from sage.all import * 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): """ 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. 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 """ numerator = lagrange_psi(x, xs)/(x - xs[k]) denominator = lagrange_denominator(k, xs) return (numerator / denominator) def lagrange_polynomial(x, xs, ys): """ Return the Lagrange form of the interpolating polynomial in `x` 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]). SETUP:: sage: from mjo.interpolation import lagrange_polynomial TESTS:: sage: xs = [ -pi/2, -pi/6, 0, pi/6, pi/2 ] 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 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(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) def divided_difference_coefficients(xs): """ Assuming some function `f`, compute the coefficients of the divided difference f[xs[0], ..., xs[n]]. SETUP:: sage: from mjo.interpolation import divided_difference_coefficients 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] """ return [ ~lagrange_denominator(k, xs) for k in range(len(xs)) ] 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. SETUP:: sage: from mjo.interpolation import divided_difference TESTS:: sage: xs = [0] sage: ys = list(map(sin, xs)) sage: divided_difference(xs, ys) 0 sage: xs = [0, pi] sage: ys = list(map(sin, xs)) sage: divided_difference(xs, ys) 0 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([x], [f(x=x)]) f(x) 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) """ 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. SETUP:: sage: from mjo.interpolation import lagrange_polynomial, newton_polynomial TESTS:: sage: xs = [ -pi/2, -pi/6, 0, pi/6, pi/2 ] sage: ys = list(map(sin, xs)) sage: L = lagrange_polynomial(x, xs, ys) sage: N = newton_polynomial(x, xs, ys) sage: bool(N == L) True """ return sum( divided_difference(xs[:k+1], ys[:k+1])*lagrange_psi(x, xs[:k]) for k in range(len(xs)) ) 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 product( (x - xj) for xj in xs )