octave: Matlab-compatible solvers
24.1.1 Matlab-compatible solvers
--------------------------------
Octave also provides a set of solvers for initial value problems for
Ordinary Differential Equations that have a MATLAB-compatible interface.
The options for this class of methods are set using the functions.
• ‘odeset’
• ‘odeget’
Currently implemented solvers are:
• Runge-Kutta methods
• ‘ode45’ Integrates a system of non–stiff ordinary differential
equations (non–stiff ODEs and DAEs) using second order
Dormand-Prince method. This is a fourth–order accurate
integrator therefore the local error normally expected is
O(h^5). This solver requires six function evaluations per
integration step.
• ‘ode23’ Integrates a system of non–stiff ordinary differential
equations (non-stiff ODEs and DAEs) using second order
Bogacki-Shampine method. This is a second-order accurate
integrator therefore the local error normally expected is
O(h^3). This solver requires three function evaluations per
integration step.
-- : [T, Y] = ode45 (FUN, TRANGE, INIT)
-- : [T, Y] = ode45 (FUN, TRANGE, INIT, ODE_OPT)
-- : [T, Y, TE, YE, IE] = ode45 (...)
-- : SOLUTION = ode45 (...)
-- : ode45 (...)
Solve a set of non-stiff Ordinary Differential Equations (non-stiff
ODEs) with the well known explicit Dormand-Prince method of order
4.
FUN is a function handle, inline function, or string containing the
name of the function that defines the ODE: ‘y' = f(t,y)’. The
function must accept two inputs where the first is time T and the
second is a column vector of unknowns Y.
TRANGE specifies the time interval over which the ODE will be
evaluated. Typically, it is a two-element vector specifying the
initial and final times (‘[tinit, tfinal]’). If there are more
than two elements then the solution will also be evaluated at these
intermediate time instances.
By default, ‘ode45’ uses an adaptive timestep with the
‘integrate_adaptive’ algorithm. The tolerance for the timestep
computation may be changed by using the options "RelTol" and
"AbsTol".
INIT contains the initial value for the unknowns. If it is a row
vector then the solution Y will be a matrix in which each column is
the solution for the corresponding initial value in INIT.
The optional fourth argument ODE_OPT specifies non-default options
to the ODE solver. It is a structure generated by ‘odeset’.
The function typically returns two outputs. Variable T is a column
vector and contains the times where the solution was found. The
output Y is a matrix in which each column refers to a different
unknown of the problem and each row corresponds to a time in T.
The output can also be returned as a structure SOLUTION which has a
field X containing a row vector of times where the solution was
evaluated and a field Y containing the solution matrix such that
each column corresponds to a time in X. Use ‘fieldnames
(SOLUTION)’ to see the other fields and additional information
returned.
If no output arguments are requested, and no ‘OutputFcn’ is
specified in ODE_OPT, then the ‘OutputFcn’ is set to ‘odeplot’ and
the results of the solver are plotted immediately.
If using the "Events" option then three additional outputs may be
returned. TE holds the time when an Event function returned a
zero. YE holds the value of the solution at time TE. IE contains
an index indicating which Event function was triggered in the case
of multiple Event functions.
Example: Solve the Van der Pol equation
fvdp = @(T,Y) [Y(2); (1 - Y(1)^2) * Y(2) - Y(1)];
[T,Y] = ode45 (fvdp, [0, 20], [2, 0]);
DONTPRINTYET See also: 
odeset XREFodeset, odeget XREFodeget, *noteDONTPRINTYET See also: odeset XREFodeset, odeget XREFodeget,
ode23 XREFode23.
-- : [T, Y] = ode23 (FUN, TRANGE, INIT)
-- : [T, Y] = ode23 (FUN, TRANGE, INIT, ODE_OPT)
-- : [T, Y, TE, YE, IE] = ode23 (...)
-- : SOLUTION = ode23 (...)
-- : ode23 (...)
Solve a set of non-stiff Ordinary Differential Equations (non-stiff
ODEs) with the well known explicit Bogacki-Shampine method of order
3. For the definition of this method see
<http://en.wikipedia.org/wiki/List_of_Runge%E2%80%93Kutta_methods>.
FUN is a function handle, inline function, or string containing the
name of the function that defines the ODE: ‘y' = f(t,y)’. The
function must accept two inputs where the first is time T and the
second is a column vector of unknowns Y.
TRANGE specifies the time interval over which the ODE will be
evaluated. Typically, it is a two-element vector specifying the
initial and final times (‘[tinit, tfinal]’). If there are more
than two elements then the solution will also be evaluated at these
intermediate time instances.
By default, ‘ode23’ uses an adaptive timestep with the
‘integrate_adaptive’ algorithm. The tolerance for the timestep
computation may be changed by using the options "RelTol" and
"AbsTol".
INIT contains the initial value for the unknowns. If it is a row
vector then the solution Y will be a matrix in which each column is
the solution for the corresponding initial value in INIT.
The optional fourth argument ODE_OPT specifies non-default options
to the ODE solver. It is a structure generated by ‘odeset’.
The function typically returns two outputs. Variable T is a column
vector and contains the times where the solution was found. The
output Y is a matrix in which each column refers to a different
unknown of the problem and each row corresponds to a time in T.
The output can also be returned as a structure SOLUTION which has a
field X containing a row vector of times where the solution was
evaluated and a field Y containing the solution matrix such that
each column corresponds to a time in X. Use ‘fieldnames
(SOLUTION)’ to see the other fields and additional information
returned.
If no output arguments are requested, and no ‘OutputFcn’ is
specified in ODE_OPT, then the ‘OutputFcn’ is set to ‘odeplot’ and
the results of the solver are plotted immediately.
If using the "Events" option then three additional outputs may be
returned. TE holds the time when an Event function returned a
zero. YE holds the value of the solution at time TE. IE contains
an index indicating which Event function was triggered in the case
of multiple Event functions.
Example: Solve the Van der Pol equation
fvdp = @(T,Y) [Y(2); (1 - Y(1)^2) * Y(2) - Y(1)];
[T,Y] = ode23 (fvdp, [0, 20], [2, 0]);
DONTPRINTYET See also: odeset XREFodeset, odeget XREFodeget, *noteDONTPRINTYET See also: odeset XREFodeset, odeget XREFodeget,
ode45 XREFode45.
-- : ODESTRUCT = odeset ()
-- : ODESTRUCT = odeset ("FIELD1", VALUE1, "FIELD2", VALUE2, ...)
-- : ODESTRUCT = odeset (OLDSTRUCT, "FIELD1", VALUE1, "FIELD2", VALUE2,
...)
-- : ODESTRUCT = odeset (OLDSTRUCT, NEWSTRUCT)
-- : odeset ()
Create or modify an ODE options structure.
When called with no input argument and one output argument, return
a new ODE options structure that contains all possible fields
initialized to their default values. If no output argument is
requested, display a list of the common ODE solver options along
with their default value.
If called with name-value input argument pairs "FIELD1", "VALUE1",
"FIELD2", "VALUE2", ... return a new ODE options structure with all
the most common option fields initialized, *and* set the values of
the fields "FIELD1", "FIELD2", ... to the values VALUE1, VALUE2,
....
If called with an input structure OLDSTRUCT then overwrite the
values of the options "FIELD1", "FIELD2", ... with new values
VALUE1, VALUE2, ... and return the modified structure.
When called with two input ODE options structures OLDSTRUCT and
NEWSTRUCT overwrite all values from the structure OLDSTRUCT with
new values from the structure NEWSTRUCT. Empty values in NEWSTRUCT
will not overwrite values in OLDSTRUCT.
The most commonly used ODE options, which are always assigned a
value by odeset, are the following:
AbsTol
Absolute error tolerance.
BDF
Events
Event function. An event function must have the form ‘[value,
isterminal, direction] = my_events_f (t, y)’
InitialSlope
Consistent initial slope vector for DAE solvers.
InitialStep
Initial time step size.
Jacobian
Jacobian matrix, specified as a constant matrix or a function
of time and state.
JConstant
Specify whether the Jacobian is a constant matrix or depends
on the state.
JPattern
If the Jacobian matrix is sparse and non-constant but
maintains a constant sparsity pattern, specify the sparsity
pattern.
Mass
Mass matrix, specified as a constant matrix or a function of
time and state.
MassSingular
Specify whether the mass matrix is singular. Accepted values
include "yes", "no", "maybe".
MaxOrder
Maximum order of formula.
MaxStep
Maximum time step value.
MStateDependence
Specify whether the mass matrix depends on the state or only
on time.
MvPattern
If the mass matrix is sparse and non-constant but maintains a
constant sparsity pattern, specify the sparsity pattern.
NonNegative
Specify elements of the state vector that are expected to
remain nonnegative during the simulation.
NormControl
Control error relative to the 2-norm of the solution, rather
than its absolute value.
OutputFcn
Function to monitor the state during the simulation. For the
form of the function to use see odeplot.
OutputSel
Indices of elements of the state vector to be passed to the
output monitoring function.
Refine
Specify whether output should be returned only at the end of
each time step or also at intermediate time instances. The
value should be a scalar indicating the number of equally
spaced time points to use within each timestep at which to
RelTol
Relative error tolerance.
Stats
Print solver statistics after simulation.
Vectorized
Specify whether odefun can be passed multiple values of the
state at once.
Field names that are not in the above list are also accepted and
added to the result structure.
See also: odeget XREFodeget.
-- : VAL = odeget (ODE_OPT, FIELD)
-- : VAL = odeget (ODE_OPT, FIELD, DEFAULT)
Query the value of the property FIELD in the ODE options structure
ODE_OPT.
If called with two input arguments and the first input argument
ODE_OPT is an ODE option structure and the second input argument
FIELD is a string specifying an option name, then return the option
value VAL corresponding to FIELD from ODE_OPT.
If called with an optional third input argument, and FIELD is not
set in the structure ODE_OPT, then return the default value DEFAULT
instead.
See also: odeset XREFodeset.
-- : STOP_SOLVE = odeplot (T, Y, FLAG)
Open a new figure window and plot the solution of an ode problem at
each time step during the integration.
The types and values of the input parameters T and Y depend on the
input FLAG that is of type string. Valid values of FLAG are:
‘"init"’
The input T must be a column vector of length 2 with the first
and last time step (‘[TFIRST TLAST]’. The input Y contains
the initial conditions for the ode problem (Y0).
‘""’
The input T must be a scalar double specifying the time for
which the solution in input Y was calculated.
‘"done"’
The inputs should be empty, but are ignored if they are
present.
‘odeplot’ always returns false, i.e., don’t stop the ode solver.
Example: solve an anonymous implementation of the "Van der Pol"
equation and display the results while solving.
fvdp = @(t,y) [y(2); (1 - y(1)^2) * y(2) - y(1)];
opt = odeset ("OutputFcn", @odeplot, "RelTol", 1e-6);
sol = ode45 (fvdp, [0 20], [2 0], opt);
Background Information: This function is called by an ode solver
function if it was specified in the "OutputFcn" property of an
options structure created with ‘odeset’. The ode solver will
initially call the function with the syntax ‘odeplot ([TFIRST,
TLAST], Y0, "init")’. The function initializes internal variables,
creates a new figure window, and sets the x limits of the plot.
Subsequently, at each time step during the integration the ode
solver calls ‘odeplot (T, Y, [])’. At the end of the solution the
ode solver calls ‘odeplot ([], [], "done")’ so that odeplot can
perform any clean-up actions required.
DONTPRINTYET See also: odeset XREFodeset, odeget XREFodeget, *noteDONTPRINTYET See also: odeset XREFodeset, odeget XREFodeget,
ode23 XREFode23, ode45 XREFode45.
Info Catalog
octave: Ordinary Differential Equations