DC TOLERANCE ANALYSIS OF NONLINEAR CIRCUITS USING SET-VALUED FUNCTIONS

In the tolerance analysis of electronic circuits, the concept of set-valued function is often useful. In this paper, an efficient algorithm is proposed for finding all solution sets of nonlinear resistive circuits described by systems of nonlinear equations containing set-valued functions termed piecewise-trapezoidal (PWT) functions. By numerical examples, the effectiveness of the proposed algorithm is confirmed from various viewpoints. It is also shown that the proposed algorithm could find all solution regions to a system of 1000 PWT equations in practical computation time.

Download Full-text

SOME HIGH ORDER ITERATIVE METHODS FOR NONLINEAR EQUATIONS BASED ON THE MODIFIED HOMOTOPY PERTURBATION METHODS

Asian-European Journal of Mathematics ◽

10.1142/s1793557110000349 ◽

2010 ◽

Vol 03 (03) ◽

pp. 395-408

Author(s):

Bilian Chen ◽

Yajun Xie ◽

Changfeng Ma

Keyword(s):

Nonlinear Equation ◽

Iterative Methods ◽

Nonlinear Equations ◽

Perturbation Methods ◽

Systems Of Nonlinear Equations ◽

Convergence Criteria ◽

Numerical Examples ◽

Homotopy Perturbation ◽

Nonlinear Equation Systems ◽

High Order Iterative Methods

In this paper, we present some efficient iterative methods for solving nonlinear equation (systems of nonlinear equations, respectively) by using modified homotopy perturbation methods. We also discuss the convergence criteria of the present methods. Some numerical examples are given to illustrate the performance and efficiency of the proposed methods.

Download Full-text

An efficient algorithm for finding all solutions of separable systems of nonlinear equations

BIT Numerical Mathematics ◽

10.1007/s10543-007-0132-1 ◽

2007 ◽

Vol 47 (3) ◽

pp. 681-691 ◽

Cited By ~ 8

Author(s):

Kiyotaka Yamamura ◽

Koki Suda

Keyword(s):

Nonlinear Equations ◽

Efficient Algorithm ◽

Systems Of Nonlinear Equations ◽

All Solutions

Download Full-text

New Multi-Step Iterative Methods for Solving Systems of Nonlinear Equations and Their Application on GNSS Pseudorange Equations

Sensors ◽

10.3390/s20215976 ◽

2020 ◽

Vol 20 (21) ◽

pp. 5976

Author(s):

Kalyanasundaram Madhu ◽

Arul Elango ◽

René Jr Landry ◽

Mo’tassem Al-arydah

Keyword(s):

Iterative Methods ◽

Nonlinear Equations ◽

Computation Time ◽

Efficiency Index ◽

Systems Of Nonlinear Equations ◽

Test Problems ◽

System Of Nonlinear Equations ◽

Additional Function ◽

Computation Efficiency ◽

Global Navigation Satellite

A two-step fifth and a multi-step 5+3r order iterative method are derived, r≥1 for finding the solution of system of nonlinear equations. The new two-step fifth order method requires two functions, two first order derivatives, and the multi-step methods needs a additional function per step. The performance of this method has been tested with finding solutions to several test problems then applied to solving pseudorange nonlinear equations on Global Navigation Satellite Signal (GNSS). To solve the problem, at least four satellite’s measurements are needed to locate the user position and receiver time offset. In this work, a number of satellites from 4 to 8 are considered such that the number of equations is more than the number of unknown variables to calculate the user position. Moreover, the Geometrical Dilution of Precision (GDOP) values are computed based on the satellite selection algorithm (fuzzy logic method) which could be able to bring the best suitable combination of satellites. We have restricted the number of satellites to 4 to 6 for solving the pseudorange equations to get better GDOP value even after increasing the number of satellites beyond six also yields a 0.4075 GDOP value. Actually, the conventional methods utilized in the position calculation module of the GNSS receiver typically converge with six iterations for finding the user position whereas the proposed method takes only three iterations which really decreases the computation time which provide quicker position calculation. A practical study was done to evaluate the computation efficiency index (CE) and efficiency index (IE) of the new model. From the simulation outcomes, it has been noted that the new method is more efficient and converges 33% faster than the conventional iterative methods with good accuracy of 92%.

Download Full-text

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

An International Journal of Optimization and Control Theories & Applications (IJOCTA) ◽

10.11121/ijocta.01.2019.00743 ◽

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.

Download Full-text

An Adaptive Nonmonotone Line Search Technique for Solving Systems of Nonlinear Equations

Journal of Mathematics ◽

10.1155/2021/7134561 ◽

2021 ◽

Vol 2021 ◽

pp. 1-6

Author(s):

Masoud Hatamian ◽

Mahmoud Paripour ◽

Farajollah Mohammadi Yaghoobi ◽

Nasrin Karamikabir

Keyword(s):

Global Convergence ◽

Nonlinear Equations ◽

Line Search ◽

Nonmonotone Line Search ◽

Systems Of Nonlinear Equations ◽

System Of Nonlinear Equations ◽

Search Technique ◽

Numerical Examples ◽

Line Search Technique ◽

Novel Algorithm

In this article, a new nonmonotone line search technique is proposed for solving a system of nonlinear equations. We attempt to answer this question how to control the degree of the nonmonotonicity of line search rules in order to reach a more efficient algorithm? Therefore, we present a novel algorithm that can avoid the increase of unsuccessful iterations. For this purpose, we show the robust behavior of the proposed algorithm by solving a few numerical examples. Under some suitable assumptions, the global convergence of our strategy is proved.

Download Full-text

Ball Convergence for a Multi-Step Harmonic Mean Newton-Like Method in Banach Space

International Journal of Computational Methods ◽

10.1142/s0219876219400188 ◽

2019 ◽

Vol 17 (05) ◽

pp. 1940018 ◽

Cited By ~ 1

Author(s):

Ramandeep Behl ◽

Ali Saleh Alshormani ◽

Ioannis K. Argyros

Keyword(s):

Banach Space ◽

Nonlinear Equations ◽

Local Convergence ◽

Harmonic Mean ◽

Order Derivative ◽

Systems Of Nonlinear Equations ◽

Numerical Examples ◽

First Order ◽

Ball Convergence ◽

Local Convergence Analysis

In this paper, we present a local convergence analysis of some iterative methods to approximate a locally unique solution of nonlinear equations in a Banach space setting. In the earlier study [Babajee et al. (2015) “On some improved harmonic mean Newton-like methods for solving systems of nonlinear equations,” Algorithms 8(4), 895–909], demonstrate convergence of their methods under hypotheses on the fourth-order derivative or even higher. However, only first-order derivative of the function appears in their proposed scheme. In this study, we have shown that the local convergence of these methods depends on hypotheses only on the first-order derivative and the Lipschitz condition. In this way, we not only expand the applicability of these methods but also proposed the theoretical radius of convergence of these methods. Finally, a variety of concrete numerical examples demonstrate that our results even apply to solve those nonlinear equations where earlier studies cannot apply.

Download Full-text