Research article · 2017 · Full text
International Journal of Statistics and Applied MathematicsComparative study of convergence and solution of algebraic and transcendental equations using zero-finding methods
Volume 2(6) · Pages 187–189
Research areasNumerical Methods and AlgorithmsAdvanced Optimization Algorithms ResearchPower System Optimization and Stability
Abstract
Author-writtenThe paper is about Newton-Raphson Method which is used to solve non-square and non-linear problems. The study also aims to comparing the rate of performance, rate of convergence, root finding. It also represents a new approach calculation using non-linear terms and this will be similar to Newton Raphson simple method. An inverse Jacobian matrix will be used for the iteration process and this will be further used for distributed power load flow calculation and will also be helpful in some of the application.
Summary
Source-checkedPlain-language reading of the full paper, prepared with AI and checked against the text. Each point links to the authors' own words.
Why it matters
- Finding the roots of equations is a common challenge in various practical applications. View evidence ↗
- The Newton-Raphson method is a well-known numerical technique named after Isaac Newton and Joseph Raphson. View evidence ↗
- The Newton-Raphson method is recognized for its ability to improve convergence properties and its fast rate of convergence. View evidence ↗
The question
- The study aims to compare root-finding performance and rates of convergence among different methods. View evidence ↗
What they did
- The Newton method's process involves a single iteration and the evaluation of the derivative during each iteration. View evidence ↗
- Calculations for finding roots using this method are performed using scientific calculators and are also adopted by MATLAB. View evidence ↗
- To demonstrate the method for finding the square root of 5, the function f(x) = x² - 5 was used on the interval [1, 3] with an initial guess of x1 = 2. View evidence ↗
What they found
- When comparing convergence rates, the Bisection method was the slowest, followed by the Newton method, with the Secant method being the fastest. View evidence ↗
- Quantitatively, the Newton method is 7.678622465 times better than the Bisection method, and the Secant method is 1.389482397 times better than the Newton method. View evidence ↗
- In the square root approximation example, the results stabilized to more than ten decimal places after only 5 iterations. View evidence ↗
- The Newton-Raphson method is characterized by quadratic convergence, as the error at iteration n+1 is proportional to the square of the error at iteration n. View evidence ↗
What it means
- The authors conclude that the Secant method is the most effective because it offers a convergence rate similar to Newton-Raphson but only requires one function evaluation per iteration. View evidence ↗
- The use of scientific calculators with the Newton-Raphson method is concluded to reduce the time complexity involved in solving non-linear equations. View evidence ↗
- The authors conclude that the Bisection method has a very slow convergence rate compared to the other methods studied. View evidence ↗
Visual summary
1 visualsIllustrations explain the paper; they do not show study data.
Convergence Rate Comparison
AI-generated conceptual illustration (A mathematical coordinate plane with a curved function line crossing a horizontal axis). It explains the idea and does not show study data.
View source evidence ↗Introduction
Here we consider the nonlinear equation and try to find the solution of it. We have to found the solution with the help of Newton -Raphson method. This method is very familiar for its fast rate of convergence and for improving the convergence property. Newton's method, named after Isaac-Newton and Joseph -Raphson. Any zero-finding method (Bisection Method, False Position Method, Newton-Raphson, etc.) can also be used to find a minimum or maximum of such a function, by finding a zero in the function's first derivative, see Newton's method has an optimization algorithm. Root finding is also one of the problem in practical applications. Newton method is very fast and efficient as compared to the other methods. Newton method requires only one iteration and the derivative evaluation per iteration. The result of comparing the rate of convergence of Bisection, Newton and Secant methods came as Bisection method < Newton method < Secant method which in terms of number is that the Newton method is 7.678622465 times better than the Bisection method whereas Secant method is 1.389482397 times better than the Newton method. Finding roots of the nonlinear equation with the help of Newton Raphson method, provides good result with fast convergence speed and Mat lab also adopted this method for finding the roots and tool used for such calculations is scientific calculator.
Formulation
In numerical analysis, Newton's method (also known as the Newton-Raphson method), named after Isaac Newton and Joseph Raphson, is a method for finding successively better approximations to the roots (or zeroes) of a real valued function.
The Newton-Raphson method in one variable is implemented as follows: Given a function ƒ defined over the reals x, and its derivative ƒ', we begin with a first guess x0 for a root of the function f. Provided the function satisfies all the assumptions made in the derivation of the formula, a better approximation x1 is x1=x0-
Geometrically, (x1, 0) is the intersection with the x-axis of the tangent to the graph of f at (x0, f (x0)).
The process is repeated as
Until a sufficiently accurate value is reached.
Applications Square root of a number
Consider the problem of finding the square root of a number. Newton's method is one of many methods of computing square roots.
Example. Let us find an approximation to 5 to ten decimal places.
Note that 5 is an irrational number. Therefore the sequence of decimals which defines 5 will not stop. Clearly r = 5 is the only zero of f(x) = x 2 -5 on the interval [1,3] .
be the successive approximations obtained through Newton's method. We have Let us start this process by taking x1 = 2.
It is quite remarkable that the results stabilize for more than ten decimal places after only 5 iterations! Therefore, Newton Raphson Method is said to have quadratic convergence.
Trigonometry function
Trigonometry function
Conclusion
From this paper we have concluded that, the convergence rate of Newton method is fast as compared to other methods. Secant method is the most effective method as it has a convergence rate close to that of the Newton Raphson method but it requires only a single function evaluating per iteration.
We have also cocluded that the convergence rate of bisection method is very slow and it is difficult to extend such kind of systems equations. So in comparison, Newton's method have a fast converging rate. The effectiveness of using scientific calculator in solving non-linear equations using Newton-Raphson method also reduces the time complexity for solving nonlinear equations. In this work a sequence of iterative methods for solving nonlinear equation f(x) = 0 with higherorder convergence is developed. The method can be continuously applied to generate an iterative scheme with arbitrarily specified order of convergence. We also concluded that the Newton Raphson method can be used very effectively to determine the intrinsic value based on its measured permittivity.
References
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