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\noindent\textbf{First name:} Christoph\\
\textbf{Last Name:} Oberauer\\
\textbf{Email:} coberau@cosy.sbg.ac.at\\
\textbf{Date:} 18.01.2003\\
\textbf{Homework number:} 5\\
\textbf{Homework Title:} Exercise 5.3


\paragraph{Problem description:\\}
Newton's method is sometimes used to implement the built-in square root function on a computer, with the initial guess supplied by a lookup table.
\begin{enumerate}
\item
What is the Newton iteration for computing the square root of a positive number $y$ (i.e., for solving the equation $f(x)=x^2-y=0$, given $y$)?
\item
If we assume the starting guess has an accuracy of 4 bits, how many iterations would be necessary to attain 24-bit accuracy? 53-bit accuracy?
\end{enumerate}


\paragraph{Problem solution:}
\begin{enumerate}
\item
Use $x_{k+1}=x_k-f(x_k)/f'(x_k)$ to calculate given case.
\item
Newton's method has a convergence rate of 2 (quadratic), which doubles the number of accurate digits every iteration.
\end{enumerate}


\paragraph{Results:}
\begin{enumerate}
\item
Iteration for this case:
$$x_{k+1}=x_k-\frac{x^2_k-y}{2x_k}$$
\item
To get 24-bit accuracy three iterations are needed (first iteration yields 8 accurate bits, second 16 and third 32).
To get 53-bit accuracy four iterations are needed (fourth iteration yields 64 accurate bits).
\end{enumerate}


\paragraph{Discussion and Comments:}
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