
<A NAME=bounds>

<TITLE>bounds</TITLE></A>
<b><a href=r37_idx.html>INDEX</a></b><p><p>



<B>BOUNDS</B> _ _ _  _ _ _  _ _ _  _ _ _ <B>operator</B><P>
<P>
 
<P>
<P>
Upper and lower bounds of a real valued function over an 

<A HREF=r37_0422.html>interval</A> or a rectangular multivariate domain are comp
uted 
by the operator <em>bounds</em>. The algorithmic basis is the computation 
with inequalities: starting from the interval(s) of the 
variables, the bounds are propagated in the expression 
using the rules for inequality computation. Some knowledge 
about the behavior of special functions like ABS, SIN, COS, EXP, LOG, 
fractional exponentials etc. is integrated and can be evaluated 
if the operator <em>bounds</em> is called with rounded mode on 
(otherwise only algebraic evaluation rules are available). 
 <P>
<P>
If <em>bounds</em> finds a singularity within an interval, the evaluation 
is stopped with an error message indicating the problem part 
of the expression. 
 <P>
<P>
 <P> <H3> 
syntax: </H3>
<em>bounds</em>(&lt;exp&gt;,&lt;var&gt;=(&lt;l&gt; .. &lt;u&gt;) 
 [,&lt;var&gt;=(&lt;l&gt; .. &lt;u&gt;) ...]) 
<P>
<P>
or 
<P>
<P>
<em>bounds</em>(&lt;exp&gt;,{&lt;var&gt;=(&lt;l&gt; .. &lt;u&gt;) 
 [,&lt;var&gt;=(&lt;l&gt; .. &lt;u&gt;) ...]}) 
<P>
<P>
<P>
<P>
where &lt;exp&gt; is the function to be investigated, 
&lt;var&gt; are the variables of &lt;exp&gt;, 
&lt;l&gt; and &lt;u&gt; specify the area as set of 
<A HREF=r37_0422.html>interval</A><em>s</em>. 
<P>
<P>
<em>bounds</em>computes upper and lower bounds for the expression in the 
given area. An 
<A HREF=r37_0422.html>interval</A> is returned. 
 <P>
<P>
 <P> <H3> 
examples: </H3>
<P><PRE><TT>
bounds(sin x,x=(1 .. 2));

  -1 .. 1


on rounded;

bounds(sin x,x=(1 .. 2));

  0.84147098481 .. 1


bounds(x**2+x,x=(-0.5 .. 0.5));

  - 0.25 .. 0.75

</TT></PRE><P>