--- name: "Math::GSL::ODEIV" section: "3pm" platform: "linux" package: "libmath-gsl-perl" version: "0.45-2+b1" release: "Debian 13 (Trixie)" platforms: ["linux"] url: "https://new.carta.tech/Math::GSL::ODEIV" --- # Math::GSL::ODEIV(3pm) functions for solving ordinary differential equation (ODE) initial value problems ## 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. - • `$gsl_odeiv_step_gear2` - M=2 implicit Gear method. For more information on the functions, we refer you to the GSL official documentation: \ ## EXAMPLE The example is taken from \. ``` use strict; use warnings; use Math::GSL::Errno qw($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 \ and Thierry Moisan \ ## COPYRIGHT AND LICENSE 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.