The numerical integrator

  Its prototype is:

   int taylor_step_ODE_NAME(MY_FLOAT *time,
                            MY_FLOAT *xvars,
                            int      direction,
                            int      step_ctrl_method,
                            double   log10abserr,
                            double   log10relerr,
                            MY_FLOAT *endtime,
                            MY_FLOAT *stepused,
                            int      *order);

The function taylor_step_ODE_NAME does one step of numerical integration of the given system of ODEs, using the control parameters passed to it. It returns 1 if endtime is reached, 0 otherwise.

{Parameters:

• time
  on input: time of the initial condition
  on output: new time
• xvars
  on input: initial condition
  on output: new condition, corresponding to the (output) time ti
• direction
  flag to integrate forward or backwards.
  -1: forward
  : backwards
  Note: this flag is ignored if step_ctrl_method is set to 0.
• step_ctrl_method
  flag for the step size control. Its possible values are:

  0:

  no step size control, so the step and order are provided by the user. The parameter ht is used as step, and the parameter order (see below) is used as the order.

  1:

  standard stepsize control. it uses an approximation to the optimal order and to the radius of convergence of the series to approximate the 'optimal' step size. It tries to keep the absolute and relative errors below the given values. See the paper for more details.

  2:

  as 1, but adding an extra condition on the stepsize h: the terms of the series - after being multiplied by the suitable power of h - cannot grow.

  3:

  user defined stepsize control. The code has to be included in the routine compute_timestep_user_defined (see the code). The user must also include code for the selection of degree, in the function compute_order_user_defined.

• log10abserr
  decimal log of the absolute accuracy required.
• log10relerr
  decimal log of the relative accuracy required.
• endtime
  if NULL, it is ignored. if step_ctrl_method is set to 0, it is also ignored. otherwise, if next step is going to be outside endtime, reduce the step size so that the new time time is exactly endtime (in that case, the function returns 1).
• ht
  on input: ignored/used as a time step (see parameter step_ctl_method)
  on output: time step used
• order

  input: this parameter is only used if step_ctrl_method is 0, or if you add the proper code for the case step_ctrl_method=3.
  If step_ctrl_method is 0, its possible values are:
  ;SPMlt;2: the program will select degree 2,
  : the program will use this degree.

  output: degree used.

Returned value:

• 0: ok.
• 1: ok, and time=endtime.