Math::GSL::ODEIV(3pm)

📦
version 0.49 · 6 versions
, in ubuntu stonking

💖 NAME 💖

Math::GSL::ODEIV — bestie, this is the module that solves ordinary differential equation (ODE) initial value problems. It's a Perl wrapper around the GSL ODEIV gang, and it lives in the Math::GSL family of manual pages (the 3pm kind, aka user-contributed Perl docs). 🦋

✨ SYNOPSIS ✨

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

That's it. One line and you've got the whole toolbox. 💅

🌸 DESCRIPTION 🌸

Okay so here is a list of all the functions in this module, no gatekeeping, no cap:

  • gsl_odeiv_step_alloc($T, $dim) — hands you back a pointer to a freshly allocated instance of a stepping function of type $T, for a system with $dim dimensions.
Heads up bestie: $T must be one of the step type constants listed further down this page. 💀
  • gsl_odeiv_step_reset($s) — resets the stepping function $s. Use this whenever the next use of $s is not a continuation of a previous step.
  • gsl_odeiv_step_free($s) — frees all the memory associated with the stepping function $s. Let it go, she's done. 🕊️
  • gsl_odeiv_step_name($s) — returns a pointer to the name of the stepping function.
  • gsl_odeiv_step_order($s) — returns the order of the stepping function on the previous step. Lowkey flexible: this order can vary if the stepping function itself is adaptive.
  • gsl_odeiv_step_apply — that's literally all the original doc says about this one. It's listed and that's it. 🙈
  • gsl_odeiv_control_alloc($T) — returns a pointer to a newly allocated instance of a control function of type $T. Fr fr, this one is only needed if you're defining new types of control functions. For most purposes the standard control functions described below are sufficient. And $T here is a gsl_odeiv_control_type.
  • gsl_odeiv_control_init($c, $eps_abs, $eps_rel, $a_y, $a_dydt) — initializes the control function $c with these parameters: $eps_abs (absolute error), $eps_rel (relative error), $a_y (scaling factor for y), and $a_dydt (scaling factor for derivatives).
  • gsl_odeiv_control_free — listed, no description given in the source. That's the whole vibe. 💀
  • gsl_odeiv_control_hadjust — also just listed, no prose attached.
  • gsl_odeiv_control_name — same situation, name only.
  • gsl_odeiv_control_standard_new($eps_abs, $eps_rel, $a_y, $a_dydt) — the standard control object is a four parameter heuristic based on the absolute and relative errors $eps_abs and $eps_rel, plus scaling factors $a_y and $a_dydt for the system state y(t) and derivatives y'(t) respectively. Okay buckle up, here's the procedure:

The step-size adjustment 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. And 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) — creates a new control object that keeps the local error on each step within an absolute error of $eps_abs and a 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) — creates a new control object that keeps the local error on each step within an absolute error of $eps_abs and a 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) — creates a new control object using the same algorithm as gsl_odeiv_control_standard_new, but with an absolute error that's 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. And the same error control heuristic is used by the Matlab ode suite, so you know it's got pedigree. 💅

  • gsl_odeiv_evolve_alloc($dim) — 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, $y) — this one advances the system ($e, $dydt) from time $t and position $y using the stepping function $step. The new time and position get 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. And on the final time-step, the value of $t will be set to $t1 exactly. Clean. ✨
  • gsl_odeiv_evolve_reset($e) — resets the evolution function $e. Use it whenever the next use of $e will not be a continuation of a previous step.
  • gsl_odeiv_evolve_free($e) — frees all the memory associated with the evolution function $e.

This module also includes the following constants, bestie:

  • $GSL_ODEIV_HADJ_INC
  • $GSL_ODEIV_HADJ_NIL
  • $GSL_ODEIV_HADJ_DEC

🎀 Step Type 🎀

The step type constants, aka the whole menu of integrators:

  • $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 a 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 — yep, it's on the list.
  • $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 official 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>. Run it, it's iconic:

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 (C) 2008-2024 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.

and that's the whole page, bestie 💖 this was perl v5.42.3, dated 2026-08-20, and it's the Math::GSL::ODEIV(3pm) manual page from the user-contributed Perl documentation. go integrate something, main character 🫶✨

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