A new method of secant-like for nonlinear equations

In this paper, a new method for solving nonlinear equations f(x) = 0 is presented. In many literatures the derivatives are used, but the new method does not use the derivatives. Like the method of secant, the first derivative is replaced with a finite difference in this new method. The new method converges not only faster than the method of secant but also Newton’s method. The fact that the new method’s convergence order is 2.618 is proved, and numerical results show that the new method is efficient.     

A new logarithmic-quadratic proximal method for nonlinear complementarity problems

In this paper, we propose a new modified logarithmic-quadratic proximal (LQP) method for solving nonlinear complementarity problems (NCP). We suggest using a prediction-correction method to solve NCP. The predictor is obtained via solving the LQP system approximately under significantly relaxed accuracy criterion and the new iterate is computed by using a new step size α_k. Under suitable conditions, we prove that the new method is globally convergent. We report preliminary computational results to illustrate the efficiency of the proposed method. This new method can be considered as a significant refinement of the previously known methods for solving nonlinear complementarity problems.     

The prediction–correction approach to nonlinear complementarity problems

This paper presents a prediction–correction approach to solving the nonlinear complementarity problem (NCP). Each iteration of the new method consists of a prediction and a correction. The predictor is produced by an inexact Logarithmic-Quadratic Proximal method; and then it is corrected by the Proximal Point Algorithm. Convergence of the new method is proved under mild assumptions. Comparison to existing methods shows the superiority of the new method. Numerical experiments including the application to traffic equilibrium problems demonstrate that the new method is attractive in practice.     

A new direct search method based on separable fractional interpolation model

In this paper, we propose a new separable fractional interpolation model which can be established by 2n interpolation points where n is the number of variables. Based on this model, a new direct search method is presented. In this method, a new iterate is determined by solving the fractional interpolation model in trust region. Under mild assumptions, the convergence results of this method are given and proved. Numerical experiments show that the new method is promising.     

Exponentially-fitted Runge-Kutta-Nyström method for the numerical solution of initial-value problems with oscillating solutions

A new approach for constructing efficient Runge-Kutta-Nyström methods is introduced in this paper. Based on this new approach a new exponentially-fitted Runge-Kutta-Nyström fourth-algebraic-order method is obtained for the numerical solution of initial-value problems with oscillating solutions. The new method has an extended interval of periodicity. Numerical illustrations on well-known initial-value problems with oscillating solutions indicate that the new method is more efficient than other ones.     

Third-order modification of Newton's method

In this paper, we present a new modification of Newton's method for solving non-linear equations. Analysis of convergence shows that the new method is cubically convergent. Numerical examples show that the new method can compete with the classical Newton's method.     

An improved Goldstein's type method for a class of variant variational inequalities

This paper aims at presenting an improved Goldstein's type method for a class of variant variational inequalities. In particular, the iterate computed by an existing Goldstein's type method [He, A Goldstein's type projection method for a class of variant variational inequalities J. Comput. Math. 17(4) (1999) 425–434]. is used to construct a descent direction, and thus the new method generates the new iterate by searching the optimal step size along the descent direction. Some restrictions on the involving functions of the existing Goldstein's type methods are relaxed, while the global convergence of the new method is proved without additional assumptions. The computational superiority of the new method is verified by the comparison to some existing methods.     

A variant of Chebyshev’s method with sixth-order convergence

In this paper, we present a new variant of Chebyshev’s method for solving non-linear equations. Analysis of convergence shows that the new method has sixth-order convergence. Per iteration the new method requires two evaluations of the function, one of its first derivative and one of its second derivative. Thus the efficiency, in term of function evaluations, of the new method is better than that of Chebyshev’s method. Numerical examples verifying the theory are given.      

A new compound Riccati equations rational expansion method and its application to the (2 + 1)-dimensional asymmetric Nizhnik–Novikov–Vesselov system

In this paper, based on a new general ansätze and symbolic computation, a new compound Riccati equations rational expansion method is proposed. Being concise and straightforward, it is applied to the (2 + 1)-dimensional asymmetric Nizhnik–Novikov–Vesselov system. It is shown that more complexiton solutions can be found by this new method. The method can be applied to other nonlinear partial differential equations in mathematical physics.     

An efficient Levenberg–Marquardt method with a new LM parameter for systems of nonlinear equations

A modified Levenberg–Marquardt method for solving singular systems of nonlinear equations was proposed by Fan [J Comput Appl Math. 2003;21;625–636]. Using trust region techniques, the global and quadratic convergence of the method were proved. In this paper, to improve this method, we decide to introduce a new Levenberg–Marquardt parameter while also incorporate a new nonmonotone technique to this method. The global and quadratic convergence of the new method is proved under the local error bound condition. Numerical results show the new algorithm is efficient and promising.     

On the Higher-Order Method for the Solution of a Nonlinear Scalar Equation

In this paper, a higher-order method for the solution of a nonlinear scalar equation is presented. It is proved that the new method is locally convergent with an order of (m+2), where m is the highest order derivative used in the iterative formula. Some numerical examples are used to demonstrate the new method.     

非线性耦合Klein-Gordon方程组和耦合Schr\"{o}dinger-Boussinesq方程组的双行波解

本文提出了一种全新复合$(\frac{G''}{G})$展开方法,运用这种新方法并借助符号计算软件构造了非线性耦合Klein-Gordon方程组和耦合Schr\"{o}dinger-Boussinesq方程组的多种双行波解,包括双双曲正切函数解,双正切函数解,双有理函数解以及它们的混合解. 复合$(\frac{G''}{G})$展开方法不但直接有效地求出了两类非线性偏微分方程的双行波解,而且扩大了解的范围.这种新方法对于研究非线性偏微分方程具有广泛的应用意义.     

Solution of Ill-Posed Problems via Adaptive Grid Regularization: Convergence Analysis

A new regularization method, adaptive grid regularization, has been presented. Numerical results there show in a convincing way that this method is a powerful tool to identify discontinuities of solutions of ill-posed problems. It is the aim of this paper to give a convergence analysis for this new method.     

An improved proximal alternating direction method for monotone variational inequalities with separable structure

To solve a class of variational inequalities with separable structure, this paper presents a new method to improve the proximal alternating direction method (PADM) in the following senses: an iterate generated by the PADM is utilized to generate a descent direction; and an appropriate step size along this descent direction is identified. Hence, a descent-like method is developed. Convergence of the new method is proved under mild assumptions. Some numerical results demonstrate that the new method is efficient.     

A fast and efficient two-grid method for solving d-dimensional poisson equations

The aim of this paper is to introduce a fast and efficient new two-grid method to solve the d-dimensional (d=1,2,3) Poisson elliptic equations. The finite difference equations at all interior grid points form a large sparse linear system, which needs to be solved efficiently. The solution cost of this sparse linear system usually dominates the total cost of solving the discretized partial differential equation. The finite difference equations are based on applying a finite difference scheme of two- and four-orders (compact finite difference method) for discretizing the spatial derivative. The obtained linear systems of Poisson elliptic equations have been solved by a new two-grid (NTG) method and we also note that the NTG method which is used for solving the large sparse linear systems is faster and more effective than that of the standard two-grid method. We utilize the local Fourier analysis to show that the spectral radius of the new two-grid method for 1D and 2D models is less than that of the standard two-grid method. As well as, we expand the corresponding algorithm to the new multi-grid method. The numerical examples show the efficiency of the new algorithms for solving the d-dimensional Poisson equations.     

Speeding up elliptic curve discrete logarithm computations with point halving

Pollard rho method and its parallelized variants are at present known as the best generic algorithms for computing elliptic curve discrete logarithms. We propose new iteration function for the rho method by exploiting the fact that point halving is more efficient than point addition for elliptic curves over binary fields. We present a careful analysis of the alternative rho method with new iteration function. Compared to the previous r-adding walk, generally the new method can achieve a significant speedup for computing elliptic curve discrete logarithms over binary fields. For instance, for certain NIST-recommended curves over binary fields, the new method is about 12–17% faster than the previous best methods.     

Semi-convergence analysis of preconditioned deteriorated PSS iteration method for singular saddle point problems

In this paper, we propose a two-parameter preconditioned variant of the deteriorated PSS iteration method (J. Comput. Appl. Math., 273, 41–60 (2015)) for solving singular saddle point problems. Semi-convergence analysis shows that the new iteration method is convergent unconditionally. The new iteration method can also be regarded as a preconditioner to accelerate the convergence of Krylov subspace methods. Eigenvalue distribution of the corresponding preconditioned matrix is presented, which is instructive for the Krylov subspace acceleration. Note that, when the leading block of the saddle point matrix is symmetric, the new iteration method will reduce to the preconditioned accelerated HSS iteration method (Numer. Algor., 63 (3), 521–535 2013), the semi-convergence conditions of which can be simplified by the results in this paper. To further improve the effectiveness of the new iteration method, a relaxed variant is given, which has much better convergence and spectral properties. Numerical experiments are presented to investigate the performance of the new iteration methods for solving singular saddle point problems.     

Parallel multisplitting two-stage iterative methods with general weighting matrices for non-symmetric positive definite systems

In this work, we propose a new parallel multisplitting iterative method for non-symmetric positive definite linear systems. Based on optimization theory, the new method has two great improvements; one is that only one splitting needs to be convergent, and the other is that the weighting matrices are not scalar and nonnegative matrices. The convergence of the new parallel multisplitting iterative method is discussed. Finally, the numerical results show that the new method is effective.     

A bias corrected nonparametric regression estimator

In this article, we propose a new method of bias reduction in nonparametric regression estimation. The proposed new estimator has asymptotic bias order h⁴, where h is a smoothing parameter, in contrast to the usual bias order h² for the local linear regression. In addition, the proposed estimator has the same order of the asymptotic variance as the local linear regression. Our proposed method is closely related to the bias reduction method for kernel density estimation proposed by Chung and Lindsay (2011). However, our method is not a direct extension of their density estimate, but a totally new one based on the bias cancelation result of their proof.     

A new LQP alternating direction method for solving variational inequality problems with separable structure

We presented a new logarithmic-quadratic proximal alternating direction scheme for the separable constrained convex programming problem. The predictor is obtained by solving series of related systems of non-linear equations in a parallel wise. The new iterate is obtained by searching the optimal step size along a new descent direction. The new direction is obtained by the linear combination of two descent directions. Global convergence of the proposed method is proved under certain assumptions. We show the O(1 / t) convergence rate for the parallel LQP alternating direction method.