Using numerical and symbolic computing software

The aim of this assignment is to get some exposure to Maple and Mathematica, two symbolic computing packages, and to Matlab, a numerical computing package. These are commercial packages available for both UNIX and Windows environments. You are tempted to ask which one is better...to this the answer is the usual one: "They are all equally terrible, but they are terrible in different ways, therefore, learn to use them all".

Symbolic Packages

Maple and mathematica can do a variety of tasks, such as algebra, calculus, graphics. They are programmable.

Matlab

A numerical package which is suitable for numerical computation, graphics. A thorough implementation allows you to run C and Fortran codes from matlab and viceversa, and also has maple embedded. It is programmable.

The platform

the interface for these programs is quite different in the UNIX and the Windows environment and this is leading to slightly different capabilities. The instructions that appear below are as platform-independent as possible. Part of the assignment is to take the instructions that appear and find their effective implementation on whatever platform you are using.

Instructions

You will need to download the data file xy.dat via anonymous ftp from the site
```
 math.arizona.edu 
```
which is in the directory
```
 /incoming/restrepo 
```
.
Run mathematica and maple, separately, (you must find how it is invoked) -- and compute:
- Define the function f(x)=x^2. Then find f(1+a).
- Integrate the function 1/x^3, from 1 to infinity
- Differentiate the function x^2 y log(x+a) with respect to x.
- Simplify the expression -1 + x - 2 xy + 2 x xy - xy^2 + x xy^2 - 2 y + 2 x y - 2 xy y + 2 x xy y - y^2 + x y^2
- 1D Graphics. Plot sin(1/x) for x between -1 and 1. Include axis labels, graph title.
- 1D Graphics. Load the data file xy.dat. It is a column-oriented list of x y pairs, plot.
- 2D Graphics. Plot sin(x)sin(y) for x and y between -pi and pi. Make a contour plot of the function as well. Include title, axis labels.
We encourage you to explore both packages and learn some of their quirks, their strengths and weaknesses.
Hand in hard copies of the 1D and 2D plots from maple and mathematica.
Invoke matlab
- At the Matlab prompt >>, type
```
 diary hw5.txt 
```
  (Every command should be followed by hitting ``return'' key). The command diary will record your MATLAB session in the file hw5.txt. Now enter
```
 help diary 
```
  to receive on-line information about diary. You can get help on any MATLAB command that you are not sure how to use. you can also use the more useful lookfor command.
- Enter
```
 a=[1,2,3] 
```
  You get a row vector or 1X 3 matrix a . Enter
```
b=[1;2;3]
```
  You get a column vector or 3X1 matrix b. Notice that rows of a matrix are separated by ``;'' but entries on the same row are separated by ``,'' or blank space. Enter
```
A = b*a 
```
  You get a 3X3 matrix A which is the product of b with a. Enter
```
S = a*b 
```
  You get a single value or a 1X1 matrix S. Note that there is only one data struture in MATLAB, that is, matrix. To calculate the determinant of A, enter
```
det(A)
```
  Now enter
```
I = [1,0,0; 0,1,0; 0,0,1]
```
  to get a 3X3 identity matrix (or you can get it by simply type
```
I = eye(3)
```
  , use ``help eye'' to find why). Now enter
```
B = A + I
```
  and find the determinant of B with
```
det(B)
```
  Is B invertible? Enter
```
x = inv(B) * b
```
  to solve the linear system of equations
```
B x = b
```
  for vector x Try also
```
 x = B\ b 
```
  What is the solution of
```
Bx = b
```
  ? Does the notation
```
B*a
```
  make sense? Why? Enter
```
 B*a
```
  and see what happens.
- You can put all the commands you entered in the previous subproblem in a file then ask MATLAB to read the file to execute all the commands. This file is called an m-file, which typically stores a sequence of commands and looks like a piece of code. Open another window and in the same directory where you invoked MATLAB, use your favorite editor to write the following in a file called ``test.m'':\
```
          a = [1,2,3];
          b = [1;2;3];
          A = b*a; 
          I = eye(3);
          B = I + A;
          x = B\b
         
```
  Save the file. In the MATLAB window, type
```
 test 
```
  without the .m. It returns x. Note the inclusion of semicolons at the end of statement suppresses the printing of the results. One can also write several statements separated by semicolons on one line.
- Enter
```
 help format 
```
  Enter
```
format short 
```
  Enter
```
97.6
```
  Enter
```
format long
```
  Enter
```
97.6
```
  Did you see any difference between short and long format? Can you explain? You can add your explanation in the file ``hw1.txt'' after this MATLAB session is over. Find one real number which has the similar phenomena and another one which does not.
- MATLAB computes everything using double precision. Here's a script to find the smallest floating point number eps such that 1+eps*1 not equal to 1, you can run the following sequence of commands (you can put them in a m-file ``epsilon.m'' or input them one by one):
```
           for i = 1 : 100        % start a "for" loop.
             x = 2^(-i);
             if (1+x) == 1        % logical "if" statement
               y = 2^(-i+1);
               disp( sprintf('EPSILON = %22.18f', y) );
                                  % display "EPSILON with 18 digits after
                                  % the decimal point.
               break;             % exit the loop once "EPSILON" is found.
             end                  % end the "if" statement
           end                    % end the "for" loop.
          
```
  The words after ``%'' are comments and ignored by MATLAB. Use MATLAB on-line help to find how to use the commands
```
 disp, sprintf, break, for, if
```
  . In fact, this eps is a MATLAB constant. Enter
```
 eps 
```
  What do you get?
- In this subproblem, we show you how to plot figures. Several MATLAB graphical commands are very useful. These are plot, subplot, loglog . Use help to find out how they can be used. Let f(x) = sqrt{1+x^2}-1 and g(x) = x^2/(sqrt{1+x^2}+1). It is easy to see these two functions are analytically the same. Use the following m-file to compute f(x), g(x) for x_k=10^{-k}, with k=1,2,...,10.
```
           for k = 1 : 10
             x(k) = 10^(-k);
             f(k) = sqrt( 1 + x(k)^2 ) - 1;
             g(k) = ( x(k)^2 ) / ( sqrt( 1 + x(k)^2 ) + 1 );
           end                    
           loglog( x, f, 'o', x, g, '+' )
           print figure.ps          % produce a PostScript file of the figure
          
```
  Here the sqrt is the square root function. Why f(x_k) not equal to g(x_k) epecially for larger k? Why do we choose to use loglog instead of plot to draw the figure? Try to use plot to draw f(x_k), g(x_k) against x_k and see why.
- Use Matlab to create a plot of the function cos(1.7x) for x between [0,2 pi]. Enter
```
x=0:0.1:2*pi;
y=cos(1.7*x)};
plot(x,y)
```
  The first command creates a vector, x , which includes the values 0, 0.1, 0.2, ... up to 2 pi. The second command creates a vector y of the same length, for which y(i)=cos(1.7x(i)). The last command creates a smooth graph which passes through all the points with coordinates (x(i),y(i)). Hence, this graph only approximates the real graph. The finer spacing we use (say, 0.01 instead of 0.1), the better approximate graph we get. Create a hard copy of your output. To find out how, enter {\bf help printing} at the Matlab prompt.
- This exercise will introduce you to one of Matlab's loop structures. Enter the following commands to sum the integers from 1 to 100:
```
 Sum=0;
 for i=1:100,
    Sum=Sum+i;
 end
 Sum
```
  Create an m-file to do the job (why did we use a capitalized Sum? try help sum ). Then generate a vector
```
 j=0:1:100; 
```
  then enter
```
 sum(j) 
```
- Finally, we learn how to import files. To enter the file xy.dat (which should be in the matlab working directory after you ftp it -see above-), you can load it by typing
```
 load xy.dat; 
```
  Type xy. What do you get? Type
```
 size(xy) 
```
  to find out how big the array is. Now, you can assign the first column of the array xy to xp by typing
```
 xp=xy(:,1); 
```
  Similarly, the second column can be assigned to yp by typing
```
 yp=xy(:,2);
```
  Type yp. What is the result? Plot the array xy and put a grid on the figure. Save the plot as a postscript file.
Print the PostScript figures. Insert your answers to questions in the previous MATLAB session into ``hw5.txt'' and print the file. Hand in the figures and hw5.txt.
Enter
```
diary off 
```
to end saving the session to file ``hw5.txt''. Then enter quit to quit MATLAB.