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<H2><A NAME="SECTION00623000000000000000"> </A><A NAME="inverse"> </A><A NAME="2017"> </A>
<BR>
Defining Information About an Inverse
</H2>
In this section we show how to define an inverse or an approximate inverse for a
mapping. We recall that the ``inverse function'' for a vector field is trivial, since
integrating the equations of motion backwards in time is the same as integrating
forward in time--except we pass in a negative time step to a numerical integrator.
Thus if we were installing a vector field, we would skip this section. Moreover, it
frequently happens that a mapping does not have an inverse, nor is there any sort of
an approximate inverse. In that case, do not edit the inverse routine at all.
<P>
For diffeomorphisms, it is sometimes difficult to iterate backwards in time.
The Reference Manual<A NAME="2018"> </A> discusses Newton's method<A NAME="2019"> </A>
and implicitly iterating a mapping backwards<A NAME="2020"> </A>;
it also briefly indicates some of the ways that Newton's method
can fail. One of the most important aspects of Newton's algorithm is its dependence on a ``good''
initial guess. In the present section, we will discuss one way to provide a good guess.
<P>
For our bouncing ball example, it turns out that we can find an explicit inverse<A NAME="2021"> </A> for <I>f</I>since
<!-- MATH: $\alpha > 0$ -->
74#74.
This inverse is the map given by
<A NAME="bball_inv"> </A>
<BR>
75#75
<BR>
<P>
To enter this inverse into DsTool, locate the section of bball_def.c which looks like
<PRE>
/* ------------------------------------------------------------------------
function used to define inverse or approximate inverse
------------------------------------------------------------------------ */
/*
int user_inv(y,x,p)
double *y,*x,*p;
{
}
*/
</PRE>
and modify this section to produce
<PRE>
/* ------------------------------------------------------------------------
function used to define inverse
------------------------------------------------------------------------ */
int bball_inv(y,x,p)
double *y,*x,*p;
{
double temp;
temp = ( p[1] * cos( x[0] ) + x[1] ) / p[0];
y[0] = x[0] - temp;
y[1] = temp;
}
</PRE>
Suppose that our map did not have an explicit inverse (or we could not calculate it).
Then we would have to resort to using Newton's method to numerically solve for the inverse iterate
at each backward step. To do so requires that we
provide an initial guess for the inverse image of the current state.
If we provide a good guess for the inverse image, then possibly we will require only a few
Newton steps to ``sharpen'' our guess and converge to a solution (See the Reference Manual<A NAME="2037"> </A>).
Suppose, for the sake of an example, that we were only interested in the bouncing ball map
<!-- MATH: $f_{\alpha, \gamma}$ -->
72#72in the special case that 17#17
is small. Then <I>f</I> can be thought of as a perturbation
of the linear map <I>g</I> defined by
<BR>
76#76
<BR>
and <I>g</I><SUP>-1</SUP> can be written as:
<BR>
77#77
<BR>
In this case, we could code <I>g</I><SUP>-1</SUP> into DsTool in place of <I>f</I><SUP>-1</SUP>.<A NAME="2056"> </A>
If we were to do this, we would modify bball_def.c to read:
<PRE>
/* ------------------------------------------------------------------------
function used to define approximate inverse
------------------------------------------------------------------------ */
int bball_approx_inv(y,x,p)
double *y,*x,*p;
{
y[0] = x[0] - x[1] / p[0];
y[1] = x[1] / p[0];
}
</PRE>
Of course, we must inform DsTool whether to interpret the procedure as a definition of the actual inverse
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<ADDRESS>
<I>John Lapeyre</I>
<BR><I>1998-09-04</I>
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