lines 9-80 of file: cpp/include/cmpad/algo/runge_kutta.hpp

{xrst_begin_parent runge_kutta}
{xrst_spell
   dy
   ns
   tf
   yf
   yi
   yt
   obj
}

A C++ Fourth Order Runge Kutta Solver
#####################################

Syntax
******
| |tab| ``# include <cmpad/algo/runge_kutta.hpp>``
| |tab| *yf* = ``cmpad::runge_kutta`` ( *fun* , *yi* , *tf*, *ns* )

Prototype
*********
{xrst_literal
   // BEGIN PROTOTYPE
   // END PROTOTYPE
}

Purpose
*******
This routine returns an approximate solution
for :math:`y( t^f )` where :math:`y(0) = y^i` and
:math:`y' (t) = f(y)` .

Vector
******
The vectors *yi* and *yf* have this
fun_obj :ref:`cpp_fun_obj@vector_type` .

fun
***
The syntax *dy* = *fun* ( *yt* ) ,
were *yt* is :math:`y(t)` ,
sets *dy* equal to the derivative :math:`y'(t)`  .
Both *yt* and *dy* are represented as *Vector* objects.

yi
**
is the value of :math:`y(t)` at :math:`t = 0` .

tf
**
is the value of *t* at which we wish to evaluate :math:`y(t)` .

ns
**
is the number of :ref:`Runge-Kutta steps<rk4_step-name>` to use.
The more steps the smaller the step size and the more accurate
the solution.

yf
**
The return value *yf* has the same size as *yi* and is the approximation
for :math:`y(t)` at *t* = *tf* .

{xrst_toc_table after
   cpp/xam/runge_kutta.cpp
}
Example
*******
:ref:`xam_runge_kutta.cpp-name`
contains an example and test of ``runge_kutta`` .

{xrst_end runge_kutta}