I. Course Syllabus
Math 1080 11-11:50 MWF, CRN 27589

Note Room Change:
Monday and Friday: Thackery Hall 525
Wednesday: Public Health A216



Office Hours during the 1/2004-4/2004 term: MWF 10-10:40,
and by appointment.
Also, if I'm in, you are welcome to come by!!!

This course in NLA is for students interested in solving scientific and engineering problems which involve lots of data and more than one dimension. Basically, any such problem reduces eventually to one in numerical linear algebra.
 
 

Text: Lecture Notes in Numerical Linear Algebra, by W Layton,
We will work out a way to let everyone get a copy efficiently. It is presently about 170 pages. About 40 typed and the rest carefully handwritten. I hope that more will be typed as the course goes on. If so, I�ll distribute these in class.

Instructor: W. Layton, Ty 603,412-624-8312,
(math office:412-624-8375) email: wjl@pitt.edu.
Grader:TBA

Programming: NLA is a �hands on�/�skills� topic. One important skill is the use of MATLAB to solve problems. Matlab is EASY! In the sophomore course , Math 0250, students are expected to learn it and use it without instruction (and they do-with success)!
Another skill is running and modifying another program in a lower level language.  I DON�T assume you know much about programming. I DO assume you can learn MATLAB on your own and that you are intelligent and can think algorithmically. I�ll teach one language in class; by the end you should feel pretty OK in modifying a working program in it.

Course Grade:= [Exam1+Exam2+Exam3+Final Exam + HW Grade]/4,
EXAM1 is: MONDAY 1/26/2004
EXAM2 is: WEDNESDAY 2/18/2004
EXAM3 is: WEDNESDAY 3/24/2004
NOTE new day for exam 3!!

letter grades will be converted to the standard scale  90=100=A, 80 89=B, 70-79=C, 60-69=D 0-59=F for averaging.

Make Up Exams- will NOT be given. Notify me BEFORE the exam if you cannot be there for a reason which is URGENT,NECESSARY and BEYOND YOUR CONTROL.
 
 

Other Books: Seeing something from different points of view can be useful. Here are some books can recommend (but only if you can find them CHEAP):
C. Cullen, An intro. to NLA, this used to be the textbook. It�s still an excellent book to read and have as a resource. (We are trying something different because it�s at a very high mathematical level nd because a more algorithmic treatment is useful.)
C. F van Loan, Intro. To Scientific Computing,excellent , about our level.
D Young and R Gregory,A survey of numerical math, vol I and II, good , higher level, older notation that some find difficult.
G Golub and C van Loan, Matrix Computation., A classic, higher level.
G Strang, Linear Algebra and its Applications, general linear algebra
G Strang, Intro to Applied Math., another interesting book of Strang that overlaps our course significantly.
D Luenberger, Optimization by vector space methods,
excellent treatment of CG methods and many topics we don�t cover
R Horn and R Johnson, Matrix Analysis, only matrix theory, excellent!
D Higham and N Higham, MATLAB guide  many like this book
 

2004 SPRING TERM (04-2)
January 5 Monday SPRING TERM CLASSES BEGIN
16 Friday SPRING TERM add/drop period ends
19 Monday Dr. Martin Luther King's birthday observance (University closed)
January 26, MONDAY, EXAM1 -no make ups so be there!
February 18, WEDNESDAY, EXAM2- be there!
March 26, FRIDAY, EXAM3, no make ups!
March 5 Friday SPRING TERM DEADLINE for students to submit Monitored Withdrawal forms to dean's office
March 7-14, incl. Sunday-Sunday Spring Recess for students (no classes)
April 16 Friday SPRING TERM:
 Last day for undergraduate day classes, Absolutely no late homework or projects accepted after this date!
April 19-24, incl. Monday-Saturday Final Exams



II. MATLAB RESOURCES

• A Matlab Primer of Professor Kendall Atkinson of the University of Iowa. (ps) or (pdf).
• A Matlab tutorial from the University of Maryland, Dept of Civil Engineering
• A Matlab tutorial from the University of New Hampshire
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III. Repositories of Scientific Software

Before developing a program from scratch, you should first do a short survey of available software for solving
the problem you're loking at.Two excellent places to search are:

http://math.nist.gov

http://www.netlib.org

Your project report should also have a short survey of other methods for solving your problem that you might find in one of these repositories!

IV. HOW TO RUN A FORTRAN PROGRAM

 Here are some basic , basic instructions on how to
(a) write a fortran program [ie, create a file].
(b) how to run it.

(a)  To create a file you need to use an editor. In unix, the pico editor
is probably easiest because it's menu driven. Dont forget though that unix
is case sensitive: large and small letters have different meanings , so
get the capitalization exactly right!

   At the prompt type:
      pico progname.for

   (the "progname " you may choose but be sure to make it .for at the
end.)

        Next paste in or type in your program. Editing it is easy-read the
commands at
the bottom of the screen. When you are done type
  'control'O

then type
   'control'X

Now you've created a file progname.for.

(b) Here's how to run progname.for

  At the prompt type
     setup sunpro

  At the next prompt type:
     f77 -C progname.for

 This compiles your program. Now run it by typing at the next prompt
    ./ a.out
 This tells the computer to dump your output to the screen.

  Try writing a simple program to add 2 numbers and output it. If it doesnt
work, check carefully the cases [small letters or caps] and the spaces in
your commands.

 Another comment: If you discover something interesting, email me and I'll
send it to the whole class.

    More Remarks:
These instructions are for our UNIXS system. There are a lot of free
FORTRAN compilers available [FORTRAN 77] as well as a lot of free C
compilers too. I'll email to you an addres on where to download one, if
you are interested in working on a PC. I suggest though that you start on
UNIXS if you've never programmed before.
 

Have fun!

V. FORTRAN Compilers-

Here are some FORTRAN resources including free and/or cheap  FORTRAN compilers for a PC.

"The Fortran Library, http://www.fortranlib.com, is a comprehensive guide
to online Fortran resources, including compiler vendors and
resellers, benchmarks, programming tools, books and articles on Fortran
and numerical methods, and commerical and public domain Fortran software.

The Fortran Market, http://www.fortran.com/fortran/ , also provides
many links to Fortran resources and is a reseller for several Fortran 90
and 95 compilers, benchmarking and test suites, and books on Fortran."

One former student installed one. Here below are his instruction on how to do this:

Subject: Fortran Compiler for PC's

First of all, the stuff has to be downloaded.

It can be gotten via anonymous ftp (using Netscape, for instance) from

ftp://ftp.bu.edu/pub/mirrors/simtelnet/gnu/djgpp/v2gnu/g770523b.zip
and you also need
ftp://ftp.bu.edu/pub/mirrors/simtelnet/gnu/djgpp/unzip386.exe

There are other things like additional documentation, but this provides the
bare essentials (I realized that it needed to be cut down for people to be
able to download it).

Then this file (g770523b.zip) needs to be unzipped.  I recommend making the
directory C:\DJGPP by typing

mkdir \DJGPP
 

at an MS-DOS prompt.  Then, go to the directory where you downloaded
g770523b.zip to, and type

move g770523b.zip \djgpp
move unzip386.exe \djgpp
cd \djgpp
unzip386 g770523.zip

You'll also need to make sure you can access the compiler, so add
C:\DJGPP\BIN to your PATH statement in your CONFIG.SYS.  If you don't know
how to do this, just type the following from an MS-DOS prompt.

echo PATH=%PATH%;C:\DJGPP\BIN >> \CONFIG.SYS

After that, you'll need to write the FORTRAN programs in something like
Notepad (do not use a Word Processor for this!).  To compile your program,
named (for instance) progname.for, type the following.
 

g77 -o progname progname.for

and to run the new program, type

progname
 

That should be it.

VI. Acessing Matlab remotely.
Dr. M. Sussman has graciously provided instructions for acessing Matlab remotely.
The first answer is: Yes, Matlab can be accessed remotely using your usual network access. I am not familiar with
"PittNet" -- is it a dialup connection (slow) or some
sort of fast connection? In either case, directions follow.

ASSUMING YOU ARE USING LINUX or another type of Unix:
Use ssh to get to unixs1 or unixs2, type "matlab": example
ssh youraccountname@unixs2.cis.pitt.edu
password
matlab
Wait a long while (1-2 minutes for a fast connection,
a looooong time for dialup) and Matlab will come up.

WARNING: if you have a (slow) dialup connection, you can
turn off the graphical stuff and run matlab in command mode:
(you get no graphics and no plotting)
ssh YoUrAcCoUnTnAmE@unixs2.cis.pitt.edu
password
DISPLAY=
matlab

If you have a dialup and you want plotting but no gui, do NOT
reset DISPLAY to nothing. Just do:
matlab -nosplash -nojvm

If you turn off the gui, you will need some sort of editor.
Either open another ssh session with your favorite editor
in it (pico, vi, emacs, etc.) or use the exclamation point
to start an editor from the Matlab command line (!pico ...)

ASSUMING YOU ARE USING WINDOWS
You either have an "X-server" program or you don't. I do
not know whether one is included in the programs they supply
to students.
ASSUMING YOU HAVE A DIALUP CONNECTION OR YOU DO NOT HAVE AN X-server
Use the Start->Run menu and type
telnet unixs2.cis.pitt.edu
(or unixs1 or unixs)
It will ask you to log in. Then just type
matlab
to get the LINE version of Matlab (no graphics, no plotting). There
is no way to do plotting without an X-server. You also will not have
the Matlab editor, but you can edit a file FROM THE MATLAB COMMAND
LINE with the command "!pico filename.m" where you don't type the
quotation marks and you put your own name where I have "filename.m".
When you exit pico, you will return to Matlab.

ASSUMING YOU HAVE BOTH AN X-server AND A FAST CONNECTION
Start the X-server first. You will need to know the "display" name
It will be something like 192.160.1.1:0 (your IP address first, then
colon zero).
Use the Start->Run menu and type
telnet unixs2.cis.pitt.edu
username, password, etc.
DISPLAY=192.160.1.1:0
EXPORT DISPLAY
matlab
There might be a telnet or rlogin or ssh command that comes with
the X-server. If so, the DISPLAY variable is automatically set.

REMARK:
There is a free Matlab clone called "octave" that runs on Windows
(If you run linux, install it from your CDs). See
http://octave.sourceforge.net/Octave_Windows.htm
It is not so pretty as Matlab, and it is mostly command-line
driven, but it will do plotting and most simple stuff.

VII. A Synopsis of FORTRAN.



A Synopsis of "5-Statement FORTRAN"
W. Layton
Mathematics Department
University of Pittsburgh




ABSTRACT
This is prepared solely for the benefit of my students. It�s intended to have 2 purposes.
1. To introduce math students who�ve never programmed before but are  used to the algorithmic
thought in rigorous proofs to a language quickly and get them working, and
2. To do the same for students who have experience programming in another language  like C, for example.

1. Introduction.

        FORTRAN is short for FORmula TRANslation language. It is a low
level language designed for doing scientific calculations and lots of
them. Although it is out of favor in CS departments, it is extremely handy
to know a bit of FORTRAN because:
        (estimated) 95% of scientific software is in FORTRAN.
Thus, if you must modify an existing program it will likely be a FORTRAN
one.
It is also useful for us because it is simple ( = easy to learn) and its
logic parallels the logic of mathematical       formulas. Generally, for
these and other reasons, if you program in another language, it is a good
idea to know how to call a FORTRAN subroutine from that language.

2. FORTRAN Statements.

        When you type a line of a FORTRAN program, different locations of
characters (i.e., which column it is types in) have different meanings.

Column 1              Column 2-5          Column 6                       Column 7-72
Comment indicator Line number  Blank or continuation Executable FORTRAN statement

Generally, try not to continue a statement from one line to another: Keep
formulas short enough that you can glance at it and see what it is doing.
Thus, try to leave column 6 blank. A character in column 1 tells the
compiler to ignore the line so you use these lines to write notes to
yourself. You can also use "C " in column 1 to de-activate a line. This is
called "commenting it out."

Types of Variables.
 
        Variable names can have no more than 7 characctters.
You should declare variables but in FORTRAN you don't need to
because there are defaults.
INTEGERS:
Integers are stored as such and added exactly. Any variable
beginning with
        I, J, K, L, M, N
is by default an integer. Thus, old-style FORTRAN would use the
(undeclared) variable ICOUNT.  Variables can also be declared as integers
initially, meaning before the first executable statement. For example,

_ _ _ _ _ _ _ _ _ INTEGER COUNT

will make COUNT an integer even though the undeclared default of a
variable COUNT is real. In integer arithmetic
        I  =1/4
Returns I = 0 while in real arithmetic
        X = 1.0/4.0
Returns X = 0.25. Don't use mixed mode arithmetic!
 

REAL NUMBERS:
These can be declared as real before the first executable
statement by, e.g.,

_ _ _ _ _ _ _ REAL X

By default any variable beginning with
        A-H or O-Z
is also real. Reals are numbers with decimal digits such as
        PI = 3.141592736
        XINDEX = 1.0
They are stored in scientific notation and carry a certain number of
significant digits. Carrying 7 digits base 10 is typical although not
universal.

DOUBLE PRECISION
variables must always be declared n before the first
executable statement, e.g.,
 

_ _ _ _ _ _ _ DOUBLE PRECISION X

They work like reals but carry more than 2x the number of significant
figures as reals (also called single precision). Double precision is the
normal mode for many scientific calculations. If you add a real to a
double precision variable, you immediately lose al the extra accuracy of
the double precision variable so:
        Avoid mixed mode arithmetic.

Arithmetic Operations.
        These are denoted by the usual symbols
                +, -, , / and *
Exponentiation is denoted by ** with differences between real and integer
exponents:
(Integer) X**2 means X * X
(Real) X**2.0 means EXP(2.0 LOG( X ) ).
The arithmetic equals (=) has a different meaning than in arithmetic. For
example, the statement
      X=X+1.0
makes perfect sense in FORTRAN. It means, look at the left hand side, find
the storage location set aside to store the variable called X and find the
value of X. Add one to that value then put the result back in the X
storage location. This requires the computer to do something; hence, it is
called an executable statement. (It's described informally by: "look in
the box marked X. Get the number there; add one to it and put it back in
the X box.")

CONTROL STATEMENTS
        Programs always end with two statements that terminate the
program:
                STOP
                END
        Another useful statement is the unformatted print statement which
just dumps the output to your computer screen. This is fine for moderate
amounts of output. (For lots of it, read about formatted print statements
in a FORTRAN book.) An unformatted print takes the form

PRINT *, variables to be printed separated by commas.

Anything put in quotation marks is printed directly as written. Examples
of this are given later. Generally, for moderate amounts of data, an
unformatted print is fine. On your terminal screen, highlight the output.
Next copy and paste it into your favorite text editor and format it and
then print it.

3. The Heart of the Program.

        A computed basically can only do 5 things:

� Input data,
� perform arithmetic calculations,
� decisions based on logical or arithmetic tests,
� output data and
� loop (meaning perform al the above iteratively).

We've seen how to perform basic arithmetic calculations. Let's now review
some of the remainder. The real power of the computer is unleashed with
loops (doing a lot of calculations) and matrices and vectors (handling a
lot of data).

INPUT, OUTPUT and CONTROL
        Unformatted input is (usually from your terminal screen)
                READ*, variables
While unformatted output is
                PRINT*, variables.
You can label the variables by putting notes to be printed in quotes as in:

                PRINT*, �Next we print the value of X�, X

This results in a printted line like:

Next we print the value of X      .34567E2

when X has the value 34.567.
 

DECISIONS.
        Decisions in FORTRAN are performed by "IF tests." There are many
kinds. One basic kind takes the form

        IF (condition is true) THEN executable statement
        ELSE executable statement
        ENDIF

For example, the following adds 1 to x if x = 0 and subtracts 1 otherwise:
       IF(X.GE.0) THEN X=X+1.0
          ELSE X=X-1.0
           ENDIF
 

The conditions are arithmetic inequalities using the following.

Math            FORTRAN                     Example
<                    .LT.                          X<Y   is written as: (X.LT.Y)
>                     .GT.                        X>Y   is written as: (X.GT.Y)
=                     .EQ.                         I=J    is written as:  (I.EQ.J)
> or =              .GE.                            is written as:  (X.GE.Y)
< or =               .LE.                           is written as:    (X.LE.Y)
not equal           .NE.                           is written as:  (I.NE.J)
 

LOOPS.
        These are "DO-Loops" in FORTRAN and "FOR-Loops" in C. If r is a
statement number, they take the form

                DO r I = beginning, end, stepsize
                executable statements
        r       CONTINUE

where "beginning", "end" and "stepsize" are all integers.For example, this
computes the sum
1+1/2+1/4+1/8++ 1/(2**1000).

         PROGRAM SUM.for
        SUM=0.0
        DO 10 I=0,1000,1
         ADDEND=1.0/(2.0**I)
          SUM=SUM+ADDEND
 10    CONTINUE
         STOP
          END

        Suppose we want to exit a loop early when a condition holds. In
this case, in FORTRAN the only way is to use a "GO TO" statement. (This is
the only valid use of GO TO's.) For example, to exit the above when SUM>
1.5 we add an if-test with a GO TO:

        PROGRAM SUM.for
        SUM=0.0
        DO 10 I=0,1000,1
         ADDEND=1.0/(2.0**I)
          SUM=SUM+ADDEND
           IF(SUM.GT.1.5) THEN GO TO 20
               ENDIF
 10    CONTINUE
  20    PRINT*, "I=",I, "and sum=", SUM
         STOP
          END

FORTRAN�s BUILT IN FUNCTIONS
        There are many built in functions in FORTRAN. If you put a "D" in
front the function returns a double precision answer. Some common ones
include:
ABS(X)=|x| ,
TAN(X) = tangent(x),
ATAN(X)= arctangent(x),
COS(X), SIN(X),TAN(X),... ,
EXP(X) = exponential function of x= e**x ,
LOG(X)= natural logarithm of x,
LOG10(X)= logarithm base 10 of x,
SQRT(X)=square root of x,
FLOAT(I) and REAL(I) convert integer I into a real to avoid mixed mode arithmetic,
DFLOAT(I) makes I a double precision real,
and so on!

FUNCTION STATEMENTS
        Function statements are very handy to code a simple formula.
IMPORTANT: MUST BE BEFORE THE FIRST EXECUTABLE STATEMENT!

Here is an example:

       PROGRAM AREA.for
       AREA( R)=4.0*ATAN(1.0)*R**2
        X=2.0
        A=AREA(X)
        PRINT*, "Radius=", X, "AREA=" , A
        STOP
        END

Note that 4.0*ATAN(1.0) is a standard way to code PI without typing in all
its digits!
        If a function uses more calculations or a longer formula then use
a function subroutine. Function subroutines are very useful!!! Here's an
example:

       PROGRAM AREA.for
        PI=4.0*ATAN(1.0)
         R=2.0
         A=AREA(R,PI)
         STOP
          END
          FUNCTION AREA(X,Y)
           AREA=Y*X**2
           RETURN
           END

Notes:
You must have "AREA = " somewhere in function subroutine AREA
And you must always have "RETURN, END" at the end.

4. Arrays = Vectors and Matrices.
        Vectors and matrices are the way to store and manipulate lots of
data. They must be set up with various dimensioning statements
BEFORE THE FIRST EXECUTABLE STATEMENT!!!
        Examples of these dimensioning statements are the following:

       REAL X(50), A(50, 50)
       INTEGER INDEX(50)

which means the same as:

      REAL X, A
      INTEGER INDEX
      DIMENSION X(50), A(50, 50), INDEX(50)

For example, this computes the first 20 Fibonacci number:

       PROGRAM Fibonacci.for
       REAL FIBO(20)
       FIBO(1)=1.0
       FIBO(2)=2.0
       DO 10 I=3,20,1
        FIBO(I)=FIBO(I-1)+FIBO(I-2)
 10   CONTINUE
         PRINT*,FIBO(20)
         STOP
          END

Remark: It's interesting to try running this changing "20" to, e.g., "50".
You will soon see why FIBO(.) is stored as a real number rather than (at
first and second consideration) the more natural integer.

5. Programming "Rules of Thumb".
        These days one is more often modifying an existing code or linking
pieces of codes than programming a new application from scratch.
Nevertheless, the following "rules of thumb" will, if followed, save you
hours of headaches in debugging:

1. Be careful and correct: every hour you spend in checking, planning and
double checking will save four hours of debugging.

2. Use pseudo code to organize and plan your program.

3. Check it twice before running it! Then run it on several test cases
with known true-solutions.

4. Use modules. Break the program into clear chunks (one way is to use
FUNCTION subroutines). Check and debug each module independently of the
whole program before testing the whole program.

5. Use the right amount of generalization!

6. Print intermediate results.

7. Include warning messages.

8. Use variable names that are easy to understand and declare all
variables.

9. Put lots of comments in!

10. Keep your loops simple!

11. Think a lot about your data structures.

12. Use arrays to store data.

13. Use built in functions and program libraries.

14. Put clarity (for people) above compactness or even speed of execution
 
 
 
 
 
  VIII. Some of Our Assigned Homeworks.



Homework 2 Due 1/26/2004
Math 1080 Numerical Linear Algebra

1. Consider the augmented matrix of a 3x3 linear system with entries:

0 2 1 4
2 1 0 0
1 1 1 1

a. Write out the system as equations for x_1, x_2 , x_3.
b.Solve using Gaussian elimination WITH PARTIAL PIVOTING
[NO CREDIT if algorithm is not followed EXACTLY!]

2.Consider the system in 3 significant digit arithmetic:
0.000100 X_1 + 1.00 x_2 = 1.00
1.00 x_1 + 1.00 x_2 = 2.00

a. Solve it using GE with 3 digit arithmetic WITHOUT PIVOTING.
b. Repeat with partial pivoting. c. Draw conclusions.

3. Consider the eqns:
u_1 - u_2 + 2 u_3 =4
u_2 -2 u_3 + u_5 =2
-u_1 + 2 u_2 - u_5 = -1
u_1 + 2 u_2 + 3 u_3 + 4 u_4 + 5 u_5 = 6

a. Write this in matrix vector form.  b. Write the system�s augmeted matrix.

4. Give examples of 2 2x2 matrices that: a. Don�t commute.  b. Commute.

5.Let A be the 3x3 matrix with rows:
1 2 3
4 5 6
6 9 12

a. Find det(A). b. Does A^(-1) exist? Explain! c. What does det(A) tell about solving Ax=b? Explain!

Math 1080 NLA
HW 3 Due Wednesday 2/4/2004

1. a. What are the floating point operation counts for NxN systems for the following: Gaussian elimination, Back substitution, Multiplying Ax, solving Ax=b for tridiagonal A.
(b) If a program takes 10 minutes to perform each of the above and you double N, about how long will it take?

2. a.Find the LU decomposition of A ; multiply them to verify =LU, where A=
4 3
2 1
b.Using LU, solve Ax = (0,1)^tr
3. For solving Ax_j = b_j, j=1,...,M, (M fairly large), What are the advantages of using the LU decomposition as a solver over just using Gaussian elimination M times?

**********

Homework 4 due 2/11/2004   Math 1080
Numerical Linear Algebra

Consider the 2 point boundary value
-u'' = f(x) , for 0<x<1, u(0)=0, u(1)=0.

Discretization of this leads to a tridiagonal system. Solution of the system lead to an approximation to u(x) whose error is expected to be O(h**2).
Your HW is to test this expectation experimentally.
1. Consider the program tridiag.for which computes the above approximation. First , run the program, exactly as it stands. Copy->paste >print the session so turn in as part of the HW.

2. Next you need to thoroughly understand the program. Add VERY EXTENSIVE comment statements to the program and turn in a copy. Explain everything! This includes:
What is the problem?What is the true solution,the mesh-width, what does each variable mean, what does each loop calculate, how is the error calculated, does the program correspond to the notes� tridiagonal GS algorithm, what is different, how does it affect the solution.
  Add all this in using comment statements.
YOU MAY DO THIS IN TRIDIAG.FOR OR IN YOUR OWN , SELF-WRITTEN PROGRAM FOR THE HW.

3. Finally test tridiag.for or your own program as follows. Begin with N=8 solve and compute error. Repeat with N=16,32,64,128,256,512...
This involves editing the file and changing : N=8   to:  N=16  , for example.
(a) Assuming:     error(h) = C h**a
you can solve for the exponent a as follows:
        e(h)/e(h/2) = C h**a / C(h/2)**a = 2**a
Take logs of both sides and solve for a.
(b)Plot e(h) vs h. Check graphically if a is roughly 2
( c ) Keep decreasing h until you see the rate of convergence degrade to some a<2. This is the point where roundoff error influences the solution accuracy. Do you run out of storage in your account first?

Homework 5 NLA 2004
Due MONDAY 3/1/2004


This HW is to study how large a discrete Laplacian matrix you can store on
whatever computer you plan to use in 1d, 2d and 3d.

I will email you a FORTRAN program. This program will load the linear
system arising from the discrete Laplacian on the unit square I.e. In 2
DIMENSIONS. It stores it as a full matrix.

1. READ the program (save it to a file, load2004.for, exit pine, then type

pico load2004.for

at the unixs prompt.)

Your task for part (a) is to understand the program by either: I.
Edit out all my comments and then add your own explaining it don�t just
retype mine) and turn in a hard-copy of the result, or ii. Write an
equivalent program yourself, with comments, in another language. Turn in a
hard-copy.

You do not have to use FORTRAN but even if you don't you can learn a
lot about FORTRAN by just reading it and adding remarks. You can get extra
credit by writing your own program in another language.If you want to just
run the program on UNIXS you need to save it to a file. Read the
instructions on my web page!

2. Your next job is to start small and keep increasing N and the
dimension statements until you get an error message telling you that you�re out of space!

REMARKS: USE UNIXS!! WHEN YOU GET THE ERROR MESSAGE remove the core file
by typing at the unixs prompt:

rm core


Interpret your results by answering the following:
Next, take that information and find:
The smallest �H� or �N� for which you can store it as a banded matrix in 1d.

Explain how you got that number!
The smallest �H� or �N� for which you can store it as a banded matrix in 2d.
Explain how you got that number!
The smallest �H� or �N� for which you can store it as a banded matrix in 3d.
Explain how you got that number!
The smallest �H� or �N� for which you can store it as a full matrix in 2d.
Explain how you got that number!

3. Make a chart of your findings and draw conclusions.

***********



Homework 6, Math 1080 NLA 2004, Due: March 19, 2004

1. To solve the model Poisson problem with h=1/N+1

a. Fill in the chart below for d=1,2,3



d= *** #Degrees of freedom *** Bandwidth *** Band Storage *** Band G.E. FLOPS

___________________________________________________

d=1

___________________________________________________

d=2

___________________________________________________

d=3

___________________________________________________

b. If the smallest mesh solvable as above in 2d is h=1/101, predict the smallest mesh attainable in 3d.

2. Consider the Model Poisson problem in 2d with h=1/2, f(x,y)=-(x+y), and

g(x,y)={0 if x=0 or if y=0, and 1 if x=1 or if y=1}

Compute by hand an approxmation to u(1/2, 1/2).

3. If A is symmetric, and lamda=lamda(A)



a. Show cond_2(A) = | lamda |_max / | lamda |_min

b. Show that cond_2(A^2) = [ cond_2(A) ]^2

4. Take the same f and g of problem 2. Take h=1/3.

a. Find, by hand, the linear system.

b. By hand, do two steps of the Jacobi method for this linear system.

5. a. State the spectral mapping theorem

b. Let f(z)=1/(3-2z). What does f(A) mean for a square matrix A?

c. Find lamda(f(A)) if A_2x2 =
1 30
0 -10