Mercurial > octave
view scripts/ode/ode45.m @ 28041:5e44268dca6f
Replace input validation relying on str2func with alternatives (bug #57351).
* ode15i.m, ode15s.m, ode23.m, ode45.m, check_default_input.m:
Replace try/catch blocks around str2func with a call to exist()
to check whether function exists before calling str2func.
author | Rik <rik@octave.org> |
---|---|
date | Mon, 03 Feb 2020 20:22:19 -0800 |
parents | bd51beb6205e |
children | 13dba3c069f8 |
line wrap: on
line source
######################################################################## ## ## Copyright (C) 2006-2020 The Octave Project Developers ## ## See the file COPYRIGHT.md in the top-level directory of this ## distribution or <https://octave.org/copyright/>. ## ## 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/>. ## ######################################################################## ## -*- texinfo -*- ## @deftypefn {} {[@var{t}, @var{y}] =} ode45 (@var{fun}, @var{trange}, @var{init}) ## @deftypefnx {} {[@var{t}, @var{y}] =} ode45 (@var{fun}, @var{trange}, @var{init}, @var{ode_opt}) ## @deftypefnx {} {[@var{t}, @var{y}, @var{te}, @var{ye}, @var{ie}] =} ode45 (@dots{}) ## @deftypefnx {} {@var{solution} =} ode45 (@dots{}) ## @deftypefnx {} {} ode45 (@dots{}) ## ## Solve a set of non-stiff Ordinary Differential Equations (non-stiff ODEs) ## with the well known explicit @nospell{Dormand-Prince} method of order 4. ## ## @var{fun} is a function handle, inline function, or string containing the ## name of the function that defines the ODE: @code{y' = f(t,y)}. The function ## must accept two inputs where the first is time @var{t} and the second is a ## column vector of unknowns @var{y}. ## ## @var{trange} specifies the time interval over which the ODE will be ## evaluated. Typically, it is a two-element vector specifying the initial and ## final times (@code{[tinit, tfinal]}). If there are more than two elements ## then the solution will also be evaluated at these intermediate time ## instances. ## ## By default, @code{ode45} uses an adaptive timestep with the ## @code{integrate_adaptive} algorithm. The tolerance for the timestep ## computation may be changed by using the options @qcode{"RelTol"} and ## @qcode{"AbsTol"}. ## ## @var{init} contains the initial value for the unknowns. If it is a row ## vector then the solution @var{y} will be a matrix in which each column is ## the solution for the corresponding initial value in @var{init}. ## ## The optional fourth argument @var{ode_opt} specifies non-default options to ## the ODE solver. It is a structure generated by @code{odeset}. ## ## The function typically returns two outputs. Variable @var{t} is a ## column vector and contains the times where the solution was found. The ## output @var{y} is a matrix in which each column refers to a different ## unknown of the problem and each row corresponds to a time in @var{t}. ## ## The output can also be returned as a structure @var{solution} which has a ## field @var{x} containing a row vector of times where the solution was ## evaluated and a field @var{y} containing the solution matrix such that each ## column corresponds to a time in @var{x}. Use ## @w{@code{fieldnames (@var{solution})}} to see the other fields and ## additional information returned. ## ## If no output arguments are requested, and no @code{OutputFcn} is specified ## in @var{ode_opt}, then the @code{OutputFcn} is set to @code{odeplot} and the ## results of the solver are plotted immediately. ## ## If using the @qcode{"Events"} option then three additional outputs may be ## returned. @var{te} holds the time when an Event function returned a zero. ## @var{ye} holds the value of the solution at time @var{te}. @var{ie} ## contains an index indicating which Event function was triggered in the case ## of multiple Event functions. ## ## Example: Solve the @nospell{Van der Pol} equation ## ## @example ## @group ## fvdp = @@(@var{t},@var{y}) [@var{y}(2); (1 - @var{y}(1)^2) * @var{y}(2) - @var{y}(1)]; ## [@var{t},@var{y}] = ode45 (fvdp, [0, 20], [2, 0]); ## @end group ## @end example ## @seealso{odeset, odeget, ode23, ode15s} ## @end deftypefn function varargout = ode45 (fun, trange, init, varargin) if (nargin < 3) print_usage (); endif order = 5; # runge_kutta_45_dorpri uses local extrapolation solver = "ode45"; if (nargin >= 4) if (! isstruct (varargin{1})) ## varargin{1:len} are parameters for fun odeopts = odeset (); funarguments = varargin; elseif (numel (varargin) > 1) ## varargin{1} is an ODE options structure opt odeopts = varargin{1}; funarguments = {varargin{2:numel (varargin)}}; else # if (isstruct (varargin{1})) odeopts = varargin{1}; funarguments = {}; endif else # nargin == 3 odeopts = odeset (); funarguments = {}; endif if (! isnumeric (trange) || ! isvector (trange)) error ("Octave:invalid-input-arg", "ode45: TRANGE must be a numeric vector"); endif if (numel (trange) < 2) error ("Octave:invalid-input-arg", "ode45: TRANGE must contain at least 2 elements"); elseif (trange(1) == trange(2)) error ("Octave:invalid-input-arg", "ode45: invalid time span, TRANGE(1) == TRANGE(2)"); else direction = sign (trange(2) - trange(1)); endif trange = trange(:); if (! isnumeric (init) || ! isvector (init)) error ("Octave:invalid-input-arg", "ode45: INIT must be a numeric vector"); endif init = init(:); if (ischar (fun)) if (! exist (fun)) error ("Octave:invalid-input-arg", [solver ": function '" fun "' not found"]); endif fun = str2func (fun); endif if (! is_function_handle (fun)) error ("Octave:invalid-input-arg", [solver ": FUN must be a valid function handle"]); endif ## Start preprocessing, have a look which options are set in odeopts, ## check if an invalid or unused option is set [defaults, classes, attributes] = odedefaults (numel (init), trange(1), trange(end)); ## FIXME: Refine is not correctly implemented yet defaults = odeset (defaults, "Refine", 4); persistent ode45_ignore_options = ... {"BDF", "InitialSlope", "Jacobian", "JPattern", "MassSingular", "MaxOrder", "MvPattern", "Vectorized"}; defaults = rmfield (defaults, ode45_ignore_options); classes = rmfield (classes, ode45_ignore_options); attributes = rmfield (attributes, ode45_ignore_options); odeopts = odemergeopts ("ode45", odeopts, defaults, classes, attributes); odeopts.funarguments = funarguments; odeopts.direction = direction; if (! isempty (odeopts.NonNegative)) if (isempty (odeopts.Mass)) odeopts.havenonnegative = true; else odeopts.havenonnegative = false; warning ("Octave:invalid-input-arg", ['ode45: option "NonNegative" is ignored', ... " when mass matrix is set\n"]); endif else odeopts.havenonnegative = false; endif if (isempty (odeopts.OutputFcn) && nargout == 0) odeopts.OutputFcn = @odeplot; odeopts.haveoutputfunction = true; else odeopts.haveoutputfunction = ! isempty (odeopts.OutputFcn); endif if (isempty (odeopts.InitialStep)) odeopts.InitialStep = odeopts.direction * ... starting_stepsize (order, fun, trange(1), init, odeopts.AbsTol, odeopts.RelTol, strcmpi (odeopts.NormControl, "on"), odeopts.funarguments); endif if (! isempty (odeopts.Mass) && isnumeric (odeopts.Mass)) havemasshandle = false; mass = odeopts.Mass; # constant mass elseif (is_function_handle (odeopts.Mass)) havemasshandle = true; # mass defined by a function handle else # no mass matrix - creating a diag-matrix of ones for mass havemasshandle = false; # mass = diag (ones (length (init), 1), 0); endif ## Starting the initialization of the core solver ode45 if (havemasshandle) # Handle only the dynamic mass matrix, if (! strcmp (odeopts.MStateDependence, "none")) ### constant mass matrices have already mass = @(t,x) odeopts.Mass (t, x, odeopts.funarguments{:}); fun = @(t,x) mass (t, x, odeopts.funarguments{:}) ... \ fun (t, x, odeopts.funarguments{:}); else mass = @(t) odeopts.Mass (t, odeopts.funarguments{:}); fun = @(t,x) mass (t, odeopts.funarguments{:}) ... \ fun (t, x, odeopts.funarguments{:}); endif endif if (nargout == 1) ## Single output requires auto-selected intermediate times, ## which is obtained by NOT specifying specific solution times. trange = [trange(1); trange(end)]; odeopts.Refine = []; # disable Refine when single output requested elseif (numel (trange) > 2) odeopts.Refine = []; # disable Refine when specific times requested endif solution = integrate_adaptive (@runge_kutta_45_dorpri, order, fun, trange, init, odeopts); ## Postprocessing, do whatever when terminating integration algorithm if (odeopts.haveoutputfunction) # Cleanup plotter feval (odeopts.OutputFcn, [], [], "done", odeopts.funarguments{:}); endif if (! isempty (odeopts.Events)) # Cleanup event function handling ode_event_handler (odeopts.Events, solution.t(end), solution.x(end,:).', "done", odeopts.funarguments{:}); endif ## Print additional information if option Stats is set if (strcmpi (odeopts.Stats, "on")) nsteps = solution.cntloop; # cntloop from 2..end nfailed = solution.cntcycles - nsteps; # cntcycl from 1..end nfevals = 6 * solution.cntcycles + 1; # number of ode evaluations ndecomps = 0; # number of LU decompositions npds = 0; # number of partial derivatives nlinsols = 0; # no. of linear systems solutions printf ("Number of successful steps: %d\n", nsteps); printf ("Number of failed attempts: %d\n", nfailed); printf ("Number of function calls: %d\n", nfevals); endif if (nargout == 2) varargout{1} = solution.t; # Time stamps are first output argument varargout{2} = solution.x; # Results are second output argument elseif (nargout == 1) varargout{1}.x = solution.t.'; # Time stamps saved in field x (row vector) varargout{1}.y = solution.x.'; # Results are saved in field y (row vector) varargout{1}.solver = solver; # Solver name is saved in field solver if (! isempty (odeopts.Events)) varargout{1}.xe = solution.event{3}; # Time info when an event occurred varargout{1}.ye = solution.event{4}; # Results when an event occurred varargout{1}.ie = solution.event{2}; # Index info which event occurred endif if (strcmpi (odeopts.Stats, "on")) varargout{1}.stats = struct (); varargout{1}.stats.nsteps = nsteps; varargout{1}.stats.nfailed = nfailed; varargout{1}.stats.nfevals = nfevals; varargout{1}.stats.npds = npds; varargout{1}.stats.ndecomps = ndecomps; varargout{1}.stats.nlinsols = nlinsols; endif elseif (nargout > 2) varargout = cell (1,5); varargout{1} = solution.t; varargout{2} = solution.x; if (! isempty (odeopts.Events)) varargout{3} = solution.event{3}; # Time info when an event occurred varargout{4} = solution.event{4}; # Results when an event occurred varargout{5} = solution.event{2}; # Index info which event occurred endif endif endfunction %!demo %! ## Demonstrate convergence order for ode45 %! tol = 1e-5 ./ 10.^[0:8]; %! for i = 1 : numel (tol) %! opt = odeset ("RelTol", tol(i), "AbsTol", realmin); %! [t, y] = ode45 (@(t, y) -y, [0, 1], 1, opt); %! h(i) = 1 / (numel (t) - 1); %! err(i) = norm (y .* exp (t) - 1, Inf); %! endfor %! %! ## Estimate order visually %! loglog (h, tol, "-ob", %! h, err, "-b", %! h, (h/h(end)) .^ 4 .* tol(end), "k--", %! h, (h/h(end)) .^ 5 .* tol(end), "k-"); %! axis tight %! xlabel ("h"); %! ylabel ("err(h)"); %! title ("Convergence plot for ode45"); %! legend ("imposed tolerance", "ode45 (relative) error", %! "order 4", "order 5", "location", "northwest"); %! %! ## Estimate order numerically %! p = diff (log (err)) ./ diff (log (h)) ## We are using the Van der Pol equation for all tests. ## Further tests also define a reference solution (computed at high accuracy) %!function ydot = fpol (t, y) # The Van der Pol ODE %! ydot = [y(2); (1 - y(1)^2) * y(2) - y(1)]; %!endfunction %!function ref = fref () # The computed reference solution %! ref = [0.32331666704577, -1.83297456798624]; %!endfunction %!function [val, trm, dir] = feve (t, y, varargin) %! val = fpol (t, y, varargin); # We use the derivatives %! trm = zeros (2,1); # that's why component 2 %! dir = ones (2,1); # does not seem to be exact %!endfunction %!function [val, trm, dir] = fevn (t, y, varargin) %! val = fpol (t, y, varargin); # We use the derivatives %! trm = ones (2,1); # that's why component 2 %! dir = ones (2,1); # does not seem to be exact %!endfunction %!function mas = fmas (t, y, varargin) %! mas = [1, 0; 0, 1]; # Dummy mass matrix for tests %!endfunction %!function mas = fmsa (t, y, varargin) %! mas = sparse ([1, 0; 0, 1]); # A sparse dummy matrix %!endfunction %!function out = fout (t, y, flag, varargin) %! out = false; %! if (strcmp (flag, "init")) %! if (! isequal (size (t), [2, 1])) %! error ('fout: step "init"'); %! endif %! elseif (isempty (flag)) %! if (! isequal (size (t), [1, 1])) %! error ('fout: step "calc"'); %! endif %! elseif (strcmp (flag, "done")) %! if (! isempty (t)) %! warning ('fout: step "done"'); %! endif %! else %! error ("fout: invalid flag <%s>", flag); %! endif %!endfunction %! %!test # two output arguments %! [t, y] = ode45 (@fpol, [0 2], [2 0]); %! assert ([t(end), y(end,:)], [2, fref], 1e-2); %!test # not too many steps %! [t, y] = ode45 (@fpol, [0 2], [2 0]); %! assert (size (t) < 20); %!test # anonymous function instead of real function %! fvdp = @(t,y) [y(2); (1 - y(1)^2) * y(2) - y(1)]; %! [t, y] = ode45 (fvdp, [0 2], [2 0]); %! assert ([t(end), y(end,:)], [2, fref], 1e-2); %!test # string instead of function %! [t, y] = ode45 ("fpol", [0 2], [2 0]); %! assert ([t(end), y(end,:)], [2, fref], 1e-2); %!test # extra input arguments passed through %! [t, y] = ode45 (@fpol, [0 2], [2 0], 12, 13, "KL"); %! assert ([t(end), y(end,:)], [2, fref], 1e-2); %!test # empty ODEOPT structure *but* extra input arguments %! opt = odeset; %! [t, y] = ode45 (@fpol, [0 2], [2 0], opt, 12, 13, "KL"); %! assert ([t(end), y(end,:)], [2, fref], 1e-2); %!test # Solve another anonymous function below zero %! vref = [0, 14.77810590694212]; %! [t, y] = ode45 (@(t,y) y, [-2 0], 2); %! assert ([t(end), y(end,:)], vref, 1e-1); %!test # InitialStep option %! opt = odeset ("InitialStep", 1e-8); %! [t, y] = ode45 (@fpol, [0 0.2], [2 0], opt); %! assert ([t(2)-t(1)], [1e-8], 1e-9); %!test # MaxStep option %! opt = odeset ("MaxStep", 1e-3); %! sol = ode45 (@fpol, [0 0.2], [2 0], opt); %! assert ([sol.x(5)-sol.x(4)], [1e-3], 1e-3); %!test # Solve with intermediate step %! [t, y] = ode45 (@fpol, [0 1 2], [2 0]); %! assert (any((t-1) == 0)); %! assert ([t(end), y(end,:)], [2, fref], 1e-3); %!test # Solve in backward direction starting at t=0 %! vref = [-1.205364552835178, 0.951542399860817]; %! sol = ode45 (@fpol, [0 -2], [2 0]); %! assert ([sol.x(end); sol.y(:,end)], [-2; vref'], 1e-2); %!test # Solve in backward direction starting at t=2 %! vref = [-1.205364552835178, 0.951542399860817]; %! sol = ode45 (@fpol, [2 -2], fref); %! assert ([sol.x(end); sol.y(:,end)], [-2; vref'], 1e-2); %!test # Solve in backward direction starting at t=2, with intermediate step %! vref = [-1.205364552835178, 0.951542399860817]; %! [t, y] = ode45 (@fpol, [2 0 -2], fref); %! idx = find(y < 0, 1, "first") - 1; %! assert ([t(idx), y(idx,:)], [0,2,0], 1e-2); %! assert ([t(end), y(end,:)], [-2, vref], 1e-2); %!test # Solve another anonymous function in backward direction %! vref = [-1, 0.367879437558975]; %! sol = ode45 (@(t,y) y, [0 -1], 1); %! assert ([sol.x(end); sol.y(:,end)], vref', 1e-3); %!test # Solve another anonymous function below zero %! vref = [0, 14.77810590694212]; %! sol = ode45 (@(t,y) y, [-2 0], 2); %! assert ([sol.x(end); sol.y(:,end)], vref', 1e-3); %!test # Solve in backward direction starting at t=0 with MaxStep option %! vref = [-1.205364552835178, 0.951542399860817]; %! opt = odeset ("MaxStep", 1e-3); %! sol = ode45 (@fpol, [0 -2], [2 0], opt); %! assert ([abs(sol.x(8)-sol.x(7))], [1e-3], 1e-3); %! assert ([sol.x(end); sol.y(:,end)], [-2; vref'], 1e-3); %!test # AbsTol option %! opt = odeset ("AbsTol", 1e-5); %! sol = ode45 (@fpol, [0 2], [2 0], opt); %! assert ([sol.x(end); sol.y(:,end)], [2; fref'], 1e-3); %!test # AbsTol and RelTol option %! opt = odeset ("AbsTol", 1e-8, "RelTol", 1e-8); %! sol = ode45 (@fpol, [0 2], [2 0], opt); %! assert ([sol.x(end); sol.y(:,end)], [2; fref'], 1e-3); %!test # RelTol and NormControl option -- higher accuracy %! opt = odeset ("RelTol", 1e-8, "NormControl", "on"); %! sol = ode45 (@fpol, [0 2], [2 0], opt); %! assert ([sol.x(end); sol.y(:,end)], [2; fref'], 1e-5); %!test # Keeps initial values while integrating %! opt = odeset ("NonNegative", 2); %! sol = ode45 (@fpol, [0 2], [2 0], opt); %! assert ([sol.x(end); sol.y(:,end)], [2; 2; 0], 0.5); %!test # Details of OutputSel and Refine can't be tested %! opt = odeset ("OutputFcn", @fout, "OutputSel", 1, "Refine", 5); %! sol = ode45 (@fpol, [0 2], [2 0], opt); %!test # Stats must add further elements in sol %! opt = odeset ("Stats", "on"); %! stat_str = evalc ("sol = ode45 (@fpol, [0 2], [2 0], opt);"); %! assert (strncmp (stat_str, "Number of successful steps:", 27)); %! assert (isfield (sol, "stats")); %! assert (isfield (sol.stats, "nsteps")); %!test # Events option add further elements in sol %! opt = odeset ("Events", @feve); %! sol = ode45 (@fpol, [0 10], [2 0], opt); %! assert (isfield (sol, "ie")); %! assert (sol.ie(1), 2); %! assert (isfield (sol, "xe")); %! assert (isfield (sol, "ye")); %!test # Events option, now stop integration %! warning ("off", "integrate_adaptive:unexpected_termination", "local"); %! opt = odeset ("Events", @fevn, "NormControl", "on"); %! sol = ode45 (@fpol, [0 10], [2 0], opt); %! assert ([sol.ie, sol.xe, sol.ye], %! [2.0, 2.496110, -0.830550, -2.677589], 6e-1); %!test # Events option, five output arguments %! warning ("off", "integrate_adaptive:unexpected_termination", "local"); %! opt = odeset ("Events", @fevn, "NormControl", "on"); %! [t, y, vxe, ye, vie] = ode45 (@fpol, [0 10], [2 0], opt); %! assert ([vie, vxe, ye], %! [2.0, 2.496110, -0.830550, -2.677589], 6e-1); %!test # Mass option as function %! opt = odeset ("Mass", @fmas); %! sol = ode45 (@fpol, [0 2], [2 0], opt); %! assert ([sol.x(end); sol.y(:,end)], [2; fref'], 1e-3); %!test # Mass option as matrix %! opt = odeset ("Mass", eye (2,2)); %! sol = ode45 (@fpol, [0 2], [2 0], opt); %! assert ([sol.x(end); sol.y(:,end)], [2; fref'], 1e-3); %!test # Mass option as sparse matrix %! opt = odeset ("Mass", sparse (eye (2,2))); %! sol = ode45 (@fpol, [0 2], [2 0], opt); %! assert ([sol.x(end); sol.y(:,end)], [2; fref'], 1e-3); %!test # Mass option as function and sparse matrix %! opt = odeset ("Mass", @fmsa); %! sol = ode45 (@fpol, [0 2], [2 0], opt); %! assert ([sol.x(end); sol.y(:,end)], [2; fref'], 1e-3); %!test # Mass option as function and MStateDependence %! opt = odeset ("Mass", @fmas, "MStateDependence", "strong"); %! sol = ode45 (@fpol, [0 2], [2 0], opt); %! assert ([sol.x(end); sol.y(:,end)], [2; fref'], 1e-3); ## Note: The following options have no effect on this solver ## therefore it makes no sense to test them here: ## ## "BDF" ## "InitialSlope" ## "JPattern" ## "Jacobian" ## "MassSingular" ## "MaxOrder" ## "MvPattern" ## "Vectorized" %!test # Check that imaginary part of solution does not get inverted %! sol = ode45 (@(x,y) 1, [0 1], 1i); %! assert (imag (sol.y), ones (size (sol.y))) %! [x, y] = ode45 (@(x,y) 1, [0 1], 1i); %! assert (imag (y), ones (size (y))) %!error ode45 () %!error ode45 (1) %!error ode45 (1,2) %!error <TRANGE must be a numeric> ode45 (@fpol, {[0 25]}, [3 15 1]) %!error <TRANGE must be a .* vector> ode45 (@fpol, [0 25; 25 0], [3 15 1]) %!error <TRANGE must contain at least 2 elements> ode45 (@fpol, [1], [3 15 1]) %!error <invalid time span> ode45 (@fpol, [1 1], [3 15 1]) %!error <INIT must be a numeric> ode45 (@fpol, [0 25], {[3 15 1]}) %!error <INIT must be a .* vector> ode45 (@fpol, [0 25], [3 15 1; 3 15 1]) %!error <FUN must be a valid function handle> ode45 (1, [0 25], [3 15 1])