Numerical Analysis of Nonlinear Systems Using Iterative Methods

Abstract

Nonlinear systems are very common in science and engineering, but solving them numerically with iterative methods is still difficult due to the various behaviours of solvers under different problem structures. This paper investigates how algebraic structures in the Jacobian matrix, such as symmetry, condition number, sparsity and spectral distribution, affect the suitability and convergence performance of typical iterative methods for solving nonlinear system equilibria. Four cross-disciplinary case studies were used: the susceptible-infected-recovered (SIR) epidemic model, the Lotka–Volterra predator–prey system, blood ethanol kinetics and steady-state heat conduction; a structure-diagnostic framework connecting Jacobian pathologies to solver selection was established. The case studies show that a singular or ill-conditioned Jacobian can make Newton steps difficult or numerically unstable, and non-symmetric linear subproblems require solvers other than conjugate gradient. Purely imaginary eigenvalues of a dynamical-system Jacobian do not by themselves imply oscillatory failure of Newton root-finding. Well-conditioned but strongly state-dependent Jacobians in the scalar blood ethanol model do not prevent convergence, and larger coupled variants may motivate Jacobian-free alternatives when derivative evaluation is costly. In contrast, symmetric positive-definite sparse Jacobians in the heat-conduction example have no structural obstruction and can be solved efficiently by successive over-relaxation or the Thomas algorithm. Based on the above results, Jacobian structural diagnosis can be used to select a suitable iterative solver beforehand, and the focus will shift from convenience to structure.