--- /dev/null
+%# Copyright (C) 2009-2012, Sebastian Schoeps <schoeps AT math DOT uni-wuppertal DOT de>
+%# 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 <http://www.gnu.org/licenses/>.
+
+%# -*- texinfo -*-
+%# @deftypefn {Function File} {[@var{}] =} odebwe (@var{@@fun}, @var{slot}, @var{init}, [@var{opt}], [@var{par1}, @var{par2}, @dots{}])
+%# @deftypefnx {Command} {[@var{sol}] =} odebwe (@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}]] =} odebwe (@var{@@fun}, @var{slot}, @var{init}, [@var{opt}], [@var{par1}, @var{par2}, @dots{}])
+%#
+%# This function file can be used to solve a set of stiff ordinary differential equations (stiff ODEs) or stiff differential algebraic equations (stiff DAEs) with the Backward Euler method.
+%#
+%# 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)];
+%# vjac = @@(vt,vy) [0, 1; -1 - 2 * vy(1) * vy(2), 1 - vy(1)^2];
+%# vopt = odeset ("RelTol", 1e-3, "AbsTol", 1e-3, \
+%# "NormControl", "on", "OutputFcn", @@odeplot, \
+%# "Jacobian",vjac);
+%# odebwe (fvdb, [0 20], [2 0], vopt);
+%# @end example
+%# @end deftypefn
+%#
+%# @seealso{odepkg}
+
+function [varargout] = odebwe (vfun, vslot, vinit, varargin)
+
+ if (nargin == 0) %# Check number and types of all input arguments
+ help ('odebwe');
+ 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}, 'odebwe');
+ vfunarguments = {varargin{2:length(varargin)}};
+
+ else %# if (isstruct (varargin{1}))
+ vodeoptions = odepkg_structure_check (varargin{1}, 'odebwe');
+ 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
+
+ %# The adaptive method require a second estimate for
+ %# the comparsion, while the fixed step size algorithm
+ %# needs only one
+ if ~vstepsizefixed
+ vestimators = 2;
+ else
+ vestimators = 1;
+ 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 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 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;
+ warning ('OdePkg:InvalidArgument', ...
+ 'Option "InitialStep" not set, new value %f is used', vodeoptions.InitialStep);
+ end
+
+ %# Implementation of the option MaxNewtonIterations has been finished. This option
+ %# can be set by the user to another value than default value.
+ if isempty (vodeoptions.MaxNewtonIterations)
+ vodeoptions.MaxNewtonIterations = 10;
+ warning ('OdePkg:InvalidArgument', ...
+ 'Option "MaxNewtonIterations" not set, new value %f is used', vodeoptions.MaxNewtonIterations);
+ end
+
+ %# Implementation of the option NewtonTol has been finished. This option
+ %# can be set by the user to another value than default value.
+ if isempty (vodeoptions.NewtonTol)
+ vodeoptions.NewtonTol = 1e-7;
+ warning ('OdePkg:InvalidArgument', ...
+ 'Option "NewtonTol" not set, new value %f is used', vodeoptions.NewtonTol);
+ 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 = (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
+
+ %# Implementation of the option Jacobian has been finished. This option
+ %# can be set by the user to another value than default value.
+ if (~isempty (vodeoptions.Jacobian) && isnumeric (vodeoptions.Jacobian))
+ vhavejachandle = false; vjac = vodeoptions.Jacobian; %# constant jac
+ elseif (isa (vodeoptions.Jacobian, 'function_handle'))
+ vhavejachandle = true; %# jac defined by a function handle
+ else %# no Jacobian - we will use numerical differentiation
+ vhavejachandle = false;
+ 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 = sparse (eye (length (vinit)) );
+ 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.NonNegative, vodetemp.NonNegative))
+ warning ('OdePkg:InvalidArgument', ...
+ 'Option "NonNegative" will be ignored by this solver');
+ end
+ if (~isequal (vodeoptions.Refine, vodetemp.Refine))
+ warning ('OdePkg:InvalidArgument', ...
+ 'Option "Refine" 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.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 odebwe
+ vtimestamp = vslot(1,1); %# timestamp = start time
+ vtimelength = length (vslot); %# length needed if fixed steps
+ vtimestop = vslot(1,vtimelength); %# stop time = last value
+ vdirection = sign (vtimestop); %# Flag for direction to solve
+
+ if (~vstepsizefixed)
+ vstepsize = vodeoptions.InitialStep;
+ 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
+
+ %# Initialize parameters and counters
+ vcntloop = 2; vcntcycles = 1; vu = vinit; vcntsave = 2;
+ vunhandledtermination = true; vpow = 1/2; vnpds = 0;
+ vcntiter = 0; vcntnewt = 0; vndecomps = 0; vnlinsols = 0;
+
+ %# the following option enables the simplified Newton method
+ %# which evaluates the Jacobian only once instead of the
+ %# standard method that updates the Jacobian in each iteration
+ vsimplified = false; % or true
+
+ %# The solver main loop - stop if the endpoint has been reached
+ while ((vdirection * (vtimestamp) < vdirection * (vtimestop)) && ...
+ (vdirection * (vstepsize) >= vdirection * (vminstepsize)))
+
+ %# Hit the endpoint of the time slot exactely
+ if ((vtimestamp + vstepsize) > vdirection * vtimestop)
+ vstepsize = vtimestop - vdirection * vtimestamp;
+ end
+
+ %# Run the time integration for each estimator
+ %# from vtimestamp -> vtimestamp+vstepsize
+ for j = 1:vestimators
+ %# Initial value (result of the previous timestep)
+ y0 = vu;
+ %# Initial guess for Newton-Raphson
+ y(j,:) = vu;
+ %# We do not use a higher order approximation for the
+ %# comparsion, but two steps by the Backward Euler
+ %# method
+ for i = 1:j
+ % Initialize the time stepping parameters
+ vthestep = vstepsize / j;
+ vthetime = vtimestamp + i*vthestep;
+ vnewtit = 1;
+ vresnrm = inf (1, vodeoptions.MaxNewtonIterations);
+
+ %# Start the Newton iteration
+ while (vnewtit < vodeoptions.MaxNewtonIterations) && ...
+ (vresnrm (vnewtit) > vodeoptions.NewtonTol)
+
+ %# Compute the Jacobian of the non-linear equation,
+ %# that is the matrix pencil of the mass matrix and
+ %# the right-hand-side's Jacobian. Perform a (sparse)
+ %# LU-Decomposition afterwards.
+ if ( (vnewtit==1) || (~vsimplified) )
+ %# Get the mass matrix from the left-hand-side
+ 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, y(j,:)', vfunarguments{:});
+ else %# if (vmassdependence == false)
+ vmass = feval ... %# then we only have the time argument
+ (vodeoptions.Mass, y(j,:)', vfunarguments{:});
+ end
+ end
+ %# Get the Jacobian of the right-hand-side's function
+ if (vhavejachandle) %# Handle only the dynamic jacobian
+ vjac = feval(vodeoptions.Jacobian, vthetime,...
+ y(j,:)', vfunarguments{:});
+ elseif isempty(vodeoptions.Jacobian) %# If no Jacobian is given
+ vjac = feval(@jacobian, vfun, vthetime,y(j,:)',...
+ vfunarguments); %# then we differentiate
+ end
+ vnpds = vnpds + 1;
+ vfulljac = vmass/vthestep - vjac;
+ %# one could do a matrix decomposition of vfulljac here,
+ %# but the choice of decomposition depends on the problem
+ %# and therefore we use the backslash-operator in row 374
+ end
+
+ %# Compute the residual
+ vres = vmass/vthestep*(y(j,:)-y0)' - feval(vfun,vthetime,y(j,:)',vfunarguments{:});
+ vresnrm(vnewtit+1) = norm(vres,inf);
+ %# Solve the linear system
+ y(j,:) = vfulljac\(-vres+vfulljac*y(j,:)');
+ %# the backslash operator decomposes the matrix
+ %# and solves the system in a single step.
+ vndecomps = vndecomps + 1;
+ vnlinsols = vnlinsols + 1;
+ %# Prepare next iteration
+ vnewtit = vnewtit + 1;
+ end %# while Newton
+
+ %# Leave inner loop if Newton diverged
+ if vresnrm(vnewtit)>vodeoptions.NewtonTol
+ break;
+ end
+ %# Save intermediate solution as initial value
+ %# for the next intermediate step
+ y0 = y(j,:);
+ %# Count all Newton iterations
+ vcntnewt = vcntnewt + (vnewtit-1);
+ end %# for steps
+
+ %# Leave outer loop if Newton diverged
+ if vresnrm(vnewtit)>vodeoptions.NewtonTol
+ break;
+ end
+ end %# for estimators
+
+ % if all Newton iterations converged
+ if vresnrm(vnewtit)<vodeoptions.NewtonTol
+ %# First order approximation using step size h
+ y1 = y(1,:);
+ %# If adaptive: first order approximation using step
+ %# size h/2, if fixed: y1=y2=y3
+ y2 = y(vestimators,:);
+ %# Second order approximation by ("Richardson")
+ %# extrapolation using h and h/2
+ y3 = y2 + (y2-y1);
+ end
+
+ %# If Newton did not converge, repeat step with reduced
+ %# step size, otherwise calculate the absolute local
+ %# truncation error and the acceptable error
+ if vresnrm(vnewtit)>vodeoptions.NewtonTol
+ vdelta = 2; vtau = 1;
+ elseif (~vstepsizefixed)
+ if (~vnormcontrol)
+ vdelta = abs (y3 - y1)';
+ vtau = max (vodeoptions.RelTol * abs (vu.'), vodeoptions.AbsTol);
+ else
+ vdelta = norm ((y3 - y1)', 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 = y2; % or y3 if we want the 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)
+ if (vhaveoutputselection)
+ vpltout = vu(vodeoptions.OutputSel);
+ else
+ vpltout = vu;
+ end
+ vpltret = feval (vodeoptions.OutputFcn, vtimestamp, ...
+ vpltout.', [], vfunarguments{:});
+ if vpltret %# Leave 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 (vresnrm(vnewtit)>vodeoptions.NewtonTol)
+ vunhandledtermination = true;
+ break;
+ elseif (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 = vcntnewt; %# number of rhs evaluations
+ if isempty(vodeoptions.Jacobian) %# additional evaluations due
+ vnfevals = vnfevals + vcntnewt*(1+length(vinit)); %# to differentiation
+ end
+ %# 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 = 'odebwe'; %# 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
+
+function df = jacobian(vfun,vthetime,vtheinput,vfunarguments);
+%# Internal function for the numerical approximation of a jacobian
+ vlen = length(vtheinput);
+ vsigma = sqrt(eps);
+ vfun0 = feval(vfun,vthetime,vtheinput,vfunarguments{:});
+ df = zeros(vlen,vlen);
+ for j = 1:vlen
+ vbuffer = vtheinput(j);
+ if (vbuffer==0)
+ vh = vsigma;
+ elseif (abs(vbuffer)>1)
+ vh = vsigma*vbuffer;
+ else
+ vh = sign(vbuffer)*vsigma;
+ end
+ vtheinput(j) = vbuffer + vh;
+ df(:,j) = (feval(vfun,vthetime,vtheinput,...
+ vfunarguments{:}) - vfun0) / vh;
+ vtheinput(j) = vbuffer;
+ 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 = odebwe (1, [0 25], [3 15 1]);
+%!error %# input argument number two
+%! B = odebwe (@fpol, 1, [3 15 1]);
+%!error %# input argument number three
+%! B = odebwe (@flor, [0 25], 1);
+%!test %# one output argument
+%! vsol = odebwe (@fpol, [0 2], [2 0]);
+%! assert ([vsol.x(end), vsol.y(end,:)], [2, fref], 1e-3);
+%! assert (isfield (vsol, 'solver'));
+%! assert (vsol.solver, 'odebwe');
+%!test %# two output arguments
+%! [vt, vy] = odebwe (@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] = odebwe (@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 = odebwe (fvdb, [0 2], [2 0]);
+%! assert ([vsol.x(end), vsol.y(end,:)], [2, fref], 1e-3);
+%!test %# extra input arguments passed trhough
+%! vsol = odebwe (@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 = odebwe (@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 = odebwe (@fpol, [0 2], [2 0], vopt);
+%!test %# Solve vdp in fixed step sizes
+%! vsol = odebwe (@fpol, [0:0.001:2], [2 0]);
+%! assert (vsol.x(:), [0:0.001:2]');
+%! assert (vsol.y(end,:), fref, 1e-2);
+%!test %# Solve in backward direction starting at t=0
+%! %# vref = [-1.2054034414, 0.9514292694];
+%! vsol = odebwe (@fpol, [0 -2], [2 0]);
+%! %# assert ([vsol.x(end), vsol.y(end,:)], [-2, fref], 1e-3);
+%!test %# Solve in backward direction starting at t=2
+%! %# vref = [-1.2154183302, 0.9433018000];
+%! vsol = odebwe (@fpol, [2 -2], [0.3233166627 -1.8329746843]);
+%! %# assert ([vsol.x(end), vsol.y(end,:)], [-2, fref], 1e-3);
+%!test %# AbsTol option
+%! vopt = odeset ('AbsTol', 1e-5);
+%! vsol = odebwe (@fpol, [0 2], [2 0], vopt);
+%! assert ([vsol.x(end), vsol.y(end,:)], [2, fref], 1e-2);
+%!test %# AbsTol and RelTol option
+%! vopt = odeset ('AbsTol', 1e-6, 'RelTol', 1e-6);
+%! vsol = odebwe (@fpol, [0 2], [2 0], vopt);
+%! assert ([vsol.x(end), vsol.y(end,:)], [2, fref], 1e-2);
+%!test %# RelTol and NormControl option -- higher accuracy
+%! vopt = odeset ('RelTol', 1e-6, 'NormControl', 'on');
+%! vsol = odebwe (@fpol, [0 2], [2 0], vopt);
+%! assert ([vsol.x(end), vsol.y(end,:)], [2, fref], 1e-3);
+%!
+%! %# test for NonNegative option is missing
+%! %# test for OutputSel and Refine option is missing
+%!
+%!test %# Details of OutputSave can't be tested
+%! vopt = odeset ('OutputSave', 1, 'OutputSel', 1);
+%! vsla = odebwe (@fpol, [0 2], [2 0], vopt);
+%! vopt = odeset ('OutputSave', 2);
+%! vslb = odebwe (@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 = odebwe (@fpol, [0 2], [2 0], vopt);
+%! assert (isfield (vsol, 'stats'));
+%! assert (isfield (vsol.stats, 'nsteps'));
+%!test %# InitialStep option
+%! vopt = odeset ('InitialStep', 1e-8);
+%! vsol = odebwe (@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 = odebwe (@fpol, [0 0.2], [2 0], vopt);
+%! assert ([vsol.x(5)-vsol.x(4)], [1e-2], 1e-2);
+%!test %# Events option add further elements in vsol
+%! vopt = odeset ('Events', @feve);
+%! vsol = odebwe (@fpol, [0 10], [2 0], vopt);
+%! assert (isfield (vsol, 'ie'));
+%! assert (vsol.ie, [2; 1; 2; 1]);
+%! assert (isfield (vsol, 'xe'));
+%! assert (isfield (vsol, 'ye'));
+%!test %# Events option, now stop integration
+%! vopt = odeset ('Events', @fevn, 'NormControl', 'on');
+%! vsol = odebwe (@fpol, [0 10], [2 0], vopt);
+%! assert ([vsol.ie, vsol.xe, vsol.ye], ...
+%! [2.0, 2.496110, -0.830550, -2.677589], 1e-3);
+%!test %# Events option, five output arguments
+%! vopt = odeset ('Events', @fevn, 'NormControl', 'on');
+%! [vt, vy, vxe, vye, vie] = odebwe (@fpol, [0 10], [2 0], vopt);
+%! assert ([vie, vxe, vye], ...
+%! [2.0, 2.496110, -0.830550, -2.677589], 1e-3);
+%!test %# Jacobian option
+%! vopt = odeset ('Jacobian', @fjac);
+%! vsol = odebwe (@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 = odebwe (@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 %# Mass option as function
+%! vopt = odeset ('Mass', @fmas);
+%! vsol = odebwe (@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 = odebwe (@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 = odebwe (@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 = odebwe (@fpol, [0 2], [2 0], vopt);
+%! assert ([vsol.x(end), vsol.y(end,:)], [2, fref], 1e-3);
+%!
+%! %# test for MStateDependence option is missing
+%! %# test for MvPattern option is missing
+%! %# test for InitialSlope option is missing
+%! %# test for MaxOrder option is missing
+%! %# test for BDF option is missing
+%!
+%! warning ('on', 'OdePkg:InvalidArgument');
+
+%# Local Variables: ***
+%# mode: octave ***
+%# End: ***
git://git.creatis.insa-lyon.fr / CreaPhase.git / blobdiff