ida SUNDIALS differential-algebraic equation solver

## Syntax

                [t,y] = ida(F,tspan,y0,yp0,options)
                [t,y,info] = ida(F,tspan,y0,yp0,options)
                sol = ida(...)
                solext = ida(sol,tfinal,options)
                ida(...)

## Arguments

F | a function, a string or a list, the residual of the differential-algebraic equation
tspan | double vector, time interval or time points
y0,yp0 | double arrays: initial state and initial state derivative
options | a sequence of optional named arguments, see the solvers options
t | vector of time points used by the solver
y | array of solution at time values in t
sol, solext | MList of `_odeSolution` type, see the solution output help page
info | MList of `_odeSolution` type with solver information, including the sensitivity of solution (if applicable), user events information (if applicable) and statistics
tfinal | final time of extended solution

## Description

`ida` computes the solution of real or complex ordinary different-algebraic equation defined by

0 = F(t,y,y'),\quad y(t_0)=y_0,\quad y'(t_0)=yp_0.

It is an interface to the ida solver of SUNDIALS library using BDF method. The simplest call of ida is when `tspan` is a two component vector:

It is an interface to the idas solver of SUNDIALS library using BDF method. The solver has a specific feature which allows to compute the sensitivity with respect to a vector of parameters present in the initial conditions and/or the residual equation (see the DAE Sensitivity help page). The simplest call takes the form

                [t,y] = ida(F,[t0 tf],y0,yp0)
                [t,y,info] = ida(F,[t0 tf],y0,yp0)

where F(t,y,y') is the right hand side usually called "residual" of the DAE, `t0, tf` are the initial and final time, `y` is the array of solution `[y(t(1)),y(t(2)),...]` at time values in t, `info.yp` is the array of solution derivative `[y'(t(1)),y'(t(2)),...]`. Concatenation is done on next dimension if `y0` is not a column vector. The time values in `t` are those used by the solver to meet default relative and absolute estimated local error (which can be changed in

options
). In the simplest case the DAE residual is computed by a Scilab function with three arguments, for example for y'+y=0 F is coded as

                    function res = F(t,y,yp)
                      res = yp+y
                    end

See the User functions help page to learn how to pass extra parameters and/or use DLL entrypoints. When `t` has more than two components:

                [t,y] = ida(F,[t0 t1 ... tf],y0,yp0)
                [t,y,info] = ida(F,[t0 t1 ... tf],y0,yp0)

the solution is computed at prescribed points with the same precision as the two components syntax. However, the result may slightly differ (within chosen tolerance) since `t1-t0` may give a different guess of the initial step used by the solver. Provided that the initial step is the same, further solver internal steps will be the same and solution at intermediate user points is computed by continuous extension formulas.

When searching for (t,y,yp) where some functions defined in `options` vanish (see the Events help page) the syntax

    [t,y,info] = ida(F,tspan,y0,yp0,options)

allows to recover (t,y,yp) values in `info.te, info.ye, info.ype` and in `info.ie` the number(s) of the vanishing function.

When solution has to be further evaluated at arbitrary points which are not known in advance, then the syntax

    sol = ida(F,[t0 tf],y0,yp0)

yields an MList of `_odeSolution` type, which can be used later as an interpolant, see the solution output help page.

When y0 and/or yp0 do not fullfill the equation at initial time they can be used as initial guesses when using the `calcIc` option, see the Initial conditions section.

When no ouput argument is given, the only way to have access to the solution at solver or user time steps is to use a user callback (see the SUNDIALS user functions help page). For example, when there could be memory issues (e.g. if the dimension of the solution vector is very large) this allows to write the solution to disk.

## Example

In the following example, we solve the equations of the SIR model. The DAE residual is computed by the function `%SUN_sir` (defined in the SUNDIALS module):

            function r = %SUN_sir(t,y,yp)
                r = [yp(1)+0.2*y(1)*y(2)
                     yp(2)-0.2*y(1)*y(2)+0.05*y(2)
                      y(1)+y(2)+y(3)-1];
            end

            y0  = [1-1e-6; 1e-6; 0];
            yp0 = [-2e-7; 1.5e-7; 5e-8];
            [t,y] = ida(%SUN_sir, [0,200], y0, yp0);
            clf
            plot(t, y)

y0=[1-1e-6;1e-6;0]; yp0=[-2e-7;1.5e-7;5e-8]; [t,y] = ida(%SUN_sir,[0,200],y0,yp0); clf plot(t,y) gcf().axes_size(2)=300

## Initial conditions

When y(0) is known but y'(0) is not known it can be computed by using the `calcIc` option with value `"y0yp0"` and `yp0` is used as an initial guess:

                    y0  = [1-1e-6; 1e-6; 0];
                    yp0 = zeros(3,1);
                    [t,y,info] = ida(%SUN_sir, [0,200], y0, yp0, calcIc="y0yp0");
                    disp(info.yp(:,1))

When the DAE is semi-explicit and of index one, this option also allows to compute purely algebraic states by using the `yIsAlgebraic = idx` option, where `idx` is a vector with the indexes of such states. When y'(0) is known but y(0) is not, use `calcIc` option with value `"y0"`. When the DAE is of index two or more these options should not be used and coherent initial conditions must be given for all states, and setting the option `suppressAlg` is recommended (see below).

## DAE with high index

When the DAE is of high index, the `suppressAlg` option can help the solver effort. If `suppressAlg = %T` then the algebraic variables (given with the `yIsAlgebraic` option) are excluded from the local error test. The use of this option (with %T value) is discouraged when solving DAE systems of index one whereas it is often the only way to obtain a solution for systems of index two or more. See the following example below (a circular pendulum DAE of index two).

                function out=res(t,y,yd)
                    x=y(1:2);
                    xd=yd(1:2);
                    u=y(3:4);
                    ud=yd(3:4);
                    lambda=y(5);
                    out=[xd-u
                         ud+x*lambda+[0;1]
                         x'*u];
                end

                // One period
                T = 7.4162987092054876;
                y0=[1;0;0;0;0];
                yd0=[0;0;0;-1;0];
                [t,y]=ida(res,T,y0,yd0,t0=0,yIsAlgebraic=5,suppressAlg=%t)

## See also

- arkode
- cvode
- User functions
- Options (ODE and DAE solvers)

## Bibliography

A. C. Hindmarsh, P. N. Brown, K. E. Grant, S. L. Lee, R. Serban, D. E. Shumaker, and C. S. Woodward, "SUNDIALS: Suite of Nonlinear and Differential/Algebraic Equation Solvers," ACM Transactions on Mathematical Software, 31(3), pp. 363-396, 2005. Also available as LLNL technical report UCRL-JP-200037.

Alan C. Hindmarsh, Radu Serban, Cody J. Balos, David J. Gardner, Daniel R. Reynolds, and Carol S. Woodward, "User Documentation for ida", 2021, v6.1.1, available online at https://sundials.readthedocs.io/en/latest/ida.