs = [ -pi/2, -pi/6, 0, pi/6, pi/2 ]
  50         sage: lagrange_coefficient(0, x, xs)
  51         1/8*(pi - 6*x)*(pi - 2*x)*(pi + 6*x)*x/pi^4
  52
  53     """
  54     numerator = lagrange_psi(x, xs)/(x - xs[k])
  55     denominator = lagrange_denominator(k, xs)
  56
  57     return (numerator / denominator)
  58
  59
  60
  61 def lagrange_polynomial(x, xs, ys):
  62     """
  63     Return the Lagrange form of the interpolation polynomial in `x` of
  64     at the points (xs[k], ys[k]).
  65
  66     INPUT:
  67
  68       - ``x`` - The independent variable of the resulting polynomial.
  69
  70       - ``xs`` - The list of points at which we interpolate `f`.
  71
  72       - ``ys`` - The function values at `xs`.
  73
  74     OUTPUT:
  75
  76     A symbolic expression (polynomial) interpolating each (xs[k], ys[k]).
  77
  78     TESTS::
  79
  80         sage: xs = [ -pi/2, -pi/6, 0, pi/6, pi/2 ]
  81         sage: ys = map(sin, xs)
  82         sage: L = lagrange_polynomial(x, xs, ys)
  83         sage: expected  = 27/16*(pi - 6*x)*(pi - 2*x)*(pi + 2*x)*x/pi^4
  84         sage: expected -= 1/8*(pi - 6*x)*(pi - 2*x)*(pi + 6*x)*x/pi^4
  85         sage: expected -= 1/8*(pi - 6*x)*(pi + 2*x)*(pi + 6*x)*x/pi^4
  86         sage: expected += 27/16*(pi - 2*x)*(pi + 2*x)*(pi + 6*x)*x/pi^4
  87         sage: bool(L == expected)
  88         True
  89
  90     """
  91     ls = [ lagrange_coefficient(k, x, xs) for k in range(0, len(xs)) ]
  92     sigma = sum([ ys[k] * ls[k] for k in range(0, len(xs)) ])
  93     return sigma
  94
  95
  96
  97 def divided_difference_coefficients(xs):
  98     """
  99     Assuming some function `f`, compute the coefficients of the
 100     divided difference f[xs[0], ..., xs[n]].
 101
 102     TESTS:
 103
 104         sage: divided_difference_coefficients([0])
 105         [1]
 106         sage: divided_difference_coefficients([0, pi])
 107         [-1/pi, 1/pi]
 108         sage: divided_difference_coefficients([0, pi, 2*pi])
 109         [1/2/pi^2, -1/pi^2, 1/2/pi^2]
 110
 111     """
 112     coeffs = [ QQ(1)/lagrange_denominator(k, xs) for k in range(0, len(xs)) ]
 113     return coeffs
 114
 115
 116 def divided_difference(xs, ys):
 117     """
 118     Return the Newton divided difference of the points (xs[k],
 119     ys[k]). Reference:
 120
 121       http://en.wikipedia.org/wiki/Divided_differences
 122
 123     INPUT:
 124
 125       - ``xs`` -- The list of x-values.
 126
 127       - ``ys`` -- The function values at `xs`.
 128
 129     OUTPUT:
 130
 131     The (possibly symbolic) divided difference function.
 132
 133     TESTS::
 134
 135         sage: xs = [0]
 136         sage: ys = map(sin, xs)
 137         sage: divided_difference(xs, ys)
 138         0
 139         sage: xs = [0, pi]
 140         sage: ys = map(sin, xs)
 141         sage: divided_difference(xs, ys)
 142         0
 143         sage: xs = [0, pi, 2*pi]
 144         sage: ys = map(sin, xs)
 145         sage: divided_difference(xs, ys)
 146         0
 147
 148     We try something entirely symbolic::
 149
 150         sage: f = function('f', x)
 151         sage: divided_difference([x], [f(x=x)])
 152         f(x)
 153         sage: x1,x2 = var('x1,x2')
 154         sage: divided_difference([x1,x2], [f(x=x1),f(x=x2)])
 155         f(x1)/(x1 - x2) - f(x2)/(x1 - x2)
 156
 157     """
 158     coeffs = divided_difference_coefficients(xs)
 159     v_cs = vector(coeffs)
 160     v_ys = vector(ys)
 161     return v_cs.dot_product(v_ys)
 162
 163
 164 def newton_polynomial(x, xs, ys):
 165     """
 166     Return the Newton form of the interpolating polynomial of the
 167     points (xs[k], ys[k]) in the variable `x`.
 168
 169     INPUT:
 170
 171       - ``x`` -- The independent variable to use for the interpolating
 172         polynomial.
 173
 174       - ``xs`` -- The list of x-values.
 175
 176       - ``ys`` -- The function values at `xs`.
 177
 178     OUTPUT:
 179
 180     A symbolic expression.
 181
 182     TESTS:
 183
 184         sage: xs = [ -pi/2, -pi/6, 0, pi/6, pi/2 ]
 185         sage: ys = map(sin, xs)
 186         sage: L = lagrange_polynomial(x, xs, ys)
 187         sage: N = newton_polynomial(x, xs, ys)
 188         sage: bool(N == L)
 189         True
 190
 191     """
 192     degree = len(xs) - 1
 193
 194     N = SR(0)
 195
 196     for k in range(0, degree+1):
 197         term  = divided_difference(xs[:k+1], ys[:k+1])
 198         term *= lagrange_psi(x, xs[:k])
 199         N += term
 200
 201     return N
 202
 203
 204 def hermite_coefficient(k, x, xs):
 205     """
 206     Return the Hermite coefficient h_{k}(x). See Atkinson, p. 160.
 207
 208     INPUT:
 209
 210       - ``k`` -- The index of the coefficient.
 211
 212       - ``x`` -- The symbolic variable to use as the argument of h_{k}.
 213
 214       - ``xs`` -- The list of points at which the function values are
 215         known.
 216
 217     OUTPUT:
 218
 219     A symbolic expression.
 220
 221     """
 222     lk = lagrange_coefficient(k, x, xs)
 223     return (1 - 2*lk.diff(x)(x=xs[k])*(x - xs[k]))*(lk**2)
 224
 225
 226 def hermite_deriv_coefficient(k, x, xs):
 227     """
 228     Return the Hermite derivative coefficient, \tilde{h}_{k}(x). See
 229     Atkinson, p. 160.
 230
 231     INPUT:
 232
 233       - ``k`` -- The index of the coefficient.
 234
 235       - ``x`` -- The symbolic variable to use as the argument of h_{k}.
 236
 237       - ``xs`` -- The list of points at which the function values are
 238         known.
 239
 240     OUTPUT:
 241
 242     A symbolic expression.
 243
 244     """
 245     lk = lagrange_coefficient(k, x, xs)
 246     return (x - xs[k])*(lk**2)
 247
 248
 249 def hermite_interpolant(x, xs, ys, y_primes):
 250     """
 251     Return the Hermite interpolant `H(x)` such that H(xs[k]) = ys[k]
 252     and H'(xs[k]) = y_primes[k] for each k.
 253
 254     Reference: Atkinson, p. 160.
 255
 256     INPUT:
 257
 258       - ``x`` -- The symbolic variable to use as the argument of H(x).
 259
 260       - ``xs`` -- The list of points at which the function values are
 261         known.
 262
 263       - ``ys`` -- The function values at the `xs`.
 264
 265       - ``y_primes`` -- The derivatives at the `xs`.
 266
 267     OUTPUT:
 268
 269     A symbolic expression.
 270
 271     TESTS:
 272
 273         sage: xs = [ 0, pi/6, pi/2 ]
 274         sage: ys = map(sin, xs)
 275         sage: y_primes = map(cos, xs)
 276         sage: H = hermite_interpolant(x, xs, ys, y_primes)
 277         sage: expected  = -27/4*(pi - 6*x)*(pi - 2*x)^2*sqrt(3)*x^2/pi^4
 278         sage: expected += (5*(pi - 2*x)/pi + 1)*(pi - 6*x)^2*x^2/pi^4
 279         sage: expected += 81/2*((pi - 6*x)/pi + 1)*(pi - 2*x)^2*x^2/pi^4
 280         sage: expected += (pi - 6*x)^2*(pi - 2*x)^2*x/pi^4
 281         sage: bool(H == expected)
 282         True
 283
 284     """
 285     s1 = sum([ ys[k] * hermite_coefficient(k, x, xs)
 286                for k in range(0, len(xs)) ])
 287
 288     s2 = sum([ y_primes[k] * hermite_deriv_coefficient(k, x, xs)
 289                for k in range(0, len(xs)) ])
 290
 291     return (s1 + s2)
 292
 293
 294 def lagrange_psi(x, xs):
 295     """
 296     The function,
 297
 298       Psi(x) = (x - xs[0])*(x - xs[1])* ... *(x - xs[-1])
 299
 300     used in Lagrange and Hermite interpolation.
 301
 302     INPUT:
 303
 304       - ``x`` -- The independent variable of the resulting expression.
 305
 306       - ``xs`` -- A list of points.
 307
 308     OUTPUT:
 309
 310     A symbolic expression in one variable, `x`.
 311
 312     """
 313
 314     return product([ (x - xj) for xj in xs ])