The panel items on the Propagation Panel will depend on the method of propagation
chosen. The top portion of the panel consists of 12#12
stack settings from which the
user may select algorithms used by the propagator. The lower left section of the panel consists
of 13#13
integer fields which are used to control integer parameters of the algorithms
chosen. The maximum number of Newton iterations used in estimating an inverse image for a
diffeomorphism is one example of an integer parameter. The lower right section of the panel
contains a list of 14#14
double precision fields which are used to control floating point
parameters of the chosen numerical algorithms. These fields may be used, for example, to
set the minimum step size allowable for variable-step integrators, or to establish convergence
criteria for Newton's method.
If a user wishes to install propagation algorithms (such as a special-purpose integrator) then
the user will also want to adapt this panel to his or her needs.
The Reference Manual shows how this is
done.
We now describe the options for fixed-step integration algorithms:
- Integrator stack setting: Allows the user to select a numerical integrator.
The current options are
- Runge-Kutta 4:
- A fourth-order Runge-Kutta algorithm. This
is the default.
- Euler:
- The forward Euler method.
- Runge-Kutta (5):
- A fifth-order Runge-Kutta algorithm with adaptive
step-size.
- Bulirsch-Stoer:
- A Bulirsh-Stoer algorithm.
See, for example, References [2,3] for more information about these algorithms.
- Newton Iter read-write text field: This integer field specifies the maximum number
of iterations allowed in determining the point along a trajectory for which some event is satisfied. The default value
is 15.
- fd Step read-write text field: When event Event Stopping stack option is selected, the algorithm uses
forward differencing to estimate the derivative of the stopping criterion function with respect to
the time step. The value contained in the fd Step field is used in this forward difference estimate.
The default value is 10-5.
- Error read-write text field: Provides an upper bound on the maximum error tolerated
during event stopping. In other words, if
the stopping condition is, say, g(x)=c then what is the maximum allowable
norm of g(x) - c.
The default value is 10-8.
- Min dt read-write text field: The smallest allowable time step.
Even for fixed-step integrators such as standard
Runge-Kutta algorithms, this value is important because once a stopping event is detected,
the time step is varied so that the trajectory satisfies the stopping condition
to within the error specified in the Error field.
We now describe the Propagation panel as it appears for mappings:
- Jacobian stack setting: Allows the user to choose how to evaluate the jacobian
for the map. If the mapping does not have an explicit inverse, then jacobian is needed in order
to compute inverse images of points, ie, for backwards iteration of the map.
The current options are
- Explicit:
- Only available if the user has specified an explicit jacobian for the map.
- Forw Diff:
- A numerical jacobian will be used, calculated using a forward difference method.
This method is 15#15
in the finite difference step h.
- Cen Diff:
- A numerical jacobian will be used, calculated using a central difference method.
This method is 16#16
in the finite difference step h.
The following items do not appear if the mapping has an explicit inverse defined.
- Initial Guess stack setting: Newton's method requires a ``seed,'' an initial guess from which to begin.
This guess may be provided by:
- Approx Inv:
-
The map may be a perturbation from a map which does have
an exact inverse. In this case, a good guess for the inverse of the perturbed system
is often given by the exact inverse of the unperturbed system. A few steps of Newton's method
is often sufficient to converge to the inverse of the perturbed system.
- Monte Carlo:
- The initial guess for Newton's method
is chosen at random from within the hypervolume defined by the coordinate values
in the Defaults Settings Window.
- #MC read-write text field: This integer field sets the number of random guesses taken
by the Monte Carlo routine.
- Newton Iter text field: This integer field specifies the maximum number
of iterations allowed in Newton's method of computing fixed points. This algorithm is used, for
example, in determining the point along a trajectory for which some event
is satisfied. The default value is 15.
- fd Step read-write text field: If the user has not included an
explicit jacobian for the current vector field, then
this field specifies the spatial step to be used for computing a finite difference jacobian.
The default value is 10-5.
- Min Step read-write text field: The default value is
10-8. Inverse images are found via Newton's method,
an iterative process. Newton's method generates a sequence of points {xi} which
(hopefully) converges to the inverse image. The difference between xi+1 and xi is
called the ith Newton step.. If the length of the Newton step is
less than the value of Min Step, then we assume that we can no longer improve our current guess,
and so we end the Newton process.
- Conv Crit read-write text field: The default
value is 10-8. We use Newton's method to compute a root of some function, say, g.
An iterative sequence {xi}0n is said to converge
to a solution if the norm of g(xi) is less than the value of
Conv Crit for some i.