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:
ode23
integrates a system of non-stiff ordinary differential
equations (ODEs) or index-1 differential-algebraic equations (DAEs). It
uses the third-order Bogacki-Shampine method and adapts the local
step size in order to satisfy a user-specified tolerance. The solver
requires three function evaluations per integration step.
ode45
integrates a system of non-stiff ODEs (or index-1 DAEs)
using the high-order, variable-step Dormand-Prince method. It
requires six function evaluations per integration step, but may
take larger steps on smooth problems than ode23
: potentially
offering improved efficiency at smaller tolerances.
ode15s
integrates a system of stiff ODEs (or index-1 DAEs)
using a variable step, variable order method based on Backward Difference
Formulas (BDF).
ode15i
integrates a system of fully-implicit ODEs (or index-1
DAEs) using the same variable step, variable order method as ode15s
.
The function decic
can be used to compute consistent initial
conditions.
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]);
Solve a set of non-stiff Ordinary Differential Equations (non-stiff ODEs) with the well known explicit Bogacki-Shampine method of order 3.
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]);
Reference: For the definition of this method see https://en.wikipedia.org/wiki/List_of_Runge%E2%80%93Kutta_methods.
Solve a set of stiff Ordinary Differential Equations (ODEs) or stiff semi-explicit index 1 Differential Algebraic Equations (DAEs).
ode15s
uses a variable step, variable order BDF (Backward
Differentiation Formula) method that ranges from order 1 to 5.
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.
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 Robertson’s equations:
function r = robertson_dae (t, y) r = [ -0.04*y(1) + 1e4*y(2)*y(3) 0.04*y(1) - 1e4*y(2)*y(3) - 3e7*y(2)^2 y(1) + y(2) + y(3) - 1 ]; endfunction opt = odeset ("Mass", [1 0 0; 0 1 0; 0 0 0], "MStateDependence", "none"); [t,y] = ode15s (@robertson_dae, [0, 1e3], [1; 0; 0], opt);
Solve a set of fully-implicit Ordinary Differential Equations (ODEs) or index 1 Differential Algebraic Equations (DAEs).
ode15i
uses a variable step, variable order BDF (Backward
Differentiation Formula) method that ranges from order 1 to 5.
fun is a function handle, inline function, or string containing the
name of the function that defines the ODE: 0 = f(t,y,yp)
. The
function must accept three inputs where the first is time t, the
second is the function value y (a column vector), and the third
is the derivative value yp (a column vector).
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.
y0 and yp0 contain the initial values for the unknowns y and yp. If they are row vectors then the solution y will be a matrix in which each column is the solution for the corresponding initial value in y0 and yp0.
y0 and yp0 must be consistent initial conditions, meaning that
f(t,y0,yp0) = 0
is satisfied. The function decic
may be used
to compute consistent initial conditions given initial guesses.
The optional fifth 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 Robertson’s equations:
function r = robertson_dae (t, y, yp) r = [ -(yp(1) + 0.04*y(1) - 1e4*y(2)*y(3)) -(yp(2) - 0.04*y(1) + 1e4*y(2)*y(3) + 3e7*y(2)^2) y(1) + y(2) + y(3) - 1 ]; endfunction [t,y] = ode15i (@robertson_dae, [0, 1e3], [1; 0; 0], [-1e-4; 1e-4; 0]);
Compute consistent implicit ODE initial conditions y0_new and yp0_new given initial guesses y0 and yp0.
A maximum of length (y0)
components between fixed_y0 and
fixed_yp0 may be chosen as fixed values.
fun is a function handle. The function must accept three inputs where the first is time t, the second is a column vector of unknowns y, and the third is a column vector of unknowns yp.
t0 is the initial time such that
fun(t0, y0_new, yp0_new) = 0
, specified as a
scalar.
y0 is a vector used as the initial guess for y.
fixed_y0 is a vector which specifies the components of y0 to
hold fixed. Choose a maximum of length (y0)
components between
fixed_y0 and fixed_yp0 as fixed values.
Set fixed_y0(i) component to 1 if you want to fix the value of
y0(i).
Set fixed_y0(i) component to 0 if you want to allow the value of
y0(i) to change.
yp0 is a vector used as the initial guess for yp.
fixed_yp0 is a vector which specifies the components of yp0 to
hold fixed. Choose a maximum of length (yp0)
components
between fixed_y0 and fixed_yp0 as fixed values.
Set fixed_yp0(i) component to 1 if you want to fix the value of
yp0(i).
Set fixed_yp0(i) component to 0 if you want to allow the value of
yp0(i) to change.
The optional seventh argument options is a structure array. Use
odeset
to generate this structure. The relevant options are
RelTol
and AbsTol
which specify the error thresholds used to
compute the initial conditions.
The function typically returns two outputs. Variable y0_new is a column vector and contains the consistent initial value of y. The output yp0_new is a column vector and contains the consistent initial value of yp.
The optional third output resnorm is the norm of the vector of
residuals. If resnorm is small, decic
has successfully
computed the initial conditions. If the value of resnorm is large,
use RelTol
and AbsTol
to adjust it.
Example: Compute initial conditions for Robertson’s equations:
function r = robertson_dae (t, y, yp) r = [ -(yp(1) + 0.04*y(1) - 1e4*y(2)*y(3)) -(yp(2) - 0.04*y(1) + 1e4*y(2)*y(3) + 3e7*y(2)^2) y(1) + y(2) + y(3) - 1 ]; endfunction
[y0_new,yp0_new] = decic (@robertson_dae, 0, [1; 0; 0], [1; 1; 0], [-1e-4; 1; 0], [0; 0; 0]);
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
: positive scalar | vector, def. 1e-6
Absolute error tolerance.
BDF
: {"off"
} | "on"
Use BDF formulas in implicit multistep methods. Note: This option is not yet implemented.
Events
: function_handleEvent function. An event function must have the form
[value, isterminal, direction] = my_events_f (t, y)
InitialSlope
: vectorConsistent initial slope vector for DAE solvers.
InitialStep
: positive scalarInitial time step size.
Jacobian
: matrix | function_handleJacobian matrix, specified as a constant matrix or a function of time and state.
JConstant
: {"off"
} | "on"
Specify whether the Jacobian is a constant matrix or depends on the state.
JPattern
: sparse matrixIf the Jacobian matrix is sparse and non-constant but maintains a constant sparsity pattern, specify the sparsity pattern.
Mass
: matrix | function_handleMass matrix, specified as a constant matrix or a function of time and state.
MassSingular
: {"maybe"
} | "yes"
| "on"
Specify whether the mass matrix is singular.
MaxOrder
: {5
} | 4
| 3
| 2
| 1
Maximum order of formula.
MaxStep
: positive scalarMaximum time step value.
MStateDependence
: {"weak"
} | "none"
| "strong"
Specify whether the mass matrix depends on the state or only on time.
MvPattern
: sparse matrixIf the mass matrix is sparse and non-constant but maintains a constant sparsity pattern, specify the sparsity pattern. Note: This option is not yet implemented.
NonNegative
: scalar | vectorSpecify elements of the state vector that are expected to remain non-negative during the simulation.
NormControl
: {"off"
} | "on"
Control error relative to the 2-norm of the solution, rather than its absolute value.
OutputFcn
: function_handleFunction to monitor the state during the simulation. For the form of
the function to use see odeplot
.
OutputSel
: scalar | vectorIndices of elements of the state vector to be passed to the output monitoring function.
Refine
: positive scalarSpecify 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 return output. Note: This option is not yet implemented.
RelTol
: positive scalarRelative error tolerance.
Stats
: {"off"
} | "on"
Print solver statistics after simulation.
Vectorized
: {"off"
} | "on"
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.
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.
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.