We first describe the Propagation window as it appears for vector fields:
The panel items in the Propagation window will change with the method of propagation
selected. The top portion of the panel consists of
stack settings from which the
user may select algorithms used by the propagator. The lower left section of the panel consists
of
numeric fields which are used to control integer parameters of the algorithms
chosen. These fields may be used, for example, to set 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
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.
We now describe the Propagation window as it appears for mappings:
The panel entries of the Propagation window will
depend upon the current dynamical system. In particular, an option will not be displayed
if the current model does not allow the method in question to be used.
- Jacobian exclusive setting:
Allows the user to choose how to evaluate the Jacobian matrix
for the map. If the mapping does not have an explicit inverse, then Jacobian matrix
is needed in order to compute inverse images of points, i.e., for backwards iteration of the map.
The options are:
- Forward difference:
- A numerical Jacobian will be used, calculated using a
forward difference method. This method is
in the finite
difference step
.
- Central difference:
- A numerical Jacobian will be used, calculated using a
central difference method. This method is
in the finite
difference step
.
- Explicit:
- The explicit Jacobian will be used. This option is only available
if the user has supplied an explicit Jacobian for the map.
- Initial guess exclusive setting:
Allows the user to choose how to pick the initial guess (``seed'') for Newton's method.
This guess may be provided by:
- Approx inv:
-
The seed is chosen from an approximate inverse.
If the map may be considered a perturbation of a map which does have
an exact inverse, 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. This option is not available if the mapping
does not have an approximate inverse defined, i.e., if the inverse_toggle
model file variable is set to either FALSE or EXPLICIT_INV.
- Monte Carlo:
-
The seed is chosen at random from within the hypervolume defined by the coordinate
values in the Defaults window.
- Inverse algorithm exclusive setting:
Allows the user to choose which type of inverse algorithm will be used to
compute approximate inverse images. The options are:
- Newton's method:
-
Newton's method is used to calculate pre-images.
- Explicit formula:
-
An explicit formula is used to calculate pre-images. This option is not
available if the mapping does not have an explicit inverse function defined,
i.e., if the inverse_toggle model file variable is set to either FALSE or
APPROX_INV.
- #MC numeric field:
Displays the maximum number of random guesses taken by the Monte Carlo routine. The
default value is
.
- Newton iter numeric field:
Displays 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
.
- Finite diff step read-write text field:
Displays the spatial step to be used for computing a finite difference Jacobian.
The default value is
. This field is only used if the Jacobian exclusive setting
is set to either Forward difference or Central difference.
- Min step read-write text field:
Displays the minimum step required to take during Newton's method.
Newton's method generates a sequence of points
which
(hopefully) converges to the inverse image. The difference between
and
is
called the
th 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. The default value is
.
- Conv crit read-write text field:
Displays the criterion used to determine when Newton's method has converged.
We use Newton's method to compute a root of some function,
say,
. An iterative sequence
is said to converge
to a solution if the norm of
is less than the value of
Conv crit for some
. The default value is
.
- Dismiss command button:
Closes the Propagation window.