We started a new problem recently, but still within the realms of preconditioning. The exact question and algorithms will remain under wraps right now, but the preliminary results I got were incredibly curious; here’s a plot of condition number below:

Note that a constant condition number as we increase order is *extremely* good, but my adviser and I were not rejoicing. It’s just that the result was too good! We were not expecting the condition number to not increase, as that’s far better than what current literature (and our intuition) suggests.

After a week of fumbling around, I’ve extended the computation and produced the following plot:

Not that the above plot is the same algorithm, just with extended order (or p). And ahhh, there it is, the growth that we were looking for.

Even though it’s a worse result that what we initially saw, we can actually prove this version!