Five-stage Milstein methods for SDEs

In this paper, we consider the problem of computing numerical solutions for Itô stochastic differential equations (SDEs). The five-stage Milstein (FSM) methods are constructed for solving SDEs driven by an m-dimensional Wiener process. The FSM methods are fully explicit methods. It is proved that the FSM methods are convergent with strong order 1 for SDEs driven by an m-dimensional Wiener process. The analysis of stability (with multidimensional Wiener process) shows that the mean-square stable regions of the FSM methods are unbounded. The analysis of stability shows that the mean-square stable regions of the methods proposed in this paper are larger than the Milstein method and three-stage Milstein methods.     

On the comparisons of the multisplitting unsymmetric aor methods for M-matrices

This paper reveals the inner links between two known frameworks of multisplitting relaxation methods as completely as possible. By meticulously investigating the specific structures of these two frameworks, the asymptotic convergence rates as well as the monotone convergence rates of them are compared theoretically. This then definitely answers the question that which converges faster between these two frameworks of parallel matrix multisplitting relaxation methods from the standpoint of pure mathematics. At last, an example further confirms the correctness of the theoretical results.     

On the local convergence of secant-type methods

In this article, we carry out a local convergence study for Secant-type methods. Our goal is to enlarge the radius of convergence, without increasing the necessary hypothesis. Finally, some numerical tests and comparisons with early results are analyzed.     

Convergence of memory gradient methods

In this paper we present a new class of memory gradient methods for unconstrained optimization problems and develop some useful global convergence properties under some mild conditions. In the new algorithms, trust region approach is used to guarantee the global convergence. Numerical results show that some memory gradient methods are stable and efficient in practical computation. In particular, some memory gradient methods can be reduced to the BB method in some special cases.     

Higher-order simultaneous methods for the determination of polynomial multiple zeros

Starting from Laguerre's method and using Newton's and Halley's corrections for a multiple zero, new simultaneous methods of Laguerre's type for finding multiple (real or complex) zeros of polynomials are constructed. The convergence order of the proposed methods is five and six, respectively. By applying the Gauss–Seidel approach, these methods are further accelerated. The lower bounds of the R-order of convergence of the improved (single-step) methods are derived. Faster convergence of all proposed methods is attained with negligible number of additional operations, which provides a high computational efficiency of these methods. A detailed convergence analysis and numerical results are given.     

Efficient two-step derivative-free iterative methods with memory and their dynamics

In this paper, we present a family of three-parameter derivative-free iterative methods with and without memory for solving nonlinear equations. The convergence order of the new method without memory is four requiring three functional evaluations. Based on the new fourth-order method without memory, we present a family of derivative-free methods with memory. Using three self-accelerating parameters, calculated by Newton interpolatory polynomials, the convergence order of the new methods with memory are increased from 4 to 7.0174 and 7.5311 without any additional calculations. Compared with the existing methods with memory, the new method with memory can obtain higher convergence order by using relatively simple self-accelerating parameters. Numerical comparisons are made with some known methods by using the basins of attraction and through numerical computations to demonstrate the efficiency and the performance of the presented methods.     

A class of iterative methods for computing polar decomposition

In this article, there is offered a parametric class of iterative methods for computing the polar decomposition of a matrix. Each iteration of this class needs only one scalar-by-matrix and three matrix-by-matrix multiplications. It is no use computing inversion, so no numerical problems can be created because of ill-conditioning. Some available methods can be included in this class by choosing a suitable value for the parameter. There are obtained conditions under which this class is always quadratically convergent. The numerical comparison performed among six quadratically convergent methods for computing polar decomposition, and a special method of this class, chosen based on a specific value for the parameter, shows that the number of iterations of the special method is considerably near that of a cubically convergent Halley's method. Ten n×n matrices with n=5, 10, 20, 50, 100 were chosen to make this comparison.     

On Computer Implementation for Comparison of Inverse Numerical Schemes for Non-Linear Equations

In this research article, we interrogate two new modifications in inverse Weierstrass iterative method for estimating all roots of non-linear equation simultaneously. These modifications enables us to accelerate the convergence order of inverse Weierstrass method from 2 to 3. Convergence analysis proves that the orders of convergence of the two newly constructed inverse methods are 3. Using computer algebra system Mathematica, we find the lower bound of the convergence order and verify it theoretically. Dynamical planes of the inverse simultaneous methods and classical iterative methods are generated using MATLAB (R2011b), to present the global convergence properties of inverse simultaneous iterative methods as compared to classical methods. Some non-linear models are taken from Physics, Chemistry and engineering to demonstrate the performance and efficiency of the newly constructed methods. Computational CPU time, and residual graphs of the methods are provided to present the dominance behavior of our newly constructed methods as compared to existing inverse and classical simultaneous iterative methods in the literature.     

Iterative methods improving newton's method by the decomposition method

In this paper, we present a sequence of iterative methods improving Newton's method for solving nonlinear equations. The Adomian decomposition method is applied to an equivalent coupled system to construct the sequence of the methods whose order of convergence increases as it progresses. The orders of convergence are derived analytically, and then rederived by applying symbolic computation of Maple. Some numerical illustrations are given.     

Convergence and stability of the balanced methods for stochastic differential equations with jumps

This paper deals with the balanced methods which are implicit methods for stochastic differential equations with Poisson-driven jumps. It is shown that the balanced methods give a strong convergence rate of at least 1/2 and can preserve the linear mean-square stability with the sufficiently small stepsize. Weak variants are also considered and their mean-square stability analysed. Some numerical experiments are given to demonstrate the conclusions.     

An acceleration of preconditioned generalized accelerated over relaxation methods for systems of linear equations

In this paper, a new type of preconditioners are proposed to accelerate the preconditioned generalized accelerated over relaxation methods presented by Zhou et al. [Preconditioned GAOR methods for solving weighted linear least squares problems, J. Comput. Appl. Math. 224 (2009), pp. 242–249] for the linear system of the generalized least-squares problem. The convergence and comparison results are obtained. The comparison results show that the convergence rates of the proposed methods are better than those of the original methods. Finally, numerical experiments are provided to confirm the results obtained in this paper.     

A trust-region algorithm combining line search filter technique for nonlinear constrained optimization

In this paper, we propose a trust-region algorithm in association with line search filter technique for solving nonlinear equality constrained programming. At current iteration, a trial step is formed as the sum of a normal step and a tangential step which is generated by trust-region subproblem and the step size is decided by interior backtracking line search together with filter methods. Then, the next iteration is determined. This is different from general trust-region methods in which the next iteration is determined by the ratio of the actual reduction to the predicted reduction. The global convergence analysis for this algorithm is presented under some reasonable assumptions and the preliminary numerical results are reported.     

Exponentially fitted variants of Newton's method with quadratic and cubic convergence

In this paper, we present some new families of Newton-type iterative methods, in which f′(x)=0 is permitted at some points. The presented approach of deriving these iterative methods is different. They have well-known geometric interpretation and admit their geometric derivation from an exponential fitted osculating parabola. Cubically convergent methods require the use of the first and second derivatives of the function as Euler's, Halley's, Chebyshev's and other classical methods do. Furthermore, new classes of third-order multipoint iterative methods free from second derivative are derived by semi-discrete modifications of cubically convergent iterative methods. Further, the approach has been extended to solve a system of non-linear equations.     

A family of Newton-type methods for solving nonlinear equations

A family of Newton-type methods free from second and higher order derivatives for solving nonlinear equations is presented. The order of the convergence of this family depends on a function. Under a condition on this function this family converge cubically and by imposing one condition more on this function one can obtain methods of order four. It has been shown that this family covers many of the available iterative methods. From this family two new iterative methods are obtained. Numerical experiments are also included.     

A mixed strategy of combining evolutionary algorithms with multigrid methods

Multigrid methods have been proven to be an efficient approach in accelerating the convergence rate of numerical algorithms for solving partial differential equations. This paper investigates whether multigrid methods are helpful to accelerate the convergence rate of evolutionary algorithms for solving global optimization problems. A novel multigrid evolutionary algorithm is proposed and its convergence is proven. The algorithm is tested on a set of 13 well-known benchmark functions. Experiment results demonstrate that multigrid methods can accelerate the convergence rate of evolutionary algorithms and improve their performance.     

An efficient modified Polak–Ribière–Polyak conjugate gradient method with global convergence properties

The conjugate gradient (CG) method is one of the most popular methods for solving large-scale unconstrained optimization problems. In this paper, a new modified version of the CG formula that was introduced by Polak, Ribière, and Polyak is proposed for problems that are bounded below and have a Lipschitz-continuous gradient. The new parameter provides global convergence properties when the strong Wolfe-Powell (SWP) line search or the weak Wolfe-Powell (WWP) line search is employed. A proof of a sufficient descent condition is provided for the SWP line search. Numerical comparisons between the proposed parameter and other recent CG modifications are made on a set of standard unconstrained optimization problems. The numerical results demonstrate the efficiency of the proposed CG parameter compared with the other CG parameters.     

Sufficient descent conjugate gradient methods for large-scale optimization problems

Sufficient descent condition is very crucial in establishing the global convergence of nonlinear conjugate gradient method. In this paper, we modified two conjugate gradient methods such that both methods satisfy this property. Under suitable conditions, we prove the global convergence of the proposed methods. Numerical results show that the proposed methods are efficient for the given test problems.     

Split-step double balanced approximation methods for stiff stochastic differential equations

In the modelling of many important problems in science and engineering we face stiff stochastic differential equations (SDEs). In this paper, a new class of split-step double balanced (SSDB) approximation methods is constructed for numerically solving systems of stiff Itô SDEs with multi-dimensional noise. In these methods, an appropriate control function has been used twice to improve the stability properties. Under global Lipschitz conditions, convergence with order one in the mean-square sense is established. Also, the mean-square stability (MS-stability) properties of the SSDB methods have been analysed for a one-dimensional linear SDE with multiplicative noise. Therefore, the MS-stability functions of SSDB methods are determined and in some special cases, their regions of MS-stability have been compared to the stability region of the original equation. Finally, simulation results confirm that the proposed methods are efficient with respect to accuracy and computational cost.     

A nonmonotone SQP-filter method for equality constrained optimization

In this paper, we propose a nonmonotone sequential quadratic programming-filter method for solving nonlinear equality constrained optimization. This new method has more flexibility for the acceptance of the trial step and requires less computational costs compared with the monotone methods. Under reasonable conditions, we give the global convergence properties. Further, the second-order correction step and nonmonotone reduction conditions are used to overcome Maratos effect so that quadratic local convergence is achieved. The numerical experiments are reported to show the effectiveness of the proposed algorithm.