mjo/interpolation.py

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