Hybrid Krylov Methods for Nonlinear Systems of Equations

Several implementations of Newton-like iteration schemes based on Krylov subspace projection methods for solving nonlinear equations are considered. The simplest such class of methods is Newton’s algorithm in which a (linear) Krylov method is used to solve the Jacobian system approximately. A method in this class is referred to as a Newton–Krylov algorithm. To improve the global convergence properties of these basic algorithms, hybrid methods based on Powell’s dogleg strategy are proposed, as well as linesearch backtracking procedures. The main advantage of the class of methods considered in this paper is that the Jacobian matrix is never needed explicitly.