MCS 471 --- Computer Problem 2 --- Spring 1996

• GENERAL INSTRUCTIONS: Write a single, well documented, program that solves the following finite precision (double precision or 64 bits = 8 bytes is required) problem. You may use WATFOR77, regular Fortran, Pascal or C for this problem. However, you must hand in both copies of your program source code and your computer output. Document your own code with a brief description of the problem, an important variables dictionary and headings for important programming steps (use programming language comment lines).
  
  Write a single program that compares the full pivoting algorithm to the partial pivoting with scaling algorithm. For the partial pivoting algorithm write a subroutine that does "virtual" scaling and "virtual" partial pivoting (i.e. uses row scaling only in the selection of each Kth-pivot, but does not actually scale rows to save on truncation errors; use a pivot vector only for referencing row position without actually doing pivot interchanges, as in A(P(I),J)). For the full pivoting algorithm write a subroutine that uses a row pivot vector and a column pivot vector without scaling, such that the largest element in absolute value remaining in the unfinished part of the coefficient matrix is sought for the current pivot. Be sure to replace all references to A(I,J) by A(PR(I),PC(J)), say. Both subroutines should save the interchange of sign for the determinant in some variable like KCS[mm], with [mm] = PP or FP denoting partial pivoting and full pivoting, respectively.
  
  
• OUTPUT: Tables, in 7 significant digits, of
  1. A, B;
     
     
  2. XPP, XFP, RELRESPP, RELRESFP, DETPP, DETFP;
     
     
  3. ||XFP - XPP||.
     
     
  Here the postfix PP denotes partial pivoting and FP denotes full pivoting, as above. Clearly label your answers. A & B are the input data, the X[mm] are the FGE/BS approximate solutions, RELRES[mm] are the relative residuals, and DET[mm] are determinants. The relative residuals and the determinants are error and conditions checks (e-notation). The relative residuals are
  
  
  RELRES[mm] = ||A*X[mm] - B||/||B||,
  
  
  where "||<vector>||" is the 1-NORM or absolute value norm or vector "<vector>" (see your notes). For each example of input below, print out a message saying which answer, PP or FP, is best based on the smaller relative residual. Note that the relative residuals may not be a reliable indicator of the best algorithms. Let TOL = 16*maceps. Be sure to use double precision version of the built-in functions.
  
  
• INPUT: Augmented matrix [A | B] examples:

       (#1)    1.0               1.0                exp(0.3)
               -exp(0.5)*1.e-13  1.0                exp(1.0)
  
   
       (#2)    -exp(0.5)*1.e-13  1.0                exp(1.0)
               1.0               1.0                exp(0.3)
   
  
       (#3)    1.0              -1.e13*exp(-0.5)   -1.e13*exp(-0.15)
               1.0               1.0                exp(0.3)
   
  
       (#4)    1.0               1.+.68e-10         1.0
               1.+.75e-9         1.0                1.+.87e-9
   
       (#5)  exp(0.7)*1.e-07  1.0  exp(2.1)   exp(0.6)*1.e+07  -1.0
             exp(1.5)*1.e-07  1.0  exp(1.2)   exp(0.5)*1.e+07  -1.0
             exp(0.5)*1.e+07  1.0  exp(1.7)   exp(1.5)*1.e-07  +1.0
             exp(0.6)*1.e+07  1.0  exp(2.2)   exp(0.7)*1.e-07  +1.0
  

  
  
  
• OCTAVE or MAPLE Check: Check you answers with either the numerical matrix laboratory Octave or the symbolic-numeric package Maple

Email Comments or Questions to hanson@math.uic.edu