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- added section "nested products"

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svn path=/trunk/libs/numeric/ublas/; revision=40098
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Gunter Winkler
2007-10-16 21:48:01 +00:00
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doc/operations_overview.htm
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@@ -207,8 +207,40 @@ in preference to <code>iterator</code>'s for the same reason. For the more darin
can be completely turned off in uBLAS by defining the configuration macro <code>BOOST_UBLAS_NO_ELEMENT_PROXIES</code>.
</p>
<h3>Controlling the complexity of nested products</h3>
<p>What is the complexity (the number of add and multiply operations) required to compute the following?
</p>
<pre>
R = prod(A, prod(B,C));
</pre>
<p>Firstly the complexity depends on matrix size. Also since prod is transitive (not commutative)
the bracket order affects the complexity.
</p>
<p>uBLAS evaluates expressions without matrix or vector temporaries and honours
the bracketing structure. However avoiding temporaries for nested product unnecessarly increases the complexity.
Conversly by explictly using temporary matrices the complexity of a nested product can be reduced.
</p>
<p>uBLAS provides 3 alternative syntaxes for this purpose:
</p>
<pre>
temp_type T = prod(B,C); R = prod(A,T); // Preferable if T is preallocated
</pre>
<pre>
prod(A, temp_type(prod(B,C));
</pre>
<pre>
prod(A, prod&lt;temp_type&gt;(B,C));
</pre>
<p>The 'temp_type' is important. Given A,B,C are all of the same type. Say
matrix&lt;float&gt;, the choice is easy. However if the value_type is mixed (int with float or double)
or the matrix type is mixed (sparse with symmetric) the best solution is not so obvious. It is up to you! It
depends on numerical properties of A and the result of the prod(B,C).
</p>
<hr />
<p>Copyright (&copy;) 2000-2004 Joerg Walter, Mathias Koch, Gunter
<p>Copyright (&copy;) 2000-2007 Joerg Walter, Mathias Koch, Gunter
Winkler, Michael Stevens<br />
Permission to copy, use, modify, sell and distribute this document
is granted provided this copyright notice appears in all copies.