Year: 2025
Type: article
Abstract: The Kaczmarz method is a way to iteratively solve a linear system of equations A x = b Ax = b . One interprets the solution x x as the point where hyperplanes intersect and then iteratively projects an approximate solution onto these hyperplanes to get better and better approximations. We note a somewhat related idea: one could take two random hyperplanes and project one into the orthogonal complement of the other. This leads to a sequence of linear systems A ( k ) x = b ( k ) A^{(k)} x = b^{(k)} which is fast to compute, preserves the original solution and whose small singular values grow like σ ℓ ( A ( k ) ) ∼ exp ( k / n 2 ) ⋅ σ ℓ ( A ) \sigma _{\ell }(A^{(k)}) \sim \exp (k/n^2) \cdot \sigma _{\ell }(A) .
Source: Quarterly of Applied Mathematics
DOI: 10.1090/qam/1711
Author: Stefan Steinerberger
Institution: University of Washington
Language: English
Topic: Matrix Theory and Algorithms
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4407536367