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<H2><A NAME="SECTION00067000000000000000"></A><A NAME="ss:simplifyingcmpmaps"></A>
<BR>
Simplifying Compound Mappings
</H2>

<P>
The ability to simplify compound Mappings is provided by the
astSimplifyastSimplify function. This function encapsulates a number of
heuristics for converting Mappings, or combinations of Mappings within
a CmpMapCmpMap, into simpler, equivalent ones. When applied to a CmpMap,
astSimplify tries to reduce the number of individual Mappings within
it by merging neighbouring component Mappings together. It will do
this with both series and parallel combinations of Mappings, or both,
and will handle CmpMaps nested to any depth
(&#167;<A HREF="node67.html#ss:complexcmpmap">6.4</A>).

<P>
To illustrate how astSimplify works, consider the combination of
   Mappings shown in Figure&nbsp;<A HREF="#fig:simplifyexample">10</A>.
   
<DIV ALIGN="CENTER"><A NAME="fig:simplifyexample"></A><A NAME="22083"></A>
<TABLE>
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 10:</STRONG>
An over-complex compound Mapping, consisting of PermMaps,
   ZoomMaps and a UnitMapUnitMap, which can be simplified to become a single
   UnitMap.  The enclosing nested CmpMaps have been omitted for clarity.</CAPTION>
<TR><TD>
<DIV ALIGN="CENTER">
[width=0.7]sun211_figures/simpexamp
   
</DIV></TD></TR>
</TABLE>
</DIV>

<P>
If this were contained in a CmpMap, it could be simplified as follows:

<P>
<SMALL CLASS="SMALL">
<BR>
<IMG
 WIDTH="265" HEIGHT="91" ALIGN="BOTTOM" BORDER="0"
 SRC="img98.png"
 ALT="\begin{terminalv}
AstMapping *simpler;
\par
...
\par
simpler = astSimplify( cmpmap );
\end{terminalv}">
<BR>
</SMALL>
<P>
In this case, the result would be a simple 3-dimensional UnitMap (the
identity MappingMapping).  To reach this conclusion, astSimplify will have
made a number of deductions, roughly as follows:

<P>

<OL>
<LI>The two 2-dimensional ZoomMaps in series are equivalent to a
single ZoomMapZoomMap with a combined ZoomZoom factor of unity. This, in turn, is
equivalent to a 2-dimensional UnitMap.

<P>
</LI>
<LI>This UnitMap in parallel with the other 1-dimensional UnitMap is
equivalent to a single 3-dimensional UnitMap. This UnitMap, sandwiched
between any other pair of Mappings, can then be eliminated.

<P>
</LI>
<LI>The remaining two PermMaps in series are equivalent to a single
3-dimensional PermMapPermMap. When these are combined, the resulting PermMap
is found to be equivalent to a 3-dimensional UnitMap.
</LI>
</OL>

<P>
This example is a little contrived, but illustrates how astSimplify
can deal with even quite complicated compound Mappings through a
series of incremental simplifications. Where possible, this will
result in either a simpler compound Mapping or, if feasible, an atomic
(non-compound) Mapping, as here. If no simplification is possible,
astSimplify will just return a pointer to the original Mapping.

<P>
Although astSimplify cannot identify every simplification that is
theoretically possible, sufficient rules are included to deal with the
most common and important cases.

<P>

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