X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Finterpolation.py;h=6e0ecb3d79c5129a2e6270177b648142085786d1;hb=ba5106550a9a614c6b6f7a2941ddce91ab592934;hp=d5011ea47e154048b79396f4141fb907597bc614;hpb=0594e23633dd2088e6ff9b566d86d4618528212f;p=sage.d.git diff --git a/mjo/interpolation.py b/mjo/interpolation.py index d5011ea..6e0ecb3 100644 --- a/mjo/interpolation.py +++ b/mjo/interpolation.py @@ -1,5 +1,25 @@ from sage.all import * -from misc import product +product = prod + + +def lagrange_denominator(k, xs): + """ + Return the denominator of the kth Lagrange coefficient. + + INPUT: + + - ``k`` -- The index of the coefficient. + + - ``xs`` -- The list of points at which the function values are + known. + + OUTPUT: + + The product of all xs[j] with j != k. + + """ + return product( xs[k] - xs[j] for j in range(len(xs)) if j != k ) + def lagrange_coefficient(k, x, xs): """ @@ -12,9 +32,9 @@ def lagrange_coefficient(k, x, xs): INPUT: - - ``k`` -- the index of the coefficient. + - ``k`` -- The index of the coefficient. - - ``x`` -- the symbolic variable to use for the first argument + - ``x`` -- The symbolic variable to use for the first argument of l_{k}. - ``xs`` -- The list of points at which the function values are @@ -22,17 +42,21 @@ def lagrange_coefficient(k, x, xs): OUTPUT: - A symbolic function of one variable. + A symbolic expression of one variable. + + SETUP:: + + sage: from mjo.interpolation import lagrange_coefficient TESTS:: sage: xs = [ -pi/2, -pi/6, 0, pi/6, pi/2 ] sage: lagrange_coefficient(0, x, xs) - 1/8*(pi - 6*x)*(pi - 2*x)*(pi + 6*x)*x/pi^4 + 1/8*(pi + 6*x)*(pi - 2*x)*(pi - 6*x)*x/pi^4 """ - numerator = product([x - xs[j] for j in range(0, len(xs)) if j != k]) - denominator = product([xs[k] - xs[j] for j in range(0, len(xs)) if j != k]) + numerator = lagrange_psi(x, xs)/(x - xs[k]) + denominator = lagrange_denominator(k, xs) return (numerator / denominator) @@ -40,7 +64,7 @@ def lagrange_coefficient(k, x, xs): def lagrange_polynomial(x, xs, ys): """ - Return the Lagrange form of the interpolation polynomial in `x` of + Return the Lagrange form of the interpolating polynomial in `x` at the points (xs[k], ys[k]). INPUT: @@ -53,12 +77,16 @@ def lagrange_polynomial(x, xs, ys): OUTPUT: - A symbolic function (polynomial) interpolating each (xs[k], ys[k]). + A symbolic expression (polynomial) interpolating each (xs[k], ys[k]). + + SETUP:: + + sage: from mjo.interpolation import lagrange_polynomial TESTS:: sage: xs = [ -pi/2, -pi/6, 0, pi/6, pi/2 ] - sage: ys = map(sin, xs) + sage: ys = list(map(sin, xs)) sage: L = lagrange_polynomial(x, xs, ys) sage: expected = 27/16*(pi - 6*x)*(pi - 2*x)*(pi + 2*x)*x/pi^4 sage: expected -= 1/8*(pi - 6*x)*(pi - 2*x)*(pi + 6*x)*x/pi^4 @@ -68,9 +96,48 @@ def lagrange_polynomial(x, xs, ys): True """ - ls = [ lagrange_coefficient(k, x, xs) for k in range(0, len(xs)) ] - sigma = sum([ ys[k] * ls[k] for k in range(0, len(xs)) ]) - return sigma + ls = [ lagrange_coefficient(k, x, xs) for k in range(len(xs)) ] + return sum( ys[k] * ls[k] for k in range(len(xs)) ) + + + +def lagrange_interpolate(f, x, xs): + """ + Interpolate the function ``f`` at the points ``xs`` using the + Lagrange form of the interpolating polynomial. + + INPUT: + + - ``f`` -- The function to interpolate. + + - ``x`` -- The independent variable of the resulting polynomial. + + - ``xs`` -- A list of points at which to interpolate ``f``. + + OUTPUT: + + A polynomial in ``x`` which interpolates ``f`` at ``xs``. + + SETUP:: + + sage: from mjo.interpolation import lagrange_interpolate + + EXAMPLES: + + We're exact on polynomials of degree `n` if we use `n+1` points:: + + sage: t = SR.symbol('t', domain='real') + sage: lagrange_interpolate(x^2, t, [-1,0,1]).simplify_rational() + t^2 + + """ + # f should be a function of one variable. + z = f.variables()[0] + # We're really just doing map(f, xs) here; the additional + # gymnastics are to avoid a warning when calling `f` with an + # unnamed argument. + ys = [ f({z: xk}) for xk in xs ] + return lagrange_polynomial(x, xs, ys) @@ -79,7 +146,11 @@ def divided_difference_coefficients(xs): Assuming some function `f`, compute the coefficients of the divided difference f[xs[0], ..., xs[n]]. - TESTS: + SETUP:: + + sage: from mjo.interpolation import divided_difference_coefficients + + TESTS:: sage: divided_difference_coefficients([0]) [1] @@ -89,96 +160,209 @@ def divided_difference_coefficients(xs): [1/2/pi^2, -1/pi^2, 1/2/pi^2] """ - coeffs = [ product([ (QQ(1) / (xj - xi)) for xi in xs if xi != xj ]) - for xj in xs ] - return coeffs + return [ ~lagrange_denominator(k, xs) for k in range(len(xs)) ] -def divided_difference(f, xs): + +def divided_difference(xs, ys): """ - Return the Newton divided difference of `f` at the points - `xs`. Reference: + Return the Newton divided difference of the points (xs[k], + ys[k]). Reference: http://en.wikipedia.org/wiki/Divided_differences INPUT: - - ``f`` -- The function whose divided difference we seek. + - ``xs`` -- The list of x-values. - - ``xs`` -- The list of points at which to compute `f`. + - ``ys`` -- The function values at `xs`. OUTPUT: - The divided difference of `f` at ``xs``. + The (possibly symbolic) divided difference function. + + SETUP:: + + sage: from mjo.interpolation import divided_difference TESTS:: - sage: divided_difference(sin, [0]) + sage: xs = [0] + sage: ys = list(map(sin, xs)) + sage: divided_difference(xs, ys) 0 - sage: divided_difference(sin, [0, pi]) + sage: xs = [0, pi] + sage: ys = list(map(sin, xs)) + sage: divided_difference(xs, ys) 0 - sage: divided_difference(sin, [0, pi, 2*pi]) + sage: xs = [0, pi, 2*pi] + sage: ys = list(map(sin, xs)) + sage: divided_difference(xs, ys) 0 We try something entirely symbolic:: - sage: f = function('f', x) - sage: divided_difference(f, [x]) + sage: f = function('f')(x) + sage: divided_difference([x], [f(x=x)]) f(x) - sage: x1,x2 = var('x1,x2') - sage: divided_difference(f, [x1,x2]) - (f(x1) - f(x2))/(x1 - x2) + sage: x1,x2 = SR.var('x1,x2') + sage: divided_difference([x1,x2], [f(x=x1),f(x=x2)]) + f(x1)/(x1 - x2) - f(x2)/(x1 - x2) """ - if (len(xs) == 1): - # Avoid that goddamned DeprecationWarning when we have a named - # argument but don't know what it is. - if len(f.variables()) == 0: - return f(xs[0]) - else: - v = f.variables()[0] - return f({ v: xs[0] }) + coeffs = divided_difference_coefficients(xs) + v_cs = vector(coeffs) + v_ys = vector(ys) + return v_cs.dot_product(v_ys) - # Use the recursive definition. - numerator = divided_difference(f, xs[1:]) - numerator -= divided_difference(f, xs[:-1]) - return numerator / (xs[-1] - xs[0]) - -def newton_polynomial(f, x, xs): +def newton_polynomial(x, xs, ys): """ - Return the Newton form of the interpolating polynomial of `f` at - the points `xs` in the variable `x`. + Return the Newton form of the interpolating polynomial of the + points (xs[k], ys[k]) in the variable `x`. INPUT: - - ``f`` -- The function to interpolate. - - ``x`` -- The independent variable to use for the interpolating polynomial. - - ``xs`` -- The list of points at which to interpolate `f`. + - ``xs`` -- The list of x-values. + + - ``ys`` -- The function values at `xs`. OUTPUT: - A symbolic function. + A symbolic expression. - TESTS: + SETUP:: + + sage: from mjo.interpolation import lagrange_polynomial, newton_polynomial + + TESTS:: sage: xs = [ -pi/2, -pi/6, 0, pi/6, pi/2 ] - sage: ys = map(sin, xs) + sage: ys = list(map(sin, xs)) sage: L = lagrange_polynomial(x, xs, ys) - sage: N = newton_polynomial(sin, x, xs) + sage: N = newton_polynomial(x, xs, ys) sage: bool(N == L) True """ - degree = len(xs) - 1 + return sum( divided_difference(xs[:k+1], ys[:k+1])*lagrange_psi(x, xs[:k]) + for k in range(len(xs)) ) - N = SR(0) - for k in range(0, degree+1): - term = divided_difference(f, xs[:k+1]) - term *= product([ x - xk for xk in xs[:k]]) - N += term +def hermite_coefficient(k, x, xs): + """ + Return the Hermite coefficient h_{k}(x). See Atkinson, p. 160. + + INPUT: + + - ``k`` -- The index of the coefficient. + + - ``x`` -- The symbolic variable to use as the argument of h_{k}. + + - ``xs`` -- The list of points at which the function values are + known. + + OUTPUT: + + A symbolic expression. + + """ + lk = lagrange_coefficient(k, x, xs) + return (1 - 2*lk.diff(x)(x=xs[k])*(x - xs[k]))*(lk**2) + + +def hermite_deriv_coefficient(k, x, xs): + """ + Return the Hermite derivative coefficient, \tilde{h}_{k}(x). See + Atkinson, p. 160. + + INPUT: + + - ``k`` -- The index of the coefficient. + + - ``x`` -- The symbolic variable to use as the argument of h_{k}. + + - ``xs`` -- The list of points at which the function values are + known. + + OUTPUT: + + A symbolic expression. + + """ + lk = lagrange_coefficient(k, x, xs) + return (x - xs[k])*(lk**2) + + +def hermite_interpolant(x, xs, ys, y_primes): + """ + Return the Hermite interpolant `H(x)` such that H(xs[k]) = ys[k] + and H'(xs[k]) = y_primes[k] for each k. + + Reference: Atkinson, p. 160. + + INPUT: + + - ``x`` -- The symbolic variable to use as the argument of H(x). + + - ``xs`` -- The list of points at which the function values are + known. + + - ``ys`` -- The function values at the `xs`. + + - ``y_primes`` -- The derivatives at the `xs`. + + OUTPUT: + + A symbolic expression. + + SETUP:: + + sage: from mjo.interpolation import hermite_interpolant + + TESTS:: + + sage: xs = [ 0, pi/6, pi/2 ] + sage: ys = list(map(sin, xs)) + sage: y_primes = list(map(cos, xs)) + sage: H = hermite_interpolant(x, xs, ys, y_primes) + sage: expected = -27/4*(pi - 6*x)*(pi - 2*x)^2*sqrt(3)*x^2/pi^4 + sage: expected += (5*(pi - 2*x)/pi + 1)*(pi - 6*x)^2*x^2/pi^4 + sage: expected += 81/2*((pi - 6*x)/pi + 1)*(pi - 2*x)^2*x^2/pi^4 + sage: expected += (pi - 6*x)^2*(pi - 2*x)^2*x/pi^4 + sage: bool(H == expected) + True + + """ + s1 = sum( ys[k] * hermite_coefficient(k, x, xs) + for k in range(len(xs)) ) + + s2 = sum( y_primes[k] * hermite_deriv_coefficient(k, x, xs) + for k in range(len(xs)) ) + + return (s1 + s2) + + +def lagrange_psi(x, xs): + """ + The function, + + Psi(x) = (x - xs[0])*(x - xs[1])* ... *(x - xs[-1]) + + used in Lagrange and Hermite interpolation. + + INPUT: + + - ``x`` -- The independent variable of the resulting expression. + + - ``xs`` -- A list of points. + + OUTPUT: + + A symbolic expression in one variable, `x`. + + """ - return N + return product( (x - xj) for xj in xs )