The trust region method (TRM) is a very important technique to solve both of linear and nonlinear systems of equations. In this work, a new modified algorithm of a TRM with adaptive radius is introduced in purpose of solving systems of nonlinear equations. At each iteration, the new algorithm changes the trust region radius (TRR) automatically to reduce the subproblems resolving number when the current radius is rejected. The global convergence results of the new procedure under some appropriate conditions is established. The numerical effects indicate that the suggested algorithm is interesting and robustness.
The projection technique is one of the famous method and highly useful to solve the optimization problems and nonlinear systems of equations. In this work, a new projection approach for solving systems of nonlinear monotone equation is proposed combining with the conjugate gradient direction because of their low storage. The new algorithm can be used to solve the large-scale nonlinear systems of equations and satisfy the sufficient descent condition. The new algorithm generates appropriate direction then employs a good line search along this direction to reach a new point. If this point solves the problem then the algorithm stops, otherwise, it constructs a suitable hyperplane that strictly separate the current point from the solution set. The next iteration is obtained by projection the new point onto the separating hyperplane. We proved that the line search of the new projection algorithm is well defined. Furthermore, we established the global convergence under some mild conditions. The numerical experiment indicates that the new method is effective and very well.
Counts data models cope with the response variable counts, where the number of times that a certain event occurs in a fixed point is called count data, its observations consists of non-negative integers values {0,1,2,}. Due to the nature of the count data, it is generally considered that response variables do not follow normal distribution. Therefore, because of the skewed distribution, linear regression is not an effective method for analyzing counting results. And hence, the use of the linear regression model to analyse count data is likely to bias the outcomes, “Negative binomial regression” is likely to be the optimal model for analyzing count data under these limitations. Researchers may sometimes count more zeros than expected. Going to count data with several Zeros gives rise to the “Zero-inflation” concept. In health, marketing, finance, econometrics, ecology, statistical quality control, geographical and environmental fields, data with abundant zeros is common when counting the incidence of certain behavioural and natural events, such as the frequency of alcohol consumption, drug consumption, the amount of cigarettes smoked, the incidence of earthquakes, rainfall, etc. The Negative Binomial, “Zero-Inflated Negative Binomial” (ZINB), and “Zero-Altered Negative Binomial” (ZANB) models were used in this paper to analyse rainfall data.
In this work a new technique of line search is proposed to solve the nonlinear monotone equations. For this purpose we combine a modified line search rule with monotone technique. The new proposed algorithm can decrease the CPU time, the number of iterations and the functions evaluations and can increase the efficiency of the approach. The global convergence result of the recommended procedures is established under some standard conditions. Preliminary numerical results indicate that the proposed algorithm are interesting and remarkably promising.
There are many algorithms are used to solve systems of nonlinear monotone equations with various advantages and disadvantages, including the line search algorithm, trust region algorithm, projection algorithm and others. In this paper we used a new projection algorithm to solve these systems. The projection methods are considered one of the effective free derivative methods to solve systems of nonlinear monotone equations. The framework of this method is that the current iterate is separated strictly from the solution set of the problem in each iteration by a suitable hyperplane which constructs by the new algorithm. Then, in order to determine the new approximation, the current iteration is projected on this hyperplane. The global convergence of the proposed algorithm is proven under standard assumptions. The numerical results showed that the suggested algorithm is very efficiency and promising.
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