Highly Accurate Golden Section Search Algorithms and Fictitious Time Integration Method for Solving Nonlinear Eigenvalue Problems

This study sets up two new merit functions,which are minimized for the detection of real eigenvalue and complex eigenvalue to address nonlinear eigenvalue problems.For each eigen-parameter the vector variable is solved from a nonhomogeneous linear system obtained by reducing the number of eigen-equation one less,where one of the nonzero components of the eigenvector is normalized to the unit and moves the column containing that component to the right-hand side as a nonzero input vector.1D and 2D golden section search algorithms are employed to minimize the merit functions to locate real and complex eigenvalues.Simultaneously,the real and complex eigenvectors can be computed very accurately.A simpler approach to the nonlinear eigenvalue problems is proposed,which implements a normalization condition for the uniqueness of the eigenvector into the eigenequation directly.The real eigenvalues can be computed by the fictitious time integration method(FTIM),which saves computational costs compared to the one-dimensional golden section search algorithm(1D GSSA).The simpler method is also combined with the Newton iterationmethod,which is convergent very fast.All the proposed methods are easily programmed to compute the eigenvalue and eigenvector with high accuracy and efficiency.

Parameter optimization of gravity density inversion based on correlation searching and the golden section algorithm

For density inversion of gravity anomaly data, once the inversion method is determined, the main factors affecting the inversion result are the inversion parameters and subdivision scheme. A set of reasonable inversion parameters and subdivision scheme can, not only improve the inversion process efficiency, but also ensure inversion result accuracy. The gravity inversion method based on correlation searching and the golden section algorithm is an effective potential field inversion method. It can be used to invert 2D and 3D physical properties with potential data observed on flat or rough surfaces. In this paper, we introduce in detail the density inversion principles based on correlation searching and the golden section algorithm. Considering that the gold section algorithm is not globally optimized. we present a heuristic method to ensure the inversion result is globally optimized. With a series of model tests, we systematically compare and analyze the inversion result efficiency and accuracy with different parameters. Based on the model test results, we conclude the selection principles for each inversion parameter with which the inversion accuracy can be obviously improved.

Application of Golden Section Method to Channel Transect Design

During the desing of channel transect,the paper brings forward golden section method,which is 0.618 methods.In order to reduce the calculation volume of the natural depth of water h 0 and bottomˉwidth b which apply trial calculation method and graphic method,and improve the calculate precision,the mathematical model has been built up,the writer combines example to explain the train of thought,the result shows that the calculation precision is high,the correctness is tested and verified by the result which is calculated by hand.It can be referred to the hydroelectric works.

Solution of rectangular bar forging with bulging of sides by strain rate vector inner-product

Abstract：ools. The effective strain rate for for determining the total pressure developed during forging of a rectangular bar forging with bulging was expressed in terms of four-dimensional strain rate vector. The inner-product of the vector was termwise integrated and summed. The integral mean value theorem was applied to determining the ratio of the strain rate components and the values of direction cosine of the vector and then an analytical solution of stress effective factor was obtained. The compression experiments of pure lead bar were performed to test the accuracy of the solution. The optimized results of total pressure by golden section search were compared with those of the indicator readings of the testing machine. It indicates that the optimized total pressures are 2.60%-10.14% higher than those measured. The solution is available and still an upper-bound solution.

Proportional Fairness-Based Energy-Efficient Power Allocation in Downlink MIMO-NOMA Systems with Statistical CSI

In this paper, proportional fairness(PF)-based energy-efficient power allocation is studied for multiple-input multiple-output(MIMO) non-orthogonal multiple access(NOMA) systems. In our schemes, statistical channel state information(CSI) is utilized for perfect CSI is impossible to achieve in practice. PF is used to balance the transmission efficiency and user fairness. Energy efficiency(EE) is formulated under basic data rate requirements and maximum transmitting power constraints. Due to the non-convex nature of EE, a two-step algorithm is proposed to obtain sub-optimal solution with a low complexity. Firstly, power allocation is determined by golden section search for fixed power. Secondly total transmitting power is determined by fractional programming method in the feasible regions. Compared to the performance of MIMO-NOMA without PF constraint, fairness is obtained at expense of decreasing of EE.

Study on maximum efficiency control strategy for induction motor

Two new techniques for efficiency-optimization control(EOC) of induction motor drives were proposed. The first method combined Loss Model and "golden section technique", which was faster than the available methods. Secondly, the low-frequency ripple torque due to decrease of rotor flux was compensated in a feedforward manner. If load torque or speed command changed, the efficiency search algorithm would be abandoned and the rated flux would be established to get the best transient response. The close agreement between the simulation and the experimental results confirmed the validity and usefulness of the proposed techniques.

Trajectory tracking control of characteristic model for nonplanar hex-rotor UAV

The nonplanar hex-rotor unmanned aerial vehicle(UAV)has much higher driving property,greater payload capacity and damage tolerance than quad-rotor UAV.It is difficult to design a highperformance controller of easy engineering implementation for strongly coupled nonlinear hex-rotorUAV system.In response to this practical problem,an adaptive trajectory tracking control based oncharacteristic model for nonplanar hex-rotor is studied.Firstly,the dynamic model for the hex-rotorUAV is devised.Secondly,according to dynamic characteristics,environmental characteristics andcontrol performance requirements,the characteristic model of the hex-rotor UAV is constructed.Then,based on the characteristic model,a golden section adaptive controller is designed to realizetrajectory tracking.Furthermore,the stability analysis of the closed loop hex-rotor system is given.Finally,the validity of the proposed trajectory tracking control method adopted in the nonplanar hex-rotor UAV is demonstrated via numerical simulations and hex-rotor prototype experiments.

Fortified Financial Forecasting Models Based on Non-Linear Searching Approaches

The paper＇s aim is how to forecast data with variations involving at times series data to get the best forecasting model. When researchers are going to forecast data with variations involving at times series data （i.e., secular trends, cyclical variations, seasonal effects, and stochastic variations）, they believe the best forecasting model is the one which realistically considers the underlying causal factors in a situational relationship and therefore has the best ＂track records＂ in generating data. Paper＇s models can be adjusted for variations in related a time series which processes a great deal of randomness, to improve the accuracy of the financial forecasts. Because of Na＇fve forecasting models are based on an extrapolation of past values for future. These models may be adjusted for seasonal, secular, and cyclical trends in related data. When a data series processes a great deal of randomness, smoothing techniques, such as moving averages and exponential smoothing, may improve the accuracy of the financial forecasts. But neither Na＇fve models nor smoothing techniques are capable of identifying major future changes in the direction of a situational data series. Hereby, nonlinear techniques, like direct and sequential search approaches, overcome those shortcomings can be used. The methodology which we have used is based on inferential analysis. To build the models to identify the major future changes in the direction of a situational data series, a comparative model building is applied. Hereby, the paper suggests using some of the nonlinear techniques, like direct and sequential search approaches, to reduce the technical shortcomings. The final result of the paper is to manipulate, to prepare, and to integrate heuristic non-linear searching methods to serve calculating adjusted factors to produce the best forecast data.

S-shaped growth curves in fermentations and golden ratio

A model of the growth curve of microorganisms was proposed,which reveals a relation-ship with the number of a‘golden section’,1.618…,for main parameters of the growth curves.The treatment mainly concerns the ratio of the maximum asymptotic value of biomass in the phase of slow growth to the real value of biomass accumulation at the end of exponential growth,which is equal to thc square of the'golden section',i.e.,2.618.There are a few relevant theorems to explain these facts.New,yet simpler,methods were considered for deterrmining the model parameters based on hyperbolic functions.A comparison was made with one of the alternative models to demonstrate the advantage of the proposed model.The proposed model should be useful to apply at various stages of fermentation in scientific and industrial units.Further,the model could give a new impetus to the development of new mathematical knowledge regarding the algebra of the‘golden section'as a whole,as well as in connection with the introduction of a new equation at decomposing of any roots with any degrees for differences between constants and/or variables.

Bivariate Splines and Golden Section Based on Theory of Elasticity

Splines are important in both mathematics and mechanics. We investigate the relationships between bivariate splines and mechanics in this paper. The mechanical meanings of some univariate splines were viewed based on the analysis of bending beams. For the 2D case, the relationships between a class of quintic bivariate splines with smoothness 3 and bending of thin plates are presented constructively. Furthermore, the variational property of bivariate splines and golden section in splines are also discussed.

An improved neural network model for battery smarter state-of-charge estimation of energy-transportation system

The safety and reliability of battery storage systems are critical to the mass roll-out of electrified transportation and new energy generation.To achieve safe management and optimal control of batteries,the state of charge(SOC)is one of the important parameters.The machine-learning based SOC estimation methods of lithium-ion batteries have attracted substantial interests in recent years.However,a common problem with these models is that their estimation performances are not always stable,which makes them difficult to use in practical applications.To address this problem,an optimized radial basis function neural network(RBF-NN)that combines the concepts of Golden Section Method(GSM)and Sparrow Search Algorithm(SSA)is proposed in this paper.Specifically,GSM is used to determine the optimum number of neurons in hidden layer of the RBF-NN model,and its parameters such as radial base center,connection weights and so on are optimized by SSA,which greatly improve the performance of RBF-NN in SOC estimation.In the experiments,data collected from different working conditions are used to demonstrate the accuracy and generalization ability of the proposed model,and the results of the experiment indicate that the maximum error of the proposed model is less than 2%.

Optimal flight parameters of unmanned helicopter for tea plantation frost protection

To determine proper flight parameters of an unmanned helicopter for tea plantation frost protection,field experiments were conducted to study the impact of flight height,speed and interval on airflow disturbance and temperature rise around tea canopies based on the analysis and simulation of frost protection with a certain helicopter.The relationship between temperature rise after flight and the above flight parameters was established through a regression orthogonal experiment,based on which the optimal combination of flight parameters was obtained through the single-factor golden section method.The results showed that wind speed around tea canopies decreased with the increase of flight height when flight speed was constant.There was a multivariate linear relationship between temperature rise and flight parameters,and the sequence of flight parameters’influence on frost protection effect was flight interval,flight height,flight speed.The optimal combination of flight parameters were flight height of 4.0 m,flight speed of 6.0 m/s and flight interval of 20 min.After the flight with the above parameters air temperature around tea canopies increased 1.6℃ when background thermal inversion strength was 3.8℃.

Optimization of aero-engine pipeline for avoiding vibration based on length adjustment of straight-line segment

In the design and troubleshooting of aero-engine pipeline,the vibration reduction of the pipeline system is often achieved by adjusting the hoop layout,provided that the shape of pipeline remains unchanged.However,in reality,the pipeline system with the best antivibration performance may be obtained only by adjusting the pipeline shape.In this paper,a typical spatial pipeline is taken as the research object,the length of straight-line segment is taken as the design variable,and an innovative optimization method of avoiding vibration of aero-engine pipeline is proposed.The relationship between straight-line segment length and parameters that determine the geometric characteristics of the pipeline,such as the position of key reference points,bending angle,and hoop position,are derived in detail.Based on this,the parametric finite element model of the pipeline system is established.Taking the maximum first-order natural frequency of pipeline as the optimization objective and introducing process constraints and vibration avoidance constraints,the optimization model of the pipeline system is established.The genetic algorithm and the golden section algorithm are selected to solve the optimization model,and the relevant solution procedure is described in detail.Finally,two kinds of pipelines with different total lengths are selected to carry out a case study.Based on the analysis of the influence of straight-line segment length on the vibration characteristics of the pipeline system,the optimization methods developed in this paper are demonstrated.Results show that the developed optimization method can obtain the optimal single value or interval of the straight-line segment length while avoiding the excitation frequency.In addition,the optimization efficiency of the golden section algorithm is remarkably higher than that of the genetic algorithm for length optimization of a single straight-line segment.

A SEARCH TECHNIQUE FOR CHOOSING OPTIMUM PARAMETERS IN NUMERICAL WEATHER PREDICTION

The steepest descent(or ascent)search is employed for finding optimum diffusion coefficients in T42L9G model,with a view to improving the model's computational stability or prediction accuracy.The method of the steepest descent search is first described,in which the golden section search is chosen as the fundamental one- dimensional search used in the multi-dimentional steepest descent search,and then the optimization of the dif- fusion coefficients is described.

ON THE OPTIMALITY IN GENERAL SENSE FOR ODD-BLOCK SEARCH

In his classical article[3], J. Kiefer introduced the Fibonacci search as a direct optimal method. The optimality was proved under the restriction: the total number of tests is given in advance and fixed. To avoid this restriction, some different concepts of optimality were proposed and some corresponding optimal methods were obtained in [1], [2], [5] and [6]. In particular, the even-block search was treated in [1]. This paper deals with the odd-block search. The main result is Theorem 1.15.