An extended iterative algorithm for solving nonlinear systems and differential equations

An extended iterative algorithm for solving nonlinear systems and ordinary differential equations (ODE) is presented in this paper which is efficient.Convergence difficulty, high computation cost and instability are the obstacles of using traditional methods like Newton method and explicit Runge-Kutta methods for stiff problems.In this approach proposed, we enhance numerical stability and bring about the convergence speed by introducing a modifed Jacobian matrix.To solve multi variable nonlinear systems, the algorithm is structured such that it adapts to a dynamically varied update step, which reduces the sensitivity toward the initial conditions.Furthermore, for ODEs we apply an implicit numerical integration combining with the modified Jacobian so that it is well suited to stiff and high dimensional systems.It is shown by theoretical analysis that the nonlinear equations converge superlinearly and numerical experiments are used to show that the methods perform better than classical approaches in terms of accuracy and efficiency.It is used for engineering simulations as well as machine learning optimization problems.The method will be extended to partial differential equations (PDEs) as future work, as will adaptively choosing the step size for further improvements in the computation.