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

Abstract

After randomly reflecting on two hyperplanes, a new iteration method is established by making use of the circumceter of the reflective points from the viewpoint of geometry. The linear combination could be non-convex when the angle between the hyperplances is small. Theoretical analysis show that the proposed method converges and the convergence rate in expectation is also addressed in detail. The relation between our method and block Kaczmarz method is well discussed. Numerical experiments further verify that the new algorithms is efficient, and outperform the existing randomized Kaczmarz methods and randomized reflection methods in terms of the number of iterations and CPU time, especially when the coefficient matrix has highly coherent rows.