I'm looking at the exercises in the book "Numerical Linear Algebra" by Trefethen and Bau. I'm quite stuck at question 22.2:
Experiment with solving $60\times60$ systems of equations $Ax=b$ by Gaussian elimination with partial pivoting, with $A$ having the form (22.4). Do you observe that the results are useless because of the growth factor of order $2^{60}$? At first attempt you may not observe this, because the integer entries of $A$ may prevent any rounding errors from occurring. If so, find a way to modify your problem slightly so that the growth factor is the same or nearly so and catastrophic rounding errors really do take place.
I coded Gaussian elimination with partial pivoting in MATLAB and everything seems fine. However, when I try to do the above exercise, I don't get the results one would expect.
For starters: there are rounding errors in the LU decomposition even when A only has integer values (since the growth factor is $2^{59}$, MATLAB can't represent the largest values in $U$ exactly).
Even weirder: when I perturb $A$ such that the values are not all integer, the solution of the system is better and not worse at all (for the same right-hand-side).
I'm not sure why this happens, or what I'm doing wrong. Hopefully one of you knows why.