A low-memory Lanczos method with rational Krylov compression for matrix functions · HackerPapers discussions

A low-memory Lanczos method with rational Krylov compression for matrix functions — mentioned by lukefleed

Tue, 11 Nov 2025 22:46:40 GMT

That's an interesting question. I don't have too much experience, but here's my two cents. For matrix function approximations, loss of orthogonality matters less than for eigenvalue computations. The three-term recurrence maintains local orthogonality reasonably well for moderate iteration counts. My experiments [1] show orthogonality loss stays below $10^{-13}$ up to k=1000 for well-conditioned problems, and only becomes significant (jumping to $10^{-6}$ and higher) around k=700-800 for ill-conditioned spectra. Since you're evaluating $f(T_k)$ rather than extracting individual eigenpairs…

Paper and discussion history. A mention is not an endorsement.

A low-memory Lanczos method with rational Krylov compression for matrix functions — mentioned by lukefleed

Tue, 11 Nov 2025 22:23:22 GMT

Sorry for the late answer. The blog post is a simplification of the actual work; you can check out the full report here [1], where I also reference the literature about this algorithm. On the cache effects: I haven't seen this "engineering" argument made explicitly in the literature either. There are other approaches to the basis storage problem, like the compression technique in [2]. Funny enough, the authors gave a seminar at my university literally this afternoon about exactly that. I'm also unfamiliar with randomised algorithms for numerical linear algebra beyond the basics. I'll dig into…

Paper and discussion history. A mention is not an endorsement.