Example 5. Use the Gaussian elimination methods to solve  ,  where is the Hilbert matrix and  .  Use the trivial, partial scaled partial and total pivoting strategies.
Solution 5.

Perform Gaussian elimination using the various pivoting strategies.

The output has been suppressed because of its size.  

Summary  Let us compare the "true solution" with the results from the various methods.

For this example, we see that the partial pivoting solution    is slightly better than the others.

Warning.  We must proceed with caution when using a Hilbert matrix because the linear system might be ill conditioned.  

    The condition number of the above system can be determined by Mathematica.

Fact.  Given the linear system  .  If     are input with machine precision then a bound for the error in the computed solution    is given by

        

where is machine epsilon for the computer.  The computed solution    loses about    decimal digits of accuracy relative to precision of input.

Caveat.  Mathematica uses extended precision sixteen digit numbers, the solution    retains some of this accuracy.  

(c) John H. Mathews 2005