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<H1><A NAME="SECTION004210000000000000000"> </A><A NAME="1117"> </A><A NAME="1118"> </A>
<BR>
The Propagation Window
</H1>
<BR>
<DIV ALIGN="CENTER"><A NAME="prop_int"> </A><A NAME="1181"> </A>
<TABLE WIDTH="50%">
<CAPTION><STRONG>Figure 2.21:</STRONG>
A Propagation window for vector fields.</CAPTION>
<TR><TD>35#35</TD></TR>
</TABLE>
</DIV>
<BR>
<BR>
<DIV ALIGN="CENTER"><A NAME="prop_iter"> </A><A NAME="1183"> </A>
<TABLE WIDTH="50%">
<CAPTION><STRONG>Figure 2.22:</STRONG>
A Propagation window for mappings.</CAPTION>
<TR><TD>36#36</TD></TR>
</TABLE>
</DIV>
<BR>
<P>
<DL>
<DD><P>
<DT><STRONG>Window title:</STRONG>
<DD>DsTool: Propagation
<DT><STRONG>Function:</STRONG>
<DD>The Propagation window allows the user to
select and control the algorithms used to compute trajectories.
<DT><STRONG>Description:</STRONG>
<DD>The Propagation window is opened by selecting the
Propagation window button located in the Orbits window.
The Propagation window contains parameters necessary to control the forward
and backward propagation of trajectories.
The window's appearance depends upon the model chosen as well as the method of propagation.
<DT><STRONG>Panel items:</STRONG>
<DD><P>
We first describe the Propagation window as it appears for vector fields:
<P>
The panel items in the Propagation window will change with the method of propagation
selected. The top portion of the panel consists of 37#37
stack settings from which the
user may select algorithms used by the propagator. The lower left section of the panel consists
of 38#38
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 39#39
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.<A NAME="1128"> </A>
<P>
<UL>
<LI>Integration algorithm stack setting:
Allows the user to choose a numerical integrator.<A NAME="1130"> </A>
The current options are:
<DL>
<DT><STRONG>Runge-Kutta 4:</STRONG>
<DD>A fourth-order Runge-Kutta algorithm. This is the default.
<DT><STRONG>Euler:</STRONG>
<DD>The forward Euler method.
<DT><STRONG>Runge-Kutta 4QC:</STRONG>
<DD>A fourth-order Runge-Kutta algorithm with fifth-order stepsize
regulation.
<DT><STRONG>Runge-Kutta-Fehlberg 78:</STRONG>
<DD>A Runge-Kutta-Fehlberg algorithm of orders seven and eight.
<DT><STRONG>Bulirsch-Stoer:</STRONG>
<DD>The Bulirsch-Stoer algorithm with Richardson extrapolation.
<DT><STRONG>Adams-Bashforth 4:</STRONG>
<DD>The fourth-order Adams-Bashforth algorithm.
</DL> See, for example, references [<A
HREF="node68.html#pressetal">4</A>,<A
HREF="node68.html#stoerbulirsch">5</A>] for more information
about these algorithms.
<LI>Newton iter numeric field:
Displays the maximum number of iterations allowed in determining the
point along a trajectory for which some event is satisfied. The default value is 15.
<LI>Finite diff step read-write text field:
Displays the finite difference step size.
When the Stopping condition stack on the Orbits window is set to Event stopping,
the algorithm uses forward differencing to estimate the derivative of the stopping
criterion function with respect to the time step. The default value is 10<SUP>-6</SUP>.
<LI>Stopping error read-write text field:
Displays the upper bound on the maximum error tolerated during event stopping.
In other words, if the stopping condition is, say, <I>g</I>(<I>x</I>)=<I>c</I> then the maximum allowable
norm of <I>g</I>(<I>x</I>) - <I>c</I> will be the value of this field. The default value is 10<SUP>-5</SUP>.
<LI>Min dt read-write text field:
Displays the smallest allowable time step. This value is important even for fixed-step integrators
such as standard Runge-Kutta algorithms, 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 by the Error field. The default value is 10<SUP>-10</SUP>.
<P>
The remainder of the panel items are determined by which algorithm is selected. Therefore, for each
integrator, we now describe the remainder of the panel.
<LI>Runge-Kutta 4, Euler, Adams-Bashforth 4:
<DL>
<DT><DD>No additional fields.
</DL>
<LI>Runge-Kutta 4QC:
<DL>
<DT><DD>Const. fractional err. / Bounded global errors exclusive setting:
Allows the user to choose between controlling the errors at each step by a
constant amount, or controlling the errors globally. The default is to have constant
fractional errors.
<DT><DD>Max step read-write text field:
Displays the value of the largest allowable time step. The default value is 0.2.
<DT><DD>Err tol read-write text field:
Displays the value of the required accuracy. The default value is 10<SUP>-5</SUP>.
<DT><DD>Pgrow read-write text field:
Displays the value of PGROW (see [<A
HREF="node68.html#pressetal">4</A>]) for the RK-4QC algorithm.
The default value is -0.2.
<DT><DD>Pshrink read-write text field:
Displays the value of PSHRNK (see [<A
HREF="node68.html#pressetal">4</A>]) for the RK-4QC algorithm.
The default value is -0.25.
<DT><DD>Fcor read-write text field:
Displays the value of FCOR (see [<A
HREF="node68.html#pressetal">4</A>]) for the RK-4QC algorithm.
The default value is 0.6666.
<DT><DD>Safety read-write text field:
Displays the value of SAFETY (see [<A
HREF="node68.html#pressetal">4</A>]) for the RK-4QC algorithm.
The default value is 0.9.
</DL>
<LI>Runge-Kutta-Fehlberg 78:
<DL>
<DT><DD>Max step read-write text field:
Displays the value of the largest allowable time step. The default value is 0.2.
<DT><DD>Err tol read-write text field:
Displays the value of the required accuracy. The default value is 10<SUP>-5</SUP>.
<DT><DD>Safety read-write text field:
Displays the factor to reduce the step size by (40#40
in [<A
HREF="node68.html#stoerbulirsch">5</A>])
for the RKF algorithm. The default value is 0.8.
</DL>
<LI>Bulirsch-Stoer:
<DL>
<DT><DD>Const. fractional err / Bounded global errors exclusive setting:
Allows the user to choose between controlling the errors at each step by a
constant amount, or controlling the errors globally. The default is to have
constant fractional errors.
<DT><DD>BS intervals numeric field:
Displays the integer which will be used by the Bulirsch-Stoer algorithm as
the number of Bulirsch-Stoer intervals (NUSE in [<A
HREF="node68.html#pressetal">4</A>]). The default
value is 7.
<DT><DD>Max step read-write text field:
Displays the value of the largest allowable time step. The default value is 0.2.
<DT><DD>Err tol read-write text field:
Displays the value of the required accuracy. The
default value is 10<SUP>-5</SUP>.
<DT><DD>Shrink read-write text field:
Displays the value of SHRINK (see [<A
HREF="node68.html#pressetal">4</A>]) for the B-S algorithm.
The default value is 0.95.
<DT><DD>Grow read-write text field:
Displays the value of GROW (see [<A
HREF="node68.html#pressetal">4</A>]) for the B-S algorithm.
The default value is 1.2.
</DL></UL>
<P>
We now describe the Propagation window as it appears for mappings:
<P>
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.
<P>
<UL>
<LI>Jacobian exclusive setting:
Allows the user to choose how to evaluate the Jacobian matrix<A NAME="1158"> </A>
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, , for backwards iteration of the map.
The options are:
<DL>
<DT><STRONG>Forward difference:</STRONG>
<DD>A numerical Jacobian will be used, calculated using a
forward difference method. This method is
<!-- MATH: ${\cal O}(h)$ -->
41#41
in the finite
difference step <I>h</I>.
<DT><STRONG>Central difference:</STRONG>
<DD>A numerical Jacobian will be used, calculated using a
central difference method. This method is
<!-- MATH: ${\cal O}(h^2)$ -->
42#42
in the finite
difference step <I>h</I>.
<DT><STRONG>Explicit:</STRONG>
<DD>The explicit Jacobian will be used. This option is only available
if the user has supplied an explicit Jacobian for the map.
</DL>
<P>
<LI>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:
<DL>
<DT><STRONG>Approx inv:</STRONG>
<DD><A NAME="1164"> </A>
The seed is chosen from an approximate inverse.
If the map may be considered a perturbation of a map which <EM>does</EM> 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, , if the inverse_toggle
model file variable is set to either FALSE or EXPLICIT_INV.
<DT><STRONG>Monte Carlo:</STRONG>
<DD><A NAME="1166"> </A>
The seed is chosen at random from within the hypervolume defined by the coordinate
values in the Defaults window.
</DL>
<LI>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:
<DL>
<DT><STRONG>Newton's method:</STRONG>
<DD><A NAME="1169"> </A>
Newton's method is used to calculate pre-images.
<DT><STRONG>Explicit formula:</STRONG>
<DD><A NAME="1170"> </A>
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,
, if the inverse_toggle model file variable is set to either FALSE or
APPROX_INV.
</DL>
<P>
<LI>#MC numeric field:
Displays the maximum number of random guesses taken by the Monte Carlo routine. The
default value is 10.
<P>
<LI>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 15.
<P>
<LI>Finite diff step read-write text field:
Displays the spatial step to be used for computing a finite difference Jacobian.
The default value is 10<SUP>-5</SUP>. This field is only used if the Jacobian exclusive setting
is set to either Forward difference or Central difference.
<P>
<LI>Min step read-write text field:<A NAME="1173"> </A>
Displays the minimum step required to take during Newton's method.
Newton's method generates a sequence of points 43#43
which
(hopefully) converges to the inverse image. The difference between <I>x</I><SUB><I>i</I>+1</SUB> and <I>x</I><SUB><I>i</I></SUB> is
called the <I>i</I>th Newton step<A NAME="1175"> </A>. 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 10<SUP>-8</SUP>.
<P>
<LI>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, <I>g</I>. An iterative sequence
<!-- MATH: $\{x_i\}_0^n$ -->
44#44
is said to converge
to a solution if the norm of <I>g</I>(<I>x</I><SUB><I>i</I></SUB>) is less than the value of
Conv crit for some <I>i</I>. The default value is 10<SUP>-8</SUP>.
<LI>Dismiss command button:
Closes the Propagation window.
</UL></DL>
<P>
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<ADDRESS>
<I>John Lapeyre</I>
<BR><I>1998-09-04</I>
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