Convergence analysis for the Secant method based on new recurrence relations

A new convergence theorem for the Secant method in Banach spaces based on new recurrence relations is established for approximating a solution of a nonlinear operator equation. It is assumed that the divided difference of order one of the nonlinear operator is Lipschitz continuous. The convergence conditions differ from some existing ones and are easily satisfied. The results of the paper are justified by numerical examples that cannot be handled by earlier works.

Deflection Measurement for Bridges Based on Secant Inclination

Deflection is a significant indicator of bridge’s strength and its whole stiffness, so the research on deflection measurement is an important aspect of bridge health monitoring. There have existed many measurement methods of bridge deflection so far, while inclination method is gradually catching more and more attention for its fair obviously comprehensive advantages. However, the inclination method at present focuses on measuring the rotation of bridge’s section at testing point, that is, the tangent angle of deflection curve. With the tangent angle, the deflection curve can be determined by the methods of curve fitting or (and) integration or conjugate beam. The methods mentioned above, are not only complicated in calculation, but also bad in accuracy. The deflection measurement method proposed by this paper is based on measuring the inclination of two points initiatively in horizontal line, that is, the secant angle of the deflection curve, and on the simple triangle function operation. The proposed method is simple in theory, but good in accuracy for either static or dynamic load. The numerical simulation suggests that the error of the proposed method is less than 1%.

A weak condition for secant method to solve systems of nonlinear equations

In this paper, a new weak condition for the convergence of secant method to solve the systems of nonlinear equations is proposed. A convergence ball with the center x0 is replaced by that with xl, the first approximation generated by the secant method with the initial data x-1 and x0. Under the bounded conditions of the divided difference, a convergence theorem is obtained and two examples to illustrate the weakness of convergence conditions are provided. Moreover, the secant method is applied to a system of nonlinear equations to demonstrate the viability and effectiveness of the results in the paper.

The convergence ball and error analysis of the two-step Secant method

Under the assumption that the nonlinear operator has Lipschitz continuous divided differences for the first order,we obtain an estimate of the radius of the convergence ball for the two-step secant method.Moreover,we also provide an error estimate that matches the convergence order of the two-step secant method.At last,we give an application of the proposed theorem.

HOMOCENTRIC CONVERGENCE BALL OF THE SECANT METHOD

A local convergence theorem and five semi-local convergence theorems of the secant method are listed in this paper. For every convergence theorem, a convergence ball is respectively introduced, where the hypothesis conditions of the corresponding theorem can be satisfied. Since all of these convergence balls have the same center x^＊, they can be viewed as a homocentric ball. Convergence theorems are sorted by the different sizes of various radii of this homocentric ball, and the sorted sequence represents the degree of weakness on the conditions of convergence theorems.

A Scaled Conjugate Gradient Method Based on New BFGS Secant Equation with Modified Nonmonotone Line Search

In this paper, we provide and analyze a new scaled conjugate gradient method and its performance, based on the modified secant equation of the Broyden-Fletcher-Goldfarb-Shanno (BFGS) method and on a new modified nonmonotone line search technique. The method incorporates the modified BFGS secant equation in an effort to include the second order information of the objective function. The new secant equation has both gradient and function value information, and its update formula inherits the positive definiteness of Hessian approximation for general convex function. In order to improve the likelihood of finding a global optimal solution, we introduce a new modified nonmonotone line search technique. It is shown that, for nonsmooth convex problems, the proposed algorithm is globally convergent. Numerical results show that this new scaled conjugate gradient algorithm is promising and efficient for solving not only convex but also some large scale nonsmooth nonconvex problems in the sense of the Dolan-Moré performance profiles.

M-TIMES SECANT-LIKE MULTI-PROJCTION METHOD FOR SPARSE MINIMIZATION PROBLEM

In this paper, we present m time secant like multi projection algorithm for sparse unconstrained minimization problem. We prove this method are all q superlinearly convergent to the solution about m≥1 . At last, we from some numerical results, discuss how to choose the number m to determine the approximating matrix properly in practical use.

CONVERGENCE PROPERTIES OF IMPROVED SECANT METHODS WITH TRUST REGION MULTIPLIER

The secant methods discussed by Fontecilla (in 1988) are considerably revised through employing a trust region multiplier strategy and introducing a nondifferentiable merit function. In this paper the secant methods are also improved by adding a dogleg typed movement which allows to overcome a phenomena similar to the Maratos effect. Furthermore, these algorithms are analyzed and global convergence theorems as well as local superlinear convergence rate are proved.

THE RELATIONSHIP BETWEEN THE SPARSE SYMMETRIC BROYDEN METHOD AND THE M-TIME SECANT-LIKE MULTI-PROJECTION METHOD

In this paper, we discuss the relationship between the sparse symmetric Broyden (SPSB) method [1, 2] and m-time secant-like multi-projection (SMP) method [3] and prove that when m goes to infinity, the SMP method is corresponding to the SPSB method.

A modified BFGS algorithm based on a hybrid secant equation

By making a convex combination of the modified secant equations proposed by Yuan and Wei et al., a hybrid secant equation and also, a modified BFGS algorithm is proposed. The hybridization parameter is effectively computed using the available information of recent iterations. Under proper conditions, it is shown that the proposed algorithm is globally, locally and superlinearly convergent. By using the performance profile introduced by Dolan and Mor6, a comparison between the implementations of the proposed algorithm and two efficient modified BFGS algorithms proposed by Yuan and Wei et al., on a set of unconstrained optimization test problems from the CUTEr collection, is done. Numerical results demonstrating the efficiency of the proposed modified BFGS algorithm are reported.

A LINE SEARCH FILTER SECANT METHOD FOR NONLINEAR EQUALITY CONSTRAINED OPTIMIZATION

This paper formulates and analyzes a line search method for general nonlinear equalityconstrained optimization based on filter methods for step acceptance and secant methods for searchdirection.The feature of the new algorithm is that the secant algorithm is used to produce a searchdirection,a backtracking line search procedure is used to generate step size,some filtered rules areused to determine step acceptance,second order correction technique is used to reduce infeasibility andovercome the Maratos effect.Global convergence properties of this method are analyzed:under mildassumptions it is showed that every limit point of the sequence of iterates generated by the algorithmis feasible,and that there exists at least one limit point that is a stationary point for the problem.Moreover,it is also established that the Maratos effect can be overcome in our new approach by addingsecond order correction steps so that fast local superlinear convergence to a second order sufficient localsolution is achieved.Finally,the results of numerical experiments are reported to show the effectivenessof the line search filter secant method.

Micro-mechanical model for predicting the elasto-plastic behavior of composites based on secant formulation method

To predict the Elasto-Plastic Behaviors(EPBs)of aligned inclusions reinforced composites,this paper develops an interpolative Mori-Tanaka/Double-Inclusion(MT-DI)homogenization model with the secant formulation,and gives the numerical implementation algorithms of the developed MT-DI model with the secant formulation.The Finite Element(FE)homogenization method is implemented to provide the "exact" EPBs of the composites and thus validate the MT-DI model with the secant formulation.The MT-DI model with the 2 nd-order secant formulation is validated to provide the more accurate predictions,while the MT-DI model with the 1 st-order secant formulation always gives the stiffer predictions.The results show that using the macro-stress and macrostrain as the inputs,the MT-DI model with the secant formulation gives the identical predictions.The predictions of the MT-DI model with the secant formulation vary between those of the MT and DI models with the secant formulation.Meanwhile,the MT-DI model with the secant formulation does not predict the accurate EPBs for the phases of the composites.

Probabilistic analysis of secant piles with random geometric imperfections

The failure to achieve minimum design overlap between secant piles compromises the ability of a structure to perform as designed,resulting in water leakage or even ground collapse.To establish a more realistic simulation and provide guidelines for designing a safe and cost-effective secant-pile wall,a three-dimensional model of a secant pile,considering the geometric imperfections of the diameter and direction of the borehole,is introduced.An ultrasonic cross-hole test was performed during the construction of secant piles in a launching shaft in Beijing,China.Based on the test results,the statistical characteristics of the pile diameters and orientation parameters were obtained.By taking the pile diameter D,inclination angleβ,and azimuth angleαas random variables,Monte Carlo simulations were performed to discuss the influence of different design parameters on the probability density functions of the overlap of secant piles.The obtained results show that the randomness of the inclination angle and pile diameter can be well described by a normal distribution,whereas the azimuth angle is more consistent with a uniform distribution.The integrity of the secant-pile wall can be overestimated without considering the uncertainty of geometric imperfections.The failure of the secant-pile wall increases substantially with increasing spatial variability in drilling inclination and diameter.A design flowchart for pile spacing under the target safety level is proposed to help engineers design a safe and economical pile wall.

The Substitution Secant/Finite Difference Method for Large Scale Sparse Unconstrained Optimization

This paper studies a substitution secant/finite difference （SSFD） method for solving large scale sparse unconstrained optimization problems. This method is a combination of a secant method and a finite difference method, which depends on a consistent partition of the columns of the lower triangular part of the Hessian matrix. A q-superlinear convergence result and an r-convergence rate estimate show that this method has good local convergence properties. The numerical results show that this method may be competitive with some currently used algorithms.

A FILTER SECANT METHOD WITH NONMONOTONE LINE SEARCH FOR EQUALITY CONSTRAINED OPTIMIZATION

This paper proposes a filter secant method with nonmonotone line search for non-linearequality constrained optimization.The Hessian of the Lagrangian is approximated using the BFGSsecant update.This new method has more flexibility for the acceptance of the trial step and requires lesscomputational costs compared with the monotone one.The global and local convergence of the proposedmethod are given under some reasonable conditions.Further,two-step Q-superlinear convergence rateis established by introducing second order correction step.The numerical experiments are reported toshow the effectiveness of the proposed algorithm.

A FAMILY OF THE LOCAL CONVERGENCE OF THE IMPROVED SECANT METHODS FOR NONLINEAR EQUALITY CONSTRAINED OPTIMIZATION SUBJECT TO BOUNDS ON VARIABLES

This paper studies a family of the local convergence of the improved secant methods for solving the nonlinear equality constrained optimization subject to bounds on variables. The Hessian of the Lagrangian is approximated using the DFP or the BFGS secant updates. The improved secant methods are used to generate a search direction. Combining with a suitable step size, each iterate switches to trial step of strict interior feasibility. When the Hessian is only positive definite in an affine null subspace, one shows that the algorithms generate the sequences converging q-linearly and two-step q-superlinearly. Yhrthermore, under some suitable assumptions, some sequences generated by the algorithms converge locally one-step q-superlinearly. Finally, some numerical results are presented to illustrate the effectiveness of the proposed algorithms.

New types of complex torsional angle transformers based on hyperbolic secant taper

The new type of torsional angle transformer-quarter-wave length hyperbolic secant taper was derived in which the maximum point of surface shear stress is inexistent. The half-wave length complex horns can be composed on the basis of the new taper and quarter-wave taper which may be cylindrical exponential, sirnilar-catenary or similar-cosinc. The resonant frequency and angle amplification ratio of some complex transformers were measured. The experimental results basically agree with the theoretical values. The complex horns have a serics of good characteristics, such as the amplification ratio is very large, the changc of input impedance with the deviation of the frequency is little, the maximum output ultrasonic power is much higher than that of other types of the same material. Therefore, they are very useful horns in high-intensity ultrasonic field.

两阶段金融衍生品清算问题的半定规划松弛方法

在不限制暂时性及永久性价格影响参数大小关系下,研究两阶段金融衍生品清算问题的半定规划(Semi-definite programming,SDP)松弛方法,其优化模型为一个带有线性和单个非凸二次约束的非凸二次规划(Quadratically constrained quadratic program,QCQP)问题。针对该非凸QCQP问题,给出了一个带有Secant割的SDP松弛,并估计了它与原问题之间的间隙。随机例子的数值结果表明该SDP松弛可以得到原问题更紧的上界,从而为寻求原问题的一个好的近似解提供方法。

Mechanical property of cylindrical sandwich shell with gradient core of entangled wire mesh

To explore the wide-frequency damping and vibration-attenuation performances in the application of aerospace components,the cylindrical sandwich shell structure with a gradient core of entangled wire mesh was proposed in this paper.Firstly,the gradient cores of entangled wire mesh in the axial and radial directions were prepared by using an in-house Numerical Control weaving machine,and the metallurgical connection between skin sheets and the gradient core was performed using vacuum brazing.Secondly,to investigate the mechanical properties of cylindrical sandwich shells with axial or radial gradient cores,quasi-static and dynamic mechanical experiments were carried out.The primary evaluations of mechanical properties include secant stiffness,natural frequency,Specific Energy Absorption(SEA),vibration acceleration level,and so on.The results suggest that the vibration-attenuation performance of the sandwich shell is remarkable when the high-density core layer is at the end of the shell or abuts the inner skin.The axial gradient material has almost no influence on the vibration frequencies of the shell,whereas the vibration frequencies increase dramatically when the high-density core layer approaches the skin.Moreover,compared to the conventional sandwich shells,the proposed functional grading cylindrical sandwich shell exhibits more potential in mass reduction,stiffness designing,and energy dissipation.

改进的分层总和法在黄土地区高填方地基沉降变形中的应用研究

黄土高填方沉降变形控制是工程建设的重点和难点。通过室内侧限压缩试验和加湿试验,总结了重塑黄土在自重应力和含水率变化下的变形特征。基于分层总和法的思想,引入了Gunary模型和割线模量法,将黄土高填方地基的沉降分为荷载引起的施工沉降和含水率变化引起的施工后沉降,并提出了黄土高填方地基沉降变形的计算方法。结合实际工程背景,预测了高填方工程的沉降变形。结果表明:黄土高填方沉降的主要来源是施工期荷载引起的沉降;随着填土高度的增加,工后沉降占总沉降的比例逐渐增大;在最优含水率条件下,当施工期压实度达到95%以上时即可完成土的排气固结,施工后期为土的排水固结;压实度的控制对高填方工程至关重要;该研究成果丰富了黄土高填方地基沉降变形的计算理论,为黄土高填方工后沉降和湿陷性沉降的研究提供了理论依据,并对黄土高填方工程的施工及工后沉降控制具有一定的参考价值。