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view extra/ode/rk8fixed.m @ 463:a0d3391e59e2 octave-forge
Update from v1.06 to v1.14
author | pkienzle |
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date | Mon, 05 Aug 2002 02:43:13 +0000 |
parents | 6b33357c7561 |
children | 673d401bf1da |
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function [tout,xout] = rk8fixed(FUN,tspan,x0,Nsteps,ode_fcn_format,trace,count) % Copyright (C) 2001, 2000 Marc Compere % This file is intended for use with Octave. % rk8fixed.m 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 2, or (at your option) % any later version. % % rk8fixed.m 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 at www.gnu.org/copyleft/gpl.html. % % -------------------------------------------------------------------- % % rk8fixed (v1.14) is an 8th order Runge-Kutta numerical integration routine. % It requires 13 function evaluations per step. This is not the most % efficient 8th order implementation. It was just the easiest to put % together as a variant from ode78.m. % % 8th-order accurate RK methods have a local error estimate of O(h^9). % % % Usage: % [tout, xout] = rk8fixed(FUN, tspan, x0, Nsteps, ode_fcn_format, trace, count) % % INPUT: % FUN - String containing name of user-supplied problem derivatives. % Call: xprime = fun(t,x) where FUN = 'fun'. % t - Time or independent variable (scalar). % x - Solution column-vector. % xprime - Returned derivative COLUMN-vector; xprime(i) = dx(i)/dt. % tspan - [ tstart, tfinal ] % x0 - Initial value COLUMN-vector. % Nsteps - number of steps used to span [ tstart, tfinal ] % ode_fcn_format - this specifies if the user-defined ode function is in % the form: xprime = fun(t,x) (ode_fcn_format=0, default) % or: xprime = fun(x,t) (ode_fcn_format=1) % Matlab's solvers comply with ode_fcn_format=0 while % Octave's lsode() and sdirk4() solvers comply with ode_fcn_format=1. % trace - If nonzero, each step is printed. (optional, default: trace = 0). % count - if nonzero, variable 'rhs_counter' is initalized, made global % and counts the number of state-dot function evaluations % 'rhs_counter' is incremented in here, not in the state-dot file % simply make 'rhs_counter' global in the file that calls rk4fixed % % OUTPUT: % tout - Returned integration time points (row-vector). % xout - Returned solution, one solution column-vector per tout-value. % % The result can be displayed by: plot(tout, xout). % % Marc Compere % CompereM@asme.org % created : 06 October 1999 % modified: 19 May 2001 if nargin < 7, count = 0; end if nargin < 6, trace = 0; end if nargin < 5, Nsteps = 50/(tspan(2)-tspan(1)); end % <-- 50 is a guess for a default, % try verifying the solution with ode78 if nargin < 4, ode_fcn_format = 0; end if count==1, global rhs_counter if ~exist('rhs_counter'),rhs_counter=0;,end end % if count alpha_ = [ 2./27., 1/9, 1/6, 5/12, 0.5, 5/6, 1/6, 2/3, 1/3, 1, 0, 1 ]'; beta_ = [ 2/27, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ; 1/36, 1/12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ; 1/24, 0, 1/8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ; 5/12, 0, -25/16, 25/16, 0, 0, 0, 0, 0, 0, 0, 0, 0 ; 0.05, 0, 0, 0.25, 0.2, 0, 0, 0, 0, 0, 0, 0, 0 ; -25/108, 0, 0, 125/108, -65/27, 125/54, 0, 0, 0, 0, 0, 0, 0 ; 31/300, 0, 0, 0, 61/225, -2/9, 13/900, 0, 0, 0, 0, 0, 0 ; 2, 0, 0, -53/6, 704/45, -107/9, 67/90, 3, 0, 0, 0, 0, 0 ; -91/108, 0, 0, 23/108, -976/135, 311/54, -19/60, 17/6, -1/12, 0, 0, 0, 0 ; 2383/4100, 0, 0, -341/164, 4496/1025, -301/82, 2133/4100, 45/82, 45/164, 18/41, 0, 0, 0 ; 3/205, 0, 0, 0, 0, -6/41, -3/205, -3/41, 3/41, 6/41, 0, 0, 0 ; -1777/4100, 0, 0, -341/164, 4496/1025, -289/82, 2193/4100, 51/82, 33/164, 12/41, 0, 1, 0 ]'; chi_ = [ 0, 0, 0, 0, 0, 34/105, 9/35, 9/35, 9/280, 9/280, 0, 41/840, 41/840]'; % Initialization t = tspan(1); h = (tspan(2)-tspan(1))/Nsteps; xout(1,:) = x0'; tout(1) = t; x = x0(:); f = x*zeros(1,13); if trace clc, t, h, x end for i=1:Nsteps, % Compute the slopes if (ode_fcn_format==0), f(:,1) = feval(FUN,t,x); for j = 1:12 f(:,j+1) = feval(FUN, t+alpha_(j)*h, x+h*f*beta_(:,j)); end else, f(:,1) = feval(FUN,x,t); for j = 1:12 f(:,j+1) = feval(FUN, x+h*f*beta_(:,j), t+alpha_(j)*h); end end % if (ode_fcn_format==0) % increment rhs_counter if count==1, rhs_counter = rhs_counter + 13; end % if t = t + h; x = x + h*f*chi_; tout = [tout; t]; xout = [xout; x.']; if trace, home, t, h, x end end