### 24.3 Matlab-compatible solvers ¶

Octave also provides a set of solvers for initial value problems for ordinary differential equations (ODEs) 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 ODEs or index-1 differential-algebraic equations (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.
  - ode23 integrates a system of non-stiff ODEs or (or index-1 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.
  - ode23s integrates a system of stiff ODEs (or index-1 DAEs) using a modified second-order Rosenbrock method.
- Linear multistep methods
  - 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`. decic can be used to compute consistent initial conditions for `ode15i`.

Detailed information on the solvers are given in L. F. Shampine and M. W. Reichelt, The MATLAB ODE Suite, SIAM Journal on Scientific Computing, Vol. 18, 1997, pp. 1–22, DOI: https://doi.org/10.1137/S1064827594276424.

`[t, y] =` ode45 `(fcn, trange, init)` ¶
`[t, y] =` ode45 `(fcn, 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.

  fcn 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]);

  See also: odeset, odeget, ode23, ode15s.

`[t, y] =` ode23 `(fcn, trange, init)` ¶
`[t, y] =` ode23 `(fcn, 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.

  fcn 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.

  See also: odeset, odeget, ode45, ode15s.

`[t, y] =` ode23s `(fcn, trange, init)` ¶
`[t, y] =` ode23s `(fcn, trange, init, ode_opt)` ¶
`[t, y] =` ode23s `(…, par1, par2, …)` ¶
`[t, y, te, ye, ie] =` ode23s `(…)` ¶
`solution =` ode23s `(…)` ¶

  Solve a set of stiff Ordinary Differential Equations (stiff ODEs) with a Rosenbrock method of order (2,3).

  fcn is a function handle, inline function, or string containing the name of the function that defines the ODE: `M 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. M is a constant mass matrix, non-singular and possibly sparse. Set the field `"Mass"` in odeopts using odeset to specify a mass matrix.

  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 using an interpolation procedure of the same order as the one of the solver.

  By default, `ode23s` 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`. `ode23s` will ignore the following options: `"BDF"`, `"InitialSlope"`, `"MassSingular"`, `"MStateDependence"`, `"MvPattern"`, `"MaxOrder"`, `"Non-negative"`.

  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. If trange specifies intermediate time steps, only those will be returned.

  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 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 stiff Van der Pol equation

      f = @(t,y) [y(2); 1000*(1 - y(1)^2) * y(2) - y(1)];
      opt = odeset ('Mass', [1 0; 0 1], 'MaxStep', 1e-1);
      [vt, vy] = ode23s (f, [0 2000], [2 0], opt);

  See also: odeset, daspk, dassl.

`[t, y] =` ode15s `(fcn, trange, y0)` ¶
`[t, y] =` ode15s `(fcn, trange, y0, ode_opt)` ¶
`[t, y, te, ye, ie] =` ode15s `(…)` ¶
`solution =` ode15s `(…)` ¶
ode15s `(…)` ¶

  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.

  fcn 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);

  See also: decic, odeset, odeget, ode23, ode45.

`[t, y] =` ode15i `(fcn, trange, y0, yp0)` ¶
`[t, y] =` ode15i `(fcn, trange, y0, yp0, ode_opt)` ¶
`[t, y, te, ye, ie] =` ode15i `(…)` ¶
`solution =` ode15i `(…)` ¶
ode15i `(…)` ¶

  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.

  fcn 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]);

  See also: decic, odeset, odeget.

`[y0_new, yp0_new] =` decic `(fcn, t0, y0, fixed_y0, yp0, fixed_yp0)` ¶
`[y0_new, yp0_new] =` decic `(fcn, t0, y0, fixed_y0, yp0, fixed_yp0, options)` ¶
`[y0_new, yp0_new, resnorm] =` decic `(…)` ¶

  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.

  fcn 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 `fcn(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]);

  See also: ode15i, odeset.

`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`: 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_handle

    Event function. An event function must have the form `[value, isterminal, direction] = my_events_f (t, y)`

  `InitialSlope`: vector

    Consistent initial slope vector for DAE solvers.

  `InitialStep`: positive scalar

    Initial time step size.

  `Jacobian`: matrix | function_handle

    Jacobian 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 matrix

    If the Jacobian matrix is sparse and non-constant but maintains a constant sparsity pattern, specify the sparsity pattern.

  `Mass`: matrix | function_handle

    Mass 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 scalar

    Maximum time step value.

  `MStateDependence`: {`"weak"`} | `"none"` | `"strong"`

    Specify whether the mass matrix depends on the state or only on time.

  `MvPattern`: sparse matrix

    If 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 | vector

    Specify 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_handle

    Function to monitor the state during the simulation. For the form of the function to use see `odeplot`.

  `OutputSel`: scalar | vector

    Indices of elements of the state vector to be passed to the output monitoring function.

  `Refine`: positive scalar

    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 return output.

  `RelTol`: positive scalar

    Relative error tolerance.

  `Stats`: {`"off"`} | `"on"`

    Print solver statistics after simulation.

  `Vectorized`: {`"off"`} | `"on"`

    Specify whether `odefcn` 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.

`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.

`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 or vector specifying the time(s) 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.

  See also: odeset, odeget, ode23, ode45.