Time-Varying ZAI (Zhang Approximation Inverse) and SE (Simple "Optimal") Alternative Proposed for Solving Nonlinear Equations

Approximation inverse and nonlinear equations are critical topics in contemporary research. In this paper, we propose and utilize the concept of time-varying Zhang approximation inverse (TV-ZAI) to approximate the inverse of time-varying Jacobian matrix, and to solve the nonlinear equation. Specifically, we propose the norm based definitions and spectral-radius (SR) based definitions to complement the TV-ZAI. The TV-ZAI is capable of dealing with both continuous and discrete time-varying problems, which makes it more efficient in computation and easier to implement in digital computers compared with analog computers. In addition, we propose a simple alternative for the time-varying optimal Zhang approximation inverse (TV-OZAI). Besides, the corresponding numerical experimental results show the feasibility and effectiveness of the TV-ZAI and TV-OZAI-alternative solutions for solving nonlinear equations.