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@node Differential Equations
@chapter Differential Equations

Octave has built-in functions for solving ordinary differential equations,
and differential-algebraic equations.
All solvers are based on reliable ODE routines written in Fortran.

@menu
* Ordinary Differential Equations::
* Differential-Algebraic Equations::
@end menu

@cindex differential equations
@cindex ODE
@cindex DAE

@node Ordinary Differential Equations
@section Ordinary Differential Equations

The function @code{lsode} can be used to solve ODEs of the form
@tex
$$
 {dx\over dt} = f (x, t)
$$
@end tex
@ifnottex

@example
@group
dx
-- = f (x, t)
dt
@end group
@end example

@end ifnottex

@noindent
using @nospell{Hindmarsh's} ODE solver @sc{lsode}.



@c lsode libinterp/corefcn/lsode.cc
@anchor{XREFlsode}
@deftypefn  {Built-in Function} {[@var{x}, @var{istate}, @var{msg}] =} lsode (@var{fcn}, @var{x_0}, @var{t})
@deftypefnx {Built-in Function} {[@var{x}, @var{istate}, @var{msg}] =} lsode (@var{fcn}, @var{x_0}, @var{t}, @var{t_crit})
Solve the set of differential equations
@tex
$$ {dx \over dt} = f (x, t) $$
with
$$ x(t_0) = x_0 $$
@end tex
@ifnottex

@example
@group
dx
-- = f (x, t)
dt
@end group
@end example

@noindent
with

@example
x(t_0) = x_0
@end example

@end ifnottex
The solution is returned in the matrix @var{x}, with each row
corresponding to an element of the vector @var{t}.  The first element
of @var{t} should be @math{t_0} and should correspond to the initial
state of the system @var{x_0}, so that the first row of the output
is @var{x_0}.

The first argument, @var{fcn}, is a string, inline, or function handle
that names the function @math{f} to call to compute the vector of right
hand sides for the set of equations.  The function must have the form

@example
@var{xdot} = f (@var{x}, @var{t})
@end example

@noindent
in which @var{xdot} and @var{x} are vectors and @var{t} is a scalar.

If @var{fcn} is a two-element string array or a two-element cell array
of strings, inline functions, or function handles, the first element names
the function @math{f} described above, and the second element names a
function to compute the Jacobian of @math{f}.  The Jacobian function
must have the form

@example
@var{jac} = j (@var{x}, @var{t})
@end example

@noindent
in which @var{jac} is the matrix of partial derivatives
@tex
$$ J = {\partial f_i \over \partial x_j} = \left[\matrix{
{\partial f_1 \over \partial x_1}
  & {\partial f_1 \over \partial x_2}
  & \cdots
  & {\partial f_1 \over \partial x_N} \cr
{\partial f_2 \over \partial x_1}
  & {\partial f_2 \over \partial x_2}
  & \cdots
  & {\partial f_2 \over \partial x_N} \cr
 \vdots & \vdots & \ddots & \vdots \cr
{\partial f_3 \over \partial x_1}
  & {\partial f_3 \over \partial x_2}
  & \cdots
  & {\partial f_3 \over \partial x_N} \cr}\right]$$
@end tex
@ifnottex

@example
@group
             | 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 |
@end group
@end example

@end ifnottex

The second and third arguments specify the initial state of the system,
@math{x_0}, and the initial value of the independent variable @math{t_0}.

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 @var{istate} will be 2
(consistent with the Fortran version of @sc{lsode}).

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

You can use the function @code{lsode_options} to set optional
parameters for @code{lsode}.
@seealso{@ref{XREFdaspk,,daspk}, @ref{XREFdassl,,dassl}, @ref{XREFdasrt,,dasrt}}
@end deftypefn


@c lsode_options libinterp/corefcn/lsode.cc
@anchor{XREFlsode_options}
@deftypefn  {Built-in Function} {} lsode_options ()
@deftypefnx {Built-in Function} {val =} lsode_options (@var{opt})
@deftypefnx {Built-in Function} {} lsode_options (@var{opt}, @var{val})
Query or set options for the function @code{lsode}.
When called with no arguments, the names of all available options and
their current values are displayed.
Given one argument, return the value of the corresponding option.
When called with two arguments, @code{lsode_options} set the option
@var{opt} to value @var{val}.

Options include

@table @code
@item @qcode{"absolute tolerance"}
Absolute tolerance.  May be either vector or scalar.  If a vector, it
must match the dimension of the state vector.

@item @qcode{"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

@example
@group
  abs (local error in x(i)) <= ...
      rtol * abs (y(i)) + atol(i)
@end group
@end example

@item @qcode{"integration method"}
A string specifying the method of integration to use to solve the ODE
system.  Valid values are

@table @asis
@item  @qcode{"adams"}
@itemx @qcode{"non-stiff"}
No Jacobian used (even if it is available).

@item  @qcode{"bdf"}
@itemx @qcode{"stiff"}
Use stiff backward differentiation formula (BDF) method.  If a
function to compute the Jacobian is not supplied, @code{lsode} will
compute a finite difference approximation of the Jacobian matrix.
@end table

@item @qcode{"initial step size"}
The step size to be attempted on the first step (default is determined
automatically).

@item @qcode{"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.

@item @qcode{"maximum step size"}
Setting the maximum stepsize will avoid passing over very large
regions  (default is not specified).

@item @qcode{"minimum step size"}
The minimum absolute step size allowed (default is 0).

@item @qcode{"step limit"}
Maximum number of steps allowed (default is 100000).
@end table
@end deftypefn


Here is an example of solving a set of three differential equations using
@code{lsode}.  Given the function

@cindex oregonator

@example
@group
## oregonator differential equation
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
@end group
@end example

@noindent
and the initial condition @code{x0 = [ 4; 1.1; 4 ]}, the set of
equations can be integrated using the command

@example
@group
t = linspace (0, 500, 1000);

y = lsode ("f", x0, t);
@end group
@end example

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

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

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

An m-file for the differential equation used above is included with the
Octave distribution in the examples directory under the name
@file{oregonator.m}.

@node Differential-Algebraic Equations
@section Differential-Algebraic Equations

The function @code{daspk} can be used to solve DAEs of the form
@tex
$$
 0 = f (\dot{x}, x, t), \qquad x(t=0) = x_0, \dot{x}(t=0) = \dot{x}_0
$$
@end tex
@ifnottex

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

@end ifnottex

@noindent
where
@tex
$\dot{x} = {dx \over dt}$
@end tex
@ifnottex
@math{x-dot}
@end ifnottex
is the derivative of @math{x}.  The equation is solved using
@nospell{Petzold's} DAE solver @sc{daspk}.

@c daspk libinterp/corefcn/daspk.cc
@anchor{XREFdaspk}
@deftypefn {Built-in Function} {[@var{x}, @var{xdot}, @var{istate}, @var{msg}] =} daspk (@var{fcn}, @var{x_0}, @var{xdot_0}, @var{t}, @var{t_crit})
Solve the set of differential-algebraic equations
@tex
$$ 0 = f (x, \dot{x}, t) $$
with
$$ x(t_0) = x_0, \dot{x}(t_0) = \dot{x}_0 $$
@end tex
@ifnottex

@example
0 = f (x, xdot, t)
@end example

@noindent
with

@example
x(t_0) = x_0, xdot(t_0) = xdot_0
@end example

@end ifnottex
The solution is returned in the matrices @var{x} and @var{xdot},
with each row in the result matrices corresponding to one of the
elements in the vector @var{t}.  The first element of @var{t}
should be @math{t_0} and correspond to the initial state of the
system @var{x_0} and its derivative @var{xdot_0}, so that the first
row of the output @var{x} is @var{x_0} and the first row
of the output @var{xdot} is @var{xdot_0}.

The first argument, @var{fcn}, is a string, inline, or function handle
that names the function @math{f} to call to compute the vector of
residuals for the set of equations.  It must have the form

@example
@var{res} = f (@var{x}, @var{xdot}, @var{t})
@end example

@noindent
in which @var{x}, @var{xdot}, and @var{res} are vectors, and @var{t} is a
scalar.

If @var{fcn} is a two-element string array or a two-element cell array
of strings, inline functions, or function handles, the first element names
the function @math{f} described above, and the second element names a
function to compute the modified Jacobian
@tex
$$
J = {\partial f \over \partial x}
  + c {\partial f \over \partial \dot{x}}
$$
@end tex
@ifnottex

@example
@group
      df       df
jac = -- + c ------
      dx     d xdot
@end group
@end example

@end ifnottex

The modified Jacobian function must have the form

@example
@group

@var{jac} = j (@var{x}, @var{xdot}, @var{t}, @var{c})

@end group
@end example

The second and third arguments to @code{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
@code{daspk_options} function to provide additional information so
that @code{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 @var{istate} will be
greater than zero (consistent with the Fortran version of @sc{daspk}).

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

You can use the function @code{daspk_options} to set optional
parameters for @code{daspk}.
@seealso{@ref{XREFdassl,,dassl}}
@end deftypefn


@c daspk_options libinterp/corefcn/daspk.cc
@anchor{XREFdaspk_options}
@deftypefn  {Built-in Function} {} daspk_options ()
@deftypefnx {Built-in Function} {val =} daspk_options (@var{opt})
@deftypefnx {Built-in Function} {} daspk_options (@var{opt}, @var{val})
Query or set options for the function @code{daspk}.
When called with no arguments, the names of all available options and
their current values are displayed.
Given one argument, return the value of the corresponding option.
When called with two arguments, @code{daspk_options} set the option
@var{opt} to value @var{val}.

Options include

@table @code
@item @qcode{"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.

@item @qcode{"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

@example
@group
  abs (local error in x(i))
       <= rtol(i) * abs (Y(i)) + atol(i)
@end group
@end example

@item @qcode{"compute consistent initial condition"}
Denoting the differential variables in the state vector by @samp{Y_d}
and the algebraic variables by @samp{Y_a}, @code{ddaspk} can solve
one of two initialization problems:

@enumerate
@item Given Y_d, calculate Y_a and Y'_d

@item Given Y', calculate Y.
@end enumerate

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 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
@qcode{"algebraic variables"} option to declare which variables in the
problem are algebraic.

@item @qcode{"use initial condition heuristics"}
Set to a nonzero value to use the initial condition heuristics options
described below.

@item @qcode{"initial condition heuristics"}
A vector of the following parameters that can be used to control the
initial condition calculation.

@table @code
@item MXNIT
Maximum number of Newton iterations (default is 5).

@item MXNJ
Maximum number of Jacobian evaluations (default is 6).

@item MXNH
Maximum number of values of the artificial stepsize parameter to be
tried if the @qcode{"compute consistent initial condition"} option has
been set to 1 (default is 5).

Note that the maximum total number of Newton iterations allowed is
@code{MXNIT*MXNJ*MXNH} if the @qcode{"compute consistent initial
condition"} option has been set to 1 and @code{MXNIT*MXNJ} if it is
set to 2.

@item LSOFF
Set to a nonzero value to disable the linesearch algorithm (default is
0).

@item STPTOL
Minimum scaled step in linesearch algorithm (default is eps^(2/3)).

@item 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 @code{EPINIT*EPCON},
where @code{EPCON} = 0.33 is the analogous test constant used in the
time steps.  The default is @code{EPINIT} = 0.01.
@end table

@item @qcode{"print initial condition info"}
Set this option to a nonzero value to display detailed information
about the initial condition calculation (default is 0).

@item @qcode{"exclude algebraic variables from error test"}
Set to a nonzero value to exclude algebraic variables from the error
test.  You must also set the @qcode{"algebraic variables"} option to
declare which variables in the problem are algebraic (default is 0).

@item @qcode{"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
@qcode{"compute consistent initial condition"} and
@qcode{"exclude algebraic variables from error test"} options.

@item @qcode{"enforce inequality constraints"}
Set to one of the following values to enforce the inequality
constraints specified by the @qcode{"inequality constraint types"}
option (default is 0).

@enumerate
@item To have constraint checking only in the initial condition calculation.

@item To enforce constraint checking during the integration.

@item To enforce both options 1 and 2.
@end enumerate

@item @qcode{"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

@table @asis
@item -2
Less than zero.

@item -1
Less than or equal to zero.

@item 0
Not constrained.

@item 1
Greater than or equal to zero.

@item 2
Greater than zero.
@end table

This option only has an effect if the
@qcode{"enforce inequality constraints"} option is nonzero.

@item @qcode{"initial step size"}
Differential-algebraic problems may occasionally 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).

@item @qcode{"maximum order"}
Restrict the maximum order of the solution method.  This option must
be between 1 and 5, inclusive (default is 5).

@item @qcode{"maximum step size"}
Setting the maximum stepsize will avoid passing over very large
regions (default is not specified).
@end table
@end deftypefn


Octave also includes @sc{dassl}, an earlier version of @sc{daspk},
and @sc{dasrt}, which can be used to solve DAEs with constraints
(stopping conditions).

@c dassl libinterp/corefcn/dassl.cc
@anchor{XREFdassl}
@deftypefn {Built-in Function} {[@var{x}, @var{xdot}, @var{istate}, @var{msg}] =} dassl (@var{fcn}, @var{x_0}, @var{xdot_0}, @var{t}, @var{t_crit})
Solve the set of differential-algebraic equations
@tex
$$ 0 = f (x, \dot{x}, t) $$
with
$$ x(t_0) = x_0, \dot{x}(t_0) = \dot{x}_0 $$
@end tex
@ifnottex

@example
0 = f (x, xdot, t)
@end example

@noindent
with

@example
x(t_0) = x_0, xdot(t_0) = xdot_0
@end example

@end ifnottex
The solution is returned in the matrices @var{x} and @var{xdot},
with each row in the result matrices corresponding to one of the
elements in the vector @var{t}.  The first element of @var{t}
should be @math{t_0} and correspond to the initial state of the
system @var{x_0} and its derivative @var{xdot_0}, so that the first
row of the output @var{x} is @var{x_0} and the first row
of the output @var{xdot} is @var{xdot_0}.

The first argument, @var{fcn}, is a string, inline, or function handle
that names the function @math{f} to call to compute the vector of
residuals for the set of equations.  It must have the form

@example
@var{res} = f (@var{x}, @var{xdot}, @var{t})
@end example

@noindent
in which @var{x}, @var{xdot}, and @var{res} are vectors, and @var{t} is a
scalar.

If @var{fcn} is a two-element string array or a two-element cell array
of strings, inline functions, or function handles, the first element names
the function @math{f} described above, and the second element names a
function to compute the modified Jacobian

@tex
$$
J = {\partial f \over \partial x}
  + c {\partial f \over \partial \dot{x}}
$$
@end tex
@ifnottex

@example
@group
      df       df
jac = -- + c ------
      dx     d xdot
@end group
@end example

@end ifnottex

The modified Jacobian function must have the form

@example
@group

@var{jac} = j (@var{x}, @var{xdot}, @var{t}, @var{c})

@end group
@end example

The second and third arguments to @code{dassl} 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, @sc{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 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 @var{istate} will be
greater than zero (consistent with the Fortran version of @sc{dassl}).

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

You can use the function @code{dassl_options} to set optional
parameters for @code{dassl}.
@seealso{@ref{XREFdaspk,,daspk}, @ref{XREFdasrt,,dasrt}, @ref{XREFlsode,,lsode}}
@end deftypefn


@c dassl_options libinterp/corefcn/dassl.cc
@anchor{XREFdassl_options}
@deftypefn  {Built-in Function} {} dassl_options ()
@deftypefnx {Built-in Function} {val =} dassl_options (@var{opt})
@deftypefnx {Built-in Function} {} dassl_options (@var{opt}, @var{val})
Query or set options for the function @code{dassl}.
When called with no arguments, the names of all available options and
their current values are displayed.
Given one argument, return the value of the corresponding option.
When called with two arguments, @code{dassl_options} set the option
@var{opt} to value @var{val}.

Options include

@table @code
@item @qcode{"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.

@item @qcode{"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

@example
@group
  abs (local error in x(i))
       <= rtol(i) * abs (Y(i)) + atol(i)
@end group
@end example

@item @qcode{"compute consistent initial condition"}
If nonzero, @code{dassl} will attempt to compute a consistent set of initial
conditions.  This is generally not reliable, so it is best to provide
a consistent set and leave this option set to zero.

@item @qcode{"enforce nonnegativity constraints"}
If you know that the solutions to your equations will always be
non-negative, it may help to set this parameter to a nonzero
value.  However, it is probably best to try leaving this option set to
zero first, and only setting it to a nonzero value if that doesn't
work very well.

@item @qcode{"initial step size"}
Differential-algebraic problems may occasionally 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.

@item @qcode{"maximum order"}
Restrict the maximum order of the solution method.  This option must
be between 1 and 5, inclusive.

@item @qcode{"maximum step size"}
Setting the maximum stepsize will avoid passing over very large
regions  (default is not specified).

@item @qcode{"step limit"}
Maximum number of integration steps to attempt on a single call to the
underlying Fortran code.
@end table
@end deftypefn


@c dasrt libinterp/corefcn/dasrt.cc
@anchor{XREFdasrt}
@deftypefn  {Built-in Function} {[@var{x}, @var{xdot}, @var{t_out}, @var{istat}, @var{msg}] =} dasrt (@var{fcn}, [], @var{x_0}, @var{xdot_0}, @var{t})
@deftypefnx {Built-in Function} {@dots{} =} dasrt (@var{fcn}, @var{g}, @var{x_0}, @var{xdot_0}, @var{t})
@deftypefnx {Built-in Function} {@dots{} =} dasrt (@var{fcn}, [], @var{x_0}, @var{xdot_0}, @var{t}, @var{t_crit})
@deftypefnx {Built-in Function} {@dots{} =} dasrt (@var{fcn}, @var{g}, @var{x_0}, @var{xdot_0}, @var{t}, @var{t_crit})
Solve the set of differential-algebraic equations
@tex
$$ 0 = f (x, \dot{x}, t) $$
with
$$ x(t_0) = x_0, \dot{x}(t_0) = \dot{x}_0 $$
@end tex
@ifnottex

@example
0 = f (x, xdot, t)
@end example

@noindent
with

@example
x(t_0) = x_0, xdot(t_0) = xdot_0
@end example

@end ifnottex
with functional stopping criteria (root solving).

The solution is returned in the matrices @var{x} and @var{xdot},
with each row in the result matrices corresponding to one of the
elements in the vector @var{t_out}.  The first element of @var{t}
should be @math{t_0} and correspond to the initial state of the
system @var{x_0} and its derivative @var{xdot_0}, so that the first
row of the output @var{x} is @var{x_0} and the first row
of the output @var{xdot} is @var{xdot_0}.

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

The first argument, @var{fcn}, is a string, inline, or function handle
that names the function @math{f} to call to compute the vector of
residuals for the set of equations.  It must have the form

@example
@var{res} = f (@var{x}, @var{xdot}, @var{t})
@end example

@noindent
in which @var{x}, @var{xdot}, and @var{res} are vectors, and @var{t} is a
scalar.

If @var{fcn} is a two-element string array or a two-element cell array
of strings, inline functions, or function handles, the first element names
the function @math{f} described above, and the second element names a
function to compute the modified Jacobian

@tex
$$
J = {\partial f \over \partial x}
  + c {\partial f \over \partial \dot{x}}
$$
@end tex
@ifnottex

@example
@group
      df       df
jac = -- + c ------
      dx     d xdot
@end group
@end example

@end ifnottex

The modified Jacobian function must have the form

@example
@group

@var{jac} = j (@var{x}, @var{xdot}, @var{t}, @var{c})

@end group
@end example

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

@example
@var{g_out} = g (@var{x}, @var{t})
@end example

@noindent
and return a vector of the constraint function values.
If the value of any of the constraint functions changes sign, @sc{dasrt}
will attempt to stop the integration at the point of the sign change.

If the name of the constraint function is omitted, @code{dasrt} solves
the same problem as @code{daspk} or @code{dassl}.

Note that because of numerical errors in the constraint functions
due to round-off and integration error, @sc{dasrt} may return false
roots, or return the same root at two or more nearly equal values of
@var{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 @sc{dasrt} should nevertheless allow integration to a
point slightly past that root, so that @sc{dasrt} can locate the root
by interpolation.

The third and fourth arguments to @code{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, @sc{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 @var{istate} will be
greater than zero (consistent with the Fortran version of @sc{dassl}).

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

You can use the function @code{dasrt_options} to set optional
parameters for @code{dasrt}.
@seealso{@ref{XREFdasrt_options,,dasrt_options}, @ref{XREFdaspk,,daspk}, @ref{XREFdasrt,,dasrt}, @ref{XREFlsode,,lsode}}
@end deftypefn


@c dasrt_options libinterp/corefcn/dasrt.cc
@anchor{XREFdasrt_options}
@deftypefn  {Built-in Function} {} dasrt_options ()
@deftypefnx {Built-in Function} {val =} dasrt_options (@var{opt})
@deftypefnx {Built-in Function} {} dasrt_options (@var{opt}, @var{val})
Query or set options for the function @code{dasrt}.
When called with no arguments, the names of all available options and
their current values are displayed.
Given one argument, return the value of the corresponding option.
When called with two arguments, @code{dasrt_options} set the option
@var{opt} to value @var{val}.

Options include

@table @code
@item @qcode{"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.

@item @qcode{"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

@example
@group
  abs (local error in x(i)) <= ...
      rtol(i) * abs (Y(i)) + atol(i)
@end group
@end example

@item @qcode{"initial step size"}
Differential-algebraic problems may occasionally 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.

@item @qcode{"maximum order"}
Restrict the maximum order of the solution method.  This option must
be between 1 and 5, inclusive.

@item @qcode{"maximum step size"}
Setting the maximum stepsize will avoid passing over very large
regions.

@item @qcode{"step limit"}
Maximum number of integration steps to attempt on a single call to the
underlying Fortran code.
@end table
@end deftypefn


See @nospell{K. E. Brenan}, et al., @cite{Numerical Solution of Initial-Value
Problems in Differential-Algebraic Equations}, North-Holland (1989) for
more information about the implementation of @sc{dassl}.
