Zhang Approximation Inverse (ZAI), Basis Matrix, and Simple Optimal ZAI Alternative Used for Solving Linear Matrix-Vector Equation (LMVE)

In this paper, a new iterative algorithm for linear matrix-vector equation (LMVE) solving on the basis of Zhang approximation inverse (ZAI) is proposed. The optimal Zhang approximation inverse (OZAI) is defined, and a simple alternative is provided. Numerical examples are presented to illustrate the efficacy of using the optimal step size with the simple alternative. Besides, the OZAI-alternative iterative algorithm is compared with classical iterative algorithms for LMVE solving through numerical experiments. In addition, the performance of ZAI iterative algorithm is studied by using different step sizes, condition number intervals, and dimensions.