\def\list#1{{\par\smallskip\leftskip=3pc\parindent=0pt\parskip=0pt
\let\\=\par #1\par\smallskip}}
\centerline{\bf CS/MATH 3414 Homework \# 13}\bigskip

\item{(10) 1.}A second order accurate finite difference approximation to
the solution $u(x)$ of the linear two-point boundary value problem
\def\7{7\pi}$$\displaylines{u''-\7\cos(\7x)u'+(\7)^2\sin(\7x)u=
-(\7)^2\sin(\7x),\qquad 0\le x\le1,\cr  u(0)=u(1)=0,\cr}$$ is given by
the solution $U$ to the linear system $AU=D$:$$\eqalign{&\bigl[
-\7\cos(\7x_i)h/2-1\bigr]U_{i-1} + \bigl[2-(\7h)^2\sin(\7x_i)\bigr]U_i
\cr &\qquad + \bigl[\7\cos(\7x_i)h/2-1\bigr]U_{i+1}=(\7h)^2\sin(\7x_i)
,\qquad i=1,\ldots,n,\cr}$$ where $$h={1\over n+1},\quad x_i=ih,\quad
U_0=U_{n+1}=0,\quad u_i=u(x_i),\quad U_i=u_i+{\cal O}(h^2).$$ 
Solve $AU=D$ for $n+1=25$, 50, 100, 200, 400, 800, and plot the solution
vectors $U$. The exact solution satisfies $u(.5)=1/e-1$. Comment (one
well written paragraph) on these plots and what you believe is the
accuracy of your computed solutions.

\item{(10) 2.}{\bf(Extrapolated finite differences.)} Consider again the
two-point boundary value problem above. An analysis of the
finite difference approximation $U$ shows that $$u_i=U_i+a_2h^2+a_4h^4
+a_6h^6+\cdots,$$ where the coefficients $a_i$ are independent of $h$.
With $n+1=50$, 100, 200, 400, solve $AU=D$ and estimate $u(.5)$ using
extrapolation to the limit. Compare the four finite difference
approximations, the extrapolated value $\hat u(.5)$, and the exact value
$u(.5)$. As before, use DGTSV (LAPACK) to factor and solve $AU=D$.

\item{(10) 3.}{\bf(Shooting.)}Consider the initial value problem (IVP)
\def\7{7\pi}$$\displaylines{u''-\7\cos(\7x)u'+(\7)^2\sin(\7x)u=
-(\7)^2\sin(\7x),\qquad 0\le x\le1,\cr  u(0)=0,\qquad u'(0)=
\alpha.\cr}$$ Denote the solution by $u(x;\alpha)$. Observe that the
two-point boundary value problem is equivalent to finding a value of
$\alpha$ such that $$f(\alpha)=u(1;\alpha)=0.$$ $f(\alpha)$ is a highly
nonlinear function defined only implicitly by solving an IVP. Using root
finding and IVP software, find $\alpha$ such that $|f(\alpha)|\le10
^{-10}$. Now compute $u(.5)$ with this $\alpha$ and compare it with the
values obtained in problem 1. Compare the time (human and computer)
required by extrapolated finite differences and shooting.

%The subroutines ROOT (HOMPACK), ZEROIN (CS3410), RKF (CS3410), and
%DDRIV2 (Kahaner) should be useful to you.
\medskip\hrule\medskip

Use GAMS to retrieve one of the ODE IVP software modules in 
search class I1a1a or I1a1b.  You will also need to use a root
solving software package such as ROOT (HOMPACK) or ZEROIN (CS3410). 
A brief description of GAMS is included with this homework assignment.

GAMS existed before the World Wide Web, and was
available to run in both X-windows and non-X-windows environments.
For historical interest, here is a description of how the preWWW
X-windows version of GAMS operated.

{\narrower\parindent=0pt
To retrieve software modules, use search to select the desired problem 
class.  Before browsing through the modules, set the search filters so
that access is free and precision is double.  This is accomplished by 
issuing the following gams commands:

\list{filter access free\\filter precision double}
\noindent
The filters need only be set once while gams is active.  Browse through
the software modules until you reach a module you wish to retrieve.
The commands 

\list{get documentation\\get fullsource}
\noindent
retrieve the software module's documentation and software files to your
home directory.  If you are using xgams, substitue the xgams versions of
these commands.

}\bigskip
GAMS is reached through the World Wide Web at $$\tt
http://gams.nist.gov\;.$$
\vfil\eject

$$\vbox{\hbox{\vrule
\vbox{\hrule\smallskip\hbox{ The GAMS Virtual Software Repository }
\smallskip\hrule}\vrule}}$$

The Guide to Available Mathematical Software (GAMS) project of the
National Institute of Standards and Technology (NIST) studies
techniques to provide scientists and engineers with improved access to
reusable computer software which is available to them for use in
mathematical modeling and statistical analysis. One of the products of
this work is an on-line cross-index of available mathematical
software. This system also operates as a virtual software repository.
That is, it provides centralized access to such items as abstracts,
documentation, and source of software modules that it catalogs;
however, rather than operate a physical repository of its own, GAMS
provides transparent access to multiple repositories operated by
others.

Currently GAMS indexes four software repositories: NIST Cray (GRANTA),
NIST Convex (TIBER), NIST Suns (CAMSUN), and NETLIB, a public-domain
software collection of Oak Ridge National Laboratory and AT\&T Bell
Labs. Among the packages cataloged in GAMS are : the IMSL, NAG, PORT,
and SLATEC libraries; the BLAS, EISPACK, FISHPAK, FNLIB, FFTPACK,
LAPACK, LINPACK, and STARPAC packages; the DATAPLOT and SAS statistical
analysis systems; as well as other collections such as the Collected
Algorithms of the ACM. Note that although GAMS catalogs both
public-domain and proprietary software, source code of proprietary
software products are not available through GAMS, although related
items such as documentation and example programs often are.

All problem-solving software modules in GAMS are assigned one or more
problem classifications from a tree-structured taxonomy of mathematical
and statistical problems. Users can browse through modules in any
given problem class. To find an appropriate class, one can utilize the
taxonomy as a decision tree, or enter keywords which are then mapped to
problem classes.  Search filters can be declared which allow users to
specify preferences such computing precision, programming language and
access type (i.e., free or proprietary).  In addition, users can browse
through all modules in a given package, or all modules with a given
name.

Complete documentation on using the GAMS system is available within the
system itself.  Please report any problems to gams@cam.nist.gov.  We
are always interested in hearing suggestions for improving the system.

\list{Computing and Applied Mathematics Laboratory
\\National Institute of Standards and Technology
\\Gaithersburg, MD 20899 USA}

Disclaimers :  (1) GAMS catalogs mathematical and statistical software
that has been made available for use by NIST staff.  Identification of
commercial products in GAMS does not imply recommendation or
endorsement by NIST.  (2) Not all software available at the
repositories indexed by GAMS is retrievable using GAMS.

\vfil\eject