solving system of nonlinear equations

It is well-known that the solution of Hamilton-Jacobi equation may have singularity i.e., the solution is non-smooth or nearly non-smooth. We construct a frozen Jacobian multi-step iterative method for solving Hamilton-Jacobi equation under the assumption that the solution is nearly singular. The frozen Jacobian iterative methods are computationally very efficient because a single instance of the iterative method uses a single inversion (in the scene of LU factorization) of the frozen Jacobian. The multi-step part enhances the convergence order by solving lower and upper triangular systems. The convergence order of our proposed iterative method is 3(m − 1) for m ≥ 3. For attaining good numerical accuracy in the solution, we use Chebyshev pseudo-spectral collocation method. Some Hamilton-Jacobi equations are solved, and numerically obtained results show high accuracy.

A frozen Jacobian iterative method is proposed for solving systems of nonlinear equations. In particular, we are interested in solving the systems of nonlinear equations associated with initial value problems (IVPs) and boundary value problems (BVPs). In a single instance of the proposed iterative method DEDF, we evaluate two Jacobians, one inversion of the Jacobian and four function evaluations. The direct inversion of the Jacobian is computationally expensive, so, for a moderate size, LU factorization is a good direct method to solve the linear system. We employed the LU factorization of the Jacobian to avoid the direct inversion. The convergence order of the proposed iterative method is at least eight, and it is nine for some particular classes of problems. The discretization of IVPs and BVPs is employed by using Jacobi-Gauss-Lobatto collocation (J-GL-C) method. A comparison of J-GL-C methods is presented in order to choose best collocation method. The validity, accuracy and the efficiency of our DEDF are shown by solving eleven IVPs and BVPs problems.

Gradient methods are the optimization methods that use gradient of function in searching for the optimal solution. They are widely used for solving nonlinear optimization problems when the objective function and constraints are continuous and mathematically well-defined. In practice, we commonly apply the finite difference formula or MATLAB's symbolic toolbox to compute the gradient of function in any form, so the problem of dealing with complicated differentiation is not that important.

Frozen Jacobian iterative methods are of practical interest to solve the system of nonlinear equations. A frozen Jacobian multi-step iterative method is presented. We divide the multi-step iterative method into two parts namely base method and multi-step part. The convergence order of the constructed frozen Jacobian iterative method is three, and we design the base method in a way that we can maximize the convergence order in the multi-step part. In the multi-step part, we utilize a single evaluation of the function, solve four systems of lower and upper triangular systems and a second frozen Jacobian. The attained convergence order per multi-step is four. Hence, the general formula for the convergence order is 3 + 4(m - 2) for m = 2 and m is the number of multi-steps. In a single instance of the iterative method, we employ only single inversion of the Jacobian in the form of LU factors that makes the method computationally cheaper because the LU factors are used to solve four syste...

It is well-known that the solution of Hamilton-Jacobi equation may have singularity i.e., the solution is non-smooth or nearly non-smooth. We construct a frozen Jacobian multi-step iterative method for solving Hamilton-Jacobi equation under the assumption that the solution is nearly singular. The frozen Jacobian iterative methods are computationally very efficient because a single instance of the iterative method uses a single inversion (in the scene of LU factorization) of the frozen Jacobian. The multi-step part enhances the convergence order by solving lower and upper triangular systems. The convergence order of our proposed iterative method is 3(m -1) for m ≥ 3. For attaining good numerical accuracy in the solution, we use Chebyshev pseudo-spectral collocation method. Some Hamilton-Jacobi equations are solved, and numerically obtained results show high accuracy.

A generalization of the Newton multi-step iterative method is presented, in the form of distinct families of methods depending on proper parameters. The proposed generalization of the Newton multi-step consists of two parts, namely the base method and the multi-step part. The multi-step part requires a single evaluation of function per step. During the multi-step phase, we have to solve systems of linear equations whose coefficient matrix is the Jacobian evaluated at the initial guess. The direct inversion of the Jacobian it is an expensive operation, and hence, for moderately large systems, the LU factorization is a reasonable choice. Once we have the LU factors of the Jacobian, starting from the base method, we only solve systems of lower and upper triangular matrices that are in fact computationally economical. The developed families involve unknown parameters, and we are interested in setting them with the goal of maximizing the convergence order of the global method. Few families are investigated in some detail. The validity and numerical accuracy of the solution of the system of nonlinear equations are presented via numerical simulations, also involving examples coming from standard approximations of ordinary differential and partial differential nonlinear equations. The obtained results show the efficiency of constructed iterative methods, under the assumption of smoothness of the nonlinear function.

We present the iterative methods of fifth and eighth order of convergence for solving systems of nonlinear equations. Fifth order method is composed of two steps namely, Newton's and Newton-like steps and requires the evaluations of two functions, two first derivatives and one matrix inversion in each iteration. The eighth order method is composed of three steps, of which the first two steps are that of the proposed fifth order method whereas the third is Newton-like step. This method requires one extra function evaluation in addition to the evaluations of fifth order method. Computational efficiency of proposed techniques is discussed and compared with the existing methods. Some numerical examples are considered to test the performance of the new methods. Moreover, theoretical results concerning order of convergence and computational efficiency are confirmed in numerical examples. Numerical results have confirmed the robust and efficient character of the proposed techniques.

The objective of this research is to propose a new multi-step method in tackling system of nonlinear equations. The constructed iterative scheme achieves a higher rate of convergence whereas only one LU decomposition per cycle is required to proceed. This makes the efficiency index to be high as well in contrast to the existing solvers. The usefulness of the presented approach for tackling differential equations of nonlinear type with partial derivatives is also given.

Derivative-free schemes are a class of competitive methods since they are one remedy in cases at which the computation of the Jacobian or higher order derivatives of multi-dimensional functions is difficult. This article studies a variant of Steffensen’s method with memory for tackling a nonlinear system of equations, to not only be independent of the Jacobian calculation but also to improve the computational efficiency. The analytical parts of the work are supported by several tests, including an application in mixed integral equations.

It is well-known that the solution of Hamilton-Jacobi equation may have singularity i.e., the solution is non-smooth or nearly non-smooth. We construct a frozen Jacobian multi-step iterative method for solving Hamilton-Jacobi equation under the assumption that the solution is nearly singular. The frozen Jacobian iterative methods are computationally very efficient because a single instance of the iterative method uses a single inversion (in the scene of LU factorization) of the frozen Jacobian. The multi-step part enhances the convergence order by solving lower and upper triangular systems. The convergence order of our proposed iterative method is 3(m − 1) for m ≥ 3. For attaining good numerical accuracy in the solution, we use Chebyshev pseudo-spectral collocation method. Some Hamilton-Jacobi equations are solved, and numerically obtained results show high accuracy.

It is well-known that the solution of Hamilton-Jacobi equation may have singularity i.e., the solution is non-smooth or nearly non-smooth. We construct a frozen Jacobian multi-step iterative method for solving Hamilton-Jacobi equation under the assumption that the solution is nearly singular. The frozen Jacobian iterative methods are computationally very efficient because a single instance of the iterative method uses a single inversion (in the scene of LU factorization) of the frozen Jacobian. The multi-step part enhances the convergence order by solving lower and upper triangular systems. The convergence order of our proposed iterative method is 3(m − 1) for m ≥ 3. For attaining good numerical accuracy in the solution, we use Chebyshev pseudo-spectral collocation method. Some Hamilton-Jacobi equations are solved, and numerically obtained results show high accuracy.

A multi-step frozen Jacobian iterative scheme for solving system of nonlinear equations associated with IVPs (initial value problems) and BVPs (boundary value problems) is constructed. The multi-step iterative schemes consist of two parts, namely base method and a multi-step part. The proposed iterative scheme uses higher order Fréchet derivatives in the base method part and offers high convergence order (CO) 3s + 1, here s is the number of steps. The increment in the CO per step is three, and we solve three upper and lower triangles systems per step in the multi-step part. A single inversion of the is not working in latexfrozen Jacobian is required and in fact, we avoid the direct inversion of the frozen Jacobian by computing the LU factors. The LU-factors are utilized in the multi-step part to solve upper and lower triangular systems repeatedly that makes the iterative scheme computationally efficient. We solve a set of IVPs and BVPs to show the validity, accuracy and efficiency of our proposed iterative scheme.

It is well-known that the solution of Hamilton-Jacobi equation may have singularity i.e., the solution is non-smooth or nearly non-smooth. We construct a frozen Jacobian multi-step iterative method for solving Hamilton-Jacobi equation under the assumption that the solution is nearly singular. The frozen Jacobian iterative methods are computationally very efficient because a single instance of the iterative method uses a single inversion (in the scene of LU factorization) of the frozen Jacobian. The multi-step part enhances the convergence order by solving lower and upper triangular systems. The convergence order of our proposed iterative method is 3(m − 1) for m ≥ 3. For attaining good numerical accuracy in the solution, we use Chebyshev pseudo-spectral collocation method. Some Hamilton-Jacobi equations are solved, and numerically obtained results show high accuracy.

Frozen Jacobian iterative methods are of practical interest to solve the system of nonlinear equations. A frozen Jacobian multi-step iterative method is presented. We divide the multi-step iterative method into two parts namely base method and multi-step part. The convergence order of the constructed frozen Jacobian iterative method is three, and we design the base method in a way that we can maximize the convergence order in the multi-step part. In the multi-step part, we utilize a single evaluation of the function, solve four systems of lower and upper triangular systems and a second frozen Jacobian. The attained convergence order per multi-step is four. Hence, the general formula for the convergence order is 3 + 4(m − 2) for m ≥ 2 and m is the number of multi-steps. In a single instance of the iterative method, we employ only single inversion of the Jacobian in the form of LU factors that makes the method computationally cheaper because the LU factors are used to solve four system of lower and upper triangular systems repeatedly. The claimed convergence order is verified by computing the computational order of convergence for a system of nonlinear equations. The efficiency and validity of the proposed iterative method are narrated by solving many nonlinear initial and boundary value problems. c 2016 All rights reserved. Keywords: Frozen Jacobian iterative methods, multi-step iterative methods, systems of nonlinear equations, nonlinear initial value problems, nonlinear boundary value problems. 2010 MSC: 65H10, 65N22.

A systematic way is presented for the construction of multi-step iterative method with frozen Jacobian. The inclusion of an auxiliary function is discussed. The presented analysis shows that how to incorporate auxiliary function in a way that we can keep the order of convergence and computational cost of Newton multi-step method. The auxiliary function provides us the way to overcome the singularity and ill-conditioning of the Jacobian. The order of convergence of proposed p-step iterative method is p + 1. Only one Jacobian inversion in the form of LU-factorization is required for a single iteration of the iterative method and in this way, it offers an efficient scheme. For the construction of our proposed iterative method, we used a decomposition technique that naturally provides different iterative schemes. We also computed the computational convergence order that confirms the claimed theoretical order of convergence. The developed iterative scheme is applied to large scale problems, and numerical results show that our iterative scheme is promising.

The quasilinearization iterative method is famous to solve boundary value problems with quadratic convergence or in other words it is equivalent to classical Newton-Raphson iterative method with second-order convergence for system of nonlinear equations. The Chebyshev-Halley method is also designed to solve system of nonlinear equations with third-order convergence but as far as we know, it is never discussed in the context of ordinary differential equations (ODEs) or partial differential equations (PDEs). In this short note, we proposed a general procedure to design higherorder iterative methods for ODEs and PDEs. In fact quasilinearization and Chebyshev-Hally method are the particular cases of our proposal.

The dynamic response of an infinite Euler-Bernoulli beam, resting on a nonlinear elastic foundation, has been investigated. The problem is treated in a general sense by defining the flexural rigidity and coefficients in nonlinear power law as a function of spatial variable which runs along the length of infinite beam. The flexural rigidity and coefficients in nonlinear power law could be discontinuous. We implemented quasilinear method (QLM) to deal with the nonlinearity of the problem which is due to the inclusion of nonlinear elastic foundation. The QLM translates the nonlinear problem in a linear associated problem that satisfies the boundary conditions defined for the nonlinear fourth-order partial differential equation (PDE). The quasilinear form (QL) is iterated until the approximate solutions converge to real solution within some tolerance. The QL-PDE is solved using Chebyshev spectral collocation method in space and time. The spatial and temporal discretization converts the QL-PDE in to system of linear algebraic equations. The validity and accuracy of proposed iterative scheme is checked by number of designed problems. The proposed iterative method has second-order convergence which is implied from QLM and shows rapid convergence in few number of iterations.

A multi-step frozen Jacobian iterative scheme for solving the system of nonlinear equations associated with IVPs (initial value problems) and BVPs (boundary value problems) is constructed. The multi-step iterative schemes consist of two parts, namely base method and a multi-step part. The proposed iterative scheme uses higher order Fréchet derivatives in the base method part and offers high convergence order (CO) 3m + 1, here s is the number of steps. The increment in the CO per step is three, and we solve three upper and lower triangles systems per step in the multi-step part. A single inversion of the frozen Jacobian is required and in fact, we avoid the direct inversion of the frozen Jacobian by computing the LU factors. The LU-factors are utilized in the multi-step part to solve upper and lower triangular systems repeatedly that makes the iterative scheme computationally efficient. We solve a set of IVPs and BVPs to show the validity, accuracy and efficiency of our proposed iterative scheme.

In this paper we analyze some issues related to the general problem of information spreading among individuals, where suitable assumptions on the information exchange are considered. In particular, starting from the scheme proposed in [5], which is based on a majority rule to treat the individualsÕ interaction, we define and introduce special individuals who play a key role in the information spreading. The latter will be addressed as opinion leaders, and have the special feature of strongly interfering with the process based on the majority rule. We consider a new model, where opinion leaders are introduced as special agents, and study its specific properties which significantly recast some conclusions worth for the model in [5]. Finally, we provide some results cocerning the dynamics of the model.