Math::GSL::ODEIV - functions for solving ordinary differential
equation (ODE) initial value problems
use Math::GSL::ODEIV qw /:all/;
Here is a list of all the functions in this module :
- "gsl_odeiv_step_alloc($T, $dim)" - This
function returns a pointer to a newly allocated instance of a stepping
function of type $T for a system of
$dim dimensions.$T must be one of the step type
constant above.
- "gsl_odeiv_step_reset($s)" - This
function resets the stepping function $s. It
should be used whenever the next use of s will not be a continuation of a
previous step.
- "gsl_odeiv_step_free($s)" - This
function frees all the memory associated with the stepping function
$s.
- "gsl_odeiv_step_name($s)" - This
function returns a pointer to the name of the stepping function.
- "gsl_odeiv_step_order($s)" - This
function returns the order of the stepping function on the previous step.
This order can vary if the stepping function itself is adaptive.
- "gsl_odeiv_step_apply "
- "gsl_odeiv_control_alloc($T)" - This
function returns a pointer to a newly allocated instance of a control
function of type $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.
$T is a gsl_odeiv_control_type.
- "gsl_odeiv_control_init($c, $eps_abs, $eps_rel,
$a_y, $a_dydt) " - This function initializes the control
function c with the parameters eps_abs (absolute error), eps_rel (relative
error), a_y (scaling factor for y) and a_dydt (scaling factor for
derivatives).
- "gsl_odeiv_control_free "
- "gsl_odeiv_control_hadjust "
- "gsl_odeiv_control_name "
- "gsl_odeiv_control_standard_new($eps_abs, $eps_rel,
$a_y, $a_dydt)" - The standard control object is a four
parameter heuristic based on absolute and relative errors
$eps_abs and $eps_rel, and
scaling factors $a_y and
$a_dydt for the system state y(t) and derivatives
y'(t) respectively. The step-size adjustment procedure for this method
begins by computing the desired error level D_i for each component, D_i =
eps_abs + eps_rel * (a_y |y_i| + a_dydt h |y'_i|) and comparing it with
the observed error E_i = |yerr_i|. If the observed error E exceeds the
desired error level D by more than 10% for any component then the method
reduces the step-size by an appropriate factor, h_new = h_old * S *
(E/D)^(-1/q) where q is the consistency order of the method (e.g. q=4 for
4(5) embedded RK), and S is a safety factor of 0.9. The ratio E/D is taken
to be the maximum of the ratios E_i/D_i. If the observed error E is less
than 50% of the desired error level D for the maximum ratio E_i/D_i then
the algorithm takes the opportunity to increase the step-size to bring the
error in line with the desired level, h_new = h_old * S * (E/D)^(-1/(q+1))
This encompasses all the standard error scaling methods. To avoid
uncontrolled changes in the stepsize, the overall scaling factor is
limited to the range 1/5 to 5.
- "gsl_odeiv_control_y_new($eps_abs,
$eps_rel)" - This function creates a new control object which
will keep the local error on each step within an absolute error of
$eps_abs and relative error of
$eps_rel with respect to the solution y_i(t). This
is equivalent to the standard control object with a_y=1 and a_dydt=0.
- "gsl_odeiv_control_yp_new($eps_abs,
$eps_rel)" - This function creates a new control object which
will keep the local error on each step within an absolute error of
$eps_abs and relative error of
$eps_rel with respect to the derivatives of the
solution y'_i(t). This is equivalent to the standard control object with
a_y=0 and a_dydt=1.
- "gsl_odeiv_control_scaled_new($eps_abs, $eps_rel,
$a_y, $a_dydt, $scale_abs, $dim) " - This function creates a
new control object which uses the same algorithm as
gsl_odeiv_control_standard_new but with an absolute error which is scaled
for each component by the array reference
$scale_abs. The formula for D_i for this control
object is, D_i = eps_abs * s_i + eps_rel * (a_y |y_i| + a_dydt h |y'_i|)
where s_i is the i-th component of the array scale_abs. The same error
control heuristic is used by the Matlab ode suite.
- "gsl_odeiv_evolve_alloc($dim)" - This
function returns a pointer to a newly allocated instance of an evolution
function for a system of $dim dimensions.
- "gsl_odeiv_evolve_apply "
- "gsl_odeiv_evolve_reset($e)" - This
function resets the evolution function $e. It
should be used whenever the next use of $e will
not be a continuation of a previous step.
- "gsl_odeiv_evolve_free($e)" - This
function frees all the memory associated with the evolution function
$e.
This module also includes the following constants :
- $GSL_ODEIV_HADJ_INC
- $GSL_ODEIV_HADJ_NIL
- $GSL_ODEIV_HADJ_DEC
- $gsl_odeiv_step_rk2 - Embedded Runge-Kutta (2, 3)
method.
- $gsl_odeiv_step_rk4 - 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.
- $gsl_odeiv_step_rkf45 - Embedded
Runge-Kutta-Fehlberg (4, 5) method. This method is a good general-purpose
integrator.
- $gsl_odeiv_step_rkck - Embedded Runge-Kutta
Cash-Karp (4, 5) method.
- $gsl_odeiv_step_rk8pd - Embedded Runge-Kutta
Prince-Dormand (8,9) method.
- $gsl_odeiv_step_rk2imp - Implicit 2nd order
Runge-Kutta at Gaussian points.
- $gsl_odeiv_step_rk2simp
- $gsl_odeiv_step_rk4imp - Implicit 4th order
Runge-Kutta at Gaussian points.
- $gsl_odeiv_step_bsimp - Implicit Bulirsch-Stoer
method of Bader and Deuflhard. This algorithm requires the Jacobian.
- $gsl_odeiv_step_gear1 - M=1 implicit Gear
method.
- $gsl_odeiv_step_gear2 - M=2 implicit Gear
method.
For more information on the functions, we refer you to the GSL
offcial documentation:
<http://www.gnu.org/software/gsl/manual/html_node/>
Jonathan "Duke" Leto <jonathan@leto.net> and
Thierry Moisan <thierry.moisan@gmail.com>
Copyright (C) 2008-2011 Jonathan "Duke" Leto and Thierry
Moisan
This program is free software; you can redistribute it and/or
modify it under the same terms as Perl itself.