ODEIV.i

   1 %module "Math::GSL::ODEIV"
   2 %{
   3     #include "gsl/gsl_odeiv.h"
   4 %}
   5
   6 %import "gsl/gsl_types.h"
   7 %include "gsl/gsl_odeiv.h"
   8
   9 %perlcode %{
  10 @EXPORT_OK = qw/
  11                gsl_odeiv_step_alloc
  12                gsl_odeiv_step_reset
  13                gsl_odeiv_step_free
  14                gsl_odeiv_step_name
  15                gsl_odeiv_step_order
  16                gsl_odeiv_step_apply
  17                gsl_odeiv_control_alloc
  18                gsl_odeiv_control_init
  19                gsl_odeiv_control_free
  20                gsl_odeiv_control_hadjust
  21                gsl_odeiv_control_name
  22                gsl_odeiv_control_standard_new
  23                gsl_odeiv_control_y_new
  24                gsl_odeiv_control_yp_new
  25                gsl_odeiv_control_scaled_new
  26                gsl_odeiv_evolve_alloc
  27                gsl_odeiv_evolve_apply
  28                gsl_odeiv_evolve_reset
  29                gsl_odeiv_evolve_free
  30                $gsl_odeiv_step_rk2
  31                $gsl_odeiv_step_rk4
  32                $gsl_odeiv_step_rkf45
  33                $gsl_odeiv_step_rkck
  34                $gsl_odeiv_step_rk8pd
  35                $gsl_odeiv_step_rk2imp
  36                $gsl_odeiv_step_rk2simp
  37                $gsl_odeiv_step_rk4imp
  38                $gsl_odeiv_step_bsimp
  39                $gsl_odeiv_step_gear1
  40                $gsl_odeiv_step_gear2
  41                $GSL_ODEIV_HADJ_INC
  42                $GSL_ODEIV_HADJ_NIL
  43                $GSL_ODEIV_HADJ_DEC
  44                $gsl_odeiv_control_standard
  45         /;
  46 %EXPORT_TAGS = ( all => [ @EXPORT_OK ] );
  47
  48 __END__
  49
  50 =head1 NAME
  51
  52 Math::GSL::ODEIV - functions for solving ordinary differential equation (ODE) initial value problems
  53
  54 =head1 SYNOPSIS
  55
  56 use Math::GSL::ODEIV qw /:all/;
  57
  58 =head1 DESCRIPTION
  59
  60 Here is a list of all the functions in this module :
  61
  62 =over
  63
  64 =item * C<gsl_odeiv_step_alloc($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.
  65
  66 =item * C<gsl_odeiv_step_reset($s)> - This function resets the stepping function $s. It should be used whenever the next use of s will not be a continuation of a previous step.
  67
  68 =item * C<gsl_odeiv_step_free($s)> - This function frees all the memory associated with the stepping function $s.
  69
  70 =item * C<gsl_odeiv_step_name($s)> - This function returns a pointer to the name of the stepping function.
  71
  72 =item * C<gsl_odeiv_step_order($s)> - This function returns the order of the stepping function on the previous step. This order can vary if the stepping function itself is adaptive.
  73
  74 =item * C<gsl_odeiv_step_apply >
  75
  76 =item * C<gsl_odeiv_control_alloc($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. $T is a gsl_odeiv_control_type.
  77
  78 =item * C<gsl_odeiv_control_init($c, $eps_abs, $eps_rel, $a_y, $a_dydt) > - This function initializes the control function c with the parameters eps_abs (absolute error), eps_rel (relative error), a_y (scaling factor for y) and a_dydt (scaling factor for derivatives).
  79
  80 =item * C<gsl_odeiv_control_free >
  81
  82 =item * C<gsl_odeiv_control_hadjust >
  83
  84 =item * C<gsl_odeiv_control_name >
  85
  86 =item * C<gsl_odeiv_control_standard_new($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 $a_dydt for the system state y(t) and derivatives y'(t) respectively. The step-size adjustment procedure for this method begins by computing the desired error level D_i for each component, D_i = eps_abs + eps_rel * (a_y |y_i| + a_dydt h |y'_i|) and comparing it with the observed error E_i = |yerr_i|. If the observed error E exceeds the desired error level D by more than 10% for any component then the method reduces the step-size by an appropriate factor, h_new = h_old * S * (E/D)^(-1/q) where q is the consistency order of the method (e.g. q=4 for 4(5) embedded RK), and S is a safety factor of 0.9. The ratio E/D is taken to be the maximum of the ratios E_i/D_i. If the observed error E is less than 50% of the desired error level D for the maximum ratio E_i/D_i then the algorithm takes the opportunity to increase the step-size to bring the error in line with the desired level, h_new = h_old * S * (E/D)^(-1/(q+1)) This encompasses all the standard error scaling methods. To avoid uncontrolled changes in the stepsize, the overall scaling factor is limited to the range 1/5 to 5.
  87
  88 =item * C<gsl_odeiv_control_y_new($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 $eps_rel with respect to the solution y_i(t). This is equivalent to the standard control object with a_y=1 and a_dydt=0.
  89
  90 =item * C<gsl_odeiv_control_yp_new($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 $eps_rel with respect to the derivatives of the solution y'_i(t). This is equivalent to the standard control object with a_y=0 and a_dydt=1.
  91
  92 =item * C<gsl_odeiv_control_scaled_new($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 $scale_abs. The formula for D_i for this control object is, D_i = eps_abs * s_i + eps_rel * (a_y |y_i| + a_dydt h |y'_i|) where s_i is the i-th component of the array scale_abs. The same error control heuristic is used by the Matlab ode suite.
  93
  94 =item * C<gsl_odeiv_evolve_alloc($dim)> - This function returns a pointer to a newly allocated instance of an evolution function for a system of $dim dimensions.
  95
  96 =item * C<gsl_odeiv_evolve_apply >
  97
  98 =item * C<gsl_odeiv_evolve_reset($e)> - This function resets the evolution function $e. It should be used whenever the next use of $e will not be a continuation of a previous step.
  99
 100 =item * C<gsl_odeiv_evolve_free($e)> - This function frees all the memory associated with the evolution function $e.
 101
 102 =back
 103
 104 This module also includes the following constants :
 105
 106 =over
 107
 108 =item * C<$GSL_ODEIV_HADJ_INC>
 109
 110 =item * C<$GSL_ODEIV_HADJ_NIL>
 111
 112 =item * C<$GSL_ODEIV_HADJ_DEC>
 113
 114 =back
 115
 116 =head2 Step Type
 117
 118 =over
 119
 120 =item * C<$gsl_odeiv_step_rk2> - Embedded Runge-Kutta (2, 3) method.
 121
 122 =item * C<$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.
 123
 124 =item * C<$gsl_odeiv_step_rkf45> - Embedded Runge-Kutta-Fehlberg (4, 5) method. This method is a good general-purpose integrator.
 125
 126 =item * C<$gsl_odeiv_step_rkck> - Embedded Runge-Kutta Cash-Karp (4, 5) method.
 127
 128 =item * C<$gsl_odeiv_step_rk8pd> - Embedded Runge-Kutta Prince-Dormand (8,9) method.
 129
 130 =item * C<$gsl_odeiv_step_rk2imp> - Implicit 2nd order Runge-Kutta at Gaussian points.
 131
 132 =item * C<$gsl_odeiv_step_rk2simp>
 133
 134 =item * C<$gsl_odeiv_step_rk4imp> - Implicit 4th order Runge-Kutta at Gaussian points.
 135
 136 =item * C<$gsl_odeiv_step_bsimp> - Implicit Bulirsch-Stoer method of Bader and Deuflhard. This algorithm requires the Jacobian.
 137
 138 =item * C<$gsl_odeiv_step_gear1> - M=1 implicit Gear method.
 139
 140 =item * C<$gsl_odeiv_step_gear2> - M=2 implicit Gear method.
 141
 142 =back
 143
 144 For more informations on the functions, we refer you to the GSL offcial
 145 documentation: L<http://www.gnu.org/software/gsl/manual/html_node/>
 146
 147 Tip : search on google: site:http://www.gnu.org/software/gsl/manual/html_node/ name_of_the_function_you_want
 148
 149
 150 =head1 AUTHORS
 151
 152 Jonathan Leto <jonathan@leto.net> and Thierry Moisan <thierry.moisan@gmail.com>
 153
 154 =head1 COPYRIGHT AND LICENSE
 155
 156 Copyright (C) 2008 Jonathan Leto and Thierry Moisan
 157
 158 This program is free software; you can redistribute it and/or modify it
 159 under the same terms as Perl itself.
 160
 161 =cut
 162
 163 %}