X-Git-Url: https://git.creatis.insa-lyon.fr/pubgit/?p=CreaPhase.git;a=blobdiff_plain;f=octave_packages%2Fodepkg-0.8.2%2Fode45.m;fp=octave_packages%2Fodepkg-0.8.2%2Fode45.m;h=7d8f49f29f5f1fae5401d67e571acb99e236dec4;hp=0000000000000000000000000000000000000000;hb=c880e8788dfc484bf23ce13fa2787f2c6bca4863;hpb=1705066eceaaea976f010f669ce8e972f3734b05 diff --git a/octave_packages/odepkg-0.8.2/ode45.m b/octave_packages/odepkg-0.8.2/ode45.m new file mode 100644 index 0000000..7d8f49f --- /dev/null +++ b/octave_packages/odepkg-0.8.2/ode45.m @@ -0,0 +1,757 @@ +%# Copyright (C) 2006-2012, Thomas Treichl +%# OdePkg - A package for solving ordinary differential equations and more +%# +%# This program 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 of the License, or +%# (at your option) any later version. +%# +%# This program 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 this program; If not, see . + +%# -*- texinfo -*- +%# @deftypefn {Function File} {[@var{}] =} ode45 (@var{@@fun}, @var{slot}, @var{init}, [@var{opt}], [@var{par1}, @var{par2}, @dots{}]) +%# @deftypefnx {Command} {[@var{sol}] =} ode45 (@var{@@fun}, @var{slot}, @var{init}, [@var{opt}], [@var{par1}, @var{par2}, @dots{}]) +%# @deftypefnx {Command} {[@var{t}, @var{y}, [@var{xe}, @var{ye}, @var{ie}]] =} ode45 (@var{@@fun}, @var{slot}, @var{init}, [@var{opt}], [@var{par1}, @var{par2}, @dots{}]) +%# +%# This function file can be used to solve a set of non--stiff ordinary differential equations (non--stiff ODEs) or non--stiff differential algebraic equations (non--stiff DAEs) with the well known explicit Runge--Kutta method of order (4,5). +%# +%# If this function is called with no return argument then plot the solution over time in a figure window while solving the set of ODEs 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{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}. +%# +%# If this function is called with one return argument then return the solution @var{sol} of type structure array after solving the set of ODEs. 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}. +%# +%# 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. +%# +%# For example, solve an anonymous implementation of the Van der Pol equation +%# +%# @example +%# fvdb = @@(vt,vy) [vy(2); (1 - vy(1)^2) * vy(2) - vy(1)]; +%# +%# vopt = odeset ("RelTol", 1e-3, "AbsTol", 1e-3, \ +%# "NormControl", "on", "OutputFcn", @@odeplot); +%# ode45 (fvdb, [0 20], [2 0], vopt); +%# @end example +%# @end deftypefn +%# +%# @seealso{odepkg} + +%# ChangeLog: +%# 20010703 the function file "ode45.m" was written by Marc Compere +%# under the GPL for the use with this software. This function has been +%# taken as a base for the following implementation. +%# 20060810, Thomas Treichl +%# This function was adapted to the new syntax that is used by the +%# new OdePkg for Octave and is compatible to Matlab's ode45. + +function [varargout] = ode45 (vfun, vslot, vinit, varargin) + + if (nargin == 0) %# Check number and types of all input arguments + help ('ode45'); + error ('OdePkg:InvalidArgument', ... + 'Number of input arguments must be greater than zero'); + + elseif (nargin < 3) + print_usage; + + elseif ~(isa (vfun, 'function_handle') || isa (vfun, 'inline')) + error ('OdePkg:InvalidArgument', ... + 'First input argument must be a valid function handle'); + + elseif (~isvector (vslot) || length (vslot) < 2) + error ('OdePkg:InvalidArgument', ... + 'Second input argument must be a valid vector'); + + elseif (~isvector (vinit) || ~isnumeric (vinit)) + error ('OdePkg:InvalidArgument', ... + 'Third input argument must be a valid numerical value'); + + elseif (nargin >= 4) + + if (~isstruct (varargin{1})) + %# varargin{1:len} are parameters for vfun + vodeoptions = odeset; + vfunarguments = varargin; + + elseif (length (varargin) > 1) + %# varargin{1} is an OdePkg options structure vopt + vodeoptions = odepkg_structure_check (varargin{1}, 'ode45'); + vfunarguments = {varargin{2:length(varargin)}}; + + else %# if (isstruct (varargin{1})) + vodeoptions = odepkg_structure_check (varargin{1}, 'ode45'); + vfunarguments = {}; + + end + + else %# if (nargin == 3) + vodeoptions = odeset; + vfunarguments = {}; + end + + %# Start preprocessing, have a look which options are set in + %# vodeoptions, check if an invalid or unused option is set + vslot = vslot(:).'; %# Create a row vector + vinit = vinit(:).'; %# Create a row vector + if (length (vslot) > 2) %# Step size checking + vstepsizefixed = true; + else + vstepsizefixed = false; + end + + %# Get the default options that can be set with 'odeset' temporarily + vodetemp = odeset; + + %# Implementation of the option RelTol has been finished. This option + %# can be set by the user to another value than default value. + if (isempty (vodeoptions.RelTol) && ~vstepsizefixed) + vodeoptions.RelTol = 1e-6; + warning ('OdePkg:InvalidArgument', ... + 'Option "RelTol" not set, new value %f is used', vodeoptions.RelTol); + elseif (~isempty (vodeoptions.RelTol) && vstepsizefixed) + warning ('OdePkg:InvalidArgument', ... + 'Option "RelTol" will be ignored if fixed time stamps are given'); + end + + %# Implementation of the option AbsTol has been finished. This option + %# can be set by the user to another value than default value. + if (isempty (vodeoptions.AbsTol) && ~vstepsizefixed) + vodeoptions.AbsTol = 1e-6; + warning ('OdePkg:InvalidArgument', ... + 'Option "AbsTol" not set, new value %f is used', vodeoptions.AbsTol); + elseif (~isempty (vodeoptions.AbsTol) && vstepsizefixed) + warning ('OdePkg:InvalidArgument', ... + 'Option "AbsTol" will be ignored if fixed time stamps are given'); + else + vodeoptions.AbsTol = vodeoptions.AbsTol(:); %# Create column vector + end + + %# Implementation of the option NormControl has been finished. This + %# option can be set by the user to another value than default value. + if (strcmp (vodeoptions.NormControl, 'on')) vnormcontrol = true; + else vnormcontrol = false; end + + %# Implementation of the option NonNegative has been finished. This + %# option can be set by the user to another value than default value. + if (~isempty (vodeoptions.NonNegative)) + if (isempty (vodeoptions.Mass)), vhavenonnegative = true; + else + vhavenonnegative = false; + warning ('OdePkg:InvalidArgument', ... + 'Option "NonNegative" will be ignored if mass matrix is set'); + end + else vhavenonnegative = false; + end + + %# Implementation of the option OutputFcn has been finished. This + %# option can be set by the user to another value than default value. + if (isempty (vodeoptions.OutputFcn) && nargout == 0) + vodeoptions.OutputFcn = @odeplot; + vhaveoutputfunction = true; + elseif (isempty (vodeoptions.OutputFcn)), vhaveoutputfunction = false; + else vhaveoutputfunction = true; + end + + %# Implementation of the option OutputSel has been finished. This + %# option can be set by the user to another value than default value. + if (~isempty (vodeoptions.OutputSel)), vhaveoutputselection = true; + else vhaveoutputselection = false; end + + %# Implementation of the option OutputSave has been finished. This + %# option can be set by the user to another value than default value. + if (isempty (vodeoptions.OutputSave)), vodeoptions.OutputSave = 1; + end + + %# Implementation of the option Refine has been finished. This option + %# can be set by the user to another value than default value. + if (vodeoptions.Refine > 0), vhaverefine = true; + else vhaverefine = false; end + + %# Implementation of the option Stats has been finished. This option + %# can be set by the user to another value than default value. + + %# Implementation of the option InitialStep has been finished. This + %# option can be set by the user to another value than default value. + if (isempty (vodeoptions.InitialStep) && ~vstepsizefixed) + vodeoptions.InitialStep = (vslot(1,2) - vslot(1,1)) / 10; + vodeoptions.InitialStep = vodeoptions.InitialStep / 10^vodeoptions.Refine; + warning ('OdePkg:InvalidArgument', ... + 'Option "InitialStep" not set, new value %f is used', vodeoptions.InitialStep); + end + + %# Implementation of the option MaxStep has been finished. This option + %# can be set by the user to another value than default value. + if (isempty (vodeoptions.MaxStep) && ~vstepsizefixed) + vodeoptions.MaxStep = abs (vslot(1,2) - vslot(1,1)) / 10; + warning ('OdePkg:InvalidArgument', ... + 'Option "MaxStep" not set, new value %f is used', vodeoptions.MaxStep); + end + + %# Implementation of the option Events has been finished. This option + %# can be set by the user to another value than default value. + if (~isempty (vodeoptions.Events)), vhaveeventfunction = true; + else vhaveeventfunction = false; end + + %# The options 'Jacobian', 'JPattern' and 'Vectorized' will be ignored + %# by this solver because this solver uses an explicit Runge-Kutta + %# method and therefore no Jacobian calculation is necessary + if (~isequal (vodeoptions.Jacobian, vodetemp.Jacobian)) + warning ('OdePkg:InvalidArgument', ... + 'Option "Jacobian" will be ignored by this solver'); + end + if (~isequal (vodeoptions.JPattern, vodetemp.JPattern)) + warning ('OdePkg:InvalidArgument', ... + 'Option "JPattern" will be ignored by this solver'); + end + if (~isequal (vodeoptions.Vectorized, vodetemp.Vectorized)) + warning ('OdePkg:InvalidArgument', ... + 'Option "Vectorized" will be ignored by this solver'); + end + if (~isequal (vodeoptions.NewtonTol, vodetemp.NewtonTol)) + warning ('OdePkg:InvalidArgument', ... + 'Option "NewtonTol" will be ignored by this solver'); + end + if (~isequal (vodeoptions.MaxNewtonIterations,... + vodetemp.MaxNewtonIterations)) + warning ('OdePkg:InvalidArgument', ... + 'Option "MaxNewtonIterations" will be ignored by this solver'); + end + + %# Implementation of the option Mass has been finished. This option + %# can be set by the user to another value than default value. + if (~isempty (vodeoptions.Mass) && isnumeric (vodeoptions.Mass)) + vhavemasshandle = false; vmass = vodeoptions.Mass; %# constant mass + elseif (isa (vodeoptions.Mass, 'function_handle')) + vhavemasshandle = true; %# mass defined by a function handle + else %# no mass matrix - creating a diag-matrix of ones for mass + vhavemasshandle = false; %# vmass = diag (ones (length (vinit), 1), 0); + end + + %# Implementation of the option MStateDependence has been finished. + %# This option can be set by the user to another value than default + %# value. + if (strcmp (vodeoptions.MStateDependence, 'none')) + vmassdependence = false; + else vmassdependence = true; + end + + %# Other options that are not used by this solver. Print a warning + %# message to tell the user that the option(s) is/are ignored. + if (~isequal (vodeoptions.MvPattern, vodetemp.MvPattern)) + warning ('OdePkg:InvalidArgument', ... + 'Option "MvPattern" will be ignored by this solver'); + end + if (~isequal (vodeoptions.MassSingular, vodetemp.MassSingular)) + warning ('OdePkg:InvalidArgument', ... + 'Option "MassSingular" will be ignored by this solver'); + end + if (~isequal (vodeoptions.InitialSlope, vodetemp.InitialSlope)) + warning ('OdePkg:InvalidArgument', ... + 'Option "InitialSlope" will be ignored by this solver'); + end + if (~isequal (vodeoptions.MaxOrder, vodetemp.MaxOrder)) + warning ('OdePkg:InvalidArgument', ... + 'Option "MaxOrder" will be ignored by this solver'); + end + if (~isequal (vodeoptions.BDF, vodetemp.BDF)) + warning ('OdePkg:InvalidArgument', ... + 'Option "BDF" will be ignored by this solver'); + end + + %# Starting the initialisation of the core solver ode45 + vtimestamp = vslot(1,1); %# timestamp = start time + vtimelength = length (vslot); %# length needed if fixed steps + vtimestop = vslot(1,vtimelength); %# stop time = last value + %# 20110611, reported by Nils Strunk + %# Make it possible to solve equations from negativ to zero, + %# eg. vres = ode45 (@(t,y) y, [-2 0], 2); + vdirection = sign (vtimestop - vtimestamp); %# Direction flag + + if (~vstepsizefixed) + if (sign (vodeoptions.InitialStep) == vdirection) + vstepsize = vodeoptions.InitialStep; + else %# Fix wrong direction of InitialStep. + vstepsize = - vodeoptions.InitialStep; + end + vminstepsize = (vtimestop - vtimestamp) / (1/eps); + else %# If step size is given then use the fixed time steps + vstepsize = vslot(1,2) - vslot(1,1); + vminstepsize = sign (vstepsize) * eps; + end + + vretvaltime = vtimestamp; %# first timestamp output + vretvalresult = vinit; %# first solution output + + %# Initialize the OutputFcn + if (vhaveoutputfunction) + if (vhaveoutputselection) vretout = vretvalresult(vodeoptions.OutputSel); + else vretout = vretvalresult; end + feval (vodeoptions.OutputFcn, vslot.', ... + vretout.', 'init', vfunarguments{:}); + end + + %# Initialize the EventFcn + if (vhaveeventfunction) + odepkg_event_handle (vodeoptions.Events, vtimestamp, ... + vretvalresult.', 'init', vfunarguments{:}); + end + + vpow = 1/5; %# 20071016, reported by Luis Randez + va = [0, 0, 0, 0, 0; %# The Runge-Kutta-Fehlberg 4(5) coefficients + 1/4, 0, 0, 0, 0; %# Coefficients proved on 20060827 + 3/32, 9/32, 0, 0, 0; %# See p.91 in Ascher & Petzold + 1932/2197, -7200/2197, 7296/2197, 0, 0; + 439/216, -8, 3680/513, -845/4104, 0; + -8/27, 2, -3544/2565, 1859/4104, -11/40]; + %# 4th and 5th order b-coefficients + vb4 = [25/216; 0; 1408/2565; 2197/4104; -1/5; 0]; + vb5 = [16/135; 0; 6656/12825; 28561/56430; -9/50; 2/55]; + vc = sum (va, 2); + + %# The solver main loop - stop if the endpoint has been reached + vcntloop = 2; vcntcycles = 1; vu = vinit; vk = vu.' * zeros(1,6); + vcntiter = 0; vunhandledtermination = true; vcntsave = 2; + while ((vdirection * (vtimestamp) < vdirection * (vtimestop)) && ... + (vdirection * (vstepsize) >= vdirection * (vminstepsize))) + + %# Hit the endpoint of the time slot exactely + if (vdirection * (vtimestamp + vstepsize) > vdirection * vtimestop) + %# vstepsize = vtimestop - vdirection * vtimestamp; + %# 20110611, reported by Nils Strunk + %# The endpoint of the time slot must be hit exactly, + %# eg. vsol = ode45 (@(t,y) y, [0 -1], 1); + vstepsize = vdirection * abs (abs (vtimestop) - abs (vtimestamp)); + end + + %# Estimate the six results when using this solver + for j = 1:6 + vthetime = vtimestamp + vc(j,1) * vstepsize; + vtheinput = vu.' + vstepsize * vk(:,1:j-1) * va(j,1:j-1).'; + if (vhavemasshandle) %# Handle only the dynamic mass matrix, + if (vmassdependence) %# constant mass matrices have already + vmass = feval ... %# been set before (if any) + (vodeoptions.Mass, vthetime, vtheinput, vfunarguments{:}); + else %# if (vmassdependence == false) + vmass = feval ... %# then we only have the time argument + (vodeoptions.Mass, vthetime, vfunarguments{:}); + end + vk(:,j) = vmass \ feval ... + (vfun, vthetime, vtheinput, vfunarguments{:}); + else + vk(:,j) = feval ... + (vfun, vthetime, vtheinput, vfunarguments{:}); + end + end + + %# Compute the 4th and the 5th order estimation + y4 = vu.' + vstepsize * (vk * vb4); + y5 = vu.' + vstepsize * (vk * vb5); + if (vhavenonnegative) + vu(vodeoptions.NonNegative) = abs (vu(vodeoptions.NonNegative)); + y4(vodeoptions.NonNegative) = abs (y4(vodeoptions.NonNegative)); + y5(vodeoptions.NonNegative) = abs (y5(vodeoptions.NonNegative)); + end + if (vhaveoutputfunction && vhaverefine) + vSaveVUForRefine = vu; + end + + %# Calculate the absolute local truncation error and the acceptable error + if (~vstepsizefixed) + if (~vnormcontrol) + vdelta = abs (y5 - y4); + vtau = max (vodeoptions.RelTol * abs (vu.'), vodeoptions.AbsTol); + else + vdelta = norm (y5 - y4, Inf); + vtau = max (vodeoptions.RelTol * max (norm (vu.', Inf), 1.0), ... + vodeoptions.AbsTol); + end + else %# if (vstepsizefixed == true) + vdelta = 1; vtau = 2; + end + + %# If the error is acceptable then update the vretval variables + if (all (vdelta <= vtau)) + vtimestamp = vtimestamp + vstepsize; + vu = y5.'; %# MC2001: the higher order estimation as "local extrapolation" + %# Save the solution every vodeoptions.OutputSave steps + if (mod (vcntloop-1,vodeoptions.OutputSave) == 0) + vretvaltime(vcntsave,:) = vtimestamp; + vretvalresult(vcntsave,:) = vu; + vcntsave = vcntsave + 1; + end + vcntloop = vcntloop + 1; vcntiter = 0; + + %# Call plot only if a valid result has been found, therefore this + %# code fragment has moved here. Stop integration if plot function + %# returns false + if (vhaveoutputfunction) + for vcnt = 0:vodeoptions.Refine %# Approximation between told and t + if (vhaverefine) %# Do interpolation + vapproxtime = (vcnt + 1) * vstepsize / (vodeoptions.Refine + 2); + vapproxvals = vSaveVUForRefine.' + vapproxtime * (vk * vb5); + vapproxtime = (vtimestamp - vstepsize) + vapproxtime; + else + vapproxvals = vu.'; + vapproxtime = vtimestamp; + end + if (vhaveoutputselection) + vapproxvals = vapproxvals(vodeoptions.OutputSel); + end + vpltret = feval (vodeoptions.OutputFcn, vapproxtime, ... + vapproxvals, [], vfunarguments{:}); + if vpltret %# Leave refinement loop + break; + end + end + if (vpltret) %# Leave main loop + vunhandledtermination = false; + break; + end + end + + %# Call event only if a valid result has been found, therefore this + %# code fragment has moved here. Stop integration if veventbreak is + %# true + if (vhaveeventfunction) + vevent = ... + odepkg_event_handle (vodeoptions.Events, vtimestamp, ... + vu(:), [], vfunarguments{:}); + if (~isempty (vevent{1}) && vevent{1} == 1) + vretvaltime(vcntloop-1,:) = vevent{3}(end,:); + vretvalresult(vcntloop-1,:) = vevent{4}(end,:); + vunhandledtermination = false; break; + end + end + end %# If the error is acceptable ... + + %# Update the step size for the next integration step + if (~vstepsizefixed) + %# 20080425, reported by Marco Caliari + %# vdelta cannot be negative (because of the absolute value that + %# has been introduced) but it could be 0, then replace the zeros + %# with the maximum value of vdelta + vdelta(find (vdelta == 0)) = max (vdelta); + %# It could happen that max (vdelta) == 0 (ie. that the original + %# vdelta was 0), in that case we double the previous vstepsize + vdelta(find (vdelta == 0)) = max (vtau) .* (0.4 ^ (1 / vpow)); + + if (vdirection == 1) + vstepsize = min (vodeoptions.MaxStep, ... + min (0.8 * vstepsize * (vtau ./ vdelta) .^ vpow)); + else + vstepsize = max (- vodeoptions.MaxStep, ... + max (0.8 * vstepsize * (vtau ./ vdelta) .^ vpow)); + end + + else %# if (vstepsizefixed) + if (vcntloop <= vtimelength) + vstepsize = vslot(vcntloop) - vslot(vcntloop-1); + else %# Get out of the main integration loop + break; + end + end + + %# Update counters that count the number of iteration cycles + vcntcycles = vcntcycles + 1; %# Needed for cost statistics + vcntiter = vcntiter + 1; %# Needed to find iteration problems + + %# Stop solving because the last 1000 steps no successful valid + %# value has been found + if (vcntiter >= 5000) + error (['Solving has not been successful. The iterative', ... + ' integration loop exited at time t = %f before endpoint at', ... + ' tend = %f was reached. This happened because the iterative', ... + ' integration loop does not find a valid solution at this time', ... + ' stamp. Try to reduce the value of "InitialStep" and/or', ... + ' "MaxStep" with the command "odeset".\n'], vtimestamp, vtimestop); + end + + end %# The main loop + + %# Check if integration of the ode has been successful + if (vdirection * vtimestamp < vdirection * vtimestop) + if (vunhandledtermination == true) + error ('OdePkg:InvalidArgument', ... + ['Solving has not been successful. The iterative', ... + ' integration loop exited at time t = %f', ... + ' before endpoint at tend = %f was reached. This may', ... + ' happen if the stepsize grows smaller than defined in', ... + ' vminstepsize. Try to reduce the value of "InitialStep" and/or', ... + ' "MaxStep" with the command "odeset".\n'], vtimestamp, vtimestop); + else + warning ('OdePkg:InvalidArgument', ... + ['Solver has been stopped by a call of "break" in', ... + ' the main iteration loop at time t = %f before endpoint at', ... + ' tend = %f was reached. This may happen because the @odeplot', ... + ' function returned "true" or the @event function returned "true".'], ... + vtimestamp, vtimestop); + end + end + + %# Postprocessing, do whatever when terminating integration algorithm + if (vhaveoutputfunction) %# Cleanup plotter + feval (vodeoptions.OutputFcn, vtimestamp, ... + vu.', 'done', vfunarguments{:}); + end + if (vhaveeventfunction) %# Cleanup event function handling + odepkg_event_handle (vodeoptions.Events, vtimestamp, ... + vu.', 'done', vfunarguments{:}); + end + %# Save the last step, if not already saved + if (mod (vcntloop-2,vodeoptions.OutputSave) ~= 0) + vretvaltime(vcntsave,:) = vtimestamp; + vretvalresult(vcntsave,:) = vu; + end + + %# Print additional information if option Stats is set + if (strcmp (vodeoptions.Stats, 'on')) + vhavestats = true; + vnsteps = vcntloop-2; %# vcntloop from 2..end + vnfailed = (vcntcycles-1)-(vcntloop-2)+1; %# vcntcycl from 1..end + vnfevals = 6*(vcntcycles-1); %# number of ode evaluations + vndecomps = 0; %# number of LU decompositions + vnpds = 0; %# number of partial derivatives + vnlinsols = 0; %# no. of solutions of linear systems + %# Print cost statistics if no output argument is given + if (nargout == 0) + vmsg = fprintf (1, 'Number of successful steps: %d\n', vnsteps); + vmsg = fprintf (1, 'Number of failed attempts: %d\n', vnfailed); + vmsg = fprintf (1, 'Number of function calls: %d\n', vnfevals); + end + else + vhavestats = false; + end + + if (nargout == 1) %# Sort output variables, depends on nargout + varargout{1}.x = vretvaltime; %# Time stamps are saved in field x + varargout{1}.y = vretvalresult; %# Results are saved in field y + varargout{1}.solver = 'ode45'; %# Solver name is saved in field solver + if (vhaveeventfunction) + varargout{1}.ie = vevent{2}; %# Index info which event occured + varargout{1}.xe = vevent{3}; %# Time info when an event occured + varargout{1}.ye = vevent{4}; %# Results when an event occured + end + if (vhavestats) + varargout{1}.stats = struct; + varargout{1}.stats.nsteps = vnsteps; + varargout{1}.stats.nfailed = vnfailed; + varargout{1}.stats.nfevals = vnfevals; + varargout{1}.stats.npds = vnpds; + varargout{1}.stats.ndecomps = vndecomps; + varargout{1}.stats.nlinsols = vnlinsols; + end + elseif (nargout == 2) + varargout{1} = vretvaltime; %# Time stamps are first output argument + varargout{2} = vretvalresult; %# Results are second output argument + elseif (nargout == 5) + varargout{1} = vretvaltime; %# Same as (nargout == 2) + varargout{2} = vretvalresult; %# Same as (nargout == 2) + varargout{3} = []; %# LabMat doesn't accept lines like + varargout{4} = []; %# varargout{3} = varargout{4} = []; + varargout{5} = []; + if (vhaveeventfunction) + varargout{3} = vevent{3}; %# Time info when an event occured + varargout{4} = vevent{4}; %# Results when an event occured + varargout{5} = vevent{2}; %# Index info which event occured + end + end +end + +%! # We are using the "Van der Pol" implementation for all tests that +%! # are done for this function. We also define a Jacobian, Events, +%! # pseudo-Mass implementation. For further tests we also define a +%! # reference solution (computed at high accuracy) and an OutputFcn +%!function [ydot] = fpol (vt, vy, varargin) %# The Van der Pol +%! ydot = [vy(2); (1 - vy(1)^2) * vy(2) - vy(1)]; +%!function [vjac] = fjac (vt, vy, varargin) %# its Jacobian +%! vjac = [0, 1; -1 - 2 * vy(1) * vy(2), 1 - vy(1)^2]; +%!function [vjac] = fjcc (vt, vy, varargin) %# sparse type +%! vjac = sparse ([0, 1; -1 - 2 * vy(1) * vy(2), 1 - vy(1)^2]); +%!function [vval, vtrm, vdir] = feve (vt, vy, varargin) +%! vval = fpol (vt, vy, varargin); %# We use the derivatives +%! vtrm = zeros (2,1); %# that's why component 2 +%! vdir = ones (2,1); %# seems to not be exact +%!function [vval, vtrm, vdir] = fevn (vt, vy, varargin) +%! vval = fpol (vt, vy, varargin); %# We use the derivatives +%! vtrm = ones (2,1); %# that's why component 2 +%! vdir = ones (2,1); %# seems to not be exact +%!function [vmas] = fmas (vt, vy) +%! vmas = [1, 0; 0, 1]; %# Dummy mass matrix for tests +%!function [vmas] = fmsa (vt, vy) +%! vmas = sparse ([1, 0; 0, 1]); %# A sparse dummy matrix +%!function [vref] = fref () %# The computed reference sol +%! vref = [0.32331666704577, -1.83297456798624]; +%!function [vout] = fout (vt, vy, vflag, varargin) +%! if (regexp (char (vflag), 'init') == 1) +%! if (any (size (vt) ~= [2, 1])) error ('"fout" step "init"'); end +%! elseif (isempty (vflag)) +%! if (any (size (vt) ~= [1, 1])) error ('"fout" step "calc"'); end +%! vout = false; +%! elseif (regexp (char (vflag), 'done') == 1) +%! if (any (size (vt) ~= [1, 1])) error ('"fout" step "done"'); end +%! else error ('"fout" invalid vflag'); +%! end +%! +%! %# Turn off output of warning messages for all tests, turn them on +%! %# again if the last test is called +%!error %# input argument number one +%! warning ('off', 'OdePkg:InvalidArgument'); +%! B = ode45 (1, [0 25], [3 15 1]); +%!error %# input argument number two +%! B = ode45 (@fpol, 1, [3 15 1]); +%!error %# input argument number three +%! B = ode45 (@flor, [0 25], 1); +%!test %# one output argument +%! vsol = ode45 (@fpol, [0 2], [2 0]); +%! assert ([vsol.x(end), vsol.y(end,:)], [2, fref], 1e-3); +%! assert (isfield (vsol, 'solver')); +%! assert (vsol.solver, 'ode45'); +%!test %# two output arguments +%! [vt, vy] = ode45 (@fpol, [0 2], [2 0]); +%! assert ([vt(end), vy(end,:)], [2, fref], 1e-3); +%!test %# five output arguments and no Events +%! [vt, vy, vxe, vye, vie] = ode45 (@fpol, [0 2], [2 0]); +%! assert ([vt(end), vy(end,:)], [2, fref], 1e-3); +%! assert ([vie, vxe, vye], []); +%!test %# anonymous function instead of real function +%! fvdb = @(vt,vy) [vy(2); (1 - vy(1)^2) * vy(2) - vy(1)]; +%! vsol = ode45 (fvdb, [0 2], [2 0]); +%! assert ([vsol.x(end), vsol.y(end,:)], [2, fref], 1e-3); +%!test %# extra input arguments passed through +%! vsol = ode45 (@fpol, [0 2], [2 0], 12, 13, 'KL'); +%! assert ([vsol.x(end), vsol.y(end,:)], [2, fref], 1e-3); +%!test %# empty OdePkg structure *but* extra input arguments +%! vopt = odeset; +%! vsol = ode45 (@fpol, [0 2], [2 0], vopt, 12, 13, 'KL'); +%! assert ([vsol.x(end), vsol.y(end,:)], [2, fref], 1e-3); +%!error %# strange OdePkg structure +%! vopt = struct ('foo', 1); +%! vsol = ode45 (@fpol, [0 2], [2 0], vopt); +%!test %# Solve vdp in fixed step sizes +%! vsol = ode45 (@fpol, [0:0.1:2], [2 0]); +%! assert (vsol.x(:), [0:0.1:2]'); +%! assert (vsol.y(end,:), fref, 1e-3); +%!test %# Solve in backward direction starting at t=0 +%! vref = [-1.205364552835178, 0.951542399860817]; +%! vsol = ode45 (@fpol, [0 -2], [2 0]); +%! assert ([vsol.x(end), vsol.y(end,:)], [-2, vref], 1e-3); +%!test %# Solve in backward direction starting at t=2 +%! vref = [-1.205364552835178, 0.951542399860817]; +%! vsol = ode45 (@fpol, [2 -2], fref); +%! assert ([vsol.x(end), vsol.y(end,:)], [-2, vref], 1e-3); +%!test %# Solve another anonymous function in backward direction +%! vref = [-1, 0.367879437558975]; +%! vsol = ode45 (@(t,y) y, [0 -1], 1); +%! assert ([vsol.x(end), vsol.y(end,:)], vref, 1e-3); +%!test %# Solve another anonymous function below zero +%! vref = [0, 14.77810590694212]; +%! vsol = ode45 (@(t,y) y, [-2 0], 2); +%! assert ([vsol.x(end), vsol.y(end,:)], vref, 1e-3); +%!test %# AbsTol option +%! vopt = odeset ('AbsTol', 1e-5); +%! vsol = ode45 (@fpol, [0 2], [2 0], vopt); +%! assert ([vsol.x(end), vsol.y(end,:)], [2, fref], 1e-3); +%!test %# AbsTol and RelTol option +%! vopt = odeset ('AbsTol', 1e-8, 'RelTol', 1e-8); +%! vsol = ode45 (@fpol, [0 2], [2 0], vopt); +%! assert ([vsol.x(end), vsol.y(end,:)], [2, fref], 1e-3); +%!test %# RelTol and NormControl option -- higher accuracy +%! vopt = odeset ('RelTol', 1e-8, 'NormControl', 'on'); +%! vsol = ode45 (@fpol, [0 2], [2 0], vopt); +%! assert ([vsol.x(end), vsol.y(end,:)], [2, fref], 1e-5); +%!test %# Keeps initial values while integrating +%! vopt = odeset ('NonNegative', 2); +%! vsol = ode45 (@fpol, [0 2], [2 0], vopt); +%! assert ([vsol.x(end), vsol.y(end,:)], [2, 2, 0], 0.5); +%!test %# Details of OutputSel and Refine can't be tested +%! vopt = odeset ('OutputFcn', @fout, 'OutputSel', 1, 'Refine', 5); +%! vsol = ode45 (@fpol, [0 2], [2 0], vopt); +%!test %# Details of OutputSave can't be tested +%! vopt = odeset ('OutputSave', 1, 'OutputSel', 1); +%! vsla = ode45 (@fpol, [0 2], [2 0], vopt); +%! vopt = odeset ('OutputSave', 2); +%! vslb = ode45 (@fpol, [0 2], [2 0], vopt); +%! assert (length (vsla.x) > length (vslb.x)) +%!test %# Stats must add further elements in vsol +%! vopt = odeset ('Stats', 'on'); +%! vsol = ode45 (@fpol, [0 2], [2 0], vopt); +%! assert (isfield (vsol, 'stats')); +%! assert (isfield (vsol.stats, 'nsteps')); +%!test %# InitialStep option +%! vopt = odeset ('InitialStep', 1e-8); +%! vsol = ode45 (@fpol, [0 0.2], [2 0], vopt); +%! assert ([vsol.x(2)-vsol.x(1)], [1e-8], 1e-9); +%!test %# MaxStep option +%! vopt = odeset ('MaxStep', 1e-2); +%! vsol = ode45 (@fpol, [0 0.2], [2 0], vopt); +%! assert ([vsol.x(5)-vsol.x(4)], [1e-2], 1e-3); +%!test %# Events option add further elements in vsol +%! vopt = odeset ('Events', @feve); +%! vsol = ode45 (@fpol, [0 10], [2 0], vopt); +%! assert (isfield (vsol, 'ie')); +%! assert (vsol.ie(1), 2); +%! assert (isfield (vsol, 'xe')); +%! assert (isfield (vsol, 'ye')); +%!test %# Events option, now stop integration +%! vopt = odeset ('Events', @fevn, 'NormControl', 'on'); +%! vsol = ode45 (@fpol, [0 10], [2 0], vopt); +%! assert ([vsol.ie, vsol.xe, vsol.ye], ... +%! [2.0, 2.496110, -0.830550, -2.677589], .5e-1); +%!test %# Events option, five output arguments +%! vopt = odeset ('Events', @fevn, 'NormControl', 'on'); +%! [vt, vy, vxe, vye, vie] = ode45 (@fpol, [0 10], [2 0], vopt); +%! assert ([vie, vxe, vye], ... +%! [2.0, 2.496110, -0.830550, -2.677589], 1e-1); +%!test %# Jacobian option +%! vopt = odeset ('Jacobian', @fjac); +%! vsol = ode45 (@fpol, [0 2], [2 0], vopt); +%! assert ([vsol.x(end), vsol.y(end,:)], [2, fref], 1e-3); +%!test %# Jacobian option and sparse return value +%! vopt = odeset ('Jacobian', @fjcc); +%! vsol = ode45 (@fpol, [0 2], [2 0], vopt); +%! assert ([vsol.x(end), vsol.y(end,:)], [2, fref], 1e-3); +%! +%! %# test for JPattern option is missing +%! %# test for Vectorized option is missing +%! %# test for NewtonTol option is missing +%! %# test for MaxNewtonIterations option is missing +%! +%!test %# Mass option as function +%! vopt = odeset ('Mass', @fmas); +%! vsol = ode45 (@fpol, [0 2], [2 0], vopt); +%! assert ([vsol.x(end), vsol.y(end,:)], [2, fref], 1e-3); +%!test %# Mass option as matrix +%! vopt = odeset ('Mass', eye (2,2)); +%! vsol = ode45 (@fpol, [0 2], [2 0], vopt); +%! assert ([vsol.x(end), vsol.y(end,:)], [2, fref], 1e-3); +%!test %# Mass option as sparse matrix +%! vopt = odeset ('Mass', sparse (eye (2,2))); +%! vsol = ode45 (@fpol, [0 2], [2 0], vopt); +%! assert ([vsol.x(end), vsol.y(end,:)], [2, fref], 1e-3); +%!test %# Mass option as function and sparse matrix +%! vopt = odeset ('Mass', @fmsa); +%! vsol = ode45 (@fpol, [0 2], [2 0], vopt); +%! assert ([vsol.x(end), vsol.y(end,:)], [2, fref], 1e-3); +%!test %# Mass option as function and MStateDependence +%! vopt = odeset ('Mass', @fmas, 'MStateDependence', 'strong'); +%! vsol = ode45 (@fpol, [0 2], [2 0], vopt); +%! assert ([vsol.x(end), vsol.y(end,:)], [2, fref], 1e-3); +%!test %# Set BDF option to something else than default +%! vopt = odeset ('BDF', 'on'); +%! [vt, vy] = ode45 (@fpol, [0 2], [2 0], vopt); +%! assert ([vt(end), vy(end,:)], [2, fref], 1e-3); +%! +%! %# test for MvPattern option is missing +%! %# test for InitialSlope option is missing +%! %# test for MaxOrder option is missing +%! +%! warning ('on', 'OdePkg:InvalidArgument'); + +%# Local Variables: *** +%# mode: octave *** +%# End: *** +