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\begin{document}

	\begin{center}
{\bf $\stackrel{\mbox{MA330 (Sathaye)}}{\mbox{\bf(170222)}}$
\hfill Quiz 1 \hfill  Name:\rule{2in}{0.01in} }
	\end{center}
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\large
\centerline {\bf The Brouncker-Wallis algorithm. (due) 2/27/17}

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



\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 $t$ at each step in a prominent place.  

Be sure to record the appropriate row and column operations at each step.

{ \bf Hint Corrected:}  There are only $7$ steps needed before the right hand matrix repeats.


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\item
At the end of the above process, write down the obtained continued fraction expansion in the usual notation $[a_0;a_1;a_2;\cdots ]$. Put a bar over the repeated part.

Further work is to be done on this continued fraction as asked.

\end{enumerate}

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

\item Copy the continued fraction expansion from the previous page.


\vspace*{0.2in}


\item
Determine  {\bf all the convergents for the $t$-list until repetition.} 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 $\sqrt{19}$.
You should find further convergents at home to appreciate the approximation.

\vspace*{4in}

\item {\bf Illustration of work:} 

Supposed that  a continued fraction is $[2;\overline{3;2;1}]$.
Start with $\binom{0}{1},\binom{1}{0},\binom{2}{1}$ as the second row and write the given $t$-values as the first row (starting in column 2).

%Then continue as usual to get:
$$ 
\left(\begin{array}{cccccc}
\\
  & 2 & 3 & 2 & 1 &\\\hline
0 & 1 & 2 & 7 & 16 & 23 \\
1 & 0 & 1 & 3 & 7 & 10 \\  \\
\end{array}
\right)$$

{\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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\end{document}



