@cindex differential equations, initial value problems
@cindex initial value problems, differential equations
@cindex ordinary differential equations, initial value problem
@cindex ODEs, initial value problems
This chapter describes functions for solving ordinary differential
equation (ODE) initial value problems.  The library provides a variety
of low-level methods, such as Runge-Kutta and Bulirsch-Stoer routines,
and higher-level components for adaptive step-size control.  The
components can be combined by the user to achieve the desired solution,
with full access to any intermediate steps.

These functions are declared in the header file @file{gsl_odeiv.h}.

@menu
* Defining the ODE System::     
* Stepping Functions::          
* Adaptive Step-size Control::  
* Evolution::                   
* ODE Example programs::        
* ODE References and Further Reading::  
@end menu

@node Defining the ODE System
@section Defining the ODE System

The routines solve the general @math{n}-dimensional first-order system,
@tex
\beforedisplay
$$
{dy_i(t) \over dt} = f_i (t, y_1(t), \dots y_n(t))
$$
\afterdisplay
@end tex
@ifinfo

@example
dy_i(t)/dt = f_i(t, y_1(t), ..., y_n(t))
@end example

@end ifinfo
@noindent
for @math{i = 1, \dots, n}.  The stepping functions rely on the vector
of derivatives @math{f_i} and the Jacobian matrix, 
@c{$J_{ij} = \partial f_i(t, y(t)) / \partial y_j$}
@math{J_@{ij@} = df_i(t,y(t)) / dy_j}. 
A system of equations is defined using the @code{gsl_odeiv_system}
datatype.

@deftp {Data Type} gsl_odeiv_system
This data type defines a general ODE system with arbitrary parameters.

@table @code
@item int (* function) (double t, const double y[], double dydt[], void * params)
This function should store the vector elements
@c{$f_i(t,y,\hbox{\it params})$}
@math{f_i(t,y,params)} in the array @var{dydt},
for arguments (@var{t},@var{y}) and parameters @var{params}.
The function should return @code{GSL_SUCCESS} if the calculation 
was completed successfully. Any other return value indicates
an error.

@item  int (* jacobian) (double t, const double y[], double * dfdy, double dfdt[], void * params);
@cindex Jacobian matrix, ODEs
This function should store the vector of derivative elements 
@c{$\partial f_i(t,y,params) / \partial t$}
@math{df_i(t,y,params)/dt} in the array @var{dfdt} and the 
Jacobian matrix @c{$J_{ij}$}
@math{J_@{ij@}} in the array
@var{dfdy}, regarded as a row-ordered matrix @code{J(i,j) = dfdy[i * dimension + j]}
where @code{dimension} is the dimension of the system.
The function should return @code{GSL_SUCCESS} if the calculation 
was completed successfully.  Any other return value indicates
an error.

Some of the simpler solver algorithms do not make use of the Jacobian
matrix, so it is not always strictly necessary to provide it (the
@code{jacobian} element of the struct can be replaced by a null pointer
for those algorithms).  However, it is useful to provide the Jacobian to allow
the solver algorithms to be interchanged---the best algorithms make use
of the Jacobian.

@item size_t dimension;
This is the dimension of the system of equations.

@item void * params
This is a pointer to the arbitrary parameters of the system.
@end table
@end deftp

@node Stepping Functions
@section Stepping Functions

The lowest level components are the @dfn{stepping functions} which
advance a solution from time @math{t} to @math{t+h} for a fixed
step-size @math{h} and estimate the resulting local error.

@deftypefun {gsl_odeiv_step *} gsl_odeiv_step_alloc (const gsl_odeiv_step_type * @var{T}, size_t @var{dim})
@tpindex gsl_odeiv_step
@tpindex gsl_odeiv_step_type
This function returns a pointer to a newly allocated instance of a
stepping function of type @var{T} for a system of @var{dim} dimensions.
@end deftypefun

@deftypefun int gsl_odeiv_step_reset (gsl_odeiv_step * @var{s})
This function resets the stepping function @var{s}.  It should be used
whenever the next use of @var{s} will not be a continuation of a
previous step.
@end deftypefun

@deftypefun void gsl_odeiv_step_free (gsl_odeiv_step * @var{s})
This function frees all the memory associated with the stepping function
@var{s}.
@end deftypefun

@deftypefun {const char *} gsl_odeiv_step_name (const gsl_odeiv_step * @var{s})
This function returns a pointer to the name of the stepping function.
For example,

@example
printf ("step method is '%s'\n",
         gsl_odeiv_step_name (s));
@end example

@noindent
would print something like @code{step method is 'rkf45'}.
@end deftypefun

@deftypefun {unsigned int} gsl_odeiv_step_order (const gsl_odeiv_step * @var{s})
This function returns the order of the stepping function on the previous
step.  This order can vary if the stepping function itself is adaptive.
@end deftypefun

@deftypefun int gsl_odeiv_step_apply (gsl_odeiv_step * @var{s}, double @var{t}, double @var{h}, double @var{y}[], double @var{yerr}[], const double @var{dydt_in}[], double @var{dydt_out}[], const gsl_odeiv_system * @var{dydt})
This function applies the stepping function @var{s} to the system of
equations defined by @var{dydt}, using the step size @var{h} to advance
the system from time @var{t} and state @var{y} to time @var{t}+@var{h}.
The new state of the system is stored in @var{y} on output, with an
estimate of the absolute error in each component stored in @var{yerr}.
If the argument @var{dydt_in} is not null it should point an array
containing the derivatives for the system at time @var{t} on input. This
is optional as the derivatives will be computed internally if they are
not provided, but allows the reuse of existing derivative information.
On output the new derivatives of the system at time @var{t}+@var{h} will
be stored in @var{dydt_out} if it is not null.

If the user-supplied functions defined in the system @var{dydt} return a
status other than @code{GSL_SUCCESS} the step will be aborted.  In this
case, the elements of @var{y} will be restored to their pre-step values
and the error code from the user-supplied function will be returned. 
The step-size @var{h} will be set to the step-size which caused the error.  
If the function is called again with a smaller step-size, e.g. @math{@var{h}/10},  
it should be possible to get closer to any singularity.
To distinguish between error codes from the user-supplied functions and
those from @code{gsl_odeiv_step_apply} itself, any user-defined return
values should be distinct from the standard GSL error codes.
@end deftypefun

The following algorithms are available,

@deffn {Step Type} gsl_odeiv_step_rk2
@cindex RK2, Runge-Kutta method
@cindex Runge-Kutta methods, ordinary differential equations
Embedded Runge-Kutta (2, 3) method.
@end deffn

@deffn {Step Type} gsl_odeiv_step_rk4
@cindex RK4, Runge-Kutta method
4th order (classical) Runge-Kutta.  The error estimate is obtained by
halving the step-size.  For more efficient estimate of the error, use the
Runge-Kutta-Fehlberg method described below.
@end deffn

@deffn {Step Type} gsl_odeiv_step_rkf45
@cindex Fehlberg method, differential equations
@cindex RKF45, Runge-Kutta-Fehlberg method
Embedded Runge-Kutta-Fehlberg (4, 5) method.  This method is a good
general-purpose integrator.
@end deffn

@deffn {Step Type} gsl_odeiv_step_rkck 
@cindex Runge-Kutta Cash-Karp method
@cindex Cash-Karp, Runge-Kutta method
Embedded Runge-Kutta Cash-Karp (4, 5) method.
@end deffn

@deffn {Step Type} gsl_odeiv_step_rk8pd  
@cindex Runge-Kutta Prince-Dormand method
@cindex Prince-Dormand, Runge-Kutta method
Embedded Runge-Kutta Prince-Dormand (8,9) method.
@end deffn

@deffn {Step Type} gsl_odeiv_step_rk2imp  
Implicit 2nd order Runge-Kutta at Gaussian points.
@end deffn

@deffn {Step Type} gsl_odeiv_step_rk4imp  
Implicit 4th order Runge-Kutta at Gaussian points.
@end deffn

@deffn {Step Type} gsl_odeiv_step_bsimp   
@cindex Bulirsch-Stoer method
@cindex Bader and Deuflhard, Bulirsch-Stoer method.
@cindex Deuflhard and Bader, Bulirsch-Stoer method.
Implicit Bulirsch-Stoer method of Bader and Deuflhard.  This algorithm
requires the Jacobian.
@end deffn

@deffn {Step Type} gsl_odeiv_step_gear1 
@cindex Gear method, differential equations
M=1 implicit Gear method.
@end deffn

@deffn {Step Type} gsl_odeiv_step_gear2 
M=2 implicit Gear method.
@end deffn

@node Adaptive Step-size Control
@section Adaptive Step-size Control
@cindex Adaptive step-size control, differential equations

The control function examines the proposed change to the solution
produced by a stepping function and attempts to determine the optimal
step-size for a user-specified level of error.

@deftypefun {gsl_odeiv_control *} gsl_odeiv_control_standard_new (double @var{eps_abs}, double @var{eps_rel}, double @var{a_y}, double @var{a_dydt})
@tpindex gsl_odeiv_control
@tpindex gsl_odeiv_control_type
The standard control object is a four parameter heuristic based on
absolute and relative errors @var{eps_abs} and @var{eps_rel}, and
scaling factors @var{a_y} and @var{a_dydt} for the system state
@math{y(t)} and derivatives @math{y'(t)} respectively.

The step-size adjustment procedure for this method begins by computing
the desired error level @math{D_i} for each component,
@tex
\beforedisplay
$$
D_i = \epsilon_{abs} + \epsilon_{rel} * (a_{y} |y_i| + a_{dydt} h |y'_i|)
$$
\afterdisplay
@end tex
@ifinfo

@example
D_i = eps_abs + eps_rel * (a_y |y_i| + a_dydt h |y'_i|)
@end example

@end ifinfo
@noindent
and comparing it with the observed error @math{E_i = |yerr_i|}.  If the
observed error @var{E} exceeds the desired error level @var{D} by more
than 10% for any component then the method reduces the step-size by an
appropriate factor,
@tex
\beforedisplay
$$
h_{new} = h_{old} * S * (E/D)^{-1/q}
$$
\afterdisplay
@end tex
@ifinfo

@example
h_new = h_old * S * (E/D)^(-1/q)
@end example

@end ifinfo
@noindent
where @math{q} is the consistency order of the method (e.g. @math{q=4} for
4(5) embedded RK), and @math{S} is a safety factor of 0.9. The ratio
@math{E/D} is taken to be the maximum of the ratios
@math{E_i/D_i}. 

If the observed error @math{E} is less than 50% of the desired error
level @var{D} for the maximum ratio @math{E_i/D_i} then the algorithm
takes the opportunity to increase the step-size to bring the error in
line with the desired level,
@tex
\beforedisplay
$$
h_{new} = h_{old} * S * (E/D)^{-1/(q+1)}
$$
\afterdisplay
@end tex
@ifinfo

@example
h_new = h_old * S * (E/D)^(-1/(q+1))
@end example

@end ifinfo
@noindent
This encompasses all the standard error scaling methods. To avoid
uncontrolled changes in the stepsize, the overall scaling factor is
limited to the range @math{1/5} to 5.
@end deftypefun

@deftypefun {gsl_odeiv_control *} gsl_odeiv_control_y_new (double @var{eps_abs}, double @var{eps_rel})
This function creates a new control object which will keep the local
error on each step within an absolute error of @var{eps_abs} and
relative error of @var{eps_rel} with respect to the solution @math{y_i(t)}.
This is equivalent to the standard control object with @var{a_y}=1 and
@var{a_dydt}=0.
@end deftypefun

@deftypefun {gsl_odeiv_control *} gsl_odeiv_control_yp_new (double @var{eps_abs}, double @var{eps_rel})
This function creates a new control object which will keep the local
error on each step within an absolute error of @var{eps_abs} and
relative error of @var{eps_rel} with respect to the derivatives of the
solution @math{y'_i(t)}.  This is equivalent to the standard control
object with @var{a_y}=0 and @var{a_dydt}=1.
@end deftypefun


@deftypefun {gsl_odeiv_control *} gsl_odeiv_control_scaled_new (double @var{eps_abs}, double @var{eps_rel}, double @var{a_y}, double @var{a_dydt}, const double @var{scale_abs}[], size_t @var{dim})
This function creates a new control object which uses the same algorithm
as @code{gsl_odeiv_control_standard_new} but with an absolute error
which is scaled for each component by the array @var{scale_abs}.
The formula for @math{D_i} for this control object is,
@tex
\beforedisplay
$$
D_i = \epsilon_{abs} s_i + \epsilon_{rel} * (a_{y} |y_i| + a_{dydt} h |y'_i|)
$$
\afterdisplay
@end tex
@ifinfo

@example
D_i = eps_abs * s_i + eps_rel * (a_y |y_i| + a_dydt h |y'_i|)
@end example

@end ifinfo
@noindent
where @math{s_i} is the @math{i}-th component of the array @var{scale_abs}.
The same error control heuristic is used by the Matlab @sc{ode} suite. 
@end deftypefun

@deftypefun {gsl_odeiv_control *} gsl_odeiv_control_alloc (const gsl_odeiv_control_type * @var{T})
This function returns a pointer to a newly allocated instance of a
control function of type @var{T}.  This function is only needed for
defining new types of control functions.  For most purposes the standard
control functions described above should be sufficient. 
@end deftypefun

@deftypefun int gsl_odeiv_control_init (gsl_odeiv_control * @var{c}, double @var{eps_abs}, double @var{eps_rel}, double @var{a_y}, double @var{a_dydt})
This function initializes the control function @var{c} with the
parameters @var{eps_abs} (absolute error), @var{eps_rel} (relative
error), @var{a_y} (scaling factor for y) and @var{a_dydt} (scaling
factor for derivatives).
@end deftypefun

@deftypefun void gsl_odeiv_control_free (gsl_odeiv_control * @var{c})
This function frees all the memory associated with the control function
@var{c}.
@end deftypefun

@deftypefun int gsl_odeiv_control_hadjust (gsl_odeiv_control * @var{c}, gsl_odeiv_step * @var{s}, const double @var{y}[], const double @var{yerr}[], const double @var{dydt}[], double * @var{h})
This function adjusts the step-size @var{h} using the control function
@var{c}, and the current values of @var{y}, @var{yerr} and @var{dydt}.
The stepping function @var{step} is also needed to determine the order
of the method.  If the error in the y-values @var{yerr} is found to be
too large then the step-size @var{h} is reduced and the function returns
@code{GSL_ODEIV_HADJ_DEC}.  If the error is sufficiently small then
@var{h} may be increased and @code{GSL_ODEIV_HADJ_INC} is returned.  The
function returns @code{GSL_ODEIV_HADJ_NIL} if the step-size is
unchanged.  The goal of the function is to estimate the largest
step-size which satisfies the user-specified accuracy requirements for
the current point.
@end deftypefun

@deftypefun {const char *} gsl_odeiv_control_name (const gsl_odeiv_control * @var{c})
This function returns a pointer to the name of the control function.
For example,

@example
printf ("control method is '%s'\n", 
        gsl_odeiv_control_name (c));
@end example

@noindent
would print something like @code{control method is 'standard'}
@end deftypefun


@node Evolution
@section Evolution

The highest level of the system is the evolution function which combines
the results of a stepping function and control function to reliably
advance the solution forward over an interval @math{(t_0, t_1)}.  If the
control function signals that the step-size should be decreased the
evolution function backs out of the current step and tries the proposed
smaller step-size.  This process is continued until an acceptable
step-size is found.

@deftypefun {gsl_odeiv_evolve *} gsl_odeiv_evolve_alloc (size_t @var{dim})
@tpindex gsl_odeiv_evolve
This function returns a pointer to a newly allocated instance of an
evolution function for a system of @var{dim} dimensions.
@end deftypefun

@deftypefun int gsl_odeiv_evolve_apply (gsl_odeiv_evolve * @var{e}, gsl_odeiv_control * @var{con}, gsl_odeiv_step * @var{step}, const gsl_odeiv_system * @var{dydt}, double * @var{t}, double @var{t1}, double * @var{h}, double @var{y}[])
This function advances the system (@var{e}, @var{dydt}) from time
@var{t} and position @var{y} using the stepping function @var{step}.
The new time and position are stored in @var{t} and @var{y} on output.
The initial step-size is taken as @var{h}, but this will be modified
using the control function @var{c} to achieve the appropriate error
bound if necessary.  The routine may make several calls to @var{step} in
order to determine the optimum step-size. An estimate of 
the local error for the step can be obtained from the components of the array @code{@var{e}->yerr[]}. 
If the step-size has been changed the value of @var{h} will be modified on output. 
The maximum time @var{t1} is guaranteed not to be exceeded by the time-step.  On the
final time-step the value of @var{t} will be set to @var{t1} exactly.

If the user-supplied functions defined in the system @var{dydt} return a
status other than @code{GSL_SUCCESS} the step will be aborted.  In this
case,  @var{t} and @var{y} will be restored to their pre-step values
and the error code from the user-supplied function will be returned.  To
distinguish between error codes from the user-supplied functions and
those from @code{gsl_odeiv_evolve_apply} itself, any user-defined return
values should be distinct from the standard GSL error codes.
@end deftypefun

@deftypefun int gsl_odeiv_evolve_reset (gsl_odeiv_evolve * @var{e})
This function resets the evolution function @var{e}.  It should be used
whenever the next use of @var{e} will not be a continuation of a
previous step.
@end deftypefun

@deftypefun void gsl_odeiv_evolve_free (gsl_odeiv_evolve * @var{e})
This function frees all the memory associated with the evolution function
@var{e}.
@end deftypefun

@cindex discontinuities, in ODE systems
Where a system has discontinuous changes in the derivatives at known
times it is advisable to evolve the system between each discontinuity
in sequence.  For example, if a step-change in an external driving
force occurs at times @math{t_a, t_b, t_c, \dots} then evolving over
the ranges @math{(t_0,t_a)}, @math{(t_a,t_b)}, @dots{},
@math{(t_c,t_1)} is more efficient than using the single range
@math{(t_0,t_1)}.  

Evolving the system directly through a discontinuity with a strict
tolerance may result in extremely small steps being taken at the edge
of the discontinuity (e.g. down to the limit of machine precision).
In this case it may be necessary to impose a minimum step size
@code{hmin} suitable for the problem:

@example
while (t < t1)
@{
   gsl_odeiv_evolve_apply (e, c, s, &sys, &t, t1, &h, y);
   if (h < hmin) @{ h = hmin; @} ;
@}
@end example
The value of @var{h} returned by @code{gsl_odeiv_evolve_apply} is
always a suggested value and can be modified whenever needed.

@node ODE Example programs
@section Examples
@cindex Van der Pol oscillator, example
The following program solves the second-order nonlinear Van der Pol
oscillator equation,
@tex
\beforedisplay
$$
x''(t) + \mu x'(t) (x(t)^2 - 1) + x(t) = 0
$$
\afterdisplay
@end tex
@ifinfo

@example
x''(t) + \mu x'(t) (x(t)^2 - 1) + x(t) = 0
@end example

@end ifinfo
@noindent
This can be converted into a first order system suitable for use with
the routines described in this chapter by introducing a separate
variable for the velocity, @math{y = x'(t)},
@tex
\beforedisplay
$$
\eqalign{
x' &= y\cr
y' &= -x + \mu y (1-x^2)
}
$$
\afterdisplay
@end tex
@ifinfo

@example
x' = y
y' = -x + \mu y (1-x^2)
@end example

@end ifinfo
@noindent
The program begins by defining functions for these derivatives and
their Jacobian,

@example
@verbatiminclude examples/ode-initval.c
@end example

@noindent
For functions with multiple parameters, the appropriate information
can be passed in through the @var{params} argument using a pointer to
a struct.

The main loop of the program evolves the solution from @math{(y, y') =
(1, 0)} at @math{t=0} to @math{t=100}.  The step-size @math{h} is
automatically adjusted by the controller to maintain an absolute
accuracy of @c{$10^{-6}$} 
@math{10^@{-6@}} in the function values @var{y}.

@iftex
@sp 1
@center @image{vdp,3.4in}
@center Numerical solution of the Van der Pol oscillator equation 
@center using Prince-Dormand 8th order Runge-Kutta.
@end iftex

@noindent
To obtain the values at user-specified positions, rather than those
chosen automatically by the control function, the main loop can be modified
to advance the solution from one chosen point to the next.  For example, the
following main loop prints the solution at the points @math{t_i = 0,
1, 2, \dots, 100},

@example
  for (i = 1; i <= 100; i++)
    @{
      double ti = i * t1 / 100.0;

      while (t < ti)
        @{
          gsl_odeiv_evolve_apply (e, c, s, 
                                  &sys, 
                                  &t, ti, &h,
                                  y);
        @}
 
      printf ("%.5e %.5e %.5e\n", t, y[0], y[1]);
    @}
@end example

@noindent
Note that arbitrary values of @math{t_i} can be used for each stage of
the integration.  The equally spaced points in this example are just
used as an illustration.

It is also possible to work with a non-adaptive integrator, using only
the stepping function itself.  The following program uses the @code{rk4}
fourth-order Runge-Kutta stepping function with a fixed stepsize of
0.01,

@example
@verbatiminclude examples/odefixed.c
@end example

@noindent
The derivatives must be initialized for the starting point @math{t=0}
before the first step is taken.  Subsequent steps use the output
derivatives @var{dydt_out} as inputs to the next step by copying their
values into @var{dydt_in}.

@node ODE References and Further Reading
@section References and Further Reading

Many of the basic Runge-Kutta formulas can be found in the Handbook of
Mathematical Functions,

@itemize @w{}
@item
Abramowitz & Stegun (eds.), @cite{Handbook of Mathematical Functions},
Section 25.5.
@end itemize

@noindent
The implicit Bulirsch-Stoer algorithm @code{bsimp} is described in the
following paper,

@itemize @w{}
@item
G. Bader and P. Deuflhard, ``A Semi-Implicit Mid-Point Rule for Stiff
Systems of Ordinary Differential Equations.'', Numer.@: Math.@: 41, 373--398,
1983.
@end itemize