A class of two‐stage iterative methods for systems of weakly nonlinear equations

The discretizations of many differential equations by the finite difference or the finite element methods can often result in a class of system of weakly nonlinear equations. In this paper, by applying the two-tage iteration technique and in accordance with the special properties of this weakly nonlinear system, we first propose a general two-tage iterative method through the two-tage splitting of the system matrix. Then, by applying the accelerated overrelaxation (AOR) technique of the linear iterative methods, we present a two-tage AOR method, which particularly uses the AOR iteration as the inner iteration and is substantially a relaxed variant of the afore-presented method. For these two classes of methods, we establish their local convergence theories, and precisely estimate their asymptotic convergence factors under some suitable assumptions when the involved nonlinear mapping is only B-differentiable. When the system matrix is either a monotone matrix or an H-matrix, and the nonlinear mapping is a P-bounded mapping, we thoroughly set up the global convergence theories of these new methods. Moreover, under the assumptions that the system matrix is monotone and the nonlinear mapping is isotone, we discuss the monotone convergence properties of the new two-tage iteration methods, and investigate the influence of the matrix splittings as well as the relaxation parameters on the convergence behaviours of these methods. Numerical computations show that our new methods are feasible and efficient for solving the system of weakly nonlinear equations. This revised version was published online in June 2006 with corrections to the Cover Date.     

A class of iterative methods for solving nonlinear projection equations

A class of globally convergent iterative methods for solving nonlinear projection equations is provided under a continuity condition of the mappingF. WhenF is pseudomonotone, a necessary and sufficient condition on the nonemptiness of the solution set is obtained.The author would like to thank two referees for their useful comments on this paper and one of them, in particular, for bringing Ref. 15 to his attention. The author also thanks Professor He for sending him Ref. 23.     

Monotonically convergent iterative methods for nonlinear systems of equations

Summary This paper deals with discrete analogues of nonlinear elliptic boundary value problems and with monotonically convergent iterative methods for their numerical solution. The discrete analogues can be written asM(u)u+H(u)=0, whereM(u) is ann%n M-matrix for eachu ⁿ andH: ^{n n} . The numerical methods considered are the natural undeerrelaxation method, the successive underrelaxation method, and the Jacobi underrelaxation method. In the linear case and without underrelaxation these methods correspond to the direct, the Gauss-Seidel, and the Jacobi method for solving the underlying system of equations, resp. For suitable starting vectors and sufficiently strong underrelaxation, the sequence of iterates generated by any of these methods is shown to converge monotonically to a solution of the underlying system.     

Preconditioned iterative methods for a class of nonlinear eigenvalue problems

This paper proposes new iterative methods for the efficient computation of the smallest eigenvalue of symmetric nonlinear matrix eigenvalue problems of large order with a monotone dependence on the spectral parameter. Monotone nonlinear eigenvalue problems for differential equations have important applications in mechanics and physics. The discretization of these eigenvalue problems leads to nonlinear eigenvalue problems with very large sparse ill-conditioned matrices monotonically depending on the spectral parameter. To compute the smallest eigenvalue of large-scale matrix nonlinear eigenvalue problems, we suggest preconditioned iterative methods: preconditioned simple iteration method, preconditioned steepest descent method, and preconditioned conjugate gradient method. These methods use only matrix-vector multiplications, preconditioner-vector multiplications, linear operations with vectors, and inner products of vectors. We investigate the convergence and derive grid-independent error estimates for these methods. Numerical experiments demonstrate the practical effectiveness of the proposed methods for a model problem.     

A family of multi-point iterative methods for solving systems of nonlinear equations

We extend to n$n$-dimensional case a known multi-point family of iterative methods for solving nonlinear equations. This family includes as particular cases some well known and also some new methods. The main advantage of these methods is they have order three or four and they do not require the evaluation of any second or higher order Fréchet derivatives. A local convergence analysis and numerical examples are provided.     

A new efficient parametric family of iterative methods for solving nonlinear systems

ABSTRACT

A bi-parametric family of iterative schemes for solving nonlinear systems is presented. We prove for any value of parameters the sixth-order of convergence of any members of the class. The efficiency and computational efficiency indices are studied for this family and compared with that of the other known schemes with similar structure. In the numerical section, we solve, after discretizating, the nonlinear boundary problem described by the Fisher's equation. This numerical example confirms the theoretical results and show the performance of the proposed schemes.     

Numerical solution of nonlinear systems by a general class of iterative methods with application to nonlinear PDEs

A general class of multi-step iterative methods for finding approximate real or complex solutions of nonlinear systems is presented. The well-known technique of undetermined coefficients is used to construct the first method of the class while the higher order schemes will be attained by a frozen Jacobian. The point of attraction theory will be taken into account to prove the convergence behavior of the main proposed iterative method. Then, it will be observed that an m-step method converges with 2m-order. A discussion of the computational efficiency index alongside numerical comparisons with the existing methods will be given. Finally, we illustrate the application of the new schemes in solving nonlinear partial differential equations.     

A class of norms of iterative methods for solving systems of linear equations

A new class of norms which generalize norms previously investigated by Young [9, 14], Sheldon [4, 5], Golub [1], Golub and Varga [2], Varga [6], Wachspress [7], Young and Kincaid [12], Young [14], and Kincaid [3] is introduced. Expressions for these norms applied to the matrices associated with various iterative methods are developed.Work on this paper was sponsored by NSF Grant GP-8442 and Army Grant DA-ARO(D)-31-124-G1050 at The University of Texas at Austin.     

An iterative numerical algorithm for modeling a class of Wiener nonlinear systems

This letter presents an iterative estimation algorithm for modeling a class of output nonlinear systems. The basic idea is to derive an estimation model and to solve an optimization problem using the gradient search. The proposed iterative numerical algorithm can estimate the parameters of a class of Wiener nonlinear systems from input–output measurement data. The proposed algorithm has faster convergence rates compared with the stochastic gradient algorithm. The numerical simulation results indicate that the proposed algorithm works well.     

Third order iterative methods free from second derivative for nonlinear systems

In this work we present a family of predictor-corrector methods free from second derivative for solving nonlinear systems. We prove that the methods of this family are of third order convergence. We also perform numerical tests that allow us to compare these methods with Newton’s method. In addition, the numerical examples improve theoretical results, showing super cubic convergence for some methods of this family.     

Parallel multisplitting two-stage iterative methods for large sparse systems of weakly nonlinear equations

The finite difference or the finite element discretizations of many differential or integral equations often result in a class of systems of weakly nonlinear equations. In this paper, by reasonably applying both the multisplitting and the two-stage iteration techniques, and in accordance with the special properties of this system of weakly nonlinear equations, we first propose a general multisplitting two-stage iteration method through the two-stage multiple splittings of the system matrix. Then, by applying the accelerated overrelaxation (AOR) technique of the linear iterative methods, we present a multisplitting two-stage AOR method, which particularly uses the AOR-like iteration as inner iteration and is substantially a relaxed variant of the afore-presented method. These two methods have a forceful parallel computing function and are much more suitable to the high-speed multiprocessor systems. For these two classes of methods, we establish their local convergence theories, and precisely estimate their asymptotic convergence factors under some suitable assumptions when the involved nonlinear mapping is only directionally differentiable. When the system matrix is either an H-matrix or a monotone matrix, and the nonlinear mapping is a P-bounded mapping, we thoroughly set up the global convergence theories of these new methods. Moreover, under the assumptions that the system matrix is monotone and the nonlinear mapping is isotone, we discuss the monotone convergence properties of the new multisplitting two-stage iteration methods, and investigate the influence of the multiple splittings as well as the relaxation parameters upon the convergence behaviours of these methods. Numerical computations show that our new methods are feasible and efficient for parallel solving of the system of weakly nonlinear equations. This revised version was published online in August 2006 with corrections to the Cover Date.     

A class of iterative methods for solving saddle point problems

Summary We consider the numerical solution of indefinite systems of linear equations arising in the calculation of saddle points. We are mainly concerned with sparse systems of this type resulting from certain discretizations of partial differential equations. We present an iterative method involving two levels of iteration, similar in some respects to the Uzawa algorithm. We relate the rates of convergence of the outer and inner iterations, proving that, under natural hypotheses, the outer iteration achieves the rate of convergence of the inner iteration. The technique is applied to finite element approximations of the Stokes equations.The work of this author was supported by the Office of Naval Research under contract N00014-82K-0197, by Avions Marcel Dassault, 78 Quai Marcel Dassault, 92214 St Cloud, France, and by Direction des Recherches Etudes et Techniques, 26 boulevard Victor, F-75996 Paris Armées, FranceThe work of this author was supported by Avions Marcel Daussault-Breguet Aviation, 78 quai Marcel Daussault, F-92214 St Cloud, France and by Direction des Recherches Etudes et Techniques, 26 boulevard Victor, F-75996 Paris Armées, FranceThe work of this author was supported by Konrad-Zuse-Zentrum für Informationstechnik Berlin, Federal Republic of Germany     

Third and fourth order iterative methods free from second derivative for nonlinear systems

In this work, we develop a family of predictor-corrector methods free from second derivative for solving systems of nonlinear equations. In general, the obtained methods have order of convergence three but, in some particular cases the order is four. We also perform different numerical tests that confirm the theoretical results and allow us to compare these methods with Newton’s classical method and with other recently published methods.     

A new family of eighth-order iterative methods for solving nonlinear equations

A family of eighth-order iterative methods with four evaluations for the solution of nonlinear equations is presented. Kung and Traub conjectured that an iteration method without memory based on n evaluations could achieve optimal convergence order 2^n-1. The new family of eighth-order methods agrees with the conjecture of Kung-Traub for the case n=4. Therefore this family of methods has efficiency index equal to 1.682. Numerical comparisons are made with several other existing methods to show the performance of the presented methods.     

On some efficient derivative-free iterative methods with memory for solving systems of nonlinear equations

We present derivative free methods with memory with increasing order of convergence for solving systems of nonlinear equations. These methods relied on the basic family of fourth order methods without memory proposed by Sharma et al. (Appl. Math. Comput. 235, 383–393, 2014). The order of convergence of new family is increased from 4 of the basic family to \(2+\sqrt {5} \approx 4.24\) by suitable variation of a free self-corrected parameter in each iterative step. In a particular case of the family even higher order of convergence \(2+\sqrt {6} \approx 4.45\) is achieved. It is shown that the new methods are more efficient in general. The presented numerical tests confirm the theoretical results.     

Numerical solution of nonlinear equations by an optimal eighth-order class of iterative methods

Solving nonlinear equations by using iterative methods is discussed in this paper. An optimally convergent class of efficient three-point three-step methods without memory is suggested. Analytical proof for the class of methods is given to show the eighth-order convergence and also reveal its consistency with the conjecture of Kung and Traub. The beauty in the proposed methods from the class can be seen because of the optimization in important effecting factors, i.e. optimality order, lesser number of functional evaluations; as well as in viewpoint of efficiency index. The accuracy of some iterative methods from the proposed derivative-involved scheme is illustrated by solving numerical test problems and comparing with the available methods in the literatures.     

Convergence theory of extrapolated iterative methods for a certain class of non-symmetric linear systems

Summary A variety of iterative methods considered in [3] are applied to linear algebraic systems of the formAu=b, where the matrixA is consistently ordered [12] and the iteration matrix of the Jacobi method is skew-symmetric. The related theory of convergence is developed and the optimum values of the involved parameters for each considered scheme are determined. It reveals that under the aforementioned assumptions the Extrapolated Successive Underrelaxation method attains a rate of convergence which is clearly superior over the Successive Underrelaxation method [5] when the Jacobi iteration matrix is non-singular.