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This paper presents a weak convergence residual algorithm for finding a fixed point of a nonexpansive mapping in a real Hilbert space. To study the numerical behavior of the algorithm it is included an extensive series of numerical experiments. Our computational experiments show that the new algorithm is computationally efficient.

A residual algorithm for solving monotone equations on a real Hilbert space is presented. The monotone equations arise in various applications, such as differential equations, as well as economics, engineering, management science, probability theory and other applied sciences. The new scheme uses in a systematic way the residual as search direction combined with a suitable step-length and a nonmonotone line search globalization strategy. A convergence analysis is described. It is also presented the application of a new approach for solving the Lyapunov matricial equation. Numerical experiences are included to highlight the efficacy of the proposed algorithm for the solution of monotone equations.

This paper presents a derivative-free algorithm for solving nonlinear equations with convex constraints. The new method uses in a systematic way the projection and the residual to generate search directions and feasible iterates. A convergence analysis is described. Extensive numerical experiences are included to highlight the efficacy of the proposed algorithm for the solution of nonlinear equations with convex constraints.

A derivative-free residual method for solving nonlinear operator equations in real Hilbert spaces is discussed. This method uses in a systematic way the residual as search direction, but it does not use first order information. Furthermore a convergence analysis and numerical results of the new method applied to nonlinear integral equations using symbolic computation are presented.

In the positive definite case, the extreme generalized eigenvalues can be obtained by solving a suitable nonlinear system of equations. In this work, we adapt and study the use of recently developed low-cost derivative-free residual schemes for nonlinear systems, to solve large-scale generalized eigenvalue problems. We demonstrate the effectiveness of our approach on some standard test problems, and also on a problem associated with the vibration analysis of large structures. In our numerical results we use preconditioning strategies based on incomplete factorizations, and we compare with and without preconditioning with a well-known available package.

In this paper, we present an extension of the Barzilai–Borwein method for the minimization of a quadratic form defined in a finite-dimensional real Euclidean space. We present a convergence analysis of the proposed method. We also present some applications in the resolution of linear matrix equations such as the Sylvester equation, which are of great interest in control theory and other engineering disciplines. Our numerical results indicate that the new method competes satisfactorily in the resolution of linear matrix equations with the function lyap of the Toolbox of Control of MATLAB.

We present a hybrid algorithm that combines a genetic algorithm with the Barzilai---Borwein gradient method. Under specific assumptions the new method guarantees the convergence to a stationary point of a continuously differentiable function, from any arbitrary initial point. Our preliminary numerical results indicate that the new methodology finds efficiently and frequently the global minimum, in comparison with the globalized Barzilai---Borwein method and the genetic algorithm of the Toolbox of Genetic Algorithms of MatLab.

A fully derivative-free spectral residual method for solving largescale nonlinear systems of equations is presented. It uses in a systematic way the residual vector as a search direction, a spectral steplength that produces a nonmonotone process and a globalization strategy that allows for this nonmonotone behavior. The global convergence analysis of the combined scheme is presented. An extensive set of numerical experiments that indicate that the new combination is competitive and frequently better than well-known Newton-Krylov methods for largescale problems is also presented.

The spectral gradient method has proved to be effective for solving large-scale optimization problems. In this work we extend the spectral approach to solve nonlinear systems of equations. We consider a strategy based on nonmonotone line search techniques to guarantee global convergence. and discuss implementation details for solving large-scale problems. We compare the performance of our new method with recent implementations of inexact Newton schemes based on Krylov subspace inner iterative methods for the linear systems. Our numerical experiments indicate that the spectral approach for solving nonlinear systems competes favorably with well-established numerical methods.

This paper presents a residual approach of the square root balanced truncation algorithm for model order reduction of continuous, linear and time-invariante compartmental systems. Specifically, the new approach uses a residual method to approximate the controllability and observability gramians, whose resolution is an essential step of the square root balanced truncation algorithm, that requires a great computational cost. Numerical experiences are included to highlight the efficacy of the proposed approach.