Nonlinear Algebraic Equations Solved by an Optimal Splitting-Linearizing Iterative Method

Abstract

How to accelerate the convergence speed and avoid computing the inversion of a Jacobian matrix is important in the solution of nonlinear algebraic equations (NAEs). This paper develops an approach with a splitting-linearizing technique based on the nonlinear term to reduce the effect of the nonlinear terms. We decompose the nonlinear terms in the NAEs through a splitting parameter and then linearize the NAEs around the values at the previous step to a linear system. Through the maximal orthogonal projection concept, to minimize a merit function within a selected interval of splitting parameters, the optimal parameters can be quickly determined. In each step, a linear system is solved by the Gaussian elimination method, and the whole iteration procedure is convergent very fast. Several numerical tests show the high performance of the optimal split-linearization iterative method (OSLIM).