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But will unums help? No matter how an ill conditioned matrix is represented, it will still be ill conditioned, and we'll still be iterating, unless I've missed something in this article.

I'm skeptical that changing how we represent the numbers will help - when the physics are icky, they're icky.

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It won't really help. You'll get the same issues with stiff DEs or ill-conditioned matrices for perfect precision or floating-point. There will be some cases where it will help, but not a lot.

I've implemented a few DE solvers and sparse matrix implementations before. I think sparse matrices is one case where it would help.

Yes, this won't change the challenging nature of solving the underlying problem.

More problematic is that when a matrix is ill-conditioned it may not be all that helpful to have "an answer" since the ill-conditioning tells you there are many solutions closely satisfying the system of equations.

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