TCS-Style Correctness Analysis of MR^3 for Tridiagonal Eigenproblems

Prove a rigorous, end-to-end correctness and convergence analysis for the Multiple Relatively Robust Representations (MR^3) algorithm on arbitrary symmetric tridiagonal matrices in floating-point arithmetic under easily verifiable input conditions.

Background

MR3 is the fastest practical algorithm for symmetric tridiagonal eigenproblems, but current analyses rely on assumptions verified only post hoc or fail in the presence of tightly clustered eigenvalues.

A formal theoretical-computer-science style proof of correctness in floating-point arithmetic would place MR3 on firmer ground and could inform extensions, such as to the bidiagonal SVD.

References

There is currently no rigorous proof (in the vein of theoretical computer science) of success for a floating-point implementation of MR$3$ when applied to an arbitrary symmetric tridiagonal matrix. Give a "TCS-style" analysis of MR$3$ which proves its correctness and convergence under easily verifiable conditions on the input.

— Linear Systems and Eigenvalue Problems: Open Questions from a Simons Workshop  (2602.05394 - Amsel et al., 5 Feb 2026) in Subsection "Revisiting MR^3" (Section 3)