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It is known that the concept of optimality is not defined for multidimensional iterative methods for solving nonlinear systems of equations. However, usually optimal fourth-order schemes (extended to the case of several variables) are employed as starting steps in order to design higher order methods for this kind of problems. In this paper, we use a non-optimal (in scalar case) iterative procedure that is specially efficient for solving nonlinear systems, as the initial steps of an eighth-order scheme that improves the computational efficiency indices of the existing methods, as far as the authors know. Moreover, the method can be modified by adding similar steps, increasing the order of convergence three times per step added. This kind of procedures can be used for solving big-sized problems, such as those obtained by applying finite differences for approximating the solution of diffusion problem, heat conduction equations, etc. Numerical comparisons are made with the same existing methods, on standard nonlinear systems and Fisher’s equation by transforming it in a nonlinear system by using finite differences. From these numerical examples, we confirm the theoretical results and show the performance of the proposed schemes. [ABSTRACT FROM AUTHOR]

The GMRES method is one of the most popular iterative schemes for the solution of large linear systems of equations with a square nonsingular matrix. GMRES-type methods also have been applied to the solution of linear discrete ill-posed problems. Computational experience indicates that for the latter problems variants of the standard GMRES method, that require the solution to live in the range of a positive power of the matrix of the linear system of equations to be solved, generally yield more accurate approximations of the desired solution than standard GMRES. This paper investigates properties of these variants of GMRES. [ABSTRACT FROM AUTHOR]

Many physics problems can only be studied by coupling various numerical codes, each modeling a subaspect of the physics problem that is addressed. In most cases, the “brute force” technique of running the codes one after the other in a loop until convergence is reached requires excessive CPU time. The present paper illustrates that re-writing the coupling as a root-finding problem, to which a quasi-Newton method–here the (Inverse) Column Updating Method–can be applied, is useful to push down the computation time, at the expense of a very modest amount of supplementary programming. A simplified version of the set of codes commonly used to describe plasma heating by radio frequency waves in a tokamak plasma is adopted for illustrating the potential of the speed-up method. It consists of a wave equation as well as a Fokker–Planck velocity space diffusion and a radial energy diffusion model. It is shown that with this approach a substantial reduction in CPU time needed for convergence can be obtained. [ABSTRACT FROM AUTHOR]