Honkela et al. (2010) is a major contribution to semiparametric mean field variational Bayes methodology and their **Algorithm** 1 uses the **nonlinear** conjugate gradient method (**Algorithm** 5) with natural gradients rather than ordinary gradients. Via both simple examples and numerical studies, they make a compelling case for the use of natural gradients for optimization of the parameters of the pre-specified parametric q-density function. 3.5 Summary of Semiparametric Mean Field Variational Bayes Ramifications In this section we have discussed several iterative numerical optimization strategies. Any of these are candidates for the updating ξ in the **Algorithm** 2 cycle. Special mention has been given to the well-known **Newton**-**Raphson** iteration since it can achieve very rapid convergence and the **nonlinear** conjugate gradient method which has been shown to be effective in semiparametric mean field variational Bayes contexts when the Riemmannian geometry adjustment of Section 3.4.1 is employed (Honkela et al., 2010).

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With the wide development of science and technology, the problem of solving **nonlinear** equations by numerical methods has gained more importance than before. In order to obtain efficient **algorithm** for the **nonlinear** equations which come from the practical problems, in this paper, we present and analyze two improved **Newton**-**Raphson** methods for solving **nonlinear** equations. The methods are free from second derivatives. Several numerical results illustrate the convergence behavior and computational efficiency of the proposed methods. Computational results demonstrate that they are more efficient and perform better than **Newton**-**Raphson** method and some existing methods.

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Abstract Accurate calculations for measuring the sacrificing rate of various materials in heat shield systems have not been obtained yet. So having a reliable numerical program that calculates the surface recession rate and interior temperature history is necessary. In this study numerical solution of governing equations for charring material ablation are presented. Thermodynamic properties , C p , k changes relative to the temperature accompany the **nonlinear** nature of the governing equations. The **Newton**-**Raphson** method along with TDMA **algorithm** is used to solve this **nonlinear** equation system. Using **Newton**- **Raphson** method is one of the advantages of solving method because it is relatively simple and it can be easily generalized to more difficult problems. The obtained results are compared with reliable sources in order to examine the accuracy of compiling code.

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In large-scale power quality studies the **Newton**- **Raphson** method has proved out to be the most successful **algorithm** with its strong convergence characteristics. In order to apply **Newton**-**Raphson** method to the power quality problem the relevant equations expressed in the form of unknown nodal voltage magnitudes and phase angles.

Abstract-- In this paper we consider numerical evaluation of **nonlinear** Hamiltonian symmetric matrix of Rank 1 in an inverse eigenvalue problem via **Newton**-**Raphson** method. The approach employed **Newton**-Raphson’s method for solving the inverse eigenvalue problem in a class of Hamiltonian matrices in the neighborhood of a related nonsingular matrix of rank 1. A few numerical examples are presented to illustrate the result.

By performing load flow analysis using **NEWTON**- **RAPHSON** method the results obtained are accurate. MI- POWER software is executed to incorporate the UPFC model in NR method. To determine the steady state performance of the UPFC in the load flow studies a IEEE 5 and IEEE-30 bus systems has considered. From the analysis by **Newton**- **Raphson** method has simple calculations and is easy to execute, In **Newton**-**Raphson** approach the number of buses does not depends upon the number of iterations.

is not satisfied. One striking problem for (almost) any practical microdata analysis in achieving representative results, therefore, is to find an aqequate representative weighting scheme. In this study a consistent solution of the microdata adjustment problem - a properly (re-) weighting of microdata to fit aggregate control data - is presented based on the Minimum Information Loss (MIL) principle. Based on information theory the MIL principle in particular satisfies the desired positivity constrained of the weighting factors to be computed. In addition, the consistent solution simultaneously adjusts hierarchical microdata, e.g. household and personal information within a household/family setting. Since microdata files contain a large number of observations and characteristics, a fast numerical method to solve the adjustment procedure is necessary. Such a fast numerical solution is proposed by a modified **Newton**-**Raphson** procedure where within a set of given steplengths a 'global exponential' modification with a variable steplength yields impressive results.

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In this paper, we present a regularized **Newton** method (M-RNM) with correction for minimizing a convex function whose Hessian matrices may be singular. At every iteration, not only a RNM step is computed but also two correction steps are computed. We show that if the objective function is LC 2 , then the method posses globally convergent. Numerical results show that the new **algorithm**

systems can be made more flexible by the use of recent developments in power electronic and computing control technology. The Unified Power Flow Controller (UPFC) provides a promising means to control power flow in modern power systems. Essentially, the performance depends on proper control setting achievable through a power flow analysis program. This paper aims to present a reliable method to meet the requirements by developing a **Newton**-**Raphson** based load flow calculation program through which control setting of UPFC can be determined directly. A MATLAB program has been developed to calculate the control setting parameters of the UPFC after the load flow is converged. Case studies have been performed on IEEE 5-bus system to show that the proposed method is effective. These studies indicate that the method maintains the basic NRLF properties such as fast computational speed, high degree of accuracy and good convergence rate.

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The past ﬁfty to sixty years have seen a considerable ad- vancement in methods for solving linear systems. Krylov subspace method is the result of the huge eﬀort by the re- searchers during the last century. It is one among the top ten algorithms of the 20th century. There exists optimal linear solvers [5]. But, still there is no optimal **nonlinear** solver or the one that we know of. Our research is in the ﬁeld of optimal solution of **nonlinear** equations generated by the discretization of the **nonlinear** partial diﬀerential equation [1; 2; 3; 4]. Let us consider the following non- linear elliptic partial diﬀerential equation [4]

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In this paper, the **algorithm** for large-scale **nonlinear** equations is designed by the following steps: (i) a conjugate gradient (CG) **algorithm** is designed as a sub-**algorithm** to obtain the initial points of the main **algorithm**, where the sub-algorithm’s initial point does not have any restrictions; (ii) a quasi-**Newton** **algorithm** with the initial points given by sub-**algorithm** is deﬁned as main **algorithm**, where a new

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The minimization of a quadratic function within an ellipsoidal trust region is an important subproblem for many **nonlinear** programming algorithms. When the number of variables is large, one of the most widely used strategies is to project the original problem into a small dimensional subspace. In this paper, we introduce an **algorithm** for solving **nonlinear** least squares problems. This **algorithm** is based on constructing a basis for the Krylov subspace in conjunction with a model trust region technique to choose the step. The computational step on the small dimensional subspace lies inside the trust region. The Krylov subspace is terminated such that the termination condition allows the gradient to be decreased on it. A convergence theory of this **algorithm** is presented. It is shown that this **algorithm** is globally convergent.

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Many numerical approaches have been suggested to solve **nonlinear** problems. Some of the methods utilize successive approximation procedure to ensure every step of computing will converge to the desired root and one of the most common problems is the improper initial values for the iterative methods. This study evaluates Palancz et.al’s. (2010) paper on solving **nonlinear** equations using linear homotopy method in Mathematica. In this paper, the **Newton**-homotopy method using start-system is implemented in Maple14, to solve several **nonlinear** problems. Comparisons of results obtained in terms of number of iterations and convergence rates show promising application of the **Newton**-homotopy method for **nonlinear** problems.

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The most widely used method is the **newton** **Raphson** method for solving simultaneous **nonlinear** algebraic equations. **Newton** ’ s method is found to be more efficient and practical. The number of iterations required to obtain a solution is independent of the system size, but more functional evaluations are required at every iteration [10]. Since in the power flow problem real power and voltage magnitude are specified for the voltage-controlled buses, the power flow equation is formulated in polar form. This equation can be rewritten in admittance matrix as

[6] W. Chu, L. Ma, J. Song, and T. Vorburger, "An iterative image registration **algorithm** by optimizing similarity measurement" Journal of Research of the National Institute of Standards and Technology, Vol. 115, No. 1, 2010.

the root to the M-estimating equations. Techniques justified by uniform convergence are used here. Uniform convergence also lends itself to the use of a graphical method of plotting "expectation curves". It can be used for either identifying the M-estimator from multiple solutions of the defining equations or in large samples (e.g. > 50) as a visual indica tion of whether the fitted model is a good approximation for the under lying mechanism. Theorems based on uniform convergence are given that show a domain of convergence (numerical analysis interpretation) for the **Newton**-**Raphson** iteration method applied to M-estimating equations for the location parameter when redescending loss functions are used.

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A simple conventional **Newton** **Raphson** method is used to solve the problem of optimal location of a single DG unit which is delivering only real power in the system [7]. A load flow based method for optimal location of dispersed generation unit delivering only real power for voltage profile impro vement has been presented in [8]. A load flow based approach for optimum allocation of DG units for voltage profile improvement and loss minimization has been suggested in [9]. A **Newton** **Raphson** load flow method for optimal sizing and placement of DG units using weighting factors has been proposed in [10]. The cost and loss factors are minimized in this paper.

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Many studies on the modeling of akin process have been reported [1-3]. From previous findings, the mathematical modeling of the hydrolysis process leads to a **nonlinear** parameter estimation problem and the model parameters of the process have been determined either using conventional graphical-based technique [4] or **nonlinear** regression method [5]. However, the usual approach to estimate model parameters of a biological process is by using **nonlinear** techniques since the graphical methods have shown inferior parameter estimates compared to those generated using **nonlinear** techniques [6]. On the other hand, the **nonlinear** techniques also have their drawbacks. This scheme often fails in the search for global optimum if the search space is non linear in parameters [7]. For a large value of least squares sum, a slow convergence often appears [8]. A common practice to deal with the local convergence problem is to test different initial guess parameters. However, the probability of finding an initial condition suitable for all parameters decreases as the number of involved parameter, increases [9]. Because of the limitations imposed by those methods, an attempt was made to estimate the model parameters of the tapioca starch hydrolysis using genetic **algorithm**.

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Power flow studies of power systems give the important computations procedures to find the present operational characteristics and loading predictions which are useful for future planning of the system according to the growing demand. **Newton** **Raphson** method is a very popular one to compute the solution of power flow problem due to its fast convergence in few iterations. Jacobins of the **Newton** **Raphson** method to be solved in a simple, faxable and reliable manner to evaluate the accurate values. This paper describes a powerful technique known as Automatic Differentiation (AD), applied to compute first order partial derivatives or Jacobin elements that occur in **Newton** **Raphson** method when solving power flow problem. Evaluation of the derivatives obtained for the expressions specified in the form of computer program, in a way such that code is differentiated rather than expression, unlike in the traditional finite differentiation. Main program of **Newton** **Raphson** power flow method is exempted from the tedious task of computing error full Jacobian partial derivatives and makes the calls to a well defined AD tool to do this task accurately whenever required. This loose coupling between main program and AD code enhances the computational performance and create a flexible solution environment to compute the load flow solutions efficiently. In this paper Automatic Differentiation technique is explained in context to power flow, and then demonstrated on a 4-bus test system and simulation results have been compared and analyzed.

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