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

\item
Section 3.2: Problem 6, 8, 11, 12.

For problem 11, you will need to write a function (or subroutine)
based on the practice in the recitation this Tuesday.
The function takes the input vectors $x$, $y$, $x^*$,
where $x \in R^d$, $y=f(x) \in R^d$ are data points
and $x* \in R^m$ is the vector
of points to be interpolated (in problem 11, $x^*$
is a single point $x^*=1+\sqrt{10}$, that is $m=1$ ). 

The function then determines $d$
by checking the lengths of $x$ and/or $y$
and outputs the interpolated values $y^* \in R^m$ as an approximation
of $f(x^*)$. To start with,
you can use direct evaluation of Lagrangian interpolating 
polynomials here (and also in problem 12) as in homework 7. You are
encouraged to revise it to Neville's method later on.

Once the function is written, you can generate $x$, $y$ from
outside with any chosen $d$ and $m$ and call the function in the main program. 
At this point, you can plot the interpolating polynomial vs the
original function with roughly $n \approx 8-12$ and $m \approx 100$.
Bring your code next
Tuesday to recitation in case of any problem or doubts.

Repeat the procedure and change the original function from
$1\over 1+ x^2$ to $\sin x$, $\cos x$ and observe the difference
in behavior of the interpolating polynomials.

Do the same with cubic spline next week.

% \item Section 3.5: Problems 12, 13, 14, 20, 26, 27.  %9th ed page 162
\item Section 3.5: Problems 12, 13, 14, 30, 34, 35.  %10th ed page 162

Remark: In problem 12: `clamped' means $S'(1)=f'(1)$, $S'(3)=f'(3)$.
In problem 13: `natural' means $S''(1)=0$, $S''(3)= 0$.

\item The requirement $S(x)$, $S'(x)$ and $S''(x)$ be continuous
at interior nodes $x_1,  \cdots, x_{n-1}$ can be conveniently
referred to as $S \in C^2([a,b])$, or $S$ is a $C^2$ spline.
We showed in class how to count the number of these (continuity)
conditions and conclude that there are 4 unknowns in each
interval $[x_j, x_{j+1}]$, hence a cubic polynomial there
and that two additional boundary conditions are required.

Do the same for $C^1$ splines. Give the degree of the piecewise
polynomial and number of additional boundary conditions.