Developing high order methods for the solution of systems of nonlinear equations

2019 ◽  
Vol 342 ◽  
pp. 178-190
Author(s):  
Changbum Chun ◽  
Beny Neta
Keyword(s):  
Nonlinear Equations ◽  
High Order ◽  
Systems Of Nonlinear Equations ◽  
High Order Methods

High-order iterations for systems of nonlinear equations

2019 ◽  
Vol 97 (8) ◽  
pp. 1704-1724 ◽  
Author(s):  
T. Zhanlav ◽  
Changbum Chun ◽  
Kh. Otgondorj ◽  
V. Ulziibayar
Keyword(s):  
Nonlinear Equations ◽  
High Order ◽  
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 Multistep High-Order Methods for Nonlinear Equations Using Padé-Like Approximants

2017 ◽  
Vol 2017 ◽  
pp. 1-6
Author(s):  
Alicia Cordero ◽  
José L. Hueso ◽  
Eulalia Martínez ◽  
Juan R. Torregrosa
Keyword(s):  
Nonlinear Equation ◽  
Iterative Methods ◽  
Nonlinear Equations ◽  
High Order ◽  
High Order Methods ◽  
Optimal Methods ◽  
Numerical Tests ◽  
Theoretical Results

We present new high-order optimal iterative methods for solving a nonlinear equation, f(x)=0, by using Padé-like approximants. We compose optimal methods of order 4 with Newton’s step and substitute the derivative by using an appropriate rational approximant, getting optimal methods of order 8. In the same way, increasing the degree of the approximant, we obtain optimal methods of order 16. We also perform different numerical tests that confirm the theoretical results.

scholarly journals Proving the convergence of the iterative methods by using symbolic computation in Maple

2012 ◽  
Vol 28 (1) ◽  
pp. 1-8
Author(s):  
GHEORGHE ARDELEAN ◽  
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Symbolic Computation ◽  
High Order ◽  
High Order Methods ◽  
Convergence Error

The proofs of the convergence for some high-order methods for solving nonlinear equations, by using symbolic computation in Maple, is presented. Also, the convergence error for some Newton-type methods is evaluated by symbolic computation.

High order boundary treatment for high order methods in computational fluid dynamics

2016 ◽  
Author(s):  
André Ribeiro de Barros Aguiar ◽  
Carlos Breviglieri ◽  
Fábio Mallaco Moreira ◽  
Eduardo Jourdan ◽  
João Luiz F. Azevedo
Keyword(s):  
Fluid Dynamics ◽  
Computational Fluid Dynamics ◽  
High Order ◽  
High Order Methods ◽  
Boundary Treatment

High-Order Methods For Wave Propagation

2008 ◽  
Author(s):  
Miguel R. Visbal ◽  
Scott E. Sherer ◽  
Michael D. White
Keyword(s):  
Wave Propagation ◽  
High Order ◽  
High Order Methods
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