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 xrange(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     SETUP::
  48
  49         sage: from mjo.interpolation import lagrange_coefficient
  50
  51     TESTS::
  52
  53         sage: xs = [ -pi/2, -pi/6, 0, pi/6, pi/2 ]
  54         sage: lagrange_coefficient(0, x, xs)
  55         1/8*(pi + 6*x)*(pi - 2*x)*(pi - 6*x)*x/pi^4
  56
  57     """
  58     numerator = lagrange_psi(x, xs)/(x - xs[k])
  59     denominator = lagrange_denominator(k, xs)
  60
  61     return (numerator / denominator)
  62
  63
  64
  65 def lagrange_polynomial(x, xs, ys):
  66     """
  67     Return the Lagrange form of the interpolating polynomial in `x`
  68     at the points (xs[k], ys[k]).
  69
  70     INPUT:
  71
  72       - ``x`` - The independent variable of the resulting polynomial.
  73
  74       - ``xs`` - The list of points at which we interpolate `f`.
  75
  76       - ``ys`` - The function values at `xs`.
  77
  78     OUTPUT:
  79
  80     A symbolic expression (polynomial) interpolating each (xs[k], ys[k]).
  81
  82     SETUP::
  83
  84         sage: from mjo.interpolation import lagrange_polynomial
  85
  86     TESTS::
  87
  88         sage: xs = [ -pi/2, -pi/6, 0, pi/6, pi/2 ]
  89         sage: ys = map(sin, xs)
  90         sage: L = lagrange_polynomial(x, xs, ys)
  91         sage: expected  = 27/16*(pi - 6*x)*(pi - 2*x)*(pi + 2*x)*x/pi^4
  92         sage: expected -= 1/8*(pi - 6*x)*(pi - 2*x)*(pi + 6*x)*x/pi^4
  93         sage: expected -= 1/8*(pi - 6*x)*(pi + 2*x)*(pi + 6*x)*x/pi^4
  94         sage: expected += 27/16*(pi - 2*x)*(pi + 2*x)*(pi + 6*x)*x/pi^4
  95         sage: bool(L == expected)
  96         True
  97
  98     """
  99     ls = [ lagrange_coefficient(k, x, xs) for k in xrange(len(xs)) ]
 100     return sum( ys[k] * ls[k] for k in xrange(len(xs)) )
 101
 102
 103
 104 def lagrange_interpolate(f, x, xs):
 105     """
 106     Interpolate the function ``f`` at the points ``xs`` using the
 107     Lagrange form of the interpolating polynomial.
 108
 109     INPUT:
 110
 111       - ``f`` -- The function to interpolate.
 112
 113       - ``x`` -- The independent variable of the resulting polynomial.
 114
 115       - ``xs`` -- A list of points at which to interpolate ``f``.
 116
 117     OUTPUT:
 118
 119     A polynomial in ``x`` which interpolates ``f`` at ``xs``.
 120
 121     SETUP::
 122
 123         sage: from mjo.interpolation import lagrange_interpolate
 124
 125     EXAMPLES:
 126
 127     We're exact on polynomials of degree `n` if we use `n+1` points::
 128
 129         sage: t = SR.symbol('t', domain='real')
 130         sage: lagrange_interpolate(x^2, t, [-1,0,1]).simplify_rational()
 131         t^2
 132
 133     """
 134     # f should be a function of one variable.
 135     z = f.variables()[0]
 136     # We're really just doing map(f, xs) here; the additional
 137     # gymnastics are to avoid a warning when calling `f` with an
 138     # unnamed argument.
 139     ys = [ f({z: xk}) for xk in xs ]
 140     return lagrange_polynomial(x, xs, ys)
 141
 142
 143
 144 def divided_difference_coefficients(xs):
 145     """
 146     Assuming some function `f`, compute the coefficients of the
 147     divided difference f[xs[0], ..., xs[n]].
 148
 149     SETUP::
 150
 151         sage: from mjo.interpolation import divided_difference_coefficients
 152
 153     TESTS::
 154
 155         sage: divided_difference_coefficients([0])
 156         [1]
 157         sage: divided_difference_coefficients([0, pi])
 158         [-1/pi, 1/pi]
 159         sage: divided_difference_coefficients([0, pi, 2*pi])
 160         [1/2/pi^2, -1/pi^2, 1/2/pi^2]
 161
 162     """
 163     return [ ~lagrange_denominator(k, xs) for k in xrange(len(xs)) ]
 164
 165
 166 def divided_difference(xs, ys):
 167     """
 168     Return the Newton divided difference of the points (xs[k],
 169     ys[k]). Reference:
 170
 171       http://en.wikipedia.org/wiki/Divided_differences
 172
 173     INPUT:
 174
 175       - ``xs`` -- The list of x-values.
 176
 177       - ``ys`` -- The function values at `xs`.
 178
 179     OUTPUT:
 180
 181     The (possibly symbolic) divided difference function.
 182
 183     SETUP::
 184
 185         sage: from mjo.interpolation import divided_difference
 186
 187     TESTS::
 188
 189         sage: xs = [0]
 190         sage: ys = map(sin, xs)
 191         sage: divided_difference(xs, ys)
 192         0
 193         sage: xs = [0, pi]
 194         sage: ys = map(sin, xs)
 195         sage: divided_difference(xs, ys)
 196         0
 197         sage: xs = [0, pi, 2*pi]
 198         sage: ys = map(sin, xs)
 199         sage: divided_difference(xs, ys)
 200         0
 201
 202     We try something entirely symbolic::
 203
 204         sage: f = function('f')(x)
 205         sage: divided_difference([x], [f(x=x)])
 206         f(x)
 207         sage: x1,x2 = SR.var('x1,x2')
 208         sage: divided_difference([x1,x2], [f(x=x1),f(x=x2)])
 209         f(x1)/(x1 - x2) - f(x2)/(x1 - x2)
 210
 211     """
 212     coeffs = divided_difference_coefficients(xs)
 213     v_cs = vector(coeffs)
 214     v_ys = vector(ys)
 215     return v_cs.dot_product(v_ys)
 216
 217
 218 def newton_polynomial(x, xs, ys):
 219     """
 220     Return the Newton form of the interpolating polynomial of the
 221     points (xs[k], ys[k]) in the variable `x`.
 222
 223     INPUT:
 224
 225       - ``x`` -- The independent variable to use for the interpolating
 226         polynomial.
 227
 228       - ``xs`` -- The list of x-values.
 229
 230       - ``ys`` -- The function values at `xs`.
 231
 232     OUTPUT:
 233
 234     A symbolic expression.
 235
 236     SETUP::
 237
 238         sage: from mjo.interpolation import lagrange_polynomial, newton_polynomial
 239
 240     TESTS::
 241
 242         sage: xs = [ -pi/2, -pi/6, 0, pi/6, pi/2 ]
 243         sage: ys = map(sin, xs)
 244         sage: L = lagrange_polynomial(x, xs, ys)
 245         sage: N = newton_polynomial(x, xs, ys)
 246         sage: bool(N == L)
 247         True
 248
 249     """
 250     return sum( divided_difference(xs[:k+1], ys[:k+1])*lagrange_psi(x, xs[:k])
 251                 for k in xrange(len(xs)) )
 252
 253
 254 def hermite_coefficient(k, x, xs):
 255     """
 256     Return the Hermite coefficient h_{k}(x). See Atkinson, p. 160.
 257
 258     INPUT:
 259
 260       - ``k`` -- The index of the coefficient.
 261
 262       - ``x`` -- The symbolic variable to use as the argument of h_{k}.
 263
 264       - ``xs`` -- The list of points at which the function values are
 265         known.
 266
 267     OUTPUT:
 268
 269     A symbolic expression.
 270
 271     """
 272     lk = lagrange_coefficient(k, x, xs)
 273     return (1 - 2*lk.diff(x)(x=xs[k])*(x - xs[k]))*(lk**2)
 274
 275
 276 def hermite_deriv_coefficient(k, x, xs):
 277     """
 278     Return the Hermite derivative coefficient, \tilde{h}_{k}(x). See
 279     Atkinson, p. 160.
 280
 281     INPUT:
 282
 283       - ``k`` -- The index of the coefficient.
 284
 285       - ``x`` -- The symbolic variable to use as the argument of h_{k}.
 286
 287       - ``xs`` -- The list of points at which the function values are
 288         known.
 289
 290     OUTPUT:
 291
 292     A symbolic expression.
 293
 294     """
 295     lk = lagrange_coefficient(k, x, xs)
 296     return (x - xs[k])*(lk**2)
 297
 298
 299 def hermite_interpolant(x, xs, ys, y_primes):
 300     """
 301     Return the Hermite interpolant `H(x)` such that H(xs[k]) = ys[k]
 302     and H'(xs[k]) = y_primes[k] for each k.
 303
 304     Reference: Atkinson, p. 160.
 305
 306     INPUT:
 307
 308       - ``x`` -- The symbolic variable to use as the argument of H(x).
 309
 310       - ``xs`` -- The list of points at which the function values are
 311         known.
 312
 313       - ``ys`` -- The function values at the `xs`.
 314
 315       - ``y_primes`` -- The derivatives at the `xs`.
 316
 317     OUTPUT:
 318
 319     A symbolic expression.
 320
 321     SETUP::
 322
 323         sage: from mjo.interpolation import hermite_interpolant
 324
 325     TESTS::
 326
 327         sage: xs = [ 0, pi/6, pi/2 ]
 328         sage: ys = map(sin, xs)
 329         sage: y_primes = map(cos, xs)
 330         sage: H = hermite_interpolant(x, xs, ys, y_primes)
 331         sage: expected  = -27/4*(pi - 6*x)*(pi - 2*x)^2*sqrt(3)*x^2/pi^4
 332         sage: expected += (5*(pi - 2*x)/pi + 1)*(pi - 6*x)^2*x^2/pi^4
 333         sage: expected += 81/2*((pi - 6*x)/pi + 1)*(pi - 2*x)^2*x^2/pi^4
 334         sage: expected += (pi - 6*x)^2*(pi - 2*x)^2*x/pi^4
 335         sage: bool(H == expected)
 336         True
 337
 338     """
 339     s1 = sum( ys[k] * hermite_coefficient(k, x, xs)
 340                for k in xrange(len(xs)) )
 341
 342     s2 = sum( y_primes[k] * hermite_deriv_coefficient(k, x, xs)
 343                for k in xrange(len(xs)) )
 344
 345     return (s1 + s2)
 346
 347
 348 def lagrange_psi(x, xs):
 349     """
 350     The function,
 351
 352       Psi(x) = (x - xs[0])*(x - xs[1])* ... *(x - xs[-1])
 353
 354     used in Lagrange and Hermite interpolation.
 355
 356     INPUT:
 357
 358       - ``x`` -- The independent variable of the resulting expression.
 359
 360       - ``xs`` -- A list of points.
 361
 362     OUTPUT:
 363
 364     A symbolic expression in one variable, `x`.
 365
 366     """
 367
 368     return product( (x - xj) for xj in xs )