A fast solver for ill-conditioned linear systems using randomized stable solutions of its blocks

Mathematics > Numerical Analysis [Submitted on 2 Oct 2025 (v1), last revised 16 Jun 2026 (this version, v2)] Title:A fast solver for ill-conditioned linear systems using randomized stable solutions of its blocks View PDF HTML (experimental)Abstract:We present an enhanced version of the row-based randomized block-Kaczmarz method to solve a linear system of equations. This improvement makes use of a regularization during block updates in the solution, and a dynamic proposal distribution based on the current residual vector and effective mutual orthogonality between all blocks. The improved method provides significant gains in solving highly ill-conditioned linear systems that are either sparse, or dense least-squares problems that are significantly over/under determined. Considering the poor guarantees in effectively preconditioning iterative solutions for such ill-conditioned problems, it may also serve as a pre-solver for accelerating other iterative numerical methods, and as an inner iteration in certain types of GMRES solvers for linear systems. Submission history From: Murugesan Venkatapathi [view email][v1] Thu, 2 Oct 2025 16:08:27 UTC (124 KB) [v2] Tue, 16 Jun 2026 10:56:41 UTC (321 KB) Current browse context: math.NA References & Citations Loading... Bibliographic and Citation Tools Bibliographic Explorer (What is the Explorer?) Connected Papers (What is Connected Papers?) Litmaps (What is Litmaps?) scite Smart Citations (What are Smart Citations?) Code, Data and Media Associated with this Article alphaXiv (What is alphaXiv?) CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub (What is DagsHub?) Gotit.pub (What is GotitPub?) Hugging Face (What is Huggingface?) ScienceCast (What is ScienceCast?) Demos Recommenders and Search Tools Influence Flower (What are Influence Flowers?) CORE Recommender (What is CORE?) arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.