Modifying Singular Values: Existence of Solutions to Systems of Nonlinear Equations Having a Possibly Singular Jacobian Matrix

1977 ◽  
Vol 31 (140) ◽  
pp. 962 ◽  
Author(s):  
David Gay
Keyword(s):  
Nonlinear Equations ◽  
Existence Of Solutions ◽  
Jacobian Matrix ◽  
Singular Values ◽  
Systems Of Nonlinear Equations ◽  
Singular Jacobian

scholarly journals Improved quasi-newton method via SR1 update for solving symmetric systems of nonlinear equations

2019 ◽  
Vol 15 (1) ◽  
pp. 117-120 ◽  
Author(s):  
Muhammad Kabir Dauda ◽  
Mustafa Mamat ◽  
Mohamad Afendee Mohamed ◽  
Mahammad Yusuf Waziri
Keyword(s):  
Nonlinear Equations ◽  
Newton Method ◽  
Large Scale ◽  
Jacobian Matrix ◽  
Systems Of Nonlinear Equations ◽  
Test Problems ◽  
Inexact Line Search ◽  
Cpu Time ◽  
Number Of Iterations ◽  
Symmetric Systems

The systems of nonlinear equations emerges from many areas of computing, scientific and engineering research applications. A variety of an iterative methods for solving such systems have been developed, this include the famous Newton method. Unfortunately, the Newton method suffers setback, which includes storing  matrix at each iteration and computing Jacobian matrix, which may be difficult or even impossible to compute. To overcome the drawbacks that bedeviling Newton method, a modification to SR1 update was proposed in this study. With the aid of inexact line search procedure by Li and Fukushima, the modification was achieved by simply approximating the inverse Hessian matrix  with an identity matrix without computing the Jacobian. Unlike the classical SR1 method, the modification neither require storing  matrix at each iteration nor needed to compute the Jacobian matrix. In finding the solution to non-linear problems of the form  40 benchmark test problems were solved. A comparison was made with other two methods based on CPU time and number of iterations. In this study, the proposed method solved 37 problems effectively in terms of number of iterations. In terms of CPU time, the proposed method also outperformed the existing methods. The contribution from the methodology yielded a method that is suitable for solving symmetric systems of nonlinear equations. The derivative-free feature of the proposed method gave its advantage to solve relatively large-scale problems (10,000 variables) compared to the existing methods. From the preliminary numerical results, the proposed method turned out to be significantly faster, effective and suitable for solving large scale symmetric nonlinear equations.

scholarly journals A New Broyden rank reduction method to solve large systems of nonlinear equations

2019 ◽  
Vol 9 (2) ◽  
pp. 176-185
Author(s):  
Ouarit Mostafa ◽  
Ali Souissi ◽  
Mohamed Ziani
Keyword(s):  
Nonlinear Equations ◽  
Reduction Method ◽  
A Priori ◽  
Singular Values ◽  
High Dimensional ◽  
Systems Of Nonlinear Equations ◽  
Rank Reduction ◽  
Numerical Examples ◽  
Limited Memory ◽  
Large Systems

We propose a modification of limited memory Broyden methods, called dynamical Broyden rank reduction method, to solve high dimensional systems of nonlinear equations. Based on a thresholding process of singular values, the proposed method determines a priori the rank of the reduced update matrix. It significantly reduces the number of singular values decomposition calls of the update matrix during the iterations. Local superlinear convergence of the method is proved and some numerical examples are displayed.

Sign in / Sign up

Export Citation Format

Share Document