\begin{enumerate}

\item 
Section 3.1: Problem 6(a). 
Write a program to
evaluate the degree three polynomial using for loops.

Hint: 
One can compute 
$$
 L_{n,k}(x) = \prod_{i=0, \atop i\neq k}^n { (x-x_i) \over (x_k - x_i)}
$$ 
with a for loop in $i$ for 
each of the $k=0,\cdots,n$,
then evaluate 
$$
P(x) = \sum_{k=0}^n f(x_k) L_{n,k}(x)
$$ 
with another for loop on $k$.
Note that, in C, the index for an array $a(i)$ starts with $i=0$ by default.
However, the index starts with $i=1$ in matlab. 
One should shift the index in the Lagrange interpolation formula accordingly.
 
{\noindent \bf Remark}: This is not the most efficient method to
evaluate the Lagrangian interpolation. The proper way
is to evaluate $P(x)$ using Neville's method in section 3.2.
An alternative way is to compute the coefficients
of $P(x)$ using Newton's divided-difference
formula in section 3.3, then evaluate $P(x)$.

% Both these methods are and will not
% appear in the exams in this class.

% \item
% Section 3.1: Problems 25.
% Use the sample file.
% What you will observe is known as Runge's phenomenum.
% This is the reason to use piecewise polymomial interpolaton,
% instead of a high order polynomial interpolation.

% Section 3.1: 6(a), 9, 10, 13(a), 17;  p114, 9th ed
% Section 3.1: 6(a), 9, 10, 13(a), 17;  p112, 10th ed
\item
Section 3.1: 
% Problem 6(a) with degree two and
% $x_i = 0.25, 0.5, 0.75$.
% Problem 8 ($n=2$), 
Problems 
9, 10, 13(a), 17 (the last sentence simply means 
$h= (10-1)/n$ for some positive integer $n$).

\item
Let $x_0, \cdots, x_n$ be uniformly spaced nodes on $[a,b]$
with $x_j = a + j h$, $h = (b-a)/n$.
\begin{enumerate}
 \item
  Show that $|(x-x_0)\cdots (x-x_n)| \le n! h^{n+1}$ on $a \le x \le b$.
 \item
  Let $P_n$ be the degree $n$ interpolating polynomial of $e^x$
  with uniformly spaced nodes on $[0,1]$. Show that
$$
\max_{0\le x \le 1} | e^x - P_n(x) |
\le C h^n  \to 0 \quad \mbox{as } n \to \infty
$$
\end{enumerate}
Note that uniform convergence of interpolating
polynomials as in (b) does not hold in general.
% The example on p158-p160 % 9th ed 
The Illustration on p154-p157 % 10th ed
is a good example.
In general, we only have
$$
\max_{a\le x \le b} | f(x) - P_n(x) |
\le C_n h^n 
$$
from Theorem 3.3.
A well known example with similar behavior is
$f(x) =  1/(1+x^2)$ on $[-5,5]$ with uniformly 
spaced nodes.


% \item Section 3.5: Problems 12, 13, 14, 20, 26, 27.
% 
% \item 
% The cubic spline with not-a-knot condition gives rise to
% to a linear system $A x = b$ where
% $A$ is an $(n+1)\times (n+1)$ matrix and
% $x = (c_0, c_1, \cdots, c_n)^T$. Write down $A$.