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# 23. Differential Equations

Octave has two built-in functions for solving differential equations. Both are based on reliable ODE solvers written in Fortran.

23.1 Ordinary Differential Equations |  |
23.2 Differential-Algebraic Equations |  |

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## 23.1 Ordinary Differential Equations

The function `lsode` can be used to solve ODEs of the form

    dx
    -- = f (x, t)
    dt

using Hindmarsh's ODE solver LSODE.

Loadable Function: [x, istate, msg] lsode (fcn, x_0, t, t_crit)
  Solve the set of differential equations

      dx
      -- = f(x, t)
      dt

  with

      x(t_0) = x_0

  The solution is returned in the matrix x, with each row corresponding to an element of the vector t. The first element of t should be and should correspond to the initial state of the system x_0, so that the first row of the output is x_0.

  The first argument, fcn, is a string that names the function to call to compute the vector of right hand sides for the set of equations. The function must have the form

      xdot = f (x, t)

  in which xdot and x are vectors and t is a scalar.

  If fcn is a two-element string array, the first element names the function described above, and the second element names a function to compute the Jacobian of . The Jacobian function must have the form

      jac = j (x, t)

  in which jac is the matrix of partial derivatives

                   | df_1  df_1       df_1 |
                   | ----  ----  ...  ---- |
                   | dx_1  dx_2       dx_N |
                   |                       |
                   | df_2  df_2       df_2 |
                   | ----  ----  ...  ---- |
            df_i   | dx_1  dx_2       dx_N |
      jac = ---- = |                       |
            dx_j   |  .    .     .    .    |
                   |  .    .      .   .    |
                   |  .    .       .  .    |
                   |                       |
                   | df_N  df_N       df_N |
                   | ----  ----  ...  ---- |
                   | dx_1  dx_2       dx_N |

  The second and third arguments specify the intial state of the system, , and the initial value of the independent variable .

  The fourth argument is optional, and may be used to specify a set of times that the ODE solver should not integrate past. It is useful for avoiding difficulties with singularities and points where there is a discontinuity in the derivative.

  After a successful computation, the value of istate will be 2 (consistent with the Fortran version of LSODE).

  If the computation is not successful, istate will be something other than 2 and msg will contain additional information.

  You can use the function `lsode_options` to set optional parameters for `lsode`.

Loadable Function: lsode_options (opt, val)
  When called with two arguments, this function allows you set options parameters for the function `lsode`. Given one argument, `lsode_options` returns the value of the corresponding option. If no arguments are supplied, the names of all the available options and their current values are displayed.

  Options include

  `"absolute tolerance"`
    Absolute tolerance. May be either vector or scalar. If a vector, it must match the dimension of the state vector.
  `"relative tolerance"`
    Relative tolerance parameter. Unlike the absolute tolerance, this parameter may only be a scalar.

    The local error test applied at each integration step is

          abs (local error in x(i)) <= rtol * abs (y(i)) + atol(i)

  `"integration method"`
    A string specifing the method of integration to use to solve the ODE system. Valid values are

    "adams"
    "non-stiff"
      No Jacobian used (even if it is available).
    "bdf"
    "stiff"
      Use stiff backward differentiation formula (BDF) method. If a function to compute the Jacobian is not supplied, `lsode` will compute a finite difference approximation of the Jacobian matrix.
  `"initial step size"`
    The step size to be attempted on the first step (default is determined automatically).
  `"maximum order"`
    Restrict the maximum order of the solution method. If using the Adams method, this option must be between 1 and 12. Otherwise, it must be between 1 and 5, inclusive.
  `"maximum step size"`
    Setting the maximum stepsize will avoid passing over very large regions (default is not specified).
  `"minimum step size"`
    The minimum absolute step size allowed (default is 0).
  `"step limit"`
    Maximum number of steps allowed (default is 100000).

Here is an example of solving a set of three differential equations using `lsode`. Given the function

    function xdot = f (x, t)

      xdot = zeros (3,1);

      xdot(1) = 77.27 * (x(2) - x(1)*x(2) + x(1) \
                - 8.375e-06*x(1)^2);
      xdot(2) = (x(3) - x(1)*x(2) - x(2)) / 77.27;
      xdot(3) = 0.161*(x(1) - x(3));

    endfunction

and the initial condition `x0 = [ 4; 1.1; 4 ]`, the set of equations can be integrated using the command

    t = linspace (0, 500, 1000);

    y = lsode ("f", x0, t);

If you try this, you will see that the value of the result changes dramatically between t = 0 and 5, and again around t = 305. A more efficient set of output points might be

    t = [0, logspace (-1, log10(303), 150), \
            logspace (log10(304), log10(500), 150)];

Octave also includes ODESSA, a modification of LSODE that allows for the computation of parametric sensitivity information simultaneously with the solution of the set of ODEs.

Loadable Function: [x, sx, istate, msg] odessa (fcn, x_0, p, sx_0, t, t_crit)
  Solve the set of differential equations

      dx
      -- = f(x, t; p)
      dt

  with

      x(t_0) = x_0

  and simultaneously compute the first-order sensitivity coefficients given by

      s'(t) = j(t)*s(t) + df/dp

  in which

      s(t)  = dx(t)/dp        (sensitivity functions)
      s'(t) = d(dx(t)/dp)/dt
      j(t)  = df(x,t;p)/dx(t) (Jacobian matrix)
      df/dp = df(x,t;p)/dp    (inhomogeneity matrix)

  The solution is returned in the matrix x, with each row corresponding to an element of the vector t. The first element of t should be and should correspond to the initial state of the system x_0, so that the first row of the output is x_0.

  The sensitivities are returned in a list of matrices, sx, with each element of the list corresponding to an element of the vector t.

  The first argument, fcn, is a string that names the function to call to compute the vector of right hand sides for the set of equations. The function must have the form

      xdot = f (x, t, p)

  in which xdot and x are vectors and t is a scalar.

  The variable p is a vector of parameters.

  The fcn argument may also be an array of strings

      ["f"; "j"; "b"]

  in which the first element names the function described above, the second element names a function to compute the Jacobian of , and the third element names a function to compute the inhomogeneity matrix.

  The Jacobian function must have the form

      jac = j (x, t, p)

  in which x, t, and p have the same meanings as above for the function f, and jac is the matrix of partial derivatives

            df_i
      jac = ----
            dx_j

  The function b must have the form

      dfdp = b (x, t, p, c)

  in which x, t, and p have the same meanings as above for the function f, c indicates which partial derivatives to return in dfdp. For example, if c is 2, you should return the vector

             df_i
      dfdp = ----,    i = 1:length(x)
             dp_2

  The second argument, x_0, specifies the intial state of the system.

  The third argument, p, specifies the set of parameters.

  The fourth argument, sx_0 specifies the initial values of the sensitivities.

  The sixth argument is optional, and may be used to specify a set of times that the ODE solver should not integrate past. It is useful for avoiding difficulties with singularities and points where there is a discontinuity in the derivative.

  After a successful computation, the value of istate will be 2 (consistent with the Fortran version of ODESSA).

  If the computation is not successful, istate will be something other than 2 and msg will contain additional information.

  You can use the function `odessa_options` to set optional parameters for `odessa`.

Loadable Function: odessa_options (opt, val)
  When called with two arguments, this function allows you set options parameters for the function `odessa`. Given one argument, `odessa_options` returns the value of the corresponding option. If no arguments are supplied, the names of all the available options and their current values are displayed.

  Options include

  `"absolute tolerance"`
    Absolute tolerance. May be either vector or scalar. If a vector, it must match the dimension of the state vector.
  `"relative tolerance"`
    Relative tolerance parameter. Unlike the absolute tolerance, this parameter may only be a scalar.

    The local error test applied at each integration step is

          abs (local error in x(i))
               <= rtol * abs (y(i)) + atol(i)

  `"integration method"`
    A string specifing the method of integration to use to solve the ODE system. Valid values are

    "adams"
    "non-stiff"
      No Jacobian used (even if it is available).
    "bdf"
    "stiff"
      Use stiff backward differentiation formula (BDF) method. If a function to compute the Jacobian is not supplied, `lsode` will compute a finite difference approximation of the Jacobian matrix.
  `"initial step size"`
    The step size to be attempted on the first step (default is determined automatically).
  `"maximum order"`
    Restrict the maximum order of the solution method. If using the Adams method, this option must be between 1 and 12. Otherwise, it must be between 1 and 5, inclusive.
  `"maximum step size"`
    Setting the maximum stepsize will avoid passing over very large regions (default is not specified).
  `"minimum step size"`
    The minimum absolute step size allowed (default is 0).
  `"step limit"`
    Maximum number of steps allowed (default is 100000).

See Alan C. Hindmarsh, ODEPACK, A Systematized Collection of ODE Solvers, in Scientific Computing, R. S. Stepleman, editor, (1983) for more information about the inner workings of `lsode`.

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## 23.2 Differential-Algebraic Equations

The function `daspk` can be used to solve DAEs of the form

    0 = f (x-dot, x, t),    x(t=0) = x_0, x-dot(t=0) = x-dot_0

using Petzold's DAE solver DASPK.

Loadable Function: [x, xdot, istate, msg] = daspk (fcn, x_0, xdot_0, t, t_crit)
  Solve the set of differential-algebraic equations

      0 = f (x, xdot, t)

  with

      x(t_0) = x_0, xdot(t_0) = xdot_0

  The solution is returned in the matrices x and xdot, with each row in the result matrices corresponding to one of the elements in the vector t. The first element of t should be and correspond to the initial state of the system x_0 and its derivative xdot_0, so that the first row of the output x is x_0 and the first row of the output xdot is xdot_0.

  The first argument, fcn, is a string that names the function to call to compute the vector of residuals for the set of equations. It must have the form

      res = f (x, xdot, t)

  in which x, xdot, and res are vectors, and t is a scalar.

  If fcn is a two-element string array, the first element names the function described above, and the second element names a function to compute the modified Jacobian

            df       df
      jac = -- + c ------
            dx     d xdot

  The modified Jacobian function must have the form

      jac = j (x, xdot, t, c)

  The second and third arguments to `daspk` specify the initial condition of the states and their derivatives, and the fourth argument specifies a vector of output times at which the solution is desired, including the time corresponding to the initial condition.

  The set of initial states and derivatives are not strictly required to be consistent. If they are not consistent, you must use the `daspk_options` function to provide additional information so that `daspk` can compute a consistent starting point.

  The fifth argument is optional, and may be used to specify a set of times that the DAE solver should not integrate past. It is useful for avoiding difficulties with singularities and points where there is a discontinuity in the derivative.

  After a successful computation, the value of istate will be greater than zero (consistent with the Fortran version of DASPK).

  If the computation is not successful, the value of istate will be less than zero and msg will contain additional information.

  You can use the function `daspk_options` to set optional parameters for `daspk`.

Loadable Function: daspk_options (opt, val)
  When called with two arguments, this function allows you set options parameters for the function `daspk`. Given one argument, `daspk_options` returns the value of the corresponding option. If no arguments are supplied, the names of all the available options and their current values are displayed.

  Options include

  `"absolute tolerance"`
    Absolute tolerance. May be either vector or scalar. If a vector, it must match the dimension of the state vector, and the relative tolerance must also be a vector of the same length.
  `"relative tolerance"`
    Relative tolerance. May be either vector or scalar. If a vector, it must match the dimension of the state vector, and the absolute tolerance must also be a vector of the same length.

    The local error test applied at each integration step is

          abs (local error in x(i))
               <= rtol(i) * abs (Y(i)) + atol(i)

  `"compute consistent initial condition"`
    Denoting the differential variables in the state vector by ``Y_d`' and the algebraic variables by ``Y_a`', `ddaspk` can solve one of two initialization problems:

    1. Given Y_d, calculate Y_a and Y'_d
    2. Given Y', calculate Y.

    In either case, initial values for the given components are input, and initial guesses for the unknown components must also be provided as input. Set this option to 1 to solve the first problem, or 2 to solve the second (the default default is 0, so you must provide a set of initial conditions that are consistent).

    If this option is set to a nonzero value, you must also set the `"algebraic variables"` option to declare which variables in the problem are algebraic.

  `"use initial condition heuristics"`
    Set to a nonzero value to use the initial condition heuristics options described below.
  `"initial condition heuristics"`
    A vector of the following parameters that can be used to control the initial condition calculation.

    `MXNIT`
      Maximum number of Newton iterations (default is 5).
    `MXNJ`
      Maximum number of Jacobian evaluations (default is 6).
    `MXNH`
      Maximum number of values of the artificial stepsize parameter to be tried if the `"compute consistent initial condition"` option has been set to 1 (default is 5).

      Note that the maximum number of Newton iterations allowed in all is `MXNIT*MXNJ*MXNH` if the `"compute consistent initial condition"` option has been set to 1 and `MXNIT*MXNJ` if it is set to 2.

    `LSOFF`
      Set to a nonzero value to disable the linesearch algorithm (default is 0).
    `STPTOL`
      Minimum scaled step in linesearch algorithm (default is eps^(2/3)).
    `EPINIT`
      Swing factor in the Newton iteration convergence test. The test is applied to the residual vector, premultiplied by the approximate Jacobian. For convergence, the weighted RMS norm of this vector (scaled by the error weights) must be less than `EPINIT*EPCON`, where `EPCON` = 0.33 is the analogous test constant used in the time steps. The default is `EPINIT` = 0.01.
  `"print initial condition info"`
    Set this option to a nonzero value to display detailed information about the initial condition calculation (default is 0).
  `"exclude algebraic variables from error test"`
    Set to a nonzero value to exclude algebraic variables from the error test. You must also set the `"algebraic variables"` option to declare which variables in the problem are algebraic (default is 0).
  `"algebraic variables"`
    A vector of the same length as the state vector. A nonzero element indicates that the corresponding element of the state vector is an algebraic variable (i.e., its derivative does not appear explicitly in the equation set.

    This option is required by the `compute consistent initial condition"` and `"exclude algebraic variables from error test"` options.

  `"enforce inequality constraints"`
    Set to one of the following values to enforce the inequality constraints specified by the `"inequality constraint types"` option (default is 0).

    1. To have constraint checking only in the initial condition calculation.
    2. To enforce constraint checking during the integration.
    3. To enforce both options 1 and 2.
  `"inequality constraint types"`
    A vector of the same length as the state specifying the type of inequality constraint. Each element of the vector corresponds to an element of the state and should be assigned one of the following codes

    -2
      Less than zero.
    -1
      Less than or equal to zero.
    0
      Not constrained.
    1
      Greater than or equal to zero.
    2
      Greater than zero.

    This option only has an effect if the `"enforce inequality constraints"` option is nonzero.

  `"initial step size"`
    Differential-algebraic problems may occaisionally suffer from severe scaling difficulties on the first step. If you know a great deal about the scaling of your problem, you can help to alleviate this problem by specifying an initial stepsize (default is computed automatically).
  `"maximum order"`
    Restrict the maximum order of the solution method. This option must be between 1 and 5, inclusive (default is 5).
  `"maximum step size"`
    Setting the maximum stepsize will avoid passing over very large regions (default is not specified).

Octave also includes DASSL, an earlier version of Daspk, and dasrt, which can be used to solve DAEs with constraints (stopping conditions).

Loadable Function: [x, xdot, t_out, istat, msg] = dasrt (fcn [, g], x_0, xdot_0, t [, t_crit])
  Solve the set of differential-algebraic equations

      0 = f (x, xdot, t)

  with

      x(t_0) = x_0, xdot(t_0) = xdot_0

  with functional stopping criteria (root solving).

  The solution is returned in the matrices x and xdot, with each row in the result matrices corresponding to one of the elements in the vector t_out. The first element of t should be and correspond to the initial state of the system x_0 and its derivative xdot_0, so that the first row of the output x is x_0 and the first row of the output xdot is xdot_0.

  The vector t provides an upper limit on the length of the integration. If the stopping condition is met, the vector t_out will be shorter than t, and the final element of t_out will be the point at which the stopping condition was met, and may not correspond to any element of the vector t.

  The first argument, fcn, is a string that names the function to call to compute the vector of residuals for the set of equations. It must have the form

      res = f (x, xdot, t)

  in which x, xdot, and res are vectors, and t is a scalar.

  If fcn is a two-element string array, the first element names the function described above, and the second element names a function to compute the modified Jacobian

            df       df
      jac = -- + c ------
            dx     d xdot

  The modified Jacobian function must have the form

      jac = j (x, xdot, t, c)

  The optional second argument names a function that defines the constraint functions whose roots are desired during the integration. This function must have the form

      g_out = g (x, t)

  and return a vector of the constraint function values. If the value of any of the constraint functions changes sign, DASRT will attempt to stop the integration at the point of the sign change.

  If the name of the constraint function is omitted, `dasrt` solves the same problem as `daspk` or `dassl`.

  Note that because of numerical errors in the constraint functions due to roundoff and integration error, DASRT may return false roots, or return the same root at two or more nearly equal values of T. If such false roots are suspected, the user should consider smaller error tolerances or higher precision in the evaluation of the constraint functions.

  If a root of some constraint function defines the end of the problem, the input to DASRT should nevertheless allow integration to a point slightly past that root, so that DASRT can locate the root by interpolation.

  The third and fourth arguments to `dasrt` specify the initial condition of the states and their derivatives, and the fourth argument specifies a vector of output times at which the solution is desired, including the time corresponding to the initial condition.

  The set of initial states and derivatives are not strictly required to be consistent. In practice, however, DASSL is not very good at determining a consistent set for you, so it is best if you ensure that the initial values result in the function evaluating to zero.

  The sixth argument is optional, and may be used to specify a set of times that the DAE solver should not integrate past. It is useful for avoiding difficulties with singularities and points where there is a discontinuity in the derivative.

  After a successful computation, the value of istate will be greater than zero (consistent with the Fortran version of DASSL).

  If the computation is not successful, the value of istate will be less than zero and msg will contain additional information.

  You can use the function `dasrt_options` to set optional parameters for `dasrt`.

Loadable Function: dasrt_options (opt, val)
  When called with two arguments, this function allows you set options parameters for the function `dasrt`. Given one argument, `dasrt_options` returns the value of the corresponding option. If no arguments are supplied, the names of all the available options and their current values are displayed.

  Options include

  `"absolute tolerance"`
    Absolute tolerance. May be either vector or scalar. If a vector, it must match the dimension of the state vector, and the relative tolerance must also be a vector of the same length.
  `"relative tolerance"`
    Relative tolerance. May be either vector or scalar. If a vector, it must match the dimension of the state vector, and the absolute tolerance must also be a vector of the same length.

    The local error test applied at each integration step is

          abs (local error in x(i)) <= rtol(i) * abs (Y(i)) + atol(i)

  `"initial step size"`
    Differential-algebraic problems may occaisionally suffer from severe scaling difficulties on the first step. If you know a great deal about the scaling of your problem, you can help to alleviate this problem by specifying an initial stepsize.
  `"maximum order"`
    Restrict the maximum order of the solution method. This option must be between 1 and 5, inclusive.
  `"maximum step size"`
    Setting the maximum stepsize will avoid passing over very large regions.
  `"step limit"`
    Maximum number of integration steps to attempt on a single call to the underlying Fortran code.

See K. E. Brenan, et al., Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations, North-Holland (1989) for more information about the implementation of DASSL.

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