

\subsection{\bf NI.2:\ \ Romberg or}
\subsection{\bf Adaptive Integration}

\begin{slide}

\begin{itemize}
\item A very convenient way of evaluating integrals
is to repeatedly apply Simpson's rule---halving interval each
time.

$$\vbox{\halign{\hfil#\hfil\tabskip=1em&\hfil#\hfil&\hfil#\hfil&
\hfil#\hfil&\hfil#\hfil&\hfil#\hfil&\hfil#\hfil&\hfil#\hfil&
\hfil#\hfil&\hfil#\tabskip=0pt\cr
a&&&&&&&&b\cr
x&&&&x&&&&x&Step 1\cr
$\bullet$&&x&&$\bullet$&&x&&$\bullet$&2\cr
$\bullet$&x&$\bullet$&x&$\bullet$&x&$\bullet$&x&$\bullet$&3\cr
\multispan9\hfil $\rightarrow$ x\hfil\cr
}}$$

\item The point is that, as diagram indicates, one
need only evaluate half as many (x in diagram) functions as
you expect because you use old values ($\bullet$ in figure).

\item As the error is proportional to $nh^5$ and $n$
doubles and $h$ halves each go, error should go down by
$\approx 1/16$ each iteration and convergence is fast.  So, in
practice, one halves $h$ until change in integral between steps
is less than some assigned value.
\end{itemize}
\end{slide}

