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# Manual Reference Pages  -  MATH::GSL::ODEIV (3)

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### 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 :
o 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.
o 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.
o gsl_odeiv_step_free(\$s) - This function frees all the memory associated with the stepping function \$s.
o gsl_odeiv_step_name(\$s) - This function returns a pointer to the name of the stepping function.
o 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.
o gsl_odeiv_step_apply
o 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.
o 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).
o gsl_odeiv_control_free
o gsl_odeiv_control_name
o 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.
o 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.
o 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.
o 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.
o 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.
o gsl_odeiv_evolve_apply
o 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.
o gsl_odeiv_evolve_free(\$e) - This function frees all the memory associated with the evolution function \$e.
This module also includes the following constants :

#### Step Type

 o \$gsl_odeiv_step_rk2 - Embedded Runge-Kutta (2, 3) method. o \$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. o \$gsl_odeiv_step_rkf45 - Embedded Runge-Kutta-Fehlberg (4, 5) method. This method is a good general-purpose integrator. o \$gsl_odeiv_step_rkck - Embedded Runge-Kutta Cash-Karp (4, 5) method. o \$gsl_odeiv_step_rk8pd - Embedded Runge-Kutta Prince-Dormand (8,9) method. o \$gsl_odeiv_step_rk2imp - Implicit 2nd order Runge-Kutta at Gaussian points. o \$gsl_odeiv_step_rk2simp o \$gsl_odeiv_step_rk4imp - Implicit 4th order Runge-Kutta at Gaussian points. o \$gsl_odeiv_step_bsimp - Implicit Bulirsch-Stoer method of Bader and Deuflhard. This algorithm requires the Jacobian. o \$gsl_odeiv_step_gear1 - M=1 implicit Gear method. o \$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>