MCS 471 --- Computer Problem 1 --- Fall 2004

Floating Point Arithmetic Machine Epsilon

Consider a MATLAB program that investigates the double precision epsilon. You may also use C (C, GNU C, C++ or other C) or Fortran (F77, F90, or other F) or Maple provided digits rounding precision are specified as 16 or more and reasonable effort made to follow the spirit of the problem. Use the computer of your choice, within reason for this problem. MATLAB can be found in the UIC PCLabs and some department computer labs. However, you must hand in copies of both your

• program source code,

• your computer output.

where (1+eps) denotes the appropriate float[1+eps]>1) generated by the computer arithmetic in the given precision.

Use the following modified "bisectioning approximation" pseudo code fragment:

      kmax = 100;
      eps0 = 1;
      for k = 1 to kmax
         epssave = eps0;  
         ks = k - 1;
         eps0 = eps0/2;                 # Bisection Step.
         epsplus1 = eps0+1;          
         print, eps0, epsplus1;
         if(epsplus1 .le. 1.e0)  break;   # ".le." means "less than or equal"
      end
      print, epssave, epssave+1;      # Actual Answer is One Prior to Break.
      pchop = 1-log(epssave)/log(2.e0);  # Base 2 Precision?
      print, pchop;
      eps0 = 0;
      deleps = 2^(-1-pchop);
      print, deleps;
      for k = 1 to 7                  # Refinement Check.
         eps0 = eps0 + deleps;
         epsplus1 = eps0+1;                
         print, eps0, epsplus1;
      end

Use the sample MATLAB code to start: m-file eps64.m However, you may simply convert it to C, F or Maple if you wish.

Please answer the 3 specific questions in the code ( eps64.m) and also answer the additional 4 questions:

1. Which is Correct, the Official Chopped (2^(1-p)) or Rounded Value (2^(1-p)/2) for Precision p? Give a Written Answer with Your Justification Here:

2. Is Rounding or Chopping Demonstrated? Give a Written Answer to Question with Your Justification Here (see kd=3 in code):

3. Can You Find A Smaller eps0 = epsilon? Write and Run Your new Code, supplying your code and output. Give your Justification Here If You Can:

4. Name of Your Computer System with Processor model (eg., Pentium IV),
5. Name of Your Programming Language Compiler version,
6. List Compiler Command Used and Options Used.
7. Briefly Summarize What You Have Learned: