Performance Comparisons

In this lab we will compare the performance of two different implementations the same mathematical model. One implementation should be a conventional programming language (Fortran, C, C++, Basic, etc. This will be your choice. I provide some suggestions below). The other implementation should be either in Maple (a CAS) or MATLAB (again, your choice).

The model is a boundary value problem for the Laplace equation, using a simple relaxation method.

The model:

The Laplace equation is the partial differential equation

We are going to look for a solution  defined on the square  ,  satisfying a boundary condition of the form

where  and  are four more or less arbitrary given functions defined on  .

The physical interpretation is that  is a square plate (say made out of metal), and the functions  and  correspond to a constant temperatures applied to the boundary. The solution  models the temperature in the entire plate after it has come to thermal equilibrium.

This is perhaps the simplest of all partial differential equations, and its solutions are well understood (they are refereed to as harmonic functions). We will use the following boundary conditions in our model

Comment: This is such a simple example it is possible to solve it exactly. Our interest here is more in the method.

which when applied twice leads to

where  for some large  .

Thus the differential equation is approximated by a difference equation

By multiplying both sides by  (twice) and solving, we obtain.

Putting

the last version of the difference equation now becomes

In other words, the solution function satisfies the condition that it's value at every point is an average of its neighboring values. This is a well known property of harmonic functions called the mean value property.

We want to solve this difference equation subject to the boundary condition(s)

and

The implementation:

Think of of  as a  matrix.

The boundary conditions (above) describe the values of the entries  for i or j from  to  .

For convenience we will start by putting all the other entries  .

In one relaxation step we define

for  to  and  to  to define a new matrix  .

Once we have computed this, we then put  (do you see why this use of two variables is necessary?)

In practice you should apply relaxation steps until  and  differ by some predetermined error that you input (for example if you want 7 decimal places you should set the error at .00000005).

Theoretical estimates of how many steps this will take can be obtained (using the theory of Markov chains), but we won't worry about that!

Assignment:
 

• Write a Maple or MATLAB program to perform this relaxation. Run the program long enough to get an answer correct to 5 significant digits for n=50. Plot the resulting function (a 3 dimensional plot).

 

 
 
 
 
 

• Write some other kind of program to implement the same method. The only rule is that it not be high level math environment like MATLAB or Maple. Some possibilities are:
• C or C++ (Visual C++ is installed in the T306 lab).
• FORTRAN. Consider using g77 which you can easily download and install, either in the lab, or on your own computer (see below)
• Basic. Visual basic is installed in the T306 lab. U-Basic is easy to download and install (but suffers from being interpreted rather than compiled)
• Use Java. The "Java Development Kit" can be downloaded for free.
• Excel. Spreadsheets can be used in a powerful way for scientific computing.
• Anything else you know or are willing to learn (see the links page for many possibilities).

 

 
 
 

If possible, use the same computer as in Part 1. Be sure to use double precision arithmetic. If possible, have your program output its answer in a form that can be read by MATLAB (think about how you type numbers input into MATLAB) and plot your output using MATLAB.

• Run the two programs for different values of  . As you run both programs try timing them (with a watch). Also keep track of how many relaxation's are necessary.

 

1. An HTML document describing what you did and summarizing your results.
2. Graphics.
3. The source files for your programs (or worksheets, spreadsheets, etc.)

Appendix: How to use 77 FORTRAN
 

1. Download g77 fortran and install it. This means downloading 2 zip files: g77exe.zip (1.54Mb) and g77lib.zip (208Kb), and extracting them. It also means adding FORTRAN to the path by typing from a dos prompt something like (depending on where you installed FORTRAN)

 

 

> PATH=c:\g77\bin;%PATH%

On your own computer you might want to add this to your "autoexec.bat" file.
 
 

2. Here is a sample FORTRAN file diff.f  that implements the numerical calculation of the derivative of f(x)-sin(x)^2+1 using the "3-point" formula f'(x)=(1/2h)(f(x-h)-f(x+h)). To compile it you would type (from the dos window containing your FOTYRAN file)

 

 

> g77 -O2 -mpentium diff.f -o diff.exe

You can use this file as a template for your own FORTRAN program for the relaxation method from this lab. The compiled program diff.exe is also available for you to try out.

I used the WebEq 2.5  "wizard" program to make this web page. which converts the formulas into jpeg files. This was my first attempt at using this and it has a few rough edges. We will talk about TeX and its relatives next class in 2 weeks.