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view scripts/ode/private/runge_kutta_interpolate.m @ 27919:1891570abac8
update Octave Project Developers copyright for the new year
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update for 2020.
author | John W. Eaton <jwe@octave.org> |
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date | Mon, 06 Jan 2020 22:29:51 -0500 |
parents | b442ec6dda5c |
children | bd51beb6205e |
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## Copyright (C) 2015-2020 The Octave Project Developers ## ## See the file COPYRIGHT.md in the top-level directory of this distribution ## or <https://octave.org/COPYRIGHT.html/>. ## ## ## This file is part of Octave. ## ## Octave is free software: you can redistribute it and/or modify it ## under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## Octave is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with Octave; see the file COPYING. If not, see ## <https://www.gnu.org/licenses/>. function u_interp = runge_kutta_interpolate (order, z, u, t, k_vals, dt, func, args) switch (order) case 1 u_interp = interp1 (z, u.', t, "linear"); case 2 if (! isempty (k_vals)) der = k_vals(:,1); else der = feval (func, z(1) , u(:,1), args); endif u_interp = quadratic_interpolation (z, u, der, t); case 3 u_interp = hermite_cubic_interpolation (z, u, k_vals, t); case 5 ## ode45 with Dormand-Prince scheme: u_interp = hermite_quartic_interpolation (z, u, k_vals, t); otherwise warning (["High order interpolation not yet implemented: ", ... "using cubic interpolation instead"]); der(:,1) = feval (func, z(1), u(:,1), args); der(:,2) = feval (func, z(2), u(:,2), args); u_interp = hermite_cubic_interpolation (z, u, der, t); endswitch endfunction ## The function below can be used in an ODE solver to interpolate the solution ## at the time t_out using 2nd order Hermite interpolation. function x_out = quadratic_interpolation (t, x, der, t_out) # coefficients of the parabola a = -(x(:,1) - x(:,2) - der(:).*(t(1)-t(2))) / (t(1) - t(2))^2; b = der(:) - 2*t(1).*a; c = x(:,1) - a*t(1)^2 - b*t(1); # evaluate in t_out x_out = a*t_out.^2 + b*t_out + c; endfunction ## The function below can be used in an ODE solver to interpolate the ## solution at the time t_out using 3rd order Hermite interpolation. function x_out = hermite_cubic_interpolation (t, x, der, t_out) dt = (t(2) - t(1)); s = (t_out - t(1)) / dt; x_out = ((1 + 2*s) .* (1-s).^2) .* x(:,1) + ... (s .* (1-s).^2 * dt ) .* der(:,1) + ... ((3-2*s) .* s.^2 ) .* x(:,end) + ... ((s-1) .* s.^2 * dt ) .* der(:,end); endfunction ## The function below can be used in an ODE solver to interpolate the ## solution at the time t_out using 4th order Hermite interpolation. function x_out = hermite_quartic_interpolation (t, x, der, t_out) persistent coefs_u_half = ... [6025192743/30085553152; 0; 51252292925/65400821598; -2691868925/45128329728; 187940372067/1594534317056; -1776094331/19743644256; 11237099/235043384]; ## 4th order approximation of y in t+dt/2 as proposed by Shampine in ## Lawrence, Shampine, "Some Practical Runge-Kutta Formulas", 1986. dt = t(2) - t(1); u_half = x(:,1) + (1/2) * dt * (der(:,1:7) * coefs_u_half); ## Rescale time on [0,1] s = (t_out - t(1)) / dt; ## Hermite basis functions ## H0 = 1 - 11*s.^2 + 18*s.^3 - 8*s.^4; ## H1 = s - 4*s.^2 + 5*s.^3 - 2*s.^4; ## H2 = 16*s.^2 - 32*s.^3 + 16*s.^4; ## H3 = - 5*s.^2 + 14*s.^3 - 8*s.^4; ## H4 = s.^2 - 3*s.^3 + 2*s.^4; x_out = (1 - 11*s.^2 + 18*s.^3 - 8*s.^4) .* x(:,1) + ... ( s - 4*s.^2 + 5*s.^3 - 2*s.^4) .* (dt * der(:,1)) + ... ( 16*s.^2 - 32*s.^3 + 16*s.^4) .* u_half + ... ( - 5*s.^2 + 14*s.^3 - 8*s.^4) .* x(:,2) + ... ( s.^2 - 3*s.^3 + 2*s.^4) .* (dt * der(:,end)); endfunction