GROUP-4 PRESENTATION ON NR-METHOD.pptx

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1. Presented by M.Sc 2 nd semester (Group-4) Dipankar Saha Soujanita Dey Nirnay Banik Id : 20IUT0080023 Id : 20IUT0080024 Id : 20IUT0080028 Debajyoti Banik Aratrika Banik Satashree Deb Id : 20IUT0080029 Id : 20IUT0080031 Id : 20IUT0080032 A brief study on Newton-Raphson Method Department of Physics, ICFAI University ,Agartala, West Tripura.

2. Acknowledgment It is great pleasure for us to undertake this presentation. we feel highly doing this presentation namely “ A brief study on Newton-Raphson Method ”. We would like to express our special thanks of gratitude to our Numerical Analysis teacher “ Mr. Subal Ranjan Sahu ” for his heartful restless assist in completing this presentation. This presentation would not have completed without his enormous help and worthy experiences. Whenever we were in need, he was there behind us always. DATE: GROUP-4 9 th June,2021 M.Sc Physics, 2 nd semester ICFAI University Tripura.

3. Overview Introduction. History Of Newton- Raphson Method. Derivation. Graphical representation. Taylor- Series Expansion. Example. Rate of convergence of N-R method. Algorithm. Flowchart. MATLAB code. Advantages & Disadvantages. Applications. Conclusion.

4. Introduction In Numerical Analysis Newton-Raphson Method , also known as the Newton’s Method , is a method for finding successively better approximations to the roots of a real valued function. In Newton’s method the root is not bracketed , In fact, only one initial guess of the root is needed to get the iterative process started to find the root of an equation. The method hence falls in the category of open methods.

5. History Of Newton- Raphson Method Heron of Alexandria (10-7- AD) described a method (called Babylonian Method) to iteratively approximate a square root. Francois Viete (1540-1603) developed a method to approximate roots of polynomials. Isaac Newton (1540-1603) in 1669 (PUBLISHED IS 1711) improved upon Viete’s method. A simplified version of Newton’s method was published by Joseph Raphson (1648-1715) in 1690. Though, Newton (and Raphson ) did not see the connection between his method and calculus. The modern treatment is due to Thomas Simpson (1710-1761). Isaac Newton

6. Derivation The Newton Raphson method is based on the principle that if the initial guess of the root is at X n , Then if one draws the tangent to the curve at f( X n ) , the point X n+1 where the tangent cross the X-axis is an improve estimate of the root (shown in the figure-1 ). From the equation of slop we have, This is the starting point at X , so we write it as Similarly ,

7. Thus we get the general formula, …………………( i ) Equation ( i ) is called the Newton- Raphson formula for solving nonlinear equation of the form f(X)=0. One can repeat this process until one find the root within a desirable tolerance.

8. Graphical representation Figure-1 : Graphical representation for N-R Method

9. Taylor Series Expansion:- we have given an equation f(x)=0 and x be the approximated root of the given equation. Let (X +h) be the actual root where ‘h’ is very small such that f(X +h)=0 ……….(ii) Now from Taylor Series expansion on expanding to f(X +h) we get, Now on neglecting higher powers of h we have, From Eqn.(ii)

10. Hence 1 st approximation X 1 =(X +h) On repeating this process we get, This is the required Newton- Raphson Method . 2 nd approximation

11. Example : Perform four iterations using Newton- Raphson method to obtain the approximate value of start with an approximation X = 2 . The value of X 4 & X 5 are repeated . So, the root is 2.5712

12. Rate of Convergence of N-R method: We have the N-R formula Let, at x=α we have the exact root of a given equation f(x)=0 & ‘ ’ be the error . On substituting X n =α+ϵ n and X n+1 = α+ϵ n+1 in equation ( i ) we get  

13. Neglecting higher power of ϵ n and noting also f(α)=0 we have [neglecting higher order term] Thus we see that the rate of convergence of N-R method is 2. ϵ n+1 proportional to ϵ n 2

14. Algorithm Define the function. Choose an initial guess X such that, [ a,b ] be any interval in which f(a)<0 & f(b)>0 , then Then If f(X 1 ) = 0, then X 1 is our exact root. For more accuracy by repetition of this process we can find X 2 ,X 3 , X 4 ………. from X 1 until we are getting the accurate value of the root.  

15. Flowchart

16. MATLAB CODE This is the code which we used while using programming clear all close all clc f=@(x) f(x) ; %define function df = @ˊ(x) fˊ(x) ; %define derivative of the function a= input( 'First guess a=' ); %initial guess b= input( 'Second guess b=' ); %initial guess n = input( 'number of iterations=' ); while f(a)*f(b)>0 disp ( 'initial guesses are not correct' ) a = input( 'a=' ); b = input( 'b=' ); end x=a;

17. MATLAB CODE fprintf ( 'iteration table for newton raphson method\n' ) fprintf ( '\n iteration no x\t f(x)\t df (x)\t\n' ) for i = 1:n x( i )= x- (f(x)/ df (x)); f(x); df (x); x=x( i ); fprintf ( '\ n%d %13.4f %13.4f %13.4f %13.4f\ n' ,i,x,f (x), df (x)) end fprintf ( 'solution of the equation =% f' ,x )

18. Advantages & Disadvantages Advantages The Newton- Raphson method is faster, more reliable and the results are accurate. Requires less number of iterations for convergence. Suitable for large system.   Disadvantages The programming logic is more. Memory requirement is more. Computational time per iteration is higher due to larger number of calculations per iteration .

19. Applications Using the Newton- Raphson formula we can find the roots of the Algebraic equations and as well as Transcendental equations in less time as well as less number of iterations. Usually used in computer science for root algorithm. Used for Multidimensional root finding. Solving practical technical problems using scientific and mathematical tools.

20. Conclusion Root finding is also one of the problems in practical applications. To solve the transcendental equations most of the times we use the iteration methods. There are so many iteration methods like Newton- Raphson method. We concluded that, the convergence rate of Newton-Raphson method is fast as compared to other methods. The effectiveness of using scientific calculator in solving non-linear equations using Newton- Raphson method also reduces the time complexity, for solving nonlinear equations. So, we also concluded that the Newton- Raphson method can be used very effectively to determine the correct & accurate root in less time as well as less number of iterations.

21. Bibliography Numerical analysis and computational procedures by S.A.MOLLAH. Introductory numerical analysis by N.DATTA & R.N.JANA. Numerical methods by M.K.Jain,S.R.K.lyengar & R.K.Jain . Introductory methods of numerical analysis by S.S.Sastry . An introduction to numerical analysis by Kendall E.Atkinson .

22. Thank YOu