1 %module
"Math::GSL::ODEIV"
2 %include
"gsl_typemaps.i"
4 #include
"gsl/gsl_odeiv.h"
5 #include
"gsl/gsl_types.h"
7 %include
"gsl/gsl_types.h"
8 %include
"gsl/gsl_odeiv.h"
18 gsl_odeiv_control_alloc
19 gsl_odeiv_control_init
20 gsl_odeiv_control_free
21 gsl_odeiv_control_hadjust
22 gsl_odeiv_control_name
23 gsl_odeiv_control_standard_new
24 gsl_odeiv_control_y_new
25 gsl_odeiv_control_yp_new
26 gsl_odeiv_control_scaled_new
27 gsl_odeiv_evolve_alloc
28 gsl_odeiv_evolve_apply
29 gsl_odeiv_evolve_reset
36 $gsl_odeiv_step_rk2imp
37 $gsl_odeiv_step_rk2simp
38 $gsl_odeiv_step_rk4imp
45 $gsl_odeiv_control_standard
47 %EXPORT_TAGS
= ( all
=> [ @EXPORT_OK
] );
53 Math
::GSL
::ODEIV
- functions for solving ordinary differential equation
(ODE
) initial value problems
57 use Math
::GSL
::ODEIV qw
/:all
/;
61 Here is a list of all the functions in this module
:
65 =item
* C
<gsl_odeiv_step_alloc
($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.
67 =item
* C
<gsl_odeiv_step_reset
($s
)> - This function resets the stepping function $s. It should be used whenever the next use of s will not be a continuation of a previous step.
69 =item
* C
<gsl_odeiv_step_free
($s
)> - This function frees all the memory associated with the stepping function $s.
71 =item
* C
<gsl_odeiv_step_name
($s
)> - This function returns a pointer to the name of the stepping function.
73 =item
* C
<gsl_odeiv_step_order
($s
)> - This function returns the order of the stepping function on the previous step. This order can vary if the stepping function itself is adaptive.
75 =item
* C
<gsl_odeiv_step_apply
>
77 =item
* C
<gsl_odeiv_control_alloc
($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. $T is a gsl_odeiv_control_type.
79 =item
* C
<gsl_odeiv_control_init
($c
, $eps_abs
, $eps_rel
, $a_y
, $a_dydt
) > - This function initializes the control function c with the parameters eps_abs
(absolute error
), eps_rel
(relative error
), a_y
(scaling factor for y
) and a_dydt
(scaling factor for derivatives
).
81 =item
* C
<gsl_odeiv_control_free
>
83 =item
* C
<gsl_odeiv_control_hadjust
>
85 =item
* C
<gsl_odeiv_control_name
>
87 =item
* C
<gsl_odeiv_control_standard_new
($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 $a_dydt for the system state y
(t
) and derivatives y'
(t
) respectively. The step-size adjustment procedure for this method begins by computing the desired error level D_i for each component
, D_i
= eps_abs
+ eps_rel
* (a_y |y_i|
+ a_dydt h |y'_i|
) and comparing it with the observed error E_i
= |yerr_i|. If the observed error E exceeds the desired error level D by more than
10% for any component then the method reduces the step-size by an appropriate factor
, h_new
= h_old
* S
* (E
/D
)^
(-1/q
) where q is the consistency order of the method
(e.g. q
=4 for
4(5) embedded RK
), and S is a safety factor of
0.9. The ratio E
/D is taken to be the maximum of the ratios E_i
/D_i. If the observed error E is less than
50% of the desired error level D for the maximum ratio E_i
/D_i then the algorithm takes the opportunity to increase the step-size to bring the error in line with the desired level
, h_new
= h_old
* S
* (E
/D
)^
(-1/(q
+1)) This encompasses all the standard error scaling methods. To avoid uncontrolled changes in the stepsize
, the overall scaling factor is limited to the range
1/5 to
5.
89 =item
* C
<gsl_odeiv_control_y_new
($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 $eps_rel with respect to the solution y_i
(t
). This is equivalent to the standard control object with a_y
=1 and a_dydt
=0.
91 =item
* C
<gsl_odeiv_control_yp_new
($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 $eps_rel with respect to the derivatives of the solution y'_i
(t
). This is equivalent to the standard control object with a_y
=0 and a_dydt
=1.
93 =item
* C
<gsl_odeiv_control_scaled_new
($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 $scale_abs. The formula for D_i for this control object is
, D_i
= eps_abs
* s_i
+ eps_rel
* (a_y |y_i|
+ a_dydt h |y'_i|
) where s_i is the i-th component of the array scale_abs. The same error control heuristic is used by the Matlab ode suite.
95 =item
* C
<gsl_odeiv_evolve_alloc
($dim
)> - This function returns a pointer to a newly allocated instance of an evolution function for a system of $dim dimensions.
97 =item
* C
<gsl_odeiv_evolve_apply
>
99 =item
* C
<gsl_odeiv_evolve_reset
($e
)> - This function resets the evolution function $e. It should be used whenever the next use of $e will not be a continuation of a previous step.
101 =item
* C
<gsl_odeiv_evolve_free
($e
)> - This function frees all the memory associated with the evolution function $e.
105 This module also includes the following constants
:
109 =item
* C
<$GSL_ODEIV_HADJ_INC
>
111 =item
* C
<$GSL_ODEIV_HADJ_NIL
>
113 =item
* C
<$GSL_ODEIV_HADJ_DEC
>
121 =item
* C
<$gsl_odeiv_step_rk2
> - Embedded Runge-Kutta
(2, 3) method.
123 =item
* C
<$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.
125 =item
* C
<$gsl_odeiv_step_rkf45
> - Embedded Runge-Kutta-Fehlberg
(4, 5) method. This method is a good general-purpose integrator.
127 =item
* C
<$gsl_odeiv_step_rkck
> - Embedded Runge-Kutta Cash-Karp
(4, 5) method.
129 =item
* C
<$gsl_odeiv_step_rk8pd
> - Embedded Runge-Kutta Prince-Dormand
(8,9) method.
131 =item
* C
<$gsl_odeiv_step_rk2imp
> - Implicit
2nd order Runge-Kutta at Gaussian points.
133 =item
* C
<$gsl_odeiv_step_rk2simp
>
135 =item
* C
<$gsl_odeiv_step_rk4imp
> - Implicit
4th order Runge-Kutta at Gaussian points.
137 =item
* C
<$gsl_odeiv_step_bsimp
> - Implicit Bulirsch-Stoer method of Bader and Deuflhard. This algorithm requires the Jacobian.
139 =item
* C
<$gsl_odeiv_step_gear1
> - M
=1 implicit Gear method.
141 =item
* C
<$gsl_odeiv_step_gear2
> - M
=2 implicit Gear method.
145 For more informations on the functions
, we refer you to the GSL offcial
146 documentation
: L
<http
://www.gnu.org
/software
/gsl
/manual
/html_node
/>
148 Tip
: search on google
: site
:http
://www.gnu.org
/software
/gsl
/manual
/html_node
/ name_of_the_function_you_want
153 Jonathan Leto
<jonathan@leto.net
> and Thierry Moisan
<thierry.moisan@gmail.com
>
155 =head1 COPYRIGHT
AND LICENSE
157 Copyright
(C
) 2008 Jonathan Leto and Thierry Moisan
159 This program is free software
; you can redistribute it and
/or modify it
160 under the same terms as Perl itself.