mjo/interpolation.py

   1 from sage.all import *
   2 product = prod
   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 lagrange_interpolate(f, x, xs):
  98     """
  99     Interpolate the function ``f`` at the points ``xs`` using the
 100     Lagrange form of the interpolating polynomial.
 101
 102     INPUT:
 103
 104       - ``f`` -- The function to interpolate.
 105
 106       - ``x`` -- The independent variable of the resulting polynomial.
 107
 108       - ``xs`` -- A list of points at which to interpolate ``f``.
 109
 110     OUTPUT:
 111
 112     A polynomial in ``x`` which interpolates ``f`` at ``xs``.
 113
 114     EXAMPLES:
 115
 116     We're exact on polynomials of degree `n` if we use `n+1` points::
 117
 118         sage: t = SR.symbol('t', domain='real')
 119         sage: lagrange_interpolate(x^2, t, [-1,0,1]).simplify_rational()
 120         t^2
 121
 122     """
 123     # f should be a function of one variable.
 124     z = f.variables()[0]
 125     # We're really just doing map(f, xs) here; the additional
 126     # gymnastics are to avoid a warning when calling `f` with an
 127     # unnamed argument.
 128     ys = [ f({z: xk}) for xk in xs ]
 129     return lagrange_polynomial(x, xs, ys)
 130
 131
 132
 133 def divided_difference_coefficients(xs):
 134     """
 135     Assuming some function `f`, compute the coefficients of the
 136     divided difference f[xs[0], ..., xs[n]].
 137
 138     TESTS:
 139
 140         sage: divided_difference_coefficients([0])
 141         [1]
 142         sage: divided_difference_coefficients([0, pi])
 143         [-1/pi, 1/pi]
 144         sage: divided_difference_coefficients([0, pi, 2*pi])
 145         [1/2/pi^2, -1/pi^2, 1/2/pi^2]
 146
 147     """
 148     coeffs = [ QQ(1)/lagrange_denominator(k, xs) for k in range(0, len(xs)) ]
 149     return coeffs
 150
 151
 152 def divided_difference(xs, ys):
 153     """
 154     Return the Newton divided difference of the points (xs[k],
 155     ys[k]). Reference:
 156
 157       http://en.wikipedia.org/wiki/Divided_differences
 158
 159     INPUT:
 160
 161       - ``xs`` -- The list of x-values.
 162
 163       - ``ys`` -- The function values at `xs`.
 164
 165     OUTPUT:
 166
 167     The (possibly symbolic) divided difference function.
 168
 169     TESTS::
 170
 171         sage: xs = [0]
 172         sage: ys = map(sin, xs)
 173         sage: divided_difference(xs, ys)
 174         0
 175         sage: xs = [0, pi]
 176         sage: ys = map(sin, xs)
 177         sage: divided_difference(xs, ys)
 178         0
 179         sage: xs = [0, pi, 2*pi]
 180         sage: ys = map(sin, xs)
 181         sage: divided_difference(xs, ys)
 182         0
 183
 184     We try something entirely symbolic::
 185
 186         sage: f = function('f', x)
 187         sage: divided_difference([x], [f(x=x)])
 188         f(x)
 189         sage: x1,x2 = SR.var('x1,x2')
 190         sage: divided_difference([x1,x2], [f(x=x1),f(x=x2)])
 191         f(x1)/(x1 - x2) - f(x2)/(x1 - x2)
 192
 193     """
 194     coeffs = divided_difference_coefficients(xs)
 195     v_cs = vector(coeffs)
 196     v_ys = vector(ys)
 197     return v_cs.dot_product(v_ys)
 198
 199
 200 def newton_polynomial(x, xs, ys):
 201     """
 202     Return the Newton form of the interpolating polynomial of the
 203     points (xs[k], ys[k]) in the variable `x`.
 204
 205     INPUT:
 206
 207       - ``x`` -- The independent variable to use for the interpolating
 208         polynomial.
 209
 210       - ``xs`` -- The list of x-values.
 211
 212       - ``ys`` -- The function values at `xs`.
 213
 214     OUTPUT:
 215
 216     A symbolic expression.
 217
 218     TESTS:
 219
 220         sage: xs = [ -pi/2, -pi/6, 0, pi/6, pi/2 ]
 221         sage: ys = map(sin, xs)
 222         sage: L = lagrange_polynomial(x, xs, ys)
 223         sage: N = newton_polynomial(x, xs, ys)
 224         sage: bool(N == L)
 225         True
 226
 227     """
 228     degree = len(xs) - 1
 229
 230     N = SR(0)
 231
 232     for k in range(0, degree+1):
 233         term  = divided_difference(xs[:k+1], ys[:k+1])
 234         term *= lagrange_psi(x, xs[:k])
 235         N += term
 236
 237     return N
 238
 239
 240 def hermite_coefficient(k, x, xs):
 241     """
 242     Return the Hermite coefficient h_{k}(x). See Atkinson, p. 160.
 243
 244     INPUT:
 245
 246       - ``k`` -- The index of the coefficient.
 247
 248       - ``x`` -- The symbolic variable to use as the argument of h_{k}.
 249
 250       - ``xs`` -- The list of points at which the function values are
 251         known.
 252
 253     OUTPUT:
 254
 255     A symbolic expression.
 256
 257     """
 258     lk = lagrange_coefficient(k, x, xs)
 259     return (1 - 2*lk.diff(x)(x=xs[k])*(x - xs[k]))*(lk**2)
 260
 261
 262 def hermite_deriv_coefficient(k, x, xs):
 263     """
 264     Return the Hermite derivative coefficient, \tilde{h}_{k}(x). See
 265     Atkinson, p. 160.
 266
 267     INPUT:
 268
 269       - ``k`` -- The index of the coefficient.
 270
 271       - ``x`` -- The symbolic variable to use as the argument of h_{k}.
 272
 273       - ``xs`` -- The list of points at which the function values are
 274         known.
 275
 276     OUTPUT:
 277
 278     A symbolic expression.
 279
 280     """
 281     lk = lagrange_coefficient(k, x, xs)
 282     return (x - xs[k])*(lk**2)
 283
 284
 285 def hermite_interpolant(x, xs, ys, y_primes):
 286     """
 287     Return the Hermite interpolant `H(x)` such that H(xs[k]) = ys[k]
 288     and H'(xs[k]) = y_primes[k] for each k.
 289
 290     Reference: Atkinson, p. 160.
 291
 292     INPUT:
 293
 294       - ``x`` -- The symbolic variable to use as the argument of H(x).
 295
 296       - ``xs`` -- The list of points at which the function values are
 297         known.
 298
 299       - ``ys`` -- The function values at the `xs`.
 300
 301       - ``y_primes`` -- The derivatives at the `xs`.
 302
 303     OUTPUT:
 304
 305     A symbolic expression.
 306
 307     TESTS:
 308
 309         sage: xs = [ 0, pi/6, pi/2 ]
 310         sage: ys = map(sin, xs)
 311         sage: y_primes = map(cos, xs)
 312         sage: H = hermite_interpolant(x, xs, ys, y_primes)
 313         sage: expected  = -27/4*(pi - 6*x)*(pi - 2*x)^2*sqrt(3)*x^2/pi^4
 314         sage: expected += (5*(pi - 2*x)/pi + 1)*(pi - 6*x)^2*x^2/pi^4
 315         sage: expected += 81/2*((pi - 6*x)/pi + 1)*(pi - 2*x)^2*x^2/pi^4
 316         sage: expected += (pi - 6*x)^2*(pi - 2*x)^2*x/pi^4
 317         sage: bool(H == expected)
 318         True
 319
 320     """
 321     s1 = sum([ ys[k] * hermite_coefficient(k, x, xs)
 322                for k in range(0, len(xs)) ])
 323
 324     s2 = sum([ y_primes[k] * hermite_deriv_coefficient(k, x, xs)
 325                for k in range(0, len(xs)) ])
 326
 327     return (s1 + s2)
 328
 329
 330 def lagrange_psi(x, xs):
 331     """
 332     The function,
 333
 334       Psi(x) = (x - xs[0])*(x - xs[1])* ... *(x - xs[-1])
 335
 336     used in Lagrange and Hermite interpolation.
 337
 338     INPUT:
 339
 340       - ``x`` -- The independent variable of the resulting expression.
 341
 342       - ``xs`` -- A list of points.
 343
 344     OUTPUT:
 345
 346     A symbolic expression in one variable, `x`.
 347
 348     """
 349
 350     return product([ (x - xj) for xj in xs ])