TITLE: Reasonable "Random" matrices to test numerical algorithms
QUESTION [7 upvotes]: Hello,
in numerical analysis, it is common to compare the behavior of different algorithms, and of different implementation of algorithms. This occurs not only on the theoretical level, but also on the concrete level of implementation - and not to forget, it serves the purpose of demonstration.
A prominent problem is the solution of linear systems, both general as well as various subcases.
To test, benchmark and profile numerical implementations, you run your work on several instances of the problem. However, it is difficult question to obtain a good set of these instances. You want to inspect pathological cases (diff. degrees of ill-conditionedness) as well as "real-life" examples (whatever this may mean). Ideally, you have an algorithm which puts out matrices with certain properties in a "reasonable" probability measure. A good notion of "reasonable" might be accessible, as most such LSE problems from physics or simulations have much more structure as is actually demanded by the algorithms in theory.
In so far, I wonder whether there are works in numerical analysis how to, given $n \in \mathbb N$ randomly produce

a sequence of ${n \times n}$-matrices
optionally constraint to be symmetric, positive definite, well-conditioned
which is reasonable in whatever sense

This is probably an interesting topic within the theory of numerical algorithms.
Thanks,
Martin

REPLY [2 votes]: One way to get useful information on a numerical algorithm is to choose by hand a well-conditioned or even more an ill-conditioned example and perturb it randomly.  In fact it is useful to perturb it randomly such that the perturbations form a sequence approaching zero; you can then look at the sequence of solutions that you get and see what you can say about error propagation and the like.