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     sigma = sum([ ys[k] * ls[k] for k in xrange(len(xs)) ])
 101     return sigma
 102
 103
 104
 105 def lagrange_interpolate(f, x, xs):
 106     """
 107     Interpolate the function ``f`` at the points ``xs`` using the
 108     Lagrange form of the interpolating polynomial.
 109
 110     INPUT:
 111
 112       - ``f`` -- The function to interpolate.
 113
 114       - ``x`` -- The independent variable of the resulting polynomial.
 115
 116       - ``xs`` -- A list of points at which to interpolate ``f``.
 117
 118     OUTPUT:
 119
 120     A polynomial in ``x`` which interpolates ``f`` at ``xs``.
 121
 122     SETUP::
 123
 124         sage: from mjo.interpolation import lagrange_interpolate
 125
 126     EXAMPLES:
 127
 128     We're exact on polynomials of degree `n` if we use `n+1` points::
 129
 130         sage: t = SR.symbol('t', domain='real')
 131         sage: lagrange_interpolate(x^2, t, [-1,0,1]).simplify_rational()
 132         t^2
 133
 134     """
 135     # f should be a function of one variable.
 136     z = f.variables()[0]
 137     # We're really just doing map(f, xs) here; the additional
 138     # gymnastics are to avoid a warning when calling `f` with an
 139     # unnamed argument.
 140     ys = [ f({z: xk}) for xk in xs ]
 141     return lagrange_polynomial(x, xs, ys)
 142
 143
 144
 145 def divided_difference_coefficients(xs):
 146     """
 147     Assuming some function `f`, compute the coefficients of the
 148     divided difference f[xs[0], ..., xs[n]].
 149
 150     SETUP::
 151
 152         sage: from mjo.interpolation import divided_difference_coefficients
 153
 154     TESTS::
 155
 156         sage: divided_difference_coefficients([0])
 157         [1]
 158         sage: divided_difference_coefficients([0, pi])
 159         [-1/pi, 1/pi]
 160         sage: divided_difference_coefficients([0, pi, 2*pi])
 161         [1/2/pi^2, -1/pi^2, 1/2/pi^2]
 162
 163     """
 164     coeffs = [ QQ(1)/lagrange_denominator(k, xs) for k in xrange(len(xs)) ]
 165     return coeffs
 166
 167
 168 def divided_difference(xs, ys):
 169     """
 170     Return the Newton divided difference of the points (xs[k],
 171     ys[k]). Reference:
 172
 173       http://en.wikipedia.org/wiki/Divided_differences
 174
 175     INPUT:
 176
 177       - ``xs`` -- The list of x-values.
 178
 179       - ``ys`` -- The function values at `xs`.
 180
 181     OUTPUT:
 182
 183     The (possibly symbolic) divided difference function.
 184
 185     SETUP::
 186
 187         sage: from mjo.interpolation import divided_difference
 188
 189     TESTS::
 190
 191         sage: xs = [0]
 192         sage: ys = map(sin, xs)
 193         sage: divided_difference(xs, ys)
 194         0
 195         sage: xs = [0, pi]
 196         sage: ys = map(sin, xs)
 197         sage: divided_difference(xs, ys)
 198         0
 199         sage: xs = [0, pi, 2*pi]
 200         sage: ys = map(sin, xs)
 201         sage: divided_difference(xs, ys)
 202         0
 203
 204     We try something entirely symbolic::
 205
 206         sage: f = function('f')(x)
 207         sage: divided_difference([x], [f(x=x)])
 208         f(x)
 209         sage: x1,x2 = SR.var('x1,x2')
 210         sage: divided_difference([x1,x2], [f(x=x1),f(x=x2)])
 211         f(x1)/(x1 - x2) - f(x2)/(x1 - x2)
 212
 213     """
 214     coeffs = divided_difference_coefficients(xs)
 215     v_cs = vector(coeffs)
 216     v_ys = vector(ys)
 217     return v_cs.dot_product(v_ys)
 218
 219
 220 def newton_polynomial(x, xs, ys):
 221     """
 222     Return the Newton form of the interpolating polynomial of the
 223     points (xs[k], ys[k]) in the variable `x`.
 224
 225     INPUT:
 226
 227       - ``x`` -- The independent variable to use for the interpolating
 228         polynomial.
 229
 230       - ``xs`` -- The list of x-values.
 231
 232       - ``ys`` -- The function values at `xs`.
 233
 234     OUTPUT:
 235
 236     A symbolic expression.
 237
 238     SETUP::
 239
 240         sage: from mjo.interpolation import lagrange_polynomial, newton_polynomial
 241
 242     TESTS::
 243
 244         sage: xs = [ -pi/2, -pi/6, 0, pi/6, pi/2 ]
 245         sage: ys = map(sin, xs)
 246         sage: L = lagrange_polynomial(x, xs, ys)
 247         sage: N = newton_polynomial(x, xs, ys)
 248         sage: bool(N == L)
 249         True
 250
 251     """
 252     degree = len(xs) - 1
 253
 254     N = SR(0)
 255
 256     for k in xrange(degree+1):
 257         term  = divided_difference(xs[:k+1], ys[:k+1])
 258         term *= lagrange_psi(x, xs[:k])
 259         N += term
 260
 261     return N
 262
 263
 264 def hermite_coefficient(k, x, xs):
 265     """
 266     Return the Hermite coefficient h_{k}(x). See Atkinson, p. 160.
 267
 268     INPUT:
 269
 270       - ``k`` -- The index of the coefficient.
 271
 272       - ``x`` -- The symbolic variable to use as the argument of h_{k}.
 273
 274       - ``xs`` -- The list of points at which the function values are
 275         known.
 276
 277     OUTPUT:
 278
 279     A symbolic expression.
 280
 281     """
 282     lk = lagrange_coefficient(k, x, xs)
 283     return (1 - 2*lk.diff(x)(x=xs[k])*(x - xs[k]))*(lk**2)
 284
 285
 286 def hermite_deriv_coefficient(k, x, xs):
 287     """
 288     Return the Hermite derivative coefficient, \tilde{h}_{k}(x). See
 289     Atkinson, p. 160.
 290
 291     INPUT:
 292
 293       - ``k`` -- The index of the coefficient.
 294
 295       - ``x`` -- The symbolic variable to use as the argument of h_{k}.
 296
 297       - ``xs`` -- The list of points at which the function values are
 298         known.
 299
 300     OUTPUT:
 301
 302     A symbolic expression.
 303
 304     """
 305     lk = lagrange_coefficient(k, x, xs)
 306     return (x - xs[k])*(lk**2)
 307
 308
 309 def hermite_interpolant(x, xs, ys, y_primes):
 310     """
 311     Return the Hermite interpolant `H(x)` such that H(xs[k]) = ys[k]
 312     and H'(xs[k]) = y_primes[k] for each k.
 313
 314     Reference: Atkinson, p. 160.
 315
 316     INPUT:
 317
 318       - ``x`` -- The symbolic variable to use as the argument of H(x).
 319
 320       - ``xs`` -- The list of points at which the function values are
 321         known.
 322
 323       - ``ys`` -- The function values at the `xs`.
 324
 325       - ``y_primes`` -- The derivatives at the `xs`.
 326
 327     OUTPUT:
 328
 329     A symbolic expression.
 330
 331     SETUP::
 332
 333         sage: from mjo.interpolation import hermite_interpolant
 334
 335     TESTS::
 336
 337         sage: xs = [ 0, pi/6, pi/2 ]
 338         sage: ys = map(sin, xs)
 339         sage: y_primes = map(cos, xs)
 340         sage: H = hermite_interpolant(x, xs, ys, y_primes)
 341         sage: expected  = -27/4*(pi - 6*x)*(pi - 2*x)^2*sqrt(3)*x^2/pi^4
 342         sage: expected += (5*(pi - 2*x)/pi + 1)*(pi - 6*x)^2*x^2/pi^4
 343         sage: expected += 81/2*((pi - 6*x)/pi + 1)*(pi - 2*x)^2*x^2/pi^4
 344         sage: expected += (pi - 6*x)^2*(pi - 2*x)^2*x/pi^4
 345         sage: bool(H == expected)
 346         True
 347
 348     """
 349     s1 = sum([ ys[k] * hermite_coefficient(k, x, xs)
 350                for k in xrange(len(xs)) ])
 351
 352     s2 = sum([ y_primes[k] * hermite_deriv_coefficient(k, x, xs)
 353                for k in xrange(len(xs)) ])
 354
 355     return (s1 + s2)
 356
 357
 358 def lagrange_psi(x, xs):
 359     """
 360     The function,
 361
 362       Psi(x) = (x - xs[0])*(x - xs[1])* ... *(x - xs[-1])
 363
 364     used in Lagrange and Hermite interpolation.
 365
 366     INPUT:
 367
 368       - ``x`` -- The independent variable of the resulting expression.
 369
 370       - ``xs`` -- A list of points.
 371
 372     OUTPUT:
 373
 374     A symbolic expression in one variable, `x`.
 375
 376     """
 377
 378     return product([ (x - xj) for xj in xs ])