# 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 informations 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.