NAME

       Math::GSL::ODEIV - functions for solving ordinary differential equation (ODE) initial
       value problems

SYNOPSIS

       use Math::GSL::ODEIV qw /:all/;

DESCRIPTION

       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

   Step Type
       •   $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/>

AUTHORS

       Jonathan "Duke" Leto <jonathan@leto.net> and Thierry Moisan <thierry.moisan@gmail.com>

COPYRIGHT AND LICENSE

       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.