1. Project on Newton’s Iteration Method Presented by Dol Nath Khanal Project Advisor- Professor Dexuan Xie 05/11/2015

2. Introduction This Project shows for solving an equation by using Newton’s Iteration Method. We have analyzed standard iteration technique(Newton-Raphson Method) which has implemented by using MATLAB computation. We use MATLAB to learn: 1. How to write function in MATLAB 2. How to make loop 3. How to use algorithm 4. How to setup code with graphic 5. How to understand the results The followings are some preliminaries which are analyzed to perform the main work.

3. Finding root of a function by Newton-Raphson method

4. Newton-Raphson Method Most widely used formula for locating roots. Can be derived using Taylor series or the geometric interpretation of the slope in the figure

5. Newton’s Method Finding a root for: We will use Newton’s Method to find the root between 2 and 3.

6. Guess: (not drawn to scale) (new guess)

7. Guess: (new guess)

8. Guess: (new guess)

9. Guess: Amazingly close to zero! This is Newton’s Method of finding roots. It is an example of an algorithm (a specific set of computational steps.) It is also called the Newton-Raphson method This is a recursive algorithm because a set of steps are repeated with the previous answer put in the next repetition. Each repetition is called is called an iteration.

10. Algorithm used for Newton- Raphson Method

11. Step 1 Evaluate symbolically.

12. Step 2 Use an initial guess of the root,, to estimate the new value of the root,, as

13. Step 3 Find the absolute relative approximate error as

14. Step 4 Compare the absolute relative approximate error with the pre-specified relative error tolerance. Also, check if the number of iterations has exceeded the maximum number of iterations allowed. If so, one needs to terminate the algorithm and notify the user. Is ? Yes No Go to Step 2 using new estimate of the root. Stop the algorithm

15. Advantages and Drawbacks of Newton Raphson Method

16. Advantages Converges fast (quadratic convergence), if it converges. Requires only one guess

17. Drawbacks 1.Divergence at inflection points Selection of the initial guess or an iteration value of the root that is close to the inflection point of the function may start diverging away from the root in the Newton-Raphson method. For example, to find the root of the equation. The Newton-Raphson method reduces to. Table 1 shows the iterated values of the root of the equation. The root starts to diverge at Iteration 6 because the previous estimate of 0.92589 is close to the inflection point of. Eventually after 12 more iterations the root converges to the exact value of

18. Drawbacks – Inflection Points Iteration Number xixi 05.0000 13.6560 22.7465 32.1084 41.6000 50.92589 6−30.119 7−19.746 180.2000 Divergence at inflection point for Table 1: Divergence near inflection point.

19. 2.Division by zero For the equation the Newton-Raphson method reduces to For, the denominator will equal zero. Drawbacks – Division by Zero Pitfall of division by zero or near a zero number

20. Results obtained from the Newton-Raphson method may oscillate about the local maximum or minimum without converging on a root but converging on the local maximum or minimum. Eventually, it may lead to division by a number close to zero and may diverge. For example for the equation has no real roots. Drawbacks – Oscillations near local maximum and minimum 3. Oscillations near local maximum and minimum

21. Drawbacks – Oscillations near local maximum and minimum Oscillations around local minima for. Iteration Number 01234567890123456789 –1.0000 0.5 –1.75 –0.30357 3.1423 1.2529 –0.17166 5.7395 2.6955 0.97678 3.00 2.25 5.063 2.092 11.874 3.570 2.029 34.942 9.266 2.954 300.00 128.571 476.47 109.66 150.80 829.88 102.99 112.93 175.96 Table 2: Oscillations near local maxima and mimima in Newton-Raphson method.

22. 4. Root Jumping In some cases where the function is oscillating and has a number of roots, one may choose an initial guess close to a root. However, the guesses may jump and converge to some other root. For example Choose It will converge to instead of Drawbacks – Root Jumping Root jumping from intended location of root for.

23. Numerical Results from MATLAB & MATLAB codes are added in separate files.

24. References Matlab demos A Very Elementary MATLAB Tutorial from the MathWorks web site. A Very Elementary MATLAB Tutorial Hypertext reference from Portland State University (55 pages) Hypertext reference from Portland State University 3-day MATLAB Tutorial from MIT (25 pages) 3-day MATLAB Tutorial from MIT Class notes Wikipedia

25. Thank You