A modified Newton method for the solution of ill-conditioned systems of nonlinear equations with application to multiple shooting

1974 ◽  
Vol 22 (4) ◽  
pp. 289-315 ◽  
Author(s):  
P. Deuflhard
Keyword(s):  
Nonlinear Equations ◽  
Newton Method ◽  
Multiple Shooting ◽  
Systems Of Nonlinear Equations ◽  
Modified Newton Method

An efficient fourth order weighted-Newton method for systems of nonlinear equations

2012 ◽  
Vol 62 (2) ◽  
pp. 307-323 ◽  
Author(s):  
Janak Raj Sharma ◽  
Rangan Kumar Guha ◽  
Rajni Sharma
Keyword(s):  
Nonlinear Equations ◽  
Newton Method ◽  
Fourth Order ◽  
Systems Of Nonlinear Equations

scholarly journals Performance analysis of a modified Newton method for parameterized dual fuzzy nonlinear equations and its application

2022 ◽  
pp. 105140
Author(s):  
Ibrahim Mohammed Sulaiman ◽  
Mustafa Mamat ◽  
Maulana Malik ◽  
Kottakkaran Sooppy Nisar ◽  
Ashraf Elfasakhany
Keyword(s):  
Performance Analysis ◽  
Nonlinear Equations ◽  
Newton Method ◽  
Modified Newton Method

scholarly journals A parameterized multi-step Newton method for solving systems of nonlinear equations

2015 ◽  
Vol 71 (3) ◽  
pp. 631-653 ◽  
Author(s):  
Fayyaz Ahmad ◽  
Emran Tohidi ◽  
Juan A. Carrasco
Keyword(s):  
Nonlinear Equations ◽  
Newton Method ◽  
Systems Of Nonlinear Equations

scholarly journals Hybrid algorithm Newton method for solving systems of nonlinear equations with block Jacobi matrix

2020 ◽  
pp. 208-217
Author(s):  
O.M. Khimich ◽  
◽  
V.A. Sydoruk ◽  
A.N. Nesterenko ◽  
◽  
...  
Keyword(s):  
Nonlinear Equations ◽  
Newton Method ◽  
Jacobi Matrix ◽  
Sparse Matrices ◽  
High Order ◽  
New Method ◽  
Systems Of Nonlinear Equations ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
High Convergence Rate

Systems of nonlinear equations often arise when modeling processes of different nature. These can be both independent problems describing physical processes and also problems arising at the intermediate stage of solving more complex mathematical problems. Usually, these are high-order tasks with the big count of un-knows, that better take into account the local features of the process or the things that are modeled. In addition, more accurate discrete models allow for more accurate solutions. Usually, the matrices of such problems have a sparse structure. Often the structure of sparse matrices is one of next: band, profile, block-diagonal with bordering, etc. In many cases, the matrices of the discrete problems are symmetric and positively defined or half-defined. The solution of systems of nonlinear equations is performed mainly by iterative methods based on the Newton method, which has a high convergence rate (quadratic) near the solution, provided that the initial approximation lies in the area of gravity of the solution. In this case, the method requires, at each iteration, to calculates the Jacobi matrix and to further solving systems of linear algebraic equations. As a consequence, the complexity of one iteration is. Using the parallel computations in the step of the solving of systems of linear algebraic equations greatly accelerates the process of finding the solution of systems of nonlinear equations. In the paper, a new method for solving systems of nonlinear high-order equations with the Jacobi block matrix is proposed. The basis of the new method is to combine the classical algorithm of the Newton method with an efficient small-tile algorithm for solving systems of linear equations with sparse matrices. The times of solving the systems of nonlinear equations of different orders on the nodes of the SKIT supercomputer are given.

scholarly journals Modified Newton method to determine multiple zeros of nonlinear equations

2016 ◽  
Vol 11 (10) ◽  
pp. 5774-5780
Author(s):  
Rajinder Thukral
Keyword(s):  
Iterative Method ◽  
Nonlinear Equations ◽  
Newton Method ◽  
New Method ◽  
Convergence Order ◽  
Multiple Zeros ◽  
Numerical Comparisons ◽  
Modified Newton Method ◽  
Iteration Methods ◽  
Better Than

New one-point iterative method for solving nonlinear equations is constructed.  It is proved that the new method has the convergence order of three. Per iteration the new method requires two evaluations of the function.  Kung and Traub conjectured that the multipoint iteration methods, without memory based on n evaluations, could achieve maximum convergence order2n-1  but, the new method produces convergence order of three, which is better than expected maximum convergence order of two.  Hence, we demonstrate that the conjecture fails for a particular set of nonlinear equations. Numerical comparisons are included to demonstrate exceptional convergence speed of the proposed method using only a few function evaluations.

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