Multi-step derivative-free preconditioned Newton method for solving systems of nonlinear equations

Preconditioning of systems of nonlinear equations modifies the associated Jacobian and provides rapid convergence. The preconditioners are introduced in a way that they do not affect the convergence order of parent iterative method. The multi-step derivative-free iterative method consists of a base method and multi-step part. In the base method, the Jacobian of the system of nonlinear equation is approximated by finite difference operator and preconditioners add an extra term to modify it. The inversion of modified finite difference operator is avoided by computing LU factors. Once we have LU factors, we repeatedly use them to solve lower and upper triangular systems in the multi-step part to enhance the convergence order. The convergence order of m-step Newton iterative method is m + 1. The claimed convergence orders are verified by computing the computational order of convergence and numerical simulations clearly show that the good selection of preconditioning provides numerical stability, accuracy and rapid convergence.Peer ReviewedPostprint (author's final draft

Equations and systems of nonlinear equations: from high order numerical methods to fast Eigensolvers for structured matrices and applications

A parametrized multi-step Newton method is constructed for widening the region of convergence of classical multi-step Newton method. The second improvement is proposed in the context of multistep Newton methods, by introducing preconditioners to enhance their accuracy, without disturbing their original order of convergence and the related computational cost (in most of the cases). To find roots with unknown multiplicities preconditioners are also effective when they are applied to the Newton method for roots with unknown multiplicities. Frozen Jacobian higher order multistep iterative method for the solution of systems of nonlinear equations are developed and the related results better than those obtained when employing the classical frozen Jacobian multi-step Newton method. To get benefit from the past information that is produced by the iterative method, we constructed iterative methods with memory for solving systems of nonlinear equations. Iterative methods with memory have a greater rate of convergence, if compared with the iterative method without memory. In terms of computational cost, iterative methods with memory are marginally superior comparatively. Numerical methods are also introduced for approximating all the eigenvalues of banded symmetric Toeplitz and preconditioned Toeplitz matrices. Our proposed numerical methods work very efficiently, when the generating symbols of the considered Toeplitz matrices are bijective

A Parameterized multi-step Newton method for solving systems of nonlinear equations

We construct a novel multi-step iterative method for solving systems of nonlinear equations by introducing a parameter. to generalize the multi-step Newton method while keeping its order of convergence and computational cost. By an appropriate selection of theta, the new method can both have faster convergence and have larger radius of convergence. The new iterative method only requires one Jacobian inversion per iteration, and therefore, can be efficiently implemented using Krylov subspace methods. The new method can be used to solve nonlinear systems of partial differential equations, such as complex generalized Zakharov systems of partial differential equations, by transforming them into systems of nonlinear equations by discretizing approaches in both spatial and temporal independent variables such as, for instance, the Chebyshev pseudo-spectral discretizing method. Quite extensive tests show that the new method can have significantly faster convergence and significantly larger radius of convergence than the multi-step Newton method.Peer ReviewedPostprint (author's final draft

Equations and systems of nonlinear equations: from high order numerical methods to fast Eigensolvers for structured matrices and applications

A parametrized multi-step Newton method is constructed for widening the region of convergence of classical multi-step Newton method. The second improvement is proposed in the context of multistep Newton methods, by introducing preconditioners to enhance their accuracy, without disturbing their original order of convergence and the related computational cost (in most of the cases). To find roots with unknown multiplicities preconditioners are also effective when they are applied to the Newton method for roots with unknown multiplicities. Frozen Jacobian higher order multistep iterative method for the solution of systems of nonlinear equations are developed and the related results better than those obtained when employing the classical frozen Jacobian multi-step Newton method. To get benefit from the past information that is produced by the iterative method, we constructed iterative methods with memory for solving systems of nonlinear equations. Iterative methods with memory have a greater rate of convergence, if compared with the iterative method without memory. In terms of computational cost, iterative methods with memory are marginally superior comparatively. Numerical methods are also introduced for approximating all the eigenvalues of banded symmetric Toeplitz and preconditioned Toeplitz matrices. Our proposed numerical methods work very efficiently, when the generating symbols of the considered Toeplitz matrices are bijective

Simultaneous detection of the nonlinear restoring and excitation of a forced nonlinear oscillation: an integral approach

We address in this article, how to calculate the restoring characteristic and the excitation of a nonlinear forced oscillating system. Under the assumption that the forced nonlinear oscillator has a periodic solution with period, we constructed a system of linear equations by introducing time-dependent multipliers. The periodicity assumption helps simplify the system of linear equations. The stability and uniqueness are also presented for the inverse problem. Numerical testing is conducted to show the effectiveness of our presented methodology.Peer ReviewedPostprint (author's final draft

Achieving Robust Self-Management for Large-Scale Distributed Applications

Autonomic managers are the main architectural building blocks for constructing self-management capabilities of computing systems and applications. One of the major challenges in developing self-managing applications is robustness of management elements which form autonomic managers. We believe that transparent handling of the effects of resource churn (joins/leaves/failures) on management should be an essential feature of a platform for self-managing large-scale dynamic distributed applications, because it facilitates the development of robust autonomic managers and hence improves robustness of self-managing applications. This feature can be achieved by providing a robust management element abstraction that hides churn from the programmer. In this paper, we present a generic approach to achieve robust services that is based on finite state machine replication with dynamic reconfiguration of replica sets. We contribute a decentralized algorithm that maintains the set of nodes hosting service replicas in the presence of churn. We use this approach to implement robust management elements as robust services that can operate despite of churn. Our proposed decentralized algorithm uses peer-to-peer replica placement schemes to automate replicated state machine migration in order to tolerate churn. Our algorithm exploits lookup and failure detection facilities of a structured overlay network for managing the set of active replicas. Using the proposed approach, we can achieve a long running and highly available service, without human intervention, in the presence of resource churn. In order to validate and evaluate our approach, we have implemented a prototype that includes the proposed algorithm

Evaluation and Improvement of students’ satisfaction in Online learning during COVID-19

With the closure of educational institutions due to COVID-19, the biggest challenge with the universities and the instructors was engaging students in virtual learning. This research aimed at supporting university students in Islamabad (Pakistan) for online learning through a collaborative approach. The university started online learning during COVID-19 and had no earlier experience of such mode of learning. The first phase was identifying the problems faced by students during online learning and seeking their suggestions for overcoming them. The next step was working on the students’ opinions with a team of instructors to modify existing instructional practices during online instruction. We measured students’ satisfaction level pre and post-modification to evaluate students’ adoption of online learning. The data for both the phases were collected through a Google Form. The post-modification data revealed students’ greater satisfaction in online learning. The findings offer useful insight related to students’ adoption of online learning and making it a more meaningful, organized, and productive medium for future learning