Structure preserving Kaczmarz methods based on relaxed greedy selection for solving quaternion linear systems

Abstract

In this paper, we study the stochastic Kaczmarz iterative method for quaternion
linear systems and propose a structure-preserving algorithm for the lax-greedy stochastic Kaczmarz
Iterative (QRGRK) method. To accelerate convergence, we propose a PmQRGRK method for structure preservation using Polyak's momentum acceleration
technique. We provide asymptotic convergence theories
for the proposed iterative methods and prove that they converge to the exact solution in the
desired direction. Numerical examples are given to illustrate the effectiveness of the proposed structure-preserving
QRGRK method for true linear systems generated
from quaternion linear systems compared to the ME-RGRK and RGRK methods. In addition, we demonstrate the numerical advantages of the PmQRGRK method
over the QRGRK method in terms of the number of iterations and computational time.