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\textsc{Lund University} \\ [25pt] % Your university, school and/or department name(s)
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\huge Exercise 2: NUMA12 \\ % The assignment title
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\author{Magnus Hansson (911130-3856), Christoffer Olsson (931108-0171)
\\and Andreas Goumas (881130-3976)} % Your name

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\begin{document}

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%	PROBLEM 1
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\section*{Task 1}
\subsection*{Question}


\subsection*{Answer}




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%	PROBLEM 2
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\section*{Task 2}
\subsection*{Question}


\subsection*{Answer}

\subsection*{Chebyshev Discussion}
We have noticed that there seems to be two  similar, but different, ways of calculating the Chebyshev points. Powell (1981), equation (4.27), on page 39 states:

\begin{equation*}
x_i = \cos \lbrace \frac{[2(n-i)+1] \pi}{2(n + 1)} \rbrace , \ \ \ \ \ i = 0, 1, ..., n
\end{equation*}

Looking at this equation the only way, e.g., to get $\cos (\pi) = -1$, is to let $n$ approach $\infty$. Otherwise, the denominator will always be larger than the numerator, hence you will get values close to $1$ and $-1$, but never those exact values. That is, you will never include the end points, if you do not have an infinite number of points.
\\

If we instead turn to Trefethen (2013) and chapter 2, we have another way to attack the problem. Here we notice, that according to Euler's identity we have:

\begin{equation*}
z = e^{i x} = \cos (x) + i sin(x)
\end{equation*}

Now we let $x$ be a vector in the interval $(0, \pi)$ and we extract the real part of $z$ to get our Chebyshev points. Using this method we will always include the end points, $-1$ and $1$, in contrast to the other method (Powell 1981).
\\

What is confusing is that in table (4.5), "The norms of some interpolation operators", where Powell compares equally spaced points with Chebyshev points, the result for $n=2$ is $1.25$ for both equally spaced points and Chebyshev points. The equally spaced points are $(-1, 1)$ (as we have confirmed), but equation (4.27) does not yield those points! Equation (4.27) yields approximately the points $(-0.75, 0.75)$ and hence the norm $1.41$. This is different from table (4.5) and it seems Powell is doing something he is not telling us about. We are currently trying to compare the two different methods and discussing advantages and disadvantages, and will rise the question during our next lecture.

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%	PROBLEM 3
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\section*{Task 3}
\subsection*{Question}


\subsection*{Answer}




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%	PROBLEM 4
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\section*{Task 4}
\subsection*{Question}


\subsection*{Answer}




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\section*{Task 5}
\subsection*{Question}


\subsection*{Answer}



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%	PROBLEM 6
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\section*{Task 6}
\subsection*{Question}


\subsection*{Answer}

\section*{Feedback}

Please send the feedback to hansson.carl.magnus@gmail.com, olschr2@gmail.com\\
and goumas.andreas@hotmail.com.

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