Comparative Study of Iterative Methods for Solving Non-Linear Equations

In numerical analysis, methods for finding roots play a pivotal role in the field of many real and practical applications. The efficiency of numerical methods depends upon the convergence rate (how fast the particular method converges). The objective of this study is to compare the Bisection method, Newton-Raphson method, and False Position Method with their limitations and also analyze them to know which of them is more preferred. Limitations of these methods have allowed presenting the latest research in the area of iterative processes for solving non-linear equations. This paper analyzes the field of iterative methods which are developed in recent years with their future scope.

Numerical Solution of Nonlinear Equations in Maple

The main purpose of this paper is to obtain the real roots of an expression using the Numerical method, bisection method, Newton's method and secant method. Root analysis is calculated using specific, precise starting points and numerical methods and is represented by Maple. In this research, we used Maple software to analyze the roots of nonlinear equations by special methods, and by showing geometric diagrams, we examined the relevant examples. In this process, the Newton-Raphson method, the algorithm for root access, is fully illustrated by Maple. Also, the secant method and the bisection method were demonstrated by Maple by solving examples and drawing graphs related to each method.

A Novel Fast-Locking ADPLL Based on Bisection Method

Based on the idea of bisection method, a new structure of All-Digital Phased-Locked Loop (ADPLL) with fast-locking is proposed. The structure and locking method are different from the traditional ADPLLs. The Control Circuit consists of frequency compare module, mode-adjust module and control module, which is responsible for adjusting the frequency control word of digital-controlled-oscillator (DCO) by Bisection method according to the result of the frequency compare between reference clock and restructure clock. With a high frequency cascade structure, the DCO achieves wide tuning range and high resolution. The proposed ADPLL was designed in SMIC 180 nm CMOS process. The measured results show a lock range of 640-to-1920 MHz with a 40 MHz reference frequency. The ADPLL core occupies 0.04 mm2, and the power consumption is 29.48 mW, with a 1.8 V supply. The longest locking time is 23 reference cycles, 575 ns, at 1.92 GHz. When the ADPLL operates at 1.28 GHz–1.6 GHz, the locking time is the shortest, only 9 reference cycles, 225 ns. Compared with the recent high-performance ADPLLs, our design shows advantages of small area, short locking time, and wide tuning range.

A NEW SECOND ORDER DERIVATIVE FREE METHOD FOR NUMERICAL SOLUTION OF NON-LINEAR ALGEBRAIC AND TRANSCENDENTAL EQUATIONS USING INTERPOLATION TECHNIQUE

Its most important task in numerical analysis to find roots of nonlinear equations, several methods already exist in literature to find roots but in this paper, we introduce a unique idea by using the interpolation technique. The proposed method derived from the newton backward interpolation technique and the convergence of the proposed method is quadratic, all types of problems (taken from literature) have been solved by this method and compared their results with another existing method (bisection method (BM), regula falsi method (RFM), secant method (SM) and newton raphson method (NRM)) it’s observed that the proposed method have fast convergence. MATLAB/C++ software is used to solve problems by different methods.