The randomized circumcentered-reflection iteration method for solving consistent linear equations

A novel iteration scheme for solving consistent linear equations is introduced by exploring the circumcenter of a n-dimensional triangle formed with a starting point and its reflections across two hyperplanes. The proposed method, akin to the block Kaczmarz method, provides optimal linear combination within two subspaces. The convergence rate in expectation is also addressed in detail if the row is selected with probability proportional to p-norm of residuals. Numerical experiments demonstrate its superiority over randomized Kaczmarz and reflection methods, particularly evident as highly coherent rows. Performance are enhanced by proper parameter-tuning of p.