On the Kaczmarz methods based on relaxed greedy selection for solving matrix equation [formula omitted]

Abstract

For solving a consistent system of linear equations, the Kaczmarz method is a popular representative among iterative algorithms due to its simplicity and efficiency. Based on the Petrov–Galerkin conditions and the relaxed greedy selection strategy, we introduce two kinds of Kaczmarz-type methods to solve matrix equation AXB=C, including randomized and deterministic versions and establish the corresponding convergence theories. We prove that our algorithms all have exponential convergence rates. Then we apply our algorithms to real-world application, such as the tensor product surface fitting in computer-aided geometric design (CAGD), it is shown that our algorithms are more effective than some often used conventional matrix equation solvers.