1 %# Copyright (C) 2008-2012, Thomas Treichl <treichl@users.sourceforge.net>
2 %# OdePkg - A package for solving ordinary differential equations and more
4 %# This program is free software; you can redistribute it and/or modify
5 %# it under the terms of the GNU General Public License as published by
6 %# the Free Software Foundation; either version 2 of the License, or
7 %# (at your option) any later version.
9 %# This program is distributed in the hope that it will be useful,
10 %# but WITHOUT ANY WARRANTY; without even the implied warranty of
11 %# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
12 %# GNU General Public License for more details.
14 %# You should have received a copy of the GNU General Public License
15 %# along with this program; If not, see <http://www.gnu.org/licenses/>.
18 %# @deftypefn {Function File} {[@var{}] =} ode54d (@var{@@fun}, @var{slot}, @var{init}, @var{lags}, @var{hist}, [@var{opt}], [@var{par1}, @var{par2}, @dots{}])
19 %# @deftypefnx {Command} {[@var{sol}] =} ode54d (@var{@@fun}, @var{slot}, @var{init}, @var{lags}, @var{hist}, [@var{opt}], [@var{par1}, @var{par2}, @dots{}])
20 %# @deftypefnx {Command} {[@var{t}, @var{y}, [@var{xe}, @var{ye}, @var{ie}]] =} ode54d (@var{@@fun}, @var{slot}, @var{init}, @var{lags}, @var{hist}, [@var{opt}], [@var{par1}, @var{par2}, @dots{}])
22 %# This function file can be used to solve a set of non--stiff delay differential equations (non--stiff DDEs) with a modified version of the well known explicit Runge--Kutta method of order (2,3).
24 %# If this function is called with no return argument then plot the solution over time in a figure window while solving the set of DDEs that are defined in a function and specified by the function handle @var{@@fun}. The second input argument @var{slot} is a double vector that defines the time slot, @var{init} is a double vector that defines the initial values of the states, @var{lags} is a double vector that describes the lags of time, @var{hist} is a double matrix and describes the history of the DDEs, @var{opt} can optionally be a structure array that keeps the options created with the command @command{odeset} and @var{par1}, @var{par2}, @dots{} can optionally be other input arguments of any type that have to be passed to the function defined by @var{@@fun}.
26 %# In other words, this function will solve a problem of the form
28 %# dy/dt = fun (t, y(t), y(t-lags(1), y(t-lags(2), @dots{})))
30 %# y(slot(1)-lags(1)) = hist(1), y(slot(1)-lags(2)) = hist(2), @dots{}
33 %# If this function is called with one return argument then return the solution @var{sol} of type structure array after solving the set of DDEs. The solution @var{sol} has the fields @var{x} of type double column vector for the steps chosen by the solver, @var{y} of type double column vector for the solutions at each time step of @var{x}, @var{solver} of type string for the solver name and optionally the extended time stamp information @var{xe}, the extended solution information @var{ye} and the extended index information @var{ie} all of type double column vector that keep the informations of the event function if an event function handle is set in the option argument @var{opt}.
35 %# If this function is called with more than one return argument then return the time stamps @var{t}, the solution values @var{y} and optionally the extended time stamp information @var{xe}, the extended solution information @var{ye} and the extended index information @var{ie} all of type double column vector.
40 %# the following code solves an anonymous implementation of a chaotic behavior
43 %# fcao = @@(vt, vy, vz) [2 * vz / (1 + vz^9.65) - vy];
45 %# vopt = odeset ("NormControl", "on", "RelTol", 1e-3);
46 %# vsol = ode54d (fcao, [0, 100], 0.5, 2, 0.5, vopt);
48 %# vlag = interp1 (vsol.x, vsol.y, vsol.x - 2);
49 %# plot (vsol.y, vlag); legend ("fcao (t,y,z)");
53 %# to solve the following problem with two delayed state variables
56 %# d y1(t)/dt = -y1(t)
57 %# d y2(t)/dt = -y2(t) + y1(t-5)
58 %# d y3(t)/dt = -y3(t) + y2(t-10)*y1(t-10)
61 %# one might do the following
64 %# function f = fun (t, y, yd)
65 %# f(1) = -y(1); %% y1' = -y1(t)
66 %# f(2) = -y(2) + yd(1,1); %% y2' = -y2(t) + y1(t-lags(1))
67 %# f(3) = -y(3) + yd(2,2)*yd(1,2); %% y3' = -y3(t) + y2(t-lags(2))*y1(t-lags(2))
70 %# res = ode54d (@@fun, T, [1;1;1], [5, 10], ones (3,2));
78 function [varargout] = ode54d (vfun, vslot, vinit, vlags, vhist, varargin)
80 if (nargin == 0) %# Check number and types of all input arguments
82 error ('OdePkg:InvalidArgument', ...
83 'Number of input arguments must be greater than zero');
88 elseif (~isa (vfun, 'function_handle'))
89 error ('OdePkg:InvalidArgument', ...
90 'First input argument must be a valid function handle');
92 elseif (~isvector (vslot) || length (vslot) < 2)
93 error ('OdePkg:InvalidArgument', ...
94 'Second input argument must be a valid vector');
96 elseif (~isvector (vinit) || ~isnumeric (vinit))
97 error ('OdePkg:InvalidArgument', ...
98 'Third input argument must be a valid numerical value');
100 elseif (~isvector (vlags) || ~isnumeric (vlags))
101 error ('OdePkg:InvalidArgument', ...
102 'Fourth input argument must be a valid numerical value');
104 elseif ~(isnumeric (vhist) || isa (vhist, 'function_handle'))
105 error ('OdePkg:InvalidArgument', ...
106 'Fifth input argument must either be numeric or a function handle');
110 if (~isstruct (varargin{1}))
111 %# varargin{1:len} are parameters for vfun
112 vodeoptions = odeset;
113 vfunarguments = varargin;
115 elseif (length (varargin) > 1)
116 %# varargin{1} is an OdePkg options structure vopt
117 vodeoptions = odepkg_structure_check (varargin{1}, 'ode54d');
118 vfunarguments = {varargin{2:length(varargin)}};
120 else %# if (isstruct (varargin{1}))
121 vodeoptions = odepkg_structure_check (varargin{1}, 'ode54d');
126 else %# if (nargin == 5)
127 vodeoptions = odeset;
131 %# Start preprocessing, have a look which options have been set in
132 %# vodeoptions. Check if an invalid or unused option has been set and
134 vslot = vslot(:)'; %# Create a row vector
135 vinit = vinit(:)'; %# Create a row vector
136 vlags = vlags(:)'; %# Create a row vector
138 %# Check if the user has given fixed points of time
139 if (length (vslot) > 2), vstepsizegiven = true; %# Step size checking
140 else vstepsizegiven = false; end
142 %# Get the default options that can be set with 'odeset' temporarily
145 %# Implementation of the option RelTol has been finished. This option
146 %# can be set by the user to another value than default value.
147 if (isempty (vodeoptions.RelTol) && ~vstepsizegiven)
148 vodeoptions.RelTol = 1e-6;
149 warning ('OdePkg:InvalidOption', ...
150 'Option "RelTol" not set, new value %f is used', vodeoptions.RelTol);
151 elseif (~isempty (vodeoptions.RelTol) && vstepsizegiven)
152 warning ('OdePkg:InvalidOption', ...
153 'Option "RelTol" will be ignored if fixed time stamps are given');
154 %# This implementation has been added to odepkg_structure_check.m
155 %# elseif (~isscalar (vodeoptions.RelTol) && ~vstepsizegiven)
156 %# error ('OdePkg:InvalidOption', ...
157 %# 'Option "RelTol" must be set to a scalar value for this solver');
160 %# Implementation of the option AbsTol has been finished. This option
161 %# can be set by the user to another value than default value.
162 if (isempty (vodeoptions.AbsTol) && ~vstepsizegiven)
163 vodeoptions.AbsTol = 1e-6;
164 warning ('OdePkg:InvalidOption', ...
165 'Option "AbsTol" not set, new value %f is used', vodeoptions.AbsTol);
166 elseif (~isempty (vodeoptions.AbsTol) && vstepsizegiven)
167 warning ('OdePkg:InvalidOption', ...
168 'Option "AbsTol" will be ignored if fixed time stamps are given');
169 else %# create column vector
170 vodeoptions.AbsTol = vodeoptions.AbsTol(:);
173 %# Implementation of the option NormControl has been finished. This
174 %# option can be set by the user to another value than default value.
175 if (strcmp (vodeoptions.NormControl, 'on')), vnormcontrol = true;
176 else vnormcontrol = false;
179 %# Implementation of the option NonNegative has been finished. This
180 %# option can be set by the user to another value than default value.
181 if (~isempty (vodeoptions.NonNegative))
182 if (isempty (vodeoptions.Mass)), vhavenonnegative = true;
184 vhavenonnegative = false;
185 warning ('OdePkg:InvalidOption', ...
186 'Option "NonNegative" will be ignored if mass matrix is set');
188 else vhavenonnegative = false;
191 %# Implementation of the option OutputFcn has been finished. This
192 %# option can be set by the user to another value than default value.
193 if (isempty (vodeoptions.OutputFcn) && nargout == 0)
194 vodeoptions.OutputFcn = @odeplot;
195 vhaveoutputfunction = true;
196 elseif (isempty (vodeoptions.OutputFcn)), vhaveoutputfunction = false;
197 else vhaveoutputfunction = true;
200 %# Implementation of the option OutputSel has been finished. This
201 %# option can be set by the user to another value than default value.
202 if (~isempty (vodeoptions.OutputSel)), vhaveoutputselection = true;
203 else vhaveoutputselection = false; end
205 %# Implementation of the option Refine has been finished. This option
206 %# can be set by the user to another value than default value.
207 if (isequal (vodeoptions.Refine, vodetemp.Refine)), vhaverefine = true;
208 else vhaverefine = false; end
210 %# Implementation of the option Stats has been finished. This option
211 %# can be set by the user to another value than default value.
213 %# Implementation of the option InitialStep has been finished. This
214 %# option can be set by the user to another value than default value.
215 if (isempty (vodeoptions.InitialStep) && ~vstepsizegiven)
216 vodeoptions.InitialStep = abs (vslot(1,1) - vslot(1,2)) / 10;
217 vodeoptions.InitialStep = vodeoptions.InitialStep / 10^vodeoptions.Refine;
218 warning ('OdePkg:InvalidOption', ...
219 'Option "InitialStep" not set, new value %f is used', vodeoptions.InitialStep);
222 %# Implementation of the option MaxStep has been finished. This option
223 %# can be set by the user to another value than default value.
224 if (isempty (vodeoptions.MaxStep) && ~vstepsizegiven)
225 vodeoptions.MaxStep = abs (vslot(1,1) - vslot(1,length (vslot))) / 10;
226 %# vodeoptions.MaxStep = vodeoptions.MaxStep / 10^vodeoptions.Refine;
227 warning ('OdePkg:InvalidOption', ...
228 'Option "MaxStep" not set, new value %f is used', vodeoptions.MaxStep);
231 %# Implementation of the option Events has been finished. This option
232 %# can be set by the user to another value than default value.
233 if (~isempty (vodeoptions.Events)), vhaveeventfunction = true;
234 else vhaveeventfunction = false; end
236 %# The options 'Jacobian', 'JPattern' and 'Vectorized' will be ignored
237 %# by this solver because this solver uses an explicit Runge-Kutta
238 %# method and therefore no Jacobian calculation is necessary
239 if (~isequal (vodeoptions.Jacobian, vodetemp.Jacobian))
240 warning ('OdePkg:InvalidOption', ...
241 'Option "Jacobian" will be ignored by this solver');
243 if (~isequal (vodeoptions.JPattern, vodetemp.JPattern))
244 warning ('OdePkg:InvalidOption', ...
245 'Option "JPattern" will be ignored by this solver');
247 if (~isequal (vodeoptions.Vectorized, vodetemp.Vectorized))
248 warning ('OdePkg:InvalidOption', ...
249 'Option "Vectorized" will be ignored by this solver');
251 if (~isequal (vodeoptions.NewtonTol, vodetemp.NewtonTol))
252 warning ('OdePkg:InvalidArgument', ...
253 'Option "NewtonTol" will be ignored by this solver');
255 if (~isequal (vodeoptions.MaxNewtonIterations,...
256 vodetemp.MaxNewtonIterations))
257 warning ('OdePkg:InvalidArgument', ...
258 'Option "MaxNewtonIterations" will be ignored by this solver');
261 %# Implementation of the option Mass has been finished. This option
262 %# can be set by the user to another value than default value.
263 if (~isempty (vodeoptions.Mass) && isnumeric (vodeoptions.Mass))
264 vhavemasshandle = false; vmass = vodeoptions.Mass; %# constant mass
265 elseif (isa (vodeoptions.Mass, 'function_handle'))
266 vhavemasshandle = true; %# mass defined by a function handle
267 else %# no mass matrix - creating a diag-matrix of ones for mass
268 vhavemasshandle = false; %# vmass = diag (ones (length (vinit), 1), 0);
271 %# Implementation of the option MStateDependence has been finished.
272 %# This option can be set by the user to another value than default
274 if (strcmp (vodeoptions.MStateDependence, 'none'))
275 vmassdependence = false;
276 else vmassdependence = true;
279 %# Other options that are not used by this solver. Print a warning
280 %# message to tell the user that the option(s) is/are ignored.
281 if (~isequal (vodeoptions.MvPattern, vodetemp.MvPattern))
282 warning ('OdePkg:InvalidOption', ...
283 'Option "MvPattern" will be ignored by this solver');
285 if (~isequal (vodeoptions.MassSingular, vodetemp.MassSingular))
286 warning ('OdePkg:InvalidOption', ...
287 'Option "MassSingular" will be ignored by this solver');
289 if (~isequal (vodeoptions.InitialSlope, vodetemp.InitialSlope))
290 warning ('OdePkg:InvalidOption', ...
291 'Option "InitialSlope" will be ignored by this solver');
293 if (~isequal (vodeoptions.MaxOrder, vodetemp.MaxOrder))
294 warning ('OdePkg:InvalidOption', ...
295 'Option "MaxOrder" will be ignored by this solver');
297 if (~isequal (vodeoptions.BDF, vodetemp.BDF))
298 warning ('OdePkg:InvalidOption', ...
299 'Option "BDF" will be ignored by this solver');
302 %# Starting the initialisation of the core solver ode54d
303 vtimestamp = vslot(1,1); %# timestamp = start time
304 vtimelength = length (vslot); %# length needed if fixed steps
305 vtimestop = vslot(1,vtimelength); %# stop time = last value
308 vstepsize = vodeoptions.InitialStep;
309 vminstepsize = (vtimestop - vtimestamp) / (1/eps);
310 else %# If step size is given then use the fixed time steps
311 vstepsize = abs (vslot(1,1) - vslot(1,2));
312 vminstepsize = eps; %# vslot(1,2) - vslot(1,1) - eps;
315 vretvaltime = vtimestamp; %# first timestamp output
316 if (vhaveoutputselection) %# first solution output
317 vretvalresult = vinit(vodeoptions.OutputSel);
318 else vretvalresult = vinit;
321 %# Initialize the OutputFcn
322 if (vhaveoutputfunction)
323 feval (vodeoptions.OutputFcn, vslot', ...
324 vretvalresult', 'init', vfunarguments{:});
327 %# Initialize the History
328 if (isnumeric (vhist))
330 vhavehistnumeric = true;
331 else %# it must be a function handle
332 for vcnt = 1:length (vlags);
333 vhmat(:,vcnt) = feval (vhist, (vslot(1)-vlags(vcnt)), vfunarguments{:});
335 vhavehistnumeric = false;
338 %# Initialize DDE variables for history calculation
339 vsaveddetime = [vtimestamp - vlags, vtimestamp]';
340 vsaveddeinput = [vhmat, vinit']';
341 vsavedderesult = [vhmat, vinit']';
343 %# Initialize the EventFcn
344 if (vhaveeventfunction)
345 odepkg_event_handle (vodeoptions.Events, vtimestamp, ...
346 {vretvalresult', vhmat}, 'init', vfunarguments{:});
349 vpow = 1/5; %# 20071016, reported by Luis Randez
350 va = [0, 0, 0, 0, 0, 0; %# The Dormand-Prince 5(4) coefficients
351 1/5, 0, 0, 0, 0, 0; %# Coefficients proved on 20060827
352 3/40, 9/40, 0, 0, 0, 0; %# See p.91 in Ascher & Petzold
353 44/45, -56/15, 32/9, 0, 0, 0;
354 19372/6561, -25360/2187, 64448/6561, -212/729, 0, 0;
355 9017/3168, -355/33, 46732/5247, 49/176, -5103/18656, 0;
356 35/384, 0, 500/1113, 125/192, -2187/6784, 11/84];
357 %# 4th and 5th order b-coefficients
358 vb4 = [35/384; 0; 500/1113; 125/192; -2187/6784; 11/84; 0];
359 vb5 = [5179/57600; 0; 7571/16695; 393/640; -92097/339200; 187/2100; 1/40];
362 %# The solver main loop - stop if the endpoint has been reached
363 vcntloop = 2; vcntcycles = 1; vu = vinit; vk = vu' * zeros(1,7);
364 vcntiter = 0; vunhandledtermination = true;
365 while ((vtimestamp < vtimestop && vstepsize >= vminstepsize))
367 %# Hit the endpoint of the time slot exactely
368 if ((vtimestamp + vstepsize) > vtimestop)
369 vstepsize = vtimestop - vtimestamp; end
371 %# Estimate the seven results when using this solver
373 vthetime = vtimestamp + vc(j,1) * vstepsize;
374 vtheinput = vu' + vstepsize * vk(:,1:j-1) * va(j,1:j-1)';
375 %# Claculate the history values (or get them from an external
376 %# function) that are needed for the next step of solving
377 if (vhavehistnumeric)
378 for vcnt = 1:length (vlags)
379 %# Direct implementation of a 'quadrature cubic Hermite interpolation'
380 %# found at the Faculty for Mathematics of the University of Stuttgart
381 %# http://mo.mathematik.uni-stuttgart.de/inhalt/aussage/aussage1269
382 vnumb = find (vthetime - vlags(vcnt) >= vsaveddetime);
383 velem = min (vnumb(end), length (vsaveddetime) - 1);
384 vstep = vsaveddetime(velem+1) - vsaveddetime(velem);
385 vdiff = (vthetime - vlags(vcnt) - vsaveddetime(velem)) / vstep;
387 %# Calculation of the coefficients for the interpolation algorithm
388 vua = (1 + 2 * vdiff) * vsubs^2;
389 vub = (3 - 2 * vdiff) * vdiff^2;
390 vva = vstep * vdiff * vsubs^2;
391 vvb = -vstep * vsubs * vdiff^2;
392 vhmat(:,vcnt) = vua * vsaveddeinput(velem,:)' + ...
393 vub * vsaveddeinput(velem+1,:)' + ...
394 vva * vsavedderesult(velem,:)' + ...
395 vvb * vsavedderesult(velem+1,:)';
397 else %# the history must be a function handle
398 for vcnt = 1:length (vlags)
399 vhmat(:,vcnt) = feval ...
400 (vhist, vthetime - vlags(vcnt), vfunarguments{:});
404 if (vhavemasshandle) %# Handle only the dynamic mass matrix,
405 if (vmassdependence) %# constant mass matrices have already
406 vmass = feval ... %# been set before (if any)
407 (vodeoptions.Mass, vthetime, vtheinput, vfunarguments{:});
408 else %# if (vmassdependence == false)
409 vmass = feval ... %# then we only have the time argument
410 (vodeoptions.Mass, vthetime, vfunarguments{:});
412 vk(:,j) = vmass \ feval ...
413 (vfun, vthetime, vtheinput, vhmat, vfunarguments{:});
416 (vfun, vthetime, vtheinput, vhmat, vfunarguments{:});
420 %# Compute the 4th and the 5th order estimation
421 y4 = vu' + vstepsize * (vk * vb4);
422 y5 = vu' + vstepsize * (vk * vb5);
423 if (vhavenonnegative)
424 vu(vodeoptions.NonNegative) = abs (vu(vodeoptions.NonNegative));
425 y4(vodeoptions.NonNegative) = abs (y4(vodeoptions.NonNegative));
426 y5(vodeoptions.NonNegative) = abs (y5(vodeoptions.NonNegative));
428 vSaveVUForRefine = vu;
430 %# Calculate the absolute local truncation error and the acceptable error
434 vtau = max (vodeoptions.RelTol * vu', vodeoptions.AbsTol);
436 vdelta = norm (y5 - y4, Inf);
437 vtau = max (vodeoptions.RelTol * max (norm (vu', Inf), 1.0), ...
440 else %# if (vstepsizegiven == true)
441 vdelta = 1; vtau = 2;
444 %# If the error is acceptable then update the vretval variables
445 if (all (vdelta <= vtau))
446 vtimestamp = vtimestamp + vstepsize;
447 vu = y5'; %# MC2001: the higher order estimation as "local extrapolation"
448 vretvaltime(vcntloop,:) = vtimestamp;
449 if (vhaveoutputselection)
450 vretvalresult(vcntloop,:) = vu(vodeoptions.OutputSel);
452 vretvalresult(vcntloop,:) = vu;
454 vcntloop = vcntloop + 1; vcntiter = 0;
456 %# Update DDE values for next history calculation
457 vsaveddetime(end+1) = vtimestamp;
458 vsaveddeinput(end+1,:) = vtheinput';
459 vsavedderesult(end+1,:) = vu;
461 %# Call plot only if a valid result has been found, therefore this
462 %# code fragment has moved here. Stop integration if plot function
464 if (vhaveoutputfunction)
465 if (vhaverefine) %# Do interpolation
466 for vcnt = 0:vodeoptions.Refine %# Approximation between told and t
467 vapproxtime = (vcnt + 1) * vstepsize / (vodeoptions.Refine + 2);
468 vapproxvals = vSaveVUForRefine' + vapproxtime * (vk * vb5);
469 if (vhaveoutputselection)
470 vapproxvals = vapproxvals(vodeoptions.OutputSel);
472 feval (vodeoptions.OutputFcn, (vtimestamp - vstepsize) + vapproxtime, ...
473 vapproxvals, [], vfunarguments{:});
476 vpltret = feval (vodeoptions.OutputFcn, vtimestamp, ...
477 vretvalresult(vcntloop-1,:)', [], vfunarguments{:});
478 if (vpltret), vunhandledtermination = false; break; end
481 %# Call event only if a valid result has been found, therefore this
482 %# code fragment has moved here. Stop integration if veventbreak is
484 if (vhaveeventfunction)
486 odepkg_event_handle (vodeoptions.Events, vtimestamp, ...
487 {vu(:), vhmat}, [], vfunarguments{:});
488 if (~isempty (vevent{1}) && vevent{1} == 1)
489 vretvaltime(vcntloop-1,:) = vevent{3}(end,:);
490 vretvalresult(vcntloop-1,:) = vevent{4}(end,:);
491 vunhandledtermination = false; break;
494 end %# If the error is acceptable ...
496 %# Update the step size for the next integration step
498 %# vdelta may be 0 or even negative - could be an iteration problem
499 vdelta = max (vdelta, eps);
500 vstepsize = min (vodeoptions.MaxStep, ...
501 min (0.8 * vstepsize * (vtau ./ vdelta) .^ vpow));
502 elseif (vstepsizegiven)
503 if (vcntloop < vtimelength)
504 vstepsize = vslot(1,vcntloop-1) - vslot(1,vcntloop-2);
508 %# Update counters that count the number of iteration cycles
509 vcntcycles = vcntcycles + 1; %# Needed for postprocessing
510 vcntiter = vcntiter + 1; %# Needed to find iteration problems
512 %# Stop solving because the last 1000 steps no successful valid
513 %# value has been found
514 if (vcntiter >= 5000)
515 error (['Solving has not been successful. The iterative', ...
516 ' integration loop exited at time t = %f before endpoint at', ...
517 ' tend = %f was reached. This happened because the iterative', ...
518 ' integration loop does not find a valid solution at this time', ...
519 ' stamp. Try to reduce the value of "InitialStep" and/or', ...
520 ' "MaxStep" with the command "odeset".\n'], vtimestamp, vtimestop);
525 %# Check if integration of the ode has been successful
526 if (vtimestamp < vtimestop)
527 if (vunhandledtermination == true)
528 error (['Solving has not been successful. The iterative', ...
529 ' integration loop exited at time t = %f', ...
530 ' before endpoint at tend = %f was reached. This may', ...
531 ' happen if the stepsize grows smaller than defined in', ...
532 ' vminstepsize. Try to reduce the value of "InitialStep" and/or', ...
533 ' "MaxStep" with the command "odeset".\n'], vtimestamp, vtimestop);
535 warning ('OdePkg:HideWarning', ...
536 ['Solver has been stopped by a call of "break" in', ...
537 ' the main iteration loop at time t = %f before endpoint at', ...
538 ' tend = %f was reached. This may happen because the @odeplot', ...
539 ' function returned "true" or the @event function returned "true".'], ...
540 vtimestamp, vtimestop);
544 %# Postprocessing, do whatever when terminating integration algorithm
545 if (vhaveoutputfunction) %# Cleanup plotter
546 feval (vodeoptions.OutputFcn, vtimestamp, ...
547 vretvalresult(vcntloop-1,:)', 'done', vfunarguments{:});
549 if (vhaveeventfunction) %# Cleanup event function handling
550 odepkg_event_handle (vodeoptions.Events, vtimestamp, ...
551 {vretvalresult(vcntloop-1,:), vhmat}, 'done', vfunarguments{:});
554 %# Print additional information if option Stats is set
555 if (strcmp (vodeoptions.Stats, 'on'))
557 vnsteps = vcntloop-2; %# vcntloop from 2..end
558 vnfailed = (vcntcycles-1)-(vcntloop-2)+1; %# vcntcycl from 1..end
559 vnfevals = 7*(vcntcycles-1); %# number of ode evaluations
560 vndecomps = 0; %# number of LU decompositions
561 vnpds = 0; %# number of partial derivatives
562 vnlinsols = 0; %# no. of solutions of linear systems
563 %# Print cost statistics if no output argument is given
565 vmsg = fprintf (1, 'Number of successful steps: %d', vnsteps);
566 vmsg = fprintf (1, 'Number of failed attempts: %d', vnfailed);
567 vmsg = fprintf (1, 'Number of function calls: %d', vnfevals);
569 else vhavestats = false;
572 if (nargout == 1) %# Sort output variables, depends on nargout
573 varargout{1}.x = vretvaltime; %# Time stamps are saved in field x
574 varargout{1}.y = vretvalresult; %# Results are saved in field y
575 varargout{1}.solver = 'ode54d'; %# Solver name is saved in field solver
576 if (vhaveeventfunction)
577 varargout{1}.ie = vevent{2}; %# Index info which event occured
578 varargout{1}.xe = vevent{3}; %# Time info when an event occured
579 varargout{1}.ye = vevent{4}; %# Results when an event occured
582 varargout{1}.stats = struct;
583 varargout{1}.stats.nsteps = vnsteps;
584 varargout{1}.stats.nfailed = vnfailed;
585 varargout{1}.stats.nfevals = vnfevals;
586 varargout{1}.stats.npds = vnpds;
587 varargout{1}.stats.ndecomps = vndecomps;
588 varargout{1}.stats.nlinsols = vnlinsols;
590 elseif (nargout == 2)
591 varargout{1} = vretvaltime; %# Time stamps are first output argument
592 varargout{2} = vretvalresult; %# Results are second output argument
593 elseif (nargout == 5)
594 varargout{1} = vretvaltime; %# Same as (nargout == 2)
595 varargout{2} = vretvalresult; %# Same as (nargout == 2)
596 varargout{3} = []; %# LabMat doesn't accept lines like
597 varargout{4} = []; %# varargout{3} = varargout{4} = [];
599 if (vhaveeventfunction)
600 varargout{3} = vevent{3}; %# Time info when an event occured
601 varargout{4} = vevent{4}; %# Results when an event occured
602 varargout{5} = vevent{2}; %# Index info which event occured
604 %# else nothing will be returned, varargout{1} undefined
607 %! # We are using a "pseudo-DDE" implementation for all tests that
608 %! # are done for this function. We also define an Events and a
609 %! # pseudo-Mass implementation. For further tests we also define a
610 %! # reference solution (computed at high accuracy) and an OutputFcn.
611 %!function [vyd] = fexp (vt, vy, vz, varargin)
612 %! vyd(1,1) = exp (- vt) - vz(1); %# The DDEs that are
613 %! vyd(2,1) = vy(1) - vz(2); %# used for all examples
614 %!function [vval, vtrm, vdir] = feve (vt, vy, vz, varargin)
615 %! vval = fexp (vt, vy, vz); %# We use the derivatives
616 %! vtrm = zeros (2,1); %# don't stop solving here
617 %! vdir = ones (2,1); %# in positive direction
618 %!function [vval, vtrm, vdir] = fevn (vt, vy, vz, varargin)
619 %! vval = fexp (vt, vy, vz); %# We use the derivatives
620 %! vtrm = ones (2,1); %# stop solving here
621 %! vdir = ones (2,1); %# in positive direction
622 %!function [vmas] = fmas (vt, vy, vz, varargin)
623 %! vmas = [1, 0; 0, 1]; %# Dummy mass matrix for tests
624 %!function [vmas] = fmsa (vt, vy, vz, varargin)
625 %! vmas = sparse ([1, 0; 0, 1]); %# A dummy sparse matrix
626 %!function [vref] = fref () %# The reference solution
627 %! vref = [0.12194462133618, 0.01652432423938];
628 %!function [vout] = fout (vt, vy, vflag, varargin)
629 %! if (regexp (char (vflag), 'init') == 1)
630 %! if (any (size (vt) ~= [2, 1])) error ('"fout" step "init"'); end
631 %! elseif (isempty (vflag))
632 %! if (any (size (vt) ~= [1, 1])) error ('"fout" step "calc"'); end
634 %! elseif (regexp (char (vflag), 'done') == 1)
635 %! if (any (size (vt) ~= [1, 1])) error ('"fout" step "done"'); end
636 %! else error ('"fout" invalid vflag');
639 %! %# Turn off output of warning messages for all tests, turn them on
640 %! %# again if the last test is called
641 %!error %# input argument number one
642 %! warning ('off', 'OdePkg:InvalidOption');
643 %! B = ode54d (1, [0 5], [1; 0], 1, [1; 0]);
644 %!error %# input argument number two
645 %! B = ode54d (@fexp, 1, [1; 0], 1, [1; 0]);
646 %!error %# input argument number three
647 %! B = ode54d (@fexp, [0 5], 1, 1, [1; 0]);
648 %!error %# input argument number four
649 %! B = ode54d (@fexp, [0 5], [1; 0], [1; 1], [1; 0]);
650 %!error %# input argument number five
651 %! B = ode54d (@fexp, [0 5], [1; 0], 1, 1);
652 %!test %# one output argument
653 %! vsol = ode54d (@fexp, [0 5], [1; 0], 1, [1; 0]);
654 %! assert ([vsol.x(end), vsol.y(end,:)], [5, fref], 1e-1);
655 %! assert (isfield (vsol, 'solver'));
656 %! assert (vsol.solver, 'ode54d');
657 %!test %# two output arguments
658 %! [vt, vy] = ode54d (@fexp, [0 5], [1; 0], 1, [1; 0]);
659 %! assert ([vt(end), vy(end,:)], [5, fref], 1e-1);
660 %!test %# five output arguments and no Events
661 %! [vt, vy, vxe, vye, vie] = ode54d (@fexp, [0 5], [1; 0], 1, [1; 0]);
662 %! assert ([vt(end), vy(end,:)], [5, fref], 1e-1);
663 %! assert ([vie, vxe, vye], []);
664 %!test %# anonymous function instead of real function
665 %! faym = @(vt, vy, vz) [exp(-vt) - vz(1); vy(1) - vz(2)];
666 %! vsol = ode54d (faym, [0 5], [1; 0], 1, [1; 0]);
667 %! assert ([vsol.x(end), vsol.y(end,:)], [5, fref], 1e-1);
668 %!test %# extra input arguments passed trhough
669 %! vsol = ode54d (@fexp, [0 5], [1; 0], 1, [1; 0], 'KL');
670 %! assert ([vsol.x(end), vsol.y(end,:)], [5, fref], 1e-1);
671 %!test %# empty OdePkg structure *but* extra input arguments
673 %! vsol = ode54d (@fexp, [0 5], [1; 0], 1, [1; 0], vopt, 12, 13, 'KL');
674 %! assert ([vsol.x(end), vsol.y(end,:)], [5, fref], 1e-1);
675 %!error %# strange OdePkg structure
676 %! vopt = struct ('foo', 1);
677 %! vsol = ode54d (@fexp, [0 5], [1; 0], 1, [1; 0], vopt);
678 %!test %# AbsTol option
679 %! vopt = odeset ('AbsTol', 1e-5);
680 %! vsol = ode54d (@fexp, [0 5], [1; 0], 1, [1; 0], vopt);
681 %! assert ([vsol.x(end), vsol.y(end,:)], [5, fref], 1e-1);
682 %!test %# AbsTol and RelTol option
683 %! vopt = odeset ('AbsTol', 1e-7, 'RelTol', 1e-7);
684 %! vsol = ode54d (@fexp, [0 5], [1; 0], 1, [1; 0], vopt);
685 %! assert ([vsol.x(end), vsol.y(end,:)], [5, fref], 1e-1);
686 %!test %# RelTol and NormControl option
687 %! vopt = odeset ('AbsTol', 1e-7, 'NormControl', 'on');
688 %! vsol = ode54d (@fexp, [0 5], [1; 0], 1, [1; 0], vopt);
689 %! assert ([vsol.x(end), vsol.y(end,:)], [5, fref], .5e-1);
690 %!test %# NonNegative for second component
691 %! vopt = odeset ('NonNegative', 1);
692 %! vsol = ode54d (@fexp, [0 2.5], [1; 0], 1, [1; 0], vopt);
693 %! assert ([vsol.x(end), vsol.y(end,:)], [2.5, 0.001, 0.237], 1e-1);
694 %!test %# Details of OutputSel and Refine can't be tested
695 %! vopt = odeset ('OutputFcn', @fout, 'OutputSel', 1, 'Refine', 5);
696 %! vsol = ode54d (@fexp, [0 2.5], [1; 0], 1, [1; 0], vopt);
697 %!test %# Stats must add further elements in vsol
698 %! vopt = odeset ('Stats', 'on');
699 %! vsol = ode54d (@fexp, [0 2.5], [1; 0], 1, [1; 0], vopt);
700 %! assert (isfield (vsol, 'stats'));
701 %! assert (isfield (vsol.stats, 'nsteps'));
702 %!test %# InitialStep option
703 %! vopt = odeset ('InitialStep', 1e-8);
704 %! vsol = ode54d (@fexp, [0 5], [1; 0], 1, [1; 0], vopt);
705 %! assert ([vsol.x(end), vsol.y(end,:)], [5, fref], 1e-1);
706 %!test %# MaxStep option
707 %! vopt = odeset ('MaxStep', 1e-2);
708 %! vsol = ode54d (@fexp, [0 5], [1; 0], 1, [1; 0], vopt);
709 %! assert ([vsol.x(end), vsol.y(end,:)], [5, fref], 1e-1);
710 %!test %# Events option add further elements in vsol
711 %! vopt = odeset ('Events', @feve);
712 %! vsol = ode54d (@fexp, [0 5], [1; 0], 1, [1; 0], vopt);
713 %! assert (isfield (vsol, 'ie'));
714 %! assert (vsol.ie, [1; 1]);
715 %! assert (isfield (vsol, 'xe'));
716 %! assert (isfield (vsol, 'ye'));
717 %!test %# Events option, now stop integration
718 %! warning ('off', 'OdePkg:HideWarning');
719 %! vopt = odeset ('Events', @fevn, 'NormControl', 'on');
720 %! vsol = ode54d (@fexp, [0 5], [1; 0], 1, [1; 0], vopt);
721 %! assert ([vsol.ie, vsol.xe, vsol.ye], ...
722 %! [1.0000, 2.9219, -0.2127, -0.2671], 1e-1);
723 %!test %# Events option, five output arguments
724 %! vopt = odeset ('Events', @fevn, 'NormControl', 'on');
725 %! [vt, vy, vxe, vye, vie] = ode54d (@fexp, [0 5], [1; 0], 1, [1; 0], vopt);
726 %! assert ([vie, vxe, vye], ...
727 %! [1.0000, 2.9219, -0.2127, -0.2671], 1e-1);
729 %! %# test for Jacobian option is missing
730 %! %# test for Jacobian (being a sparse matrix) is missing
731 %! %# test for JPattern option is missing
732 %! %# test for Vectorized option is missing
733 %! %# test for NewtonTol option is missing
734 %! %# test for MaxNewtonIterations option is missing
736 %!test %# Mass option as function
737 %! vopt = odeset ('Mass', eye (2,2));
738 %! vsol = ode54d (@fexp, [0 5], [1; 0], 1, [1; 0], vopt);
739 %! assert ([vsol.x(end), vsol.y(end,:)], [5, fref], 1e-1);
740 %!test %# Mass option as matrix
741 %! vopt = odeset ('Mass', eye (2,2));
742 %! vsol = ode54d (@fexp, [0 5], [1; 0], 1, [1; 0], vopt);
743 %! assert ([vsol.x(end), vsol.y(end,:)], [5, fref], 1e-1);
744 %!test %# Mass option as sparse matrix
745 %! vopt = odeset ('Mass', sparse (eye (2,2)));
746 %! vsol = ode54d (@fexp, [0 5], [1; 0], 1, [1; 0], vopt);
747 %! assert ([vsol.x(end), vsol.y(end,:)], [5, fref], 1e-1);
748 %!test %# Mass option as function and sparse matrix
749 %! vopt = odeset ('Mass', @fmsa);
750 %! vsol = ode54d (@fexp, [0 5], [1; 0], 1, [1; 0], vopt);
751 %! assert ([vsol.x(end), vsol.y(end,:)], [5, fref], 1e-1);
752 %!test %# Mass option as function and MStateDependence
753 %! vopt = odeset ('Mass', @fmas, 'MStateDependence', 'strong');
754 %! vsol = ode54d (@fexp, [0 5], [1; 0], 1, [1; 0], vopt);
755 %! assert ([vsol.x(end), vsol.y(end,:)], [5, fref], 1e-1);
756 %!test %# Set BDF option to something else than default
757 %! vopt = odeset ('BDF', 'on');
758 %! [vt, vy] = ode54d (@fexp, [0 5], [1; 0], 1, [1; 0], vopt);
759 %! assert ([vt(end), vy(end,:)], [5, fref], 0.5);
761 %! %# test for MvPattern option is missing
762 %! %# test for InitialSlope option is missing
763 %! %# test for MaxOrder option is missing
765 %! warning ('on', 'OdePkg:InvalidOption');
767 %# Local Variables: ***