Variable $s$-Step Technique for New Conjugate Residual Algorithms for Solving Non-Square Linear Systems Arising in Control Problems

This paper introduces two algorithms based on the conjugate residual method and variable $s$-step for solving linear systems with non-square coefficient matrices. The introduced algorithms demonstrate their effectiveness through numerical comparisons with other methods, specifically CGNE and CGNR, to solve non-square linear systems. The approach aims to enhance the efficiency of solving problems related to control systems and imaging underground layers, particularly in the context of seismic tomography.