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.\" ========================================================================
.\"
.IX Title "Math::GSL::ODEIV 3pm"
.TH Math::GSL::ODEIV 3pm "2020-11-09" "perl v5.32.0" "User Contributed Perl Documentation"
.\" For nroff, turn off justification.  Always turn off hyphenation; it makes
.\" way too many mistakes in technical documents.
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.nh
.SH "NAME"
Math::GSL::ODEIV \- functions for solving ordinary differential equation (ODE) initial value problems
.SH "SYNOPSIS"
.IX Header "SYNOPSIS"
.Vb 1
\& use Math::GSL::ODEIV qw /:all/;
.Ve
.SH "DESCRIPTION"
.IX Header "DESCRIPTION"
Here is a list of all the functions in this module :
.IP "\(bu" 4
\&\f(CW\*(C`gsl_odeiv_step_alloc($T, $dim)\*(C'\fR \- This function returns a pointer to a newly allocated instance of a stepping function of type \f(CW$T\fR for a system of \f(CW$dim\fR dimensions.$T must be one of the step type constant above.
.IP "\(bu" 4
\&\f(CW\*(C`gsl_odeiv_step_reset($s)\*(C'\fR \- This function resets the stepping function \f(CW$s\fR. It should be used whenever the next use of s will not be a continuation of a previous step.
.IP "\(bu" 4
\&\f(CW\*(C`gsl_odeiv_step_free($s)\*(C'\fR \- This function frees all the memory associated with the stepping function \f(CW$s\fR.
.IP "\(bu" 4
\&\f(CW\*(C`gsl_odeiv_step_name($s)\*(C'\fR \- This function returns a pointer to the name of the stepping function.
.IP "\(bu" 4
\&\f(CW\*(C`gsl_odeiv_step_order($s)\*(C'\fR \- This function returns the order of the stepping function on the previous step. This order can vary if the stepping function itself is adaptive.
.IP "\(bu" 4
\&\f(CW\*(C`gsl_odeiv_step_apply \*(C'\fR
.IP "\(bu" 4
\&\f(CW\*(C`gsl_odeiv_control_alloc($T)\*(C'\fR \- This function returns a pointer to a newly allocated instance of a control function of type \f(CW$T\fR. This function is only needed for defining new types of control functions. For most purposes the standard control functions described above should be sufficient. \f(CW$T\fR is a gsl_odeiv_control_type.
.IP "\(bu" 4
\&\f(CW\*(C`gsl_odeiv_control_init($c, $eps_abs, $eps_rel, $a_y, $a_dydt) \*(C'\fR \- 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).
.IP "\(bu" 4
\&\f(CW\*(C`gsl_odeiv_control_free \*(C'\fR
.IP "\(bu" 4
\&\f(CW\*(C`gsl_odeiv_control_hadjust \*(C'\fR
.IP "\(bu" 4
\&\f(CW\*(C`gsl_odeiv_control_name \*(C'\fR
.IP "\(bu" 4
\&\f(CW\*(C`gsl_odeiv_control_standard_new($eps_abs, $eps_rel, $a_y, $a_dydt)\*(C'\fR \- The standard control object is a four parameter heuristic based on absolute and relative errors \f(CW$eps_abs\fR and \f(CW$eps_rel\fR, and scaling factors \f(CW$a_y\fR and \f(CW$a_dydt\fR 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 \s-1RK\s0), 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.
.IP "\(bu" 4
\&\f(CW\*(C`gsl_odeiv_control_y_new($eps_abs, $eps_rel)\*(C'\fR \- This function creates a new control object which will keep the local error on each step within an absolute error of \f(CW$eps_abs\fR and relative error of \f(CW$eps_rel\fR with respect to the solution y_i(t). This is equivalent to the standard control object with a_y=1 and a_dydt=0.
.IP "\(bu" 4
\&\f(CW\*(C`gsl_odeiv_control_yp_new($eps_abs, $eps_rel)\*(C'\fR \- This function creates a new control object which will keep the local error on each step within an absolute error of \f(CW$eps_abs\fR and relative error of \f(CW$eps_rel\fR 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.
.IP "\(bu" 4
\&\f(CW\*(C`gsl_odeiv_control_scaled_new($eps_abs, $eps_rel, $a_y, $a_dydt, $scale_abs, $dim) \*(C'\fR \- This function creates a new control object which uses the same algorithm as gsl_odeiv_control_standard_new but with an absolute error which is scaled for each component by the array reference \f(CW$scale_abs\fR. 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.
.IP "\(bu" 4
\&\f(CW\*(C`gsl_odeiv_evolve_alloc($dim)\*(C'\fR \- This function returns a pointer to a newly allocated instance of an evolution function for a system of \f(CW$dim\fR dimensions.
.IP "\(bu" 4
\&\f(CW\*(C`gsl_odeiv_evolve_apply($e, $c, $step, $dydt, \e$t, $t1, \e$h, $y)\*(C'\fR \- This function advances the system ($e, \f(CW$dydt\fR) from time \f(CW$t\fR and position \f(CW$y\fR using the stepping function \f(CW$step\fR. The new time and position are stored in \f(CW$t\fR and \f(CW$y\fR on output. The initial step-size is taken as \f(CW$h\fR, but this will be modified using the control function \f(CW$c\fR 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 \f(CW$h\fR will be modified on output. The maximum time \f(CW$t1\fR is guaranteed not to be exceeded by the time-step. On the final time-step the value of \f(CW$t\fR will be set to \f(CW$t1\fR exactly.
.IP "\(bu" 4
\&\f(CW\*(C`gsl_odeiv_evolve_reset($e)\*(C'\fR \- This function resets the evolution function \f(CW$e\fR. It should be used whenever the next use of \f(CW$e\fR will not be a continuation of a previous step.
.IP "\(bu" 4
\&\f(CW\*(C`gsl_odeiv_evolve_free($e)\*(C'\fR \- This function frees all the memory associated with the evolution function \f(CW$e\fR.
.PP
This module also includes the following constants :
.IP "\(bu" 4
\&\f(CW$GSL_ODEIV_HADJ_INC\fR
.IP "\(bu" 4
\&\f(CW$GSL_ODEIV_HADJ_NIL\fR
.IP "\(bu" 4
\&\f(CW$GSL_ODEIV_HADJ_DEC\fR
.SS "Step Type"
.IX Subsection "Step Type"
.IP "\(bu" 4
\&\f(CW$gsl_odeiv_step_rk2\fR \- Embedded Runge-Kutta (2, 3) method.
.IP "\(bu" 4
\&\f(CW$gsl_odeiv_step_rk4\fR \- 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.
.IP "\(bu" 4
\&\f(CW$gsl_odeiv_step_rkf45\fR \- Embedded Runge-Kutta-Fehlberg (4, 5) method. This method is a good general-purpose integrator.
.IP "\(bu" 4
\&\f(CW$gsl_odeiv_step_rkck\fR \- Embedded Runge-Kutta Cash-Karp (4, 5) method.
.IP "\(bu" 4
\&\f(CW$gsl_odeiv_step_rk8pd\fR \- Embedded Runge-Kutta Prince-Dormand (8,9) method.
.IP "\(bu" 4
\&\f(CW$gsl_odeiv_step_rk2imp\fR \- Implicit 2nd order Runge-Kutta at Gaussian points.
.IP "\(bu" 4
\&\f(CW$gsl_odeiv_step_rk2simp\fR
.IP "\(bu" 4
\&\f(CW$gsl_odeiv_step_rk4imp\fR \- Implicit 4th order Runge-Kutta at Gaussian points.
.IP "\(bu" 4
\&\f(CW$gsl_odeiv_step_bsimp\fR \- Implicit Bulirsch-Stoer method of Bader and Deuflhard. This algorithm requires the Jacobian.
.IP "\(bu" 4
\&\f(CW$gsl_odeiv_step_gear1\fR \- M=1 implicit Gear method.
.IP "\(bu" 4
\&\f(CW$gsl_odeiv_step_gear2\fR \- M=2 implicit Gear method.
.PP
For more information on the functions, we refer you to the \s-1GSL\s0 offcial
documentation: <http://www.gnu.org/software/gsl/manual/html_node/>
.SH "EXAMPLE"
.IX Header "EXAMPLE"
The example is taken from <https://www.math.utah.edu/software/gsl/gsl\-ref_367.html>.
.PP
.Vb 6
\& use strict;
\& use warnings;
\& use Math::GSL::Errno qw($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(\e&func, \e&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, $sys, \e$t, $t1, \e$h, $y);
\&     last if $status != $GSL_SUCCESS;
\&     printf "%.5e %.5e %.5e\en", $t, $y\->[0], $y\->[1];
\& }
\& gsl_odeiv_evolve_free($e);
\& gsl_odeiv_control_free($c);
\& gsl_odeiv_step_free($s);
.Ve
.SH "AUTHORS"
.IX Header "AUTHORS"
Jonathan \*(L"Duke\*(R" Leto <jonathan@leto.net> and Thierry Moisan <thierry.moisan@gmail.com>
.SH "COPYRIGHT AND LICENSE"
.IX Header "COPYRIGHT AND LICENSE"
Copyright (C) 2008\-2020 Jonathan \*(L"Duke\*(R" Leto and Thierry Moisan
.PP
This program is free software; you can redistribute it and/or modify it
under the same terms as Perl itself.