In this paper,we consider the maximal positive definite solution of the nonlinear matrix equation.By using the idea of Algorithm 2.1 in ZHANG(2013),a new inversion-free method with a stepsize parameter is proposed to ...In this paper,we consider the maximal positive definite solution of the nonlinear matrix equation.By using the idea of Algorithm 2.1 in ZHANG(2013),a new inversion-free method with a stepsize parameter is proposed to obtain the maximal positive definite solution of nonlinear matrix equation X+A^(*)X|^(-α)A=Q with the case 0<α≤1.Based on this method,a new iterative algorithm is developed,and its convergence proof is given.Finally,two numerical examples are provided to show the effectiveness of the proposed method.展开更多
In this paper,a new technique is introduced to construct higher-order iterative methods for solving nonlinear systems.The order of convergence of some iterative methods can be improved by three at the cost of introduc...In this paper,a new technique is introduced to construct higher-order iterative methods for solving nonlinear systems.The order of convergence of some iterative methods can be improved by three at the cost of introducing only one additional evaluation of the function in each step.Furthermore,some new efficient methods with a higher-order of convergence are obtained by using only a single matrix inversion in each iteration.Analyses of convergence properties and computational efficiency of these new methods are made and testified by several numerical problems.By comparison,the new schemes are more efficient than the corresponding existing ones,particularly for large problem sizes.展开更多
The complex vibration directly affects the dynamic safety of drill string in ultra-deep wells and extra-deep wells.It is important to understand the dynamic characteristics of drill string to ensure the safety of dril...The complex vibration directly affects the dynamic safety of drill string in ultra-deep wells and extra-deep wells.It is important to understand the dynamic characteristics of drill string to ensure the safety of drill string.Due to the super slenderness ratio of drill string,strong nonlinearity implied in dynamic analysis and the complex load environment,dynamic simulation of drill string faces great challenges.At present,many simulation methods have been developed to analyze drill string dynamics,and node iteration method is one of them.The node iteration method has a unique advantage in dealing with the contact characteristics between drill string and borehole wall,but its drawback is that the calculation consumes a considerable amount of time.This paper presents a dynamic simulation method of drilling string in extra-deep well based on successive over-relaxation node iterative method(SOR node iteration method).Through theoretical analysis and numerical examples,the correctness and validity of this method were verified,and the dynamics characteristics of drill string in extra-deep wells were calculated and analyzed.The results demonstrate that,in contrast to the conventional node iteration method,the SOR node iteration method can increase the computational efficiency by 48.2%while achieving comparable results.And the whirl trajectory of the extra-deep well drill string is extremely complicated,the maximum rotational speed downhole is approximately twice the rotational speed on the ground.The dynamic torque increases rapidly at the position of the bottom stabilizer,and the lateral vibration in the middle and lower parts of drill string is relatively intense.展开更多
A class of coupled system for the E1 Nifio-Southern Oscillation （ENSO） mechanism is studied. Using the method of variational iteration for perturbation theory, the asymptotic expansions of the solution for ENSO mode...A class of coupled system for the E1 Nifio-Southern Oscillation （ENSO） mechanism is studied. Using the method of variational iteration for perturbation theory, the asymptotic expansions of the solution for ENSO model are obtained and the asymptotic behaviour of solution for corresponding problem is considered.展开更多
A class of E1 Niйo atmospheric physics oscillation model is considered. The E1 Niйo atmospheric physics oscillation is an abnormal phenomenon involved in the tropical Pacific ocean-atmosphere interactions. The conce...A class of E1 Niйo atmospheric physics oscillation model is considered. The E1 Niйo atmospheric physics oscillation is an abnormal phenomenon involved in the tropical Pacific ocean-atmosphere interactions. The conceptual oscillator model should consider the variations of both the eastern and western Pacific anomaly patterns. An E1 Niйo atmospheric physics model is proposed using a method for the variational iteration theory. Using the variational iteration method, the approximate expansions of the solution of corresponding problem are constructed. That is, firstly, introducing a set of functional and accounting their variationals, the Lagrange multiplicators are counted, and then the variational iteration is defined, finally, the approximate solution is obtained. From approximate expansions of the solution, the zonal sea surface temperature anomaly in the equatorial eastern Pacific and the thermocline depth anomaly of the sea-air oscillation for E1 Niйo atmospheric physics model can be analyzed. E1 Niйo is a very complicated natural phenomenon. Hence basic models need to be reduced for the sea-air oscillator and are solved. The variational iteration is a simple and valid approximate method.展开更多
Under suitable conditions,the monotone convergence about the projected iteration method for solving linear complementarity problem is proved and the influence of the involved parameter matrix on the convergence rate o...Under suitable conditions,the monotone convergence about the projected iteration method for solving linear complementarity problem is proved and the influence of the involved parameter matrix on the convergence rate of this method is investigated.展开更多
In this paper, we present and analyze a family of fifth-order iterative methods free from second derivative for solving nonlinear equations. It is established that the family of iterative methods has convergence order...In this paper, we present and analyze a family of fifth-order iterative methods free from second derivative for solving nonlinear equations. It is established that the family of iterative methods has convergence order five. Numerical examples show that the new methods are comparable with the well known existing methods and give better results in many aspects.展开更多
The preconditioned Gauss-Seidel type iterative method for solving linear systems, with the proper choice of the preconditioner, is presented. Convergence of the preconditioned method applied to Z-matrices is discussed...The preconditioned Gauss-Seidel type iterative method for solving linear systems, with the proper choice of the preconditioner, is presented. Convergence of the preconditioned method applied to Z-matrices is discussed. Also the optimal parameter is presented. Numerical results show that the proper choice of the preconditioner can lead to effective by the preconditioned Gauss-Seidel type iterative methods for solving linear systems.展开更多
In this paper, the asynchronous versions of classical iterative methods for solving linear systems of equations are considered. Sufficient conditions for convergence of asynchronous relaxed processes are given for H-m...In this paper, the asynchronous versions of classical iterative methods for solving linear systems of equations are considered. Sufficient conditions for convergence of asynchronous relaxed processes are given for H-matrix by which nor only the requirements of [3] on coefficient matrix are lowered, but also a larger region of convergence than that in [3] is obtained.展开更多
Using the monotone iterative method and Monch Fixed point theorem, the existence of solutions and coupled minimal and maximal quasisolutions of initial value problems for mixed monotone second-order integro-differenti...Using the monotone iterative method and Monch Fixed point theorem, the existence of solutions and coupled minimal and maximal quasisolutions of initial value problems for mixed monotone second-order integro-differential equations in Banach spaces are studied. Some existence theorems of solutions and coupled minimal and maximal quasisolutions are obtained.展开更多
For over half a century,numerical integration methods based on finite difference,such as the Runge-Kutta method and the Euler method,have been popular and widely used for solving orbit dynamic problems.In general,a sm...For over half a century,numerical integration methods based on finite difference,such as the Runge-Kutta method and the Euler method,have been popular and widely used for solving orbit dynamic problems.In general,a small integration step size is always required to suppress the increase of the accumulated computation error,which leads to a relatively slow computation speed.Recently,a collocation iteration method,approximating the solutions of orbit dynamic problems iteratively,has been developed.This method achieves high computation accuracy with extremely large step size.Although efficient,the collocation iteration method suffers from two limitations:(A)the computational error limit of the approximate solution is not clear;(B)extensive trials and errors are always required in tuning parameters.To overcome these problems,the influence mechanism of how the dynamic problems and parameters affect the error limit of the collocation iteration method is explored.On this basis,a parameter adjustment method known as the“polishing method”is proposed to improve the computation speed.The method proposed is demonstrated in three typical orbit dynamic problems in aerospace engineering:a low Earth orbit propagation problem,a Molniya orbit propagation problem,and a geostationary orbit propagation problem.Numerical simulations show that the proposed polishing method is faster and more accurate than the finite-difference-based method and the most advanced collocation iteration method.展开更多
In this article, we introduce a new viscosity iterative method for two nonexpansive mappings in Hilbert spaces. We also prove, without commutativity assumption, that the iterates converge to a common fixed point of th...In this article, we introduce a new viscosity iterative method for two nonexpansive mappings in Hilbert spaces. We also prove, without commutativity assumption, that the iterates converge to a common fixed point of the mappings which solves some variational inequality. The results presented extend the corresponding results of Shimizu and Takahashi IT. Shimizu, W. Takahashi, Strong convergence to common fixed point of families of nonexpansive mappings, J. Math. Anal. Appl. 211 （1997）, 71-83], and Yao and Chen [Y. Yao, R. Chert, Convergence to common fixed points of average mappings without commutativity assumption in Hilbert spaces, Nonlinear Analysis 67（2007）, 1758-1763].展开更多
In this paper, a modified Newton type iterative method is considered for ap- proximately solving ill-posed nonlinear operator equations involving m-accretive mappings in Banach space. Convergence rate of the method is...In this paper, a modified Newton type iterative method is considered for ap- proximately solving ill-posed nonlinear operator equations involving m-accretive mappings in Banach space. Convergence rate of the method is obtained based on an a priori choice of the regularization parameter. Our analysis is not based on the sequential continuity of the normalized duality mapping.展开更多
Two kinds of iterative methods are designed to solve the linear system of equations, we obtain a new interpretation in terms of a geometric concept. Therefore, we have a better insight into the essence of the iterativ...Two kinds of iterative methods are designed to solve the linear system of equations, we obtain a new interpretation in terms of a geometric concept. Therefore, we have a better insight into the essence of the iterative methods and provide a reference for further study and design. Finally, a new iterative method is designed named as the diverse relaxation parameter of the SOR method which, in particular, demonstrates the geometric characteristics. Many examples prove that the method is quite effective.展开更多
Downward continuation is a key step in processing airborne geomagnetic data. However,downward continuation is a typically ill-posed problem because its computation is unstable; thus, regularization methods are needed ...Downward continuation is a key step in processing airborne geomagnetic data. However,downward continuation is a typically ill-posed problem because its computation is unstable; thus, regularization methods are needed to realize effective continuation. According to the Poisson integral plane approximate relationship between observation and continuation data, the computation formulae combined with the fast Fourier transform（FFT）algorithm are transformed to a frequency domain for accelerating the computational speed. The iterative Tikhonov regularization method and the iterative Landweber regularization method are used in this paper to overcome instability and improve the precision of the results. The availability of these two iterative regularization methods in the frequency domain is validated by simulated geomagnetic data, and the continuation results show good precision.展开更多
Massive multiple-input multiple-output(MIMO) system is capable of substantially improving the spectral efficiency as well as the capacity of wireless networks relying on equipping a large number of antenna elements at...Massive multiple-input multiple-output(MIMO) system is capable of substantially improving the spectral efficiency as well as the capacity of wireless networks relying on equipping a large number of antenna elements at the base stations. However, the excessively high computational complexity of the signal detection in massive MIMO systems imposes a significant challenge for practical hardware implementations. In this paper, we propose a novel minimum mean square error(MMSE) signal detection using the accelerated overrelaxation(AOR) iterative method without complicated matrix inversion, which is capable of reducing the overall complexity of the classical MMSE algorithm by an order of magnitude. Simulation results show that the proposed AOR-based method can approach the conventional MMSE signal detection with significant complexity reduction.展开更多
In this paper, the existence of solutions for discontinuous nonlinear parabolic differential IBVP is proved by using a more generalized monotone iterative method. Moreover, the convergence of this method is discussed.
The dynamic differential equation of a multibody System can be presented inthe form of Aq= B. Calculating the inverse of matrix A is a simple way to solve this kindof differential equatbo. Matrix A will be in the ill ...The dynamic differential equation of a multibody System can be presented inthe form of Aq= B. Calculating the inverse of matrix A is a simple way to solve this kindof differential equatbo. Matrix A will be in the ill condition if the system is configured asa main they with small mass appendages. A hierarchical iteration method is given in thispaper to avoid the problem of the inverse of an ill condition matrix calculatbo. It is pont-ed out that the stability of the system input and output is the suffcient condition of itera-tion convergence. The method omits a series formula expanding step. It is also useful toreduce the immuence of the stiff problem. The calculation progress i8 modular and structural.展开更多
Motivated by the study of regularization for sparse problems,we propose a new regularization method for sparse vector recovery.We derive sufficient conditions on the well-posedness of the new regularization,and design...Motivated by the study of regularization for sparse problems,we propose a new regularization method for sparse vector recovery.We derive sufficient conditions on the well-posedness of the new regularization,and design an iterative algorithm,namely the iteratively reweighted algorithm(IR-algorithm),for efficiently computing the sparse solutions to the proposed regularization model.The convergence of the IR-algorithm and the setting of the regularization parameters are analyzed at length.Finally,we present numerical examples to illustrate the features of the new regularization and algorithm.展开更多
In this paper, we establish two new iterative methods of order four and five by using modified homotopy perturbation technique. We also present the convergence analysis of these iterative methods. To assess the validi...In this paper, we establish two new iterative methods of order four and five by using modified homotopy perturbation technique. We also present the convergence analysis of these iterative methods. To assess the validity and performance of these iterative methods, we have applied to solve some nonlinear problems.展开更多
基金Supported in part by Natural Science Foundation of Guangxi(2023GXNSFAA026246)in part by the Central Government's Guide to Local Science and Technology Development Fund(GuikeZY23055044)in part by the National Natural Science Foundation of China(62363003)。
文摘In this paper,we consider the maximal positive definite solution of the nonlinear matrix equation.By using the idea of Algorithm 2.1 in ZHANG(2013),a new inversion-free method with a stepsize parameter is proposed to obtain the maximal positive definite solution of nonlinear matrix equation X+A^(*)X|^(-α)A=Q with the case 0<α≤1.Based on this method,a new iterative algorithm is developed,and its convergence proof is given.Finally,two numerical examples are provided to show the effectiveness of the proposed method.
基金Supported by the National Natural Science Foundation of China(12061048)NSF of Jiangxi Province(20232BAB201026,20232BAB201018)。
文摘In this paper,a new technique is introduced to construct higher-order iterative methods for solving nonlinear systems.The order of convergence of some iterative methods can be improved by three at the cost of introducing only one additional evaluation of the function in each step.Furthermore,some new efficient methods with a higher-order of convergence are obtained by using only a single matrix inversion in each iteration.Analyses of convergence properties and computational efficiency of these new methods are made and testified by several numerical problems.By comparison,the new schemes are more efficient than the corresponding existing ones,particularly for large problem sizes.
基金supported by the National Natural Science Foundation of China(52174003,52374008).
文摘The complex vibration directly affects the dynamic safety of drill string in ultra-deep wells and extra-deep wells.It is important to understand the dynamic characteristics of drill string to ensure the safety of drill string.Due to the super slenderness ratio of drill string,strong nonlinearity implied in dynamic analysis and the complex load environment,dynamic simulation of drill string faces great challenges.At present,many simulation methods have been developed to analyze drill string dynamics,and node iteration method is one of them.The node iteration method has a unique advantage in dealing with the contact characteristics between drill string and borehole wall,but its drawback is that the calculation consumes a considerable amount of time.This paper presents a dynamic simulation method of drilling string in extra-deep well based on successive over-relaxation node iterative method(SOR node iteration method).Through theoretical analysis and numerical examples,the correctness and validity of this method were verified,and the dynamics characteristics of drill string in extra-deep wells were calculated and analyzed.The results demonstrate that,in contrast to the conventional node iteration method,the SOR node iteration method can increase the computational efficiency by 48.2%while achieving comparable results.And the whirl trajectory of the extra-deep well drill string is extremely complicated,the maximum rotational speed downhole is approximately twice the rotational speed on the ground.The dynamic torque increases rapidly at the position of the bottom stabilizer,and the lateral vibration in the middle and lower parts of drill string is relatively intense.
基金Project supported by the National Natural Science Foundation of China （Grant Nos 90111011 and 10471039）, the National Key Basic Research Special Foundation of China （Grant Nos 2003CB415101-03 and 2004CB418304）, the Key Basic Research Foundation of the Chinese Academy of Sciences （Grant No KZCX3-SW-221） and in part by E-Institutes of Shanghai Municipal Education Commission （Grant No N.E03004）.
文摘A class of coupled system for the E1 Nifio-Southern Oscillation （ENSO） mechanism is studied. Using the method of variational iteration for perturbation theory, the asymptotic expansions of the solution for ENSO model are obtained and the asymptotic behaviour of solution for corresponding problem is considered.
文摘A class of E1 Niйo atmospheric physics oscillation model is considered. The E1 Niйo atmospheric physics oscillation is an abnormal phenomenon involved in the tropical Pacific ocean-atmosphere interactions. The conceptual oscillator model should consider the variations of both the eastern and western Pacific anomaly patterns. An E1 Niйo atmospheric physics model is proposed using a method for the variational iteration theory. Using the variational iteration method, the approximate expansions of the solution of corresponding problem are constructed. That is, firstly, introducing a set of functional and accounting their variationals, the Lagrange multiplicators are counted, and then the variational iteration is defined, finally, the approximate solution is obtained. From approximate expansions of the solution, the zonal sea surface temperature anomaly in the equatorial eastern Pacific and the thermocline depth anomaly of the sea-air oscillation for E1 Niйo atmospheric physics model can be analyzed. E1 Niйo is a very complicated natural phenomenon. Hence basic models need to be reduced for the sea-air oscillator and are solved. The variational iteration is a simple and valid approximate method.
文摘Under suitable conditions,the monotone convergence about the projected iteration method for solving linear complementarity problem is proved and the influence of the involved parameter matrix on the convergence rate of this method is investigated.
文摘In this paper, we present and analyze a family of fifth-order iterative methods free from second derivative for solving nonlinear equations. It is established that the family of iterative methods has convergence order five. Numerical examples show that the new methods are comparable with the well known existing methods and give better results in many aspects.
基金Project supported by MOE's 2004 New Century Excellent Talent Program (NCET)the Applied Basic Research Foundations of Sichuan Province (No.05JY029-068-2)
文摘The preconditioned Gauss-Seidel type iterative method for solving linear systems, with the proper choice of the preconditioner, is presented. Convergence of the preconditioned method applied to Z-matrices is discussed. Also the optimal parameter is presented. Numerical results show that the proper choice of the preconditioner can lead to effective by the preconditioned Gauss-Seidel type iterative methods for solving linear systems.
文摘In this paper, the asynchronous versions of classical iterative methods for solving linear systems of equations are considered. Sufficient conditions for convergence of asynchronous relaxed processes are given for H-matrix by which nor only the requirements of [3] on coefficient matrix are lowered, but also a larger region of convergence than that in [3] is obtained.
文摘Using the monotone iterative method and Monch Fixed point theorem, the existence of solutions and coupled minimal and maximal quasisolutions of initial value problems for mixed monotone second-order integro-differential equations in Banach spaces are studied. Some existence theorems of solutions and coupled minimal and maximal quasisolutions are obtained.
基金This study was co-supported by the National Key Research and Development Program of China(No.2021YFA0717100)the National Natural Science Foundation of China(Nos.12072270,U2013206).
文摘For over half a century,numerical integration methods based on finite difference,such as the Runge-Kutta method and the Euler method,have been popular and widely used for solving orbit dynamic problems.In general,a small integration step size is always required to suppress the increase of the accumulated computation error,which leads to a relatively slow computation speed.Recently,a collocation iteration method,approximating the solutions of orbit dynamic problems iteratively,has been developed.This method achieves high computation accuracy with extremely large step size.Although efficient,the collocation iteration method suffers from two limitations:(A)the computational error limit of the approximate solution is not clear;(B)extensive trials and errors are always required in tuning parameters.To overcome these problems,the influence mechanism of how the dynamic problems and parameters affect the error limit of the collocation iteration method is explored.On this basis,a parameter adjustment method known as the“polishing method”is proposed to improve the computation speed.The method proposed is demonstrated in three typical orbit dynamic problems in aerospace engineering:a low Earth orbit propagation problem,a Molniya orbit propagation problem,and a geostationary orbit propagation problem.Numerical simulations show that the proposed polishing method is faster and more accurate than the finite-difference-based method and the most advanced collocation iteration method.
基金the Thailand Research Fund for financial support under Grant BRG5280016
文摘In this article, we introduce a new viscosity iterative method for two nonexpansive mappings in Hilbert spaces. We also prove, without commutativity assumption, that the iterates converge to a common fixed point of the mappings which solves some variational inequality. The results presented extend the corresponding results of Shimizu and Takahashi IT. Shimizu, W. Takahashi, Strong convergence to common fixed point of families of nonexpansive mappings, J. Math. Anal. Appl. 211 （1997）, 71-83], and Yao and Chen [Y. Yao, R. Chert, Convergence to common fixed points of average mappings without commutativity assumption in Hilbert spaces, Nonlinear Analysis 67（2007）, 1758-1763].
文摘In this paper, a modified Newton type iterative method is considered for ap- proximately solving ill-posed nonlinear operator equations involving m-accretive mappings in Banach space. Convergence rate of the method is obtained based on an a priori choice of the regularization parameter. Our analysis is not based on the sequential continuity of the normalized duality mapping.
基金Supported by the National Natural Science Foundation of China(61272300)
文摘Two kinds of iterative methods are designed to solve the linear system of equations, we obtain a new interpretation in terms of a geometric concept. Therefore, we have a better insight into the essence of the iterative methods and provide a reference for further study and design. Finally, a new iterative method is designed named as the diverse relaxation parameter of the SOR method which, in particular, demonstrates the geometric characteristics. Many examples prove that the method is quite effective.
基金supported by the National Natural Science Foundation of China(41304022,41174026,41104047)the National 973 Foundation(61322201,2013CB733303)+1 种基金the Key laboratory Foundation of Geo-space Environment and Geodesy of the Ministry of Education(13-01-08)the Youth Innovation Foundation of High Resolution Earth Observation(GFZX04060103-5-12)
文摘Downward continuation is a key step in processing airborne geomagnetic data. However,downward continuation is a typically ill-posed problem because its computation is unstable; thus, regularization methods are needed to realize effective continuation. According to the Poisson integral plane approximate relationship between observation and continuation data, the computation formulae combined with the fast Fourier transform（FFT）algorithm are transformed to a frequency domain for accelerating the computational speed. The iterative Tikhonov regularization method and the iterative Landweber regularization method are used in this paper to overcome instability and improve the precision of the results. The availability of these two iterative regularization methods in the frequency domain is validated by simulated geomagnetic data, and the continuation results show good precision.
基金supported by the key project of the National Natural Science Foundation of China (No. 61431001)Huawei Innovation Research Program, the 5G research program of China Mobile Research Institute (Grant No. [2015] 0615)+2 种基金the open research fund of National Mobile Communications Research Laboratory Southeast University (No.2017D02)Key Laboratory of Cognitive Radio and Information Processing, Ministry of Education (Guilin University of Electronic Technology)the Foundation of Beijing Engineering and Technology Center for Convergence Networks and Ubiquitous Services, and Keysight
文摘Massive multiple-input multiple-output(MIMO) system is capable of substantially improving the spectral efficiency as well as the capacity of wireless networks relying on equipping a large number of antenna elements at the base stations. However, the excessively high computational complexity of the signal detection in massive MIMO systems imposes a significant challenge for practical hardware implementations. In this paper, we propose a novel minimum mean square error(MMSE) signal detection using the accelerated overrelaxation(AOR) iterative method without complicated matrix inversion, which is capable of reducing the overall complexity of the classical MMSE algorithm by an order of magnitude. Simulation results show that the proposed AOR-based method can approach the conventional MMSE signal detection with significant complexity reduction.
文摘In this paper, the existence of solutions for discontinuous nonlinear parabolic differential IBVP is proved by using a more generalized monotone iterative method. Moreover, the convergence of this method is discussed.
文摘The dynamic differential equation of a multibody System can be presented inthe form of Aq= B. Calculating the inverse of matrix A is a simple way to solve this kindof differential equatbo. Matrix A will be in the ill condition if the system is configured asa main they with small mass appendages. A hierarchical iteration method is given in thispaper to avoid the problem of the inverse of an ill condition matrix calculatbo. It is pont-ed out that the stability of the system input and output is the suffcient condition of itera-tion convergence. The method omits a series formula expanding step. It is also useful toreduce the immuence of the stiff problem. The calculation progress i8 modular and structural.
基金Project supported by the National Natural Science Foundation of China(No.61603322)the Research Foundation of Education Bureau of Hunan Province of China(No.16C1542)
文摘Motivated by the study of regularization for sparse problems,we propose a new regularization method for sparse vector recovery.We derive sufficient conditions on the well-posedness of the new regularization,and design an iterative algorithm,namely the iteratively reweighted algorithm(IR-algorithm),for efficiently computing the sparse solutions to the proposed regularization model.The convergence of the IR-algorithm and the setting of the regularization parameters are analyzed at length.Finally,we present numerical examples to illustrate the features of the new regularization and algorithm.
文摘In this paper, we establish two new iterative methods of order four and five by using modified homotopy perturbation technique. We also present the convergence analysis of these iterative methods. To assess the validity and performance of these iterative methods, we have applied to solve some nonlinear problems.
