1. Lecture 8 Numerical Analysis

2. Solution of Non-Linear Equations Chapter 2

3. Introduction Bisection Method Regula-Falsi Method Method of iteration Newton - Raphson Method Secant Method Muller’s Method Graeffe’s Root Squaring Method

4. Muller’s Method

5. Suppose, be any three distinct approximations to a root of f (x) = 0.

6. Noting that any three distinct points in the (x, y)-plane uniquely, determine a polynomial of second degree. A general polynomial of second degree is given by

7. Suppose, it passes through the points then the following equations will be satisfied

8. Fig: Quadratic polynomial.

9. Eliminating a, b, c, we obtain

10. which can be written as

11. The above equation can be written as That was a second degree polynomial. Now, introducing the notation

12. The above equation can be written as

13. We further define

14. With these substitution we get a simplified Equation as

15. Or

16. To compute set f = 0, we obtain where A direct solution will lead to loss of accuracy and therefore to obtain max accuracy we rewrite as:

17. so that, or Here, the positive sign must be so chosen that the denominator becomes largest in magnitude.

18. we can get a better approximation to the root, by using

19. Example Do two iterations of Muller’s method to solve starting with

20. Solution

22. Take

23. = 0.8333

24. For third iteration take,

25. Graeffe’s Root Squaring Method

26. GRAEFFE’S ROOT SQUARING METHOD is particularly attractive for finding all the roots of a polynomial equation. Consider a polynomial of third degree

28. The roots of this equation are squares or 2 i (i = 1), powers of the original roots. Here i = 1 indicates that squaring is done once. The same equation can again be squared and this squaring process is repeated as many times as required. After each squaring, the coefficients become large and overflow is possible as i increases.

29. Suppose, we have squared the given polynomial ‘ i’ times, then we can estimate the value of the roots by evaluating 2 i root of where n is the degree of the given polynomial.

30. The proper sign of each root can be determined by recalling the original equation. This method fails, when the roots of the given polynomial are repeated.

31. Example Using Graeffe root squaring method, find all the roots of the equation

32. Solution Using Graeffe root squaring method, the first three squared polynomials are as under: For i = 1, the polynomial is

33. For i = 2, the polynomial is For i = 3, the polynomial is

34. The roots of polynomial are Similarly, the roots of polynomial (2) are

35. Still better estimates of the roots obtained from polynomial (3) are The exact values of the roots of the given polynomial are 1, 2 and 3.

36. Lecture 8 Numerical Analysis

37. Revision Example Obtain the Newton-Raphson extended formula for finding the root of the equation f (x) = 0.

38. Solution Expanding f (x) by Taylor’s series, in the neighborhood of x 0, we obtain after retaining the first order term only Which gives This is the first approximation to the root. Therefore,

39. Expanding f (x) by Taylor’s series and retaining up to second order term, Therefore,

40. This can also be written as Thus, the Newton-Raphson extended formula is given by This is also known as Chebyshev’s formula of third order

41. Lecture 8 Numerical Analysis