@c DO NOT EDIT!  Generated automatically by munge-texi.

@c Copyright (C) 1996, 1997 John W. Eaton
@c This is part of the Octave manual.
@c For copying conditions, see the file gpl.texi.

@node Differential Equations
@chapter Differential Equations

Octave has two built-in functions for solving differential equations.
Both are based on reliable ODE solvers 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
@iftex
@tex
$$
 {dx\over dt} = f (x, t)
$$
@end tex
@end iftex
@ifinfo

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

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

@anchor{doc-lsode}
@deftypefn {Loadable 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
@ifinfo

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

with

@example
x(t_0) = x_0
@end example

@end ifinfo
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 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

@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, 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

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
@ifinfo

@example
             | 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 example

@end ifinfo

The second and third arguments specify the intial 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}.
@end deftypefn
@seealso{daspk, dassl, dasrt, odessa}


@anchor{doc-lsode_options}
@deftypefn {Loadable Function} {} lsode_options (@var{opt}, @var{val})
When called with two arguments, this function
allows you set options parameters for the function @code{lsode}.
Given one argument, @code{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

@table @code
@item "absolute tolerance"
Absolute tolerance.  May be either vector or scalar.  If a vector, it
must match the dimension of the state vector.
@item "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
  abs (local error in x(i)) <= rtol * abs (y(i)) + atol(i)
@end example
@item "integration method"
A string specifing the method of integration to use to solve the ODE
system.  Valid values are

@table @asis
@item "adams"
@itemx "non-stiff"
No Jacobian used (even if it is available).
@item "bdf"
@item "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 "initial step size"
The step size to be attempted on the first step (default is determined
automatically).
@item "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 "maximum step size"
Setting the maximum stepsize will avoid passing over very large
regions  (default is not specified).
@item "minimum step size"
The minimum absolute step size allowed (default is 0).
@item "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
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 Alan C. Hindmarsh, @cite{ODEPACK, A Systematized Collection of ODE
Solvers}, in Scientific Computing, R. S. Stepleman, editor, (1983) for
more information about the inner workings of @code{lsode}.

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

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

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

@noindent
using Petzold's DAE solver @sc{Daspk}.

@anchor{doc-daspk}
@deftypefn {Loadable 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
@ifinfo

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

with

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

@end ifinfo
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 that names the function 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, 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
@ifinfo

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

The modified Jacobian function must have the form

@example

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

@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}.
@end deftypefn
@seealso{dassl}


@anchor{doc-daspk_options}
@deftypefn {Loadable Function} {} daspk_options (@var{opt}, @var{val})
When called with two arguments, this function
allows you set options parameters for the function @code{daspk}.
Given one argument, @code{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

@table @code
@item "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 "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
  abs (local error in x(i))
       <= rtol(i) * abs (Y(i)) + atol(i)
@end example
@item "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 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
@code{"algebraic variables"} option to declare which variables in the
problem are algebraic.
@item "use initial condition heuristics"
Set to a nonzero value to use the initial condition heuristics options
described below.
@item "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 @code{"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
@code{MXNIT*MXNJ*MXNH} if the @code{"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 "print initial condition info"
Set this option to a nonzero value to display detailed information
about the initial condition calculation (default is 0).
@item "exclude algebraic variables from error test"
Set to a nonzero value to exclude algebraic variables from the error
test.  You must also set the @code{"algebraic variables"} option to
declare which variables in the problem are algebraic (default is 0).
@item "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
@code{compute consistent initial condition"} and
@code{"exclude algebraic variables from error test"} options.
@item "enforce inequality constraints"
Set to one of the following values to enforce the inequality
constraints specified by the @code{"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 "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
@code{"enforce inequality constraints"} option is nonzero.
@item "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).
@item "maximum order"
Restrict the maximum order of the solution method.  This option must
be between 1 and 5, inclusive (default is 5).
@item "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 @var{Daspk},
and @var{dasrt}, which can be used to solve DAEs with constraints
(stopping conditions).

@anchor{doc-dasrt}
@deftypefn {Loadable Function} {[@var{x}, @var{xdot}, @var{t_out}, @var{istat}, @var{msg}] =} 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
@ifinfo

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

with

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

@end ifinfo
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 that names the function 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, 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
@ifinfo

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

@end ifinfo

The modified Jacobian function must have the form

@example

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

@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

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 roundoff 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}.
@end deftypefn
@seealso{daspk, dasrt, lsode, odessa}


@anchor{doc-dasrt_options}
@deftypefn {Loadable Function} {} dasrt_options (@var{opt}, @var{val})
When called with two arguments, this function
allows you set options parameters for the function @code{dasrt}.
Given one argument, @code{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

@table @code
@item "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 "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
  abs (local error in x(i)) <= rtol(i) * abs (Y(i)) + atol(i)
@end example
@item "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.
@item "maximum order"
Restrict the maximum order of the solution method.  This option must
be between 1 and 5, inclusive.
@item "maximum step size"
Setting the maximum stepsize will avoid passing over very large
regions.
@item "step limit"
Maximum number of integration steps to attempt on a single call to the
underlying Fortran code.
@end table
@end deftypefn


See 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}.