Quasi-Newton Iterative Method for Solving Nonlinear Equation Systems

The Quasi-Newton iterative method is an enhanced numerical optimization algorithm specifically designed for solving nonlinear equation systems. Compared to traditional Newton's method, its primary advantage lies in avoiding the computationally intensive operations of calculating derivatives and performing matrix inversions during each iteration.

The fundamental approach of this method simplifies computational processes by approximating either the Hessian matrix or its inverse matrix. During iteration, Quasi-Newton methods update this approximation matrix using current point function values and gradient information, rather than recalculating the complete derivative matrix every time. In code implementation, this typically involves maintaining a matrix approximation that gets updated using rank-one or rank-two updates like the Broyden-class formulas.

Classic Quasi-Newton algorithms such as BFGS (Broyden-Fletcher-Goldfarb-Shanno) and DFP (Davidon-Fletcher-Powell) methods employ different strategies for maintaining and updating this approximation matrix. The BFGS method, for instance, uses a specific update formula that preserves matrix symmetry and positive definiteness, while DFP employs an alternative approach for inverse Hessian approximation. These methods maintain convergence rates comparable to Newton's method while significantly reducing computational complexity per iteration.

The Quasi-Newton method is particularly suitable for nonlinear problems where derivative calculation is difficult or computationally expensive. In practical applications, it has become one of the preferred methods for solving medium to large-scale nonlinear optimization problems. Implementation typically involves line search techniques alongside the matrix updates to ensure convergence. This method finds widespread application in engineering computations, economic modeling, and machine learning domains, where efficient optimization of complex nonlinear systems is crucial.