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<H2><A NAME="SECTION00622000000000000000"> </A><A NAME="dfdx"> </A><A NAME="1991"> </A>
<BR>
Defining Derivative Information
</H2>
If available, DsTool can use analytic information in
certain computational routines.
Jacobian information is used in root-finding algorithms
and for implicitly iterating diffeomorphisms backwards (when an explicit inverse
does not exist). Detailed information about these algorithms is found in the DsTool Reference
Manual<A NAME="1992"> </A>.
<P>
If derivative information is not provided by the
user, then DsTool will use finite difference-approximations. This section will describe
how to write a function which provides an explicit Jacobian for Equation <A HREF="node53.html#bball"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
SRC="/usr/lib/latex2html/icons.gif/cross_ref_motif.gif"></A>.
Because time is discrete for maps, it does not make sense to define a derivative with
respect to time.
Currently, DsTool does not make use of derivatives with respect to time and parameters,
but we include them for the convenience of the user who wishes to extend the capabilities
of DsTool.
<P>
We now continue with the bouncing ball example by defining the Jacobian of <I>f</I>.
At the <I>j</I>th instant of time, the Jacobian is
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH: \begin{displaymath}
\left( \begin{array}{cc}
\partial{f_1}/\partial{\phi_j} & \partial{f_1}/\partial{v_j} \\
\partial{f_2}/\partial{\phi_j} & \partial{f_2}/\partial{v_j} \\
\end{array} \right) = \left( \begin{array}{cc}
1 & 1 \\
\gamma \sin( \phi_j + v_j) & \alpha + \gamma \sin(\phi_j + v_j)
\end{array} \right)
\end{displaymath} -->
71#71
</DIV>
<BR CLEAR="ALL">
<P></P>
Find the section of the file bball_def.c which reads
<PRE>
/* ------------------------------------------------------------------------
function used to define the Jacobian
------------------------------------------------------------------------ */
/*
int user_jac(m,x,p)
double **m, *x, *p;
{
}
*/
</PRE>
Using a text editor, modify this code segment to read
<PRE>
/* ------------------------------------------------------------------------
function used to define the Jacobian
------------------------------------------------------------------------ */
int bball_jac(m,x,p)
double **m, *x, *p;
{
double temp;
temp = p[1] * sin( x[0] + x[1] );
m[0][0] = 1.0;
m[0][1] = 1.0;
m[1][0] = temp;
m[1][1] = p[0] + temp;
}
</PRE>
<P>
The routine which calls bball_jac() sends in a matrix and two arrays which contain the
current state and the current parameters for
<!-- MATH: $f_{\alpha, \gamma}$ -->
72#72.
When bball_jac() returns, it has filled
the matrix with the numerical Jacobian for
<!-- MATH: $Df_{\alpha, \gamma}$ -->
73#73
evaluated at the current state.
As remarked previously, writing a Jacobian routine is optional.
<P>
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<ADDRESS>
<I>John Lapeyre</I>
<BR><I>1998-09-04</I>
</ADDRESS>
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