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\begin{document}

	\begin{center}
{\bf $\stackrel{\mbox{MA330 (Sathaye)}}{\mbox{\bf(161005)}}$
\hfill  Quiz 1\hfill    Name:\rule{2in}{0.01in}  }
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\large
\centerline {\bf The Brouncker-Wallis algorithm.}

The aim of this exercise is to completely determine the (regular) continued fraction expansion of $\sqrt{19}$ using the algorithm discussed in the class.

This should be worked at home and submitted in class on \mrd{Wednesday 10/12/16.}

\begin{enumerate}
\item Start with the augmented matrix 
$$M_0~=~\left(
\begin{array}{rr|rr}
1 & 0 & 1 & 0 \\
0 & 1 & 0 & -19
\end{array}
\right)
\mbox{ so that the associated form is $f(x,y)=x^2 - 19y^2$.  }$$



\item
Carry out the algorithm steps to produce new matrices $M_1, M_2, \cdots $ etc. until the second part of the matrices repeats. 
{\bf be sure to } explicitly record the value of $t$ in a prominent place.  

Be sure to give complete details of how you are transforming one matrix to the next {\bf in at least one step.} The remaining steps may be done without comment.

{ \bf Hint:}  There will be $7$ steps and the $8$th step will repeat the right hand side matrix of step $2$.


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\end{enumerate}

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\rightline{ Please turn over.}
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\begin{enumerate}


\item
At the end of the above process, write down the obtained continued fraction expansion in the usual notation $[a_0;a_1;\cdots a_r]$.
Further work is to be done on this continued fraction as asked below.

\vspace*{0.2in}


\item
Determine  {\bf all the convergents.} See illustration at the bottom of this page.

Using the calculator, get a decimal value for each convergent and compare these with the calculator value of the $\sqrt{19}$.

\vspace*{4.2in}

\item {\bf Illustration:} 

Supposed that  a continued fraction is $[2;3;2;1;3]$, then you first start with $\binom{0}{1},\binom{1}{0}$.
Then continue as usual to get:
$$ 
\begin{array}{c|cccccc}
\\
\binom{0}{1} & \binom{1}{0}&\binom{2}{1}&\binom{7}{3}&\binom{16}{7}&\binom{23}{10}&\binom{85}{37}\\\\
  & 2 & 3 & 2 & 1 & 3\\
\end{array}$$

{\bf Explanation: The pair $\binom{7}{3}$ is obtained by $3\binom{2}{1}+ \binom{1}{0}$.
The pair $\binom{16}{7}$ is $2\binom{7}{3}+\binom{2}{1}$. Continue. }
Make fractions from these to get the convergents (i.e. $2/1, 7/3, 16/7$ etc.)


\end{enumerate}

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