\item % 4.9: Improper Integrals (7)
Section 4.9: For the integral in Problem 1(a),
verify numerically the order of convergence for 
standard 2nd order (midpoint, trapezoidal) 
and 4th order (Simpson's rule, Gaussian quadrature
with 2 quadrature points) composite quadratures.

Take composite midpoint rule for example, 
find and $p_{\rm M}$ in $I-I_{h, \rm M} = O(h^{p_{\rm M}})$ numerically.
Do this for at least one 2nd order method and one 4th order method.

\item % 4.9: Improper Integrals (7)
Repeat previous problem with any
desingularizing method of your choice and
verify numerically that the result 
indeed has 2nd/4th order convergence.

Remark: simple change of variable $x=t^{-p}$ does not perform well 
(not enough to restore theoretical order of convergence)
for Problem 3 (c,d) due to the
oscillatory nature of the integrands.
A more subtle subtraction of singular part is required
to desingularize the integrand and
recover theoretical order of convergence. 
 
\item % 4.9: Improper Integrals (7)
Section 4.9:   Problems 4(a).

Hint: 
Without splitting the domain into 
$\int_0^1 + \int_1^\infty $,
the change of variable $t = {1 \over x+1}$ will
do the trick and is recommended whenever
the integrand has no singularity at $x=0^+$.

\item
% Section 6.1: Linear System of Equations (14)
% Section 6.1: Problems 10, 15(a), 16(a). % 9th ed, p368
% for reference Section 6.1: Problems 20. % 9th ed, p368
                Section 6.1: Problems 12 (a,b). %10th ed, p371

For part (b), change it to $K(x,t)=-{1\over 2} e^{-|x-t|}$
and implement Algorithm 6.1 to solve for $u$.
For this problem with the new $K$, row interchanging is not needed.
% next week
% \item
% Derive a linear system of equation corresponding to the following
% boundary value problem
% $$
% \begin{array}{l}
% u''(x) + u'(x) = f(x), \quad x \in (0,1) \\
% u(0) = \alpha, \; u(1) = \beta 
% \end{array}
% $$
% with uniformly spaced grids $0=x_0 < x_1 < \cdots <x_N=1$,
% $x_i - x_{i-1}= h = 1/N$, using second order finite
% difference method.
% That is, given $\alpha$, $\beta$,
% and $f_i$, $i=1,2,\cdots, N-1$,
% try to derive a linear system of equations to solve for $u_i$,
% $i=1,2,\cdots, N-1$.
% You don't have to solve the linear system this week (but soon will).

% Section 6.1: Linear System of Equations (14)
\item
Consider an $N^2 \times N^2$ matrix $A$ with $a_{ij}=0$ except for
$i-j=0, \pm 1,  \pm 2$ (only five diagonals have nonzero entries).
Estimate total number of multiplication needed for Gaussian elimination
without pivoting on $A$ and backward substitution, respectively. 
Give the leading order of the number of multiplication
as $K N^p$. Find $K$ and $p$.

\item
Do the same for an $N^2 \times N^2$ matrix
$B$ with $b_{ij} = 0 $ except for $|i-j| \le N$
(only $2N+1$ diagonals have nonzero entries).

\item
Do the same for an $N^2 \times N^2$ matrix
$C$ with $c_{ij} = 0 $ except for $i-j = 0, \pm 1$ and $\pm N$
(only $5$ separated diagonals have nonzero entries).

% \item
% Do the same for an $N^2 \times N^2$ matrix
% $C$ with $c_{ij} = 0 $ except for $i-j = 0, \pm N$ and 
% $i=N^2/2$, $j= N^2/4$.
% (only $3$ separated diagonals and another entry have nonzero entries).