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.