scholarly journals A generalized multivariable Newton method

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Regina S. Burachik ◽  
Bethany I. Caldwell ◽  
C. Yalçın Kaya
Keyword(s):  
Newton Method ◽  
Quadratic Convergence ◽  
Polynomial Systems ◽  
Initial Guess ◽  
Experimental Methodology ◽  
Test Problems ◽  
Convergence Region ◽  
New Family ◽  
Asymptotic Error Constant ◽  
Search Domain

AbstractIt is well known that the Newton method may not converge when the initial guess does not belong to a specific quadratic convergence region. We propose a family of new variants of the Newton method with the potential advantage of having a larger convergence region as well as more desirable properties near a solution. We prove quadratic convergence of the new family, and provide specific bounds for the asymptotic error constant. We illustrate the advantages of the new methods by means of test problems, including two and six variable polynomial systems, as well as a challenging signal processing example. We present a numerical experimental methodology which uses a large number of randomized initial guesses for a number of methods from the new family, in turn providing advice as to which of the methods employed is preferable to use in a particular search domain.

scholarly journals A Generalized Inexact Newton Method for Inverse Eigenvalue Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Weiping Shen
Keyword(s):  
Newton Method ◽  
Quadratic Convergence ◽  
Eigenvalue Problems ◽  
Generalized Newton Method ◽  
Inexact Newton Method ◽  
Numerical Tests ◽  
Inverse Eigenvalue Problems ◽  
Inverse Eigenvalue ◽  
Generalized Newton ◽  
Special Case

We propose a generalized inexact Newton method for solving the inverse eigenvalue problems, which includes the generalized Newton method as a special case. Under the nonsingularity assumption of the Jacobian matrices at the solutionc*, a convergence analysis covering both the distinct and multiple eigenvalue cases is provided and the quadratic convergence property is proved. Moreover, numerical tests are given in the last section and comparisons with the generalized Newton method are made.

scholarly journals Orthonormal Ultraspherical Operational Matrix Algorithm for Fractal–Fractional Riccati Equation with Generalized Caputo Derivative

2021 ◽  
Vol 5 (3) ◽  
pp. 100
Author(s):  
Youssri Hassan Youssri
Keyword(s):  
Differential Equation ◽  
Operational Matrix ◽  
Initial Guess ◽  
Test Problems ◽  
Tau Method ◽  
Algebraic Equations ◽  
Differential Problem ◽  
Riccati Differential Equation ◽  
Ultraspherical Polynomials ◽  
Expansion Coefficients

Herein, we developed and analyzed a new fractal–fractional (FF) operational matrix for orthonormal normalized ultraspherical polynomials. We used this matrix to handle the FF Riccati differential equation with the new generalized Caputo FF derivative. Based on the developed operational matrix and the spectral Tau method, the nonlinear differential problem was reduced to a system of algebraic equations in the unknown expansion coefficients. Accordingly, the resulting system was solved by Newton’s solver with a small initial guess. The efficiency, accuracy, and applicability of the developed numerical method were checked by exhibiting various test problems. The obtained results were also compared with other recent methods, based on the available literature.

scholarly journals Convergence Region of Newton Iterative Power Flow Method: Numerical Studies

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Jiao-Jiao Deng ◽  
Hsiao-Dong Chiang
Keyword(s):  
Power Flow ◽  
Test System ◽  
Initial Guess ◽  
Fractal Boundary ◽  
Convergence Region ◽  
Nonlinear Load ◽  
Flow Equations ◽  
Flow Method ◽  
Numerical Studies ◽  
Power Flow Method

Power flow study plays a fundamental role in the process of power system operation and planning. Of the several methods in commercial power flow package, the Newton-Raphson (NR) method is the most popular one. In this paper, we numerically study the convergence region of each power flow solution under the NR method. This study of convergence region provides insights of the complexity of the NR method in finding power flow solutions. Our numerical studies confirm that the convergence region of NR method has a fractal boundary and find that this fractal boundary of convergence regions persists under different loading conditions. In addition, the convergence regions of NR method for power flow equations with different nonlinear load models are also fractal. This fractal property highlights the importance of choosing initial guesses since a small variation of an initial guess near the convergence boundary leads to two different power flow solutions. One vital variation of Newton method popular in power industry is the fast decoupled power flow method whose convergence region is also numerically studied on an IEEE 14-bus test system which is of 22-dimensional in state space.

scholarly journals Two-Step Modified Newton Method for Nonlinear Lavrentiev Regularization

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Santhosh George ◽  
Suresan Pareth
Keyword(s):  
Newton Method ◽  
Regularization Parameter ◽  
Adaptive Method ◽  
Initial Guess ◽  
Optimal Order ◽  
Lavrentiev Regularization ◽  
Modified Newton Method ◽  
Actual Solution ◽  
Ill Posed ◽  
General Source

A two step modified Newton method is considered for obtaining an approximate solution for the nonlinear ill-posed equation F(x)=f when the available data are fδ with ‖f−fδ‖≤δ and the operator F is monotone. The derived error estimate under a general source condition on x0−x^ is of optimal order; here x0 is the initial guess and x^ is the actual solution. The regularization parameter is chosen according to the adaptive method considered by Perverzev and Schock (2005). The computational results provided endorse the reliability and effectiveness of our method.

scholarly journals A Nonlinear Shooting Method and Its Application to Nonlinear Rayleigh-Bénard Convection

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Jitender Singh
Keyword(s):  
Newton Method ◽  
Quadratic Convergence ◽  
Shooting Method ◽  
Fluid Layer ◽  
Algebraic Equations ◽  
Nonlinear Odes ◽  
Bénard Convection ◽  
Benard Convection ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

The simple shooting method is revisited in order to solve nonlinear two-point BVP numerically. The BVP of the type is considered where components of are known at one of the boundaries and components of are specified at the other boundary. The map is assumed to be smooth and satisfies the Lipschitz condition. The two-point BVP is transformed into a system of nonlinear algebraic equations in several variables which, is solved numerically using the Newton method. Unlike the one-dimensional case, the Newton method does not always have quadratic convergence in general. However, we prove that the rate of convergence of the Newton iterative scheme associated with the BVPs of present type is at least quadratic. This indeed justifies and generalizes the shooting method of Ha (2001) to the BVPs arising in the higher order nonlinear ODEs. With at least quadratic convergence of Newton's method, an explicit application in solving nonlinear Rayleigh-Bénard convection in a horizontal fluid layer heated from the below is discussed where rapid convergence in nonlinear shooting essentially plays an important role.

The initial guess estimation newton method for power flow in distribution systems

2017 ◽  
Vol 4 (2) ◽  
pp. 231-242 ◽  
Author(s):  
Qiuye Sun ◽  
Ling Liu ◽  
Dazhong Ma ◽  
Huaguang Zhang
Keyword(s):  
Newton Method ◽  
Power Flow ◽  
Distribution Systems ◽  
Initial Guess

A Hybrid Fuzzy Simplex Genetic Algorithm

2000 ◽  
Author(s):  
Mohamed B. Trabia
Keyword(s):  
Genetic Algorithm ◽  
Hybrid Algorithm ◽  
Simplex Algorithm ◽  
Standard Test ◽  
Initial Guess ◽  
Test Problems ◽  
Programming Technique ◽  
Design Problems ◽  
Engineering Design Problems ◽  
Novel Algorithm

Abstract Nelder and Mead Simplex (NMS) algorithm is an effective nonlinear programming technique. Trabia and Lu (1999) recently presented a novel algorithm, Fuzzy Simplex (FS), which improved the efficiency of Nelder and Mead Simplex by using fuzzy logic to determine the orientation and size of the simplex. While Fuzzy Simplex algorithm can be successfully used to search a wide variety of functions, it suffers, as other simplex algorithms, from its dependence on the initial guess and the original simplex size. This paper addresses this problem by combining the Fuzzy Simplex with Genetic Algorithm (GA) in a hybrid algorithm. Standard test problems are used to evaluate the efficiency of the algorithm. The algorithm is also applied successfully to several engineering design problems. The Hybrid GA Fuzzy Simplex algorithm generally results in a faster convergence.

Newton Method and Trajectory-Based Method for Solving Power Flow Problems: Nonlinear Studies

2015 ◽  
Vol 25 (06) ◽  
pp. 1530018 ◽  
Author(s):  
Jiao-Jiao Deng ◽  
Hsiao-Dong Chiang ◽  
Tian-Qi Zhao
Keyword(s):  
Dynamic System ◽  
Newton Method ◽  
Stability Region ◽  
Power Flow ◽  
Nonlinear Dynamic System ◽  
Convergence Properties ◽  
Convergence Region ◽  
Flow Problems ◽  
Connected Set ◽  
The Stability

This paper analyzes the convergence properties and convergence region of a class of trajectory-based power flow methods. The convergence region of the trajectory-based method is a connected set and possesses the near-by property. The convergence region of Newton method and trajectory-based method in solving power flow problems are numerically investigated. Since the convergence region of trajectory-based method corresponds to the stability region of the nonlinear dynamic system, the stability regions of two dynamic systems are computed. The numerical results indicate that the stability region of the dynamic system possesses better geometry features than the convergence region of Newton method. These properties make the trajectory-based power flow method robust, especially on heavy loading conditions.

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