

\subsection{\bf Congruential Pseudorandom Numbers}

\begin{slide}
\begin{itemize}
\item Rather, one usually employs a deterministic method which gives
``essentially'' random numbers.  These are called pseudorandom.  

\item Most ``standard'' library random number generators use the so
called linear congruential generators which generate a sequence of
integers $I_j$ by
\[
I_{j+1} = \left\lbrack a I_j + c \right\rbrack {\rm mod} (m)
\]

\item Here, $m$ is called the modulus, $a$ is called the multiplier,
and $c$ is called the increment.

\item This is deterministic---given any seed $I_0$, one generates an
identical sequence $I_0$, $I_1$, $I_2 \ldots$

\item There are most $m$ possible random numbers, but there may be
less.  One could generate a cycle if
\[
I_{j_1} = I_{j_2} \hspace{2em} j_1 > j_2
\]

\item Then $I_{j_1+1} = I_{j_2+1}$, etc.  Also, $I_{j_1-1} = I_{j_2-1}$
and in particular $I_{j_1-j_2} = I_0$.
\end{itemize}
\end{slide}

