\begin{enumerate}
% \item 
% Section 3.1: 
% Learn how to construct Lagrange interpolating
% polynomials and practice on implementing it.
% Direct evaluation will do, but Neville's method is encouraged
% (extra credit).

\item 
% Section 3.1:
% Study the error formula (identity) for Lagrange interpolation
% and how to obtain an error bound (inequality).
% Take for granted that $|(x-x_0)\cdots (x-x_n)| \le n! h^{n+1}$ 
% for all $0 \le x \le 1$.
Let $P_n$ be the degree $n$ interpolating polynomial of $\cos(2x)$
on the uniformly spaced nodes $x_0, \cdots, x_n$ on $[0,1]$
with $x_j = j h$, $h = 1/n$. Is it true that
$$
\max_{0\le x \le 1} | \cos(2 x) - P_n(x) | \to 0 \quad \mbox{as } n \to \infty ?
$$
Explain.

\item 
% Section 3.2:
% Study how to obtain $P_{0,1,\cdots,k}$
% from
% $P_{0,1,\cdots,j-1,j+1,\cdots,k}$
% and
% $P_{0,1,\cdots,i-1,i+1,\cdots,k}$.
Denote by 
$P_{0,1,\cdots,k}(x)$ the Lagrange interpolating polynomial
on the data set \newline $(x_0,f(x_0)), (x_1,f(x_1), \cdots
(x_k,f(x_k))$.
Express $P_{0,1,\cdots,k}$
in terms of 
$P_{0,1,\cdots,j-1,j+1,\cdots,k}$
and
$P_{0,1,\cdots,i-1,i+1,\cdots,k}$.
Then verify your answer indeed is 
the Lagrange interpolating polynomial.

\item  
% Section 3.5:
% Study the meaning of cubic spline and how to match the
% coefficients at $x_j$ (such as problems 12, 13, 14).
% Memorize the meaning of natural and clamped boundary conditions.
A natural cubic spline $S$ on $[0,2]$ is defined by 
$$
S(x) = \left\{ \begin{array}{ll}
                S_0(x) = 1 + 2x - x^3, & 0\le x \le 1                 \\
                S_1(x) = 2 + b(x-1) + c(x-1)^2 + d(x-1)^3, & 1\le x \le 2 
               \end{array} \right .
$$
Find $b, c, d$.

%% \item  Section 3.5:
%% Study how to obtain the degree of the piecewise polynomial
%% and number of boundary condition needed for a $C^k$ spline. 
\item 
Suppose that we are to construct a piecewise polynomial interpolation $S(x)$
on the data
$(x_0, f(x_0)),
(x_1, f(x_1)),
\cdots,
(x_n, f(x_n))$,
with additional continuity conditions for $S'$, $S''$ and $S'''$
on the interior nodes $x_1, \cdots, x_{n-1}$.
If we use polynomials of the same degree on
each of the interval $[x_0,x_1], \cdots, [x_{n-1}, x_n]$,
what is the minimal degree needed in each interval?
How many additional end conditions are needed? 
Count carefully and explain.

\item 
% Section 3.2:
% Study how to solve nonlinear equations with Inverse Interpolation.
Given four data $(x_i,\exp(-2 x_i))$:
$(0.3, 0.5488)$,
$(0.4, 0.4493)$,
$(0.5, 0.3679)$ and
$(0.6, 0.3012)$
(you should generate the data yourself to avoid typo in inputting data).
Use Inverse Interpolation to find the root of $x=\exp(-2x)$.
You can use any algorithm for Lagrange interpolation.
Extra credits for Neville's method (or divided difference, if anyone knows it).
After finding $x$, check yourself that 
$x=\exp(-2x)$ in indeed satisfied in case of a bug in your code.
Need not show the last part.

Hand in code, put all data within the code so that it can be
executed immediately.

\end{enumerate}

{\noindent Name your codes in the same format as $s104000001.m$ or
$s103000002.c$ and make sure it is executable/compilable.}