Computer Methodologies for the Comparison of Some Efficient Derivative FreeSimultaneous Iterative Methods for Finding Roots of Non-Linear Equations

In this article,we construct the most powerful family of simultaneous iterative method with global convergence behavior among all the existing methods in literature for finding all roots of non-linear equations.Convergence analysis proved that the order of convergence of the family of derivative free simultaneous iterative method is nine.Our main aim is to check out the most regularly used simultaneous iterative methods for finding all roots of non-linear equations by studying their dynamical planes,numerical experiments and CPU time-methodology.Dynamical planes of iterative methods are drawn by using MATLAB for the comparison of global convergence properties of simultaneous iterative methods.Convergence behavior of the higher order simultaneous iterative methods are also illustrated by residual graph obtained from some numerical test examples.Numerical test examples,dynamical behavior and computational efficiency are provided to present the performance and dominant efficiency of the newly constructed derivative free family of simultaneous iterative method over existing higher order simultaneous methods in literature.

Simultaneous Determination of Iron and Manganese in Water Using Artificial Neural Network Catalytic Spectrophotometric Method

A new analytical method using Back-Propagation (BP) artificial neural network and kinetic spectrophotometry for simultaneous determination of iron and magnesium in tap water, the Yellow River water and seawater is established. By conditional experiments, the optimum analytical conditions and parameters are obtained. Levenberg-Marquart (L-M) algorithm is used for calculation in BP neural network. The topological structure of three-layer BP ANN network architecture is chosen as 15-16-2 (nodes). The initial value of gradient coefficient μ is fixed at 0.001 and the increase factor and reduction factor of μ take the default values of the system. The data are processed by computers with our own programs written in MATLAB 7.0. The relative standard deviation of the calculated results for iron and manganese is 2.30% and 2.67% respectively. The results of standard addition method show that for the tap water, the recoveries of iron and manganese are in the ranges of 98.0%-104.3% and 96.5%-104.5%, and the RSD is in the range of 0.23%-0.98%; for the Yellow River water (Lijin district of Shandong Province), the recoveries of iron and manganese are in the ranges of 96.0%-101.0% and 98.7%-104.2%, and the RSD is in the range of 0.13%-2.52%; for the seawater in Qingdao offshore, the recoveries of iron and manganese are in the ranges of 95.3%-104.8% and 95.3%-104.7%, and the RSD is in the range of 0.14%-2.66%. It is found that 21 common cations and anions do not interfere with the determination of iron and manganese under the optimum experimental conditions. This method exhibits good reproducibility and high accuracy in the determination of iron and manganese and can be used for the simultaneous determination of iron and manganese in tap water and natural water. By using the established ANN- catalytic spectrophotometric method, the iron and manganese concentrations of the surface seawater at 11 sites in Qingdao offshore are determined and the level distribution maps of iron and manganese are drawn.

SIMULTANEOUS DETERMINATION OF MULTICOMPONENT IN MIXED ACIDS BY MATHEMATICAL METHOD

A new method for the determination of components in mixed acids has been developed.The mathematical model is obtained from samples of known composition and is then used to predict the concentrations of components in unknown sample.The practical utility of this method is demonstrated for simultaneous determination of two systems of ternary mixed acids and the results are satisfactory.

Elastic solutions for partially embedded single piles subjected to simultaneous axial and lateral loading

In order to improve the design level of partially embedded single piles under simultaneous axial and lateral loads, the differential solutions were deduced, in which the soil was treated as an ideal, elastic, homogeneous, semi-infinite isotropic medium. A comparison was made between model test results and the obtained solutions to show their validity. The calculation results indicate that the horizontal displacement and bending moment of the pile increase with increases of the axial and lateral loads. The maximum horizontal displacement and bending moment decrease by 37.9% and 13.9%, respectively, when the elastic modulus of soil increases from 4 MPa to 20 MPa. The Poisson ratio of soil plays a marginal role in pile responses. There is a critical pile length under the ground, beyond which the pile behaves as though it was infinitely long. The presented solutions can make allowance for the continuous nature of soil, and if condition permits, they can approach exact ones.

Computer Geometries for Finding All Real Zeros of Polynomial Equations Simultaneously

In this research article,we construct a family of derivative free simultaneous numerical schemes to approximate all real zero of non-linear polynomial equation.We make a comparative analysis of the newly constructed numerical schemes with a well-known existing simultaneous method for determining all the distinct real zeros of polynomial equations using computer algebra system Mat Lab.Lower bound of convergence of simultaneous schemes is calculated using Mathematica.Global convergence property of the numerical schemes is presented by taking random starting initial approximation and their convergence history are graphically presented.Some real life engineering applications along with some higher degree polynomials are considered as numerical test problems to show performance and efficiency of the derivative free family of numerical methods with comparison of an existing method of same order in literature.Local computational order of convergence,CPU time,graph of computational order of convergence and residual error graphs elaborate efficiency,robustness and authentication of the suggested family of numerical methods in its domain.

Household Income and Expenditure Relationships： A Simultaneous Equation Approach

Income and expenditure are highlighted in this study as they currently affect the quality of life of the Malaysian society. In the Ninth Malaysia Plan （2006-2010） poverty eradication was emphasized as one of the socio-economic targets to uplift those poor households from the fast moving modern sectors of the economy. The primary objective of this paper is therefore to identify income and expenditure relationships among the households of the east coast of Peninsular Malaysia during the current economic downturn. Attempts to analyze household consumption as a share of total income and to identify factors contributing to household income and expenditure have become a topic of interest in literature. A substantial portion of household income devoted to productive loans characterizes the majority of Malaysians as the economy progresses to a more advance stage. A significant and strong relationship between income and types of household expenditure is explored from the socio-economic perspectives. Using primary data a system of simultaneous equations, namely the two stage least squares （2SLS） was applied. This paper identifies crucial variables associated with the income and expenditure. Some of these variables showing high correlation with the total expenditure are the housing loan, automobile loan and educational expenditure.

Physiochemical and Biological Properties of Modified Collagen Sponge from Porcine Skin

The aim of the present study was to compare one-step method to EDC/NHS crosslinking （EDC/NHS group） and one-step simultaneous method to EDC/NHS crosslinking and heparin immobilization （EDC/NHS- Heparin group） in improving physiochemical and biological properties of native collagen sponge （Control group）. Modified collagen sponge overcome the disadvantages of native collagen sponge. IR spectra suggest the change of the functional groups. DSC data indicate that the stability of caloric transformation in EDC/NHS group is slightly higher than that of EDC/NHS-Heparin group. The crosslinking degree, stability against enzymes, stability in morphologically and biomechanical properties of EDC/NHS-Heparin group are higher than those of EDC/NHS group, whereas, the water-binding capacity in EDC/NHS-Heparin group is lower than that of EDC/NHS group. HUVECs in EDC/NHS-Heparin group scaffold proliferate fast, migrate well and distribute uniformly. One-step simultaneous method gains the better effects in above aspects, heparinized collagen matrices increase in angiogenic potential and suit for defect repairing and tissue engineering.

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.

Computer Oriented Numerical Scheme for Solving Engineering Problems

In this study,we construct a family of single root finding method of optimal order four and then generalize this family for estimating of all roots of non-linear equation simultaneously.Convergence analysis proves that the local order of convergence is four in case of single root finding iterative method and six for simultaneous determination of all roots of non-linear equation.Some non-linear equations are taken from physics,chemistry and engineering to present the performance and efficiency of the newly constructed method.Some real world applications are taken from fluid mechanics,i.e.,fluid permeability in biogels and biomedical engineering which includes blood Rheology-Model which as an intermediate result give some nonlinear equations.These non-linear equations are then solved using newly developed simultaneous iterative schemes.Newly developed simultaneous iterative schemes reach to exact values on initial guessed values within given tolerance,using very less number of function evaluations in each step.Local convergence order of single root finding method is computed using CAS-Maple.Local computational order of convergence,CPU-time,absolute residuals errors are calculated to elaborate the efficiency,robustness and authentication of the iterative simultaneous method in its domain.

Optimal Design of Water Utilization Network with Energy Integration in Process Industries

Effective utilization of water and energy is the key factor of sustainable development in process industries, and also an important science and technology problem to be solved in systems engineering. In this paper,two new methods of optimal design of water utilization network with energy integration in process industries are presented, that is, stepwise and simultaneous optimization methods. They are suitable for both single contaminant and multi-contaminant systems, and the integration of energy can be carried out in the whole process system, not only limited in water network, so that energy can be utilized effectively. The two methods are illustrated by case study.

Optimization of two-factor(pH and ion concentration)simultaneous selection in HPLC using a statistical method

A computer-assisted method is presented for simultaneous optimization of pH and ion con- centration selection for the optimal separation in reversed-phase HPLC.The method is based on a polynomial estimation from nine preliminary experiments according to two-factor rectangular design. This is followed by a two-dimension computer scanning technique.Resolution is used as the selection criterion.Good agreement was obtained between predicted data and experimental results.

A comprehensive review of recent advancements and developments in heat exchanger network synthesis techniques

The rapid development of computational technology and the increasing energy demand have improved heat exchanger network(HEN)synthesis.The HEN synthesis involves several optimizations of matches,distributions of heat loads,and stream splitting of heat units.Thus,obtaining good results at high efficiency has been the main standard for evaluating the techniques in the research area of HEN synthesis.This paper first summarizes and analyzes the main contributions of the existing HEN synthesis techniques.To compare related data quantitively,information on ten typical cases is presented in this paper.Furthermore,recently improved solutions for commonly encountered existing literature cases demonstrate the evolution and competition trends in the field of HEN synthesis.The comparison data presented in this paper not only provide a useful reference for future research but also present the optimization directions.Based on the findings of this study,it is noted that there is still a large room for improvement,and current approaches are incapable of dealing with all HEN cases.Moreover,it is still difficult to escape a local optimum and overcome structural constraints when seeking the global optimum.As a follow-up to the current work,the parallel computing mode and adaptively coordinating the ratio of global and local searching abilities are major development trends for future investigation.