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Author image Håkon Hægland
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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 :

  • 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($e, $c, $step, $dydt, \$t, $t1, \$h, $y) - This function advances the system ($e, $dydt) from time $t and position $y using the stepping function $step. The new time and position are stored in $t and $y on output. The initial step-size is taken as $h, but this will be modified using the control function $c to achieve the appropriate error bound if necessary. The routine may make several calls to step in order to determine the optimum step-size. If the step-size has been changed the value of $h will be modified on output. The maximum time $t1 is guaranteed not to be exceeded by the time-step. On the final time-step the value of $t will be set to $t1 exactly.

  • 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/

EXAMPLE

The example is taken from https://www.math.utah.edu/software/gsl/gsl-ref_367.html.

 use strict;
 use warnings;
 use Math::GSL::Errno qw($GSL_SUCCESS);
 use Math::GSL::ODEIV qw/ :all /;
 use Math::GSL::Matrix qw/:all/;
 use Math::GSL::IEEEUtils qw/ :all /;
 
 sub func {
     my ($t, $y, $dydt, $params) = @_;
     my $mu = $params->{mu};
     $dydt->[0] = $y->[1];
     $dydt->[1] = -$y->[0] - $mu*$y->[1]*(($y->[0])**2 - 1);
     return $GSL_SUCCESS;
 }
 
 sub jac {
     my ($t, $y, $dfdy, $dfdt, $params) = @_;

     my $mu = $params->{mu};
     my $m = gsl_matrix_view_array($dfdy, 2, 2);
     gsl_matrix_set( $m, 0, 0, 0.0 );
     gsl_matrix_set( $m, 0, 1, 1.0 );
     gsl_matrix_set( $m, 1, 0, (-2.0 * $mu * $y->[0] * $y->[1]) - 1.0 );
     gsl_matrix_set( $m, 1, 1, -$mu * (($y->[0])**2 - 1.0) );
     $dfdt->[0] = 0.0;
     $dfdt->[1] = 0.0;
     return $GSL_SUCCESS;
 }
 
 my $T = $gsl_odeiv_step_rk8pd;
 my $s = gsl_odeiv_step_alloc($T, 2);
 my $c = gsl_odeiv_control_y_new(1e-6, 0.0);
 my $e = gsl_odeiv_evolve_alloc(2);
 my $params = { mu => 10 };
 my $sys = Math::GSL::ODEIV::gsl_odeiv_system->new(\&func, \&jac, 2, $params );
 my $t = 0.0;
 my $t1 = 100.0;
 my $h = 1e-6;
 my $y = [ 1.0, 0.0 ];
 gsl_ieee_env_setup;
 while ($t < $t1) {
     my $status = gsl_odeiv_evolve_apply ($e, $c, $s, $sys, \$t, $t1, \$h, $y);
     last if $status != $GSL_SUCCESS;
     printf "%.5e %.5e %.5e\n", $t, $y->[0], $y->[1];
 }
 gsl_odeiv_evolve_free($e);
 gsl_odeiv_control_free($c);
 gsl_odeiv_step_free($s);

AUTHORS

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

COPYRIGHT AND LICENSE

Copyright (C) 2008-2020 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.