from sage.all import * from misc import product def lagrange_coefficient(k, x, xs): """ Returns the coefficient function l_{k}(variable) of y_{k} in the Lagrange polynomial of f. See, http://en.wikipedia.org/wiki/Lagrange_polynomial for more information. INPUT: - ``k`` -- the index of the coefficient. - ``x`` -- the symbolic variable to use for the first argument of l_{k}. - ``xs`` -- The list of points at which the function values are known. OUTPUT: A symbolic expression of one variable. 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 """ 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]) return (numerator / denominator) def lagrange_polynomial(x, xs, ys): """ Return the Lagrange form of the interpolation polynomial in `x` of at the points (xs[k], ys[k]). INPUT: - ``x`` - The independent variable of the resulting polynomial. - ``xs`` - The list of points at which we interpolate `f`. - ``ys`` - The function values at `xs`. OUTPUT: A symbolic expression (polynomial) interpolating each (xs[k], ys[k]). TESTS:: sage: xs = [ -pi/2, -pi/6, 0, pi/6, pi/2 ] sage: ys = 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 sage: expected -= 1/8*(pi - 6*x)*(pi + 2*x)*(pi + 6*x)*x/pi^4 sage: expected += 27/16*(pi - 2*x)*(pi + 2*x)*(pi + 6*x)*x/pi^4 sage: bool(L == expected) 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 def divided_difference_coefficients(xs): """ Assuming some function `f`, compute the coefficients of the divided difference f[xs[0], ..., xs[n]]. TESTS: sage: divided_difference_coefficients([0]) [1] sage: divided_difference_coefficients([0, pi]) [-1/pi, 1/pi] sage: divided_difference_coefficients([0, pi, 2*pi]) [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 def divided_difference(xs, ys): """ Return the Newton divided difference of the points (xs[k], ys[k]). Reference: http://en.wikipedia.org/wiki/Divided_differences INPUT: - ``xs`` -- The list of x-values. - ``ys`` -- The function values at `xs`. OUTPUT: The (possibly symbolic) divided difference function. TESTS:: sage: xs = [0] sage: ys = map(sin, xs) sage: divided_difference(xs, ys) 0 sage: xs = [0, pi] sage: ys = map(sin, xs) sage: divided_difference(xs, ys) 0 sage: xs = [0, pi, 2*pi] sage: ys = map(sin, xs) sage: divided_difference(xs, ys) 0 We try something entirely symbolic:: sage: f = function('f', x) sage: divided_difference([x], [f(x=x)]) f(x) sage: x1,x2 = var('x1,x2') sage: divided_difference([x1,x2], [f(x=x1),f(x=x2)]) f(x1)/(x1 - x2) - f(x2)/(x1 - x2) """ coeffs = divided_difference_coefficients(xs) v_cs = vector(coeffs) v_ys = vector(ys) return v_cs.dot_product(v_ys) def newton_polynomial(x, xs, ys): """ Return the Newton form of the interpolating polynomial of the points (xs[k], ys[k]) in the variable `x`. INPUT: - ``x`` -- The independent variable to use for the interpolating polynomial. - ``xs`` -- The list of x-values. - ``ys`` -- The function values at `xs`. OUTPUT: A symbolic expression. TESTS: sage: xs = [ -pi/2, -pi/6, 0, pi/6, pi/2 ] sage: ys = map(sin, xs) sage: L = lagrange_polynomial(x, xs, ys) sage: N = newton_polynomial(x, xs, ys) sage: bool(N == L) True """ degree = len(xs) - 1 N = SR(0) for k in range(0, degree+1): term = divided_difference(xs[:k+1], ys[:k+1]) term *= product([ x - xk for xk in xs[:k]]) N += term return N 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. TESTS: sage: xs = [ 0, pi/6, pi/2 ] sage: ys = map(sin, xs) sage: y_primes = 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(0, len(xs)) ]) s2 = sum([ y_primes[k] * hermite_deriv_coefficient(k, x, xs) for k in range(0, 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 product([ (x - xj) for xj in xs ])