A math trick from the 1900s just fixed a stability glitch that has haunted computer simulations for decades.
April 20, 2026
Original Paper
Towards Universal Convergence of Backward Error in Linear System Solvers
arXiv · 2604.16075
The Takeaway
Richardson iteration achieves a universal convergence rate for backward error in any positive semidefinite linear system. This method is immune to the condition number, which usually makes these problems impossible to solve on a computer. Most solvers become unstable or incredibly slow when dealing with poorly conditioned data. This discovery means we can now solve core mathematical problems with total reliability regardless of the input's complexity. Scientific simulations and engineering models will become significantly faster and more accurate across the board.
From the abstract
The quest for an algorithm that solves an $n\times n$ linear system in $O(n^2)$ time complexity, or $O(n^2 \text{poly}(1/\epsilon))$ when solving up to $\epsilon$ relative error, is a long-standing open problem in numerical linear algebra and theoretical computer science. There are two predominant paradigms for measuring relative error: forward error (i.e., distance from the output to the optimum solution) and backward error (i.e., distance to the nearest problem solved by the output). In most p