1. Numerical Analysis
Lecture 5

2. Solution of Non-Linear Equations
Chapter 2
Solution of Non-Linear Equations

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

4. Bisection Method (Bolzano)

5. Method of false position
Regula-Falsi Method
Method of false position

6. Here, we choose two points xn and xn -1 such that f (xn) and f (xn-1) are of opposite signs. Intermediate value property suggests that the graph of y = f (x) crosses the x-axis between these two points and therefore, the root lies between these two points.

7. Example
Using Regula-Falsi method, find the real root of the following equation correct, to three decimal places:
x log 10 x =1.2

8. Solution Let f (x) = x log10x – 1.2 f (2) = – 0.5979, f (3) = 0.2314.
Since f (2) and f (3) are of opposite signs, the real root lies between x1 = 2, x2 = 3.

9. The first approximation is obtained from

10. Since f (x2) and f (x3) are of opposite signs, the root of f (x) = 0 lies between x2 and x3. Now, the second approximation is given by

11. Answer !
Thus, the root of the given equation correct to three decimal places is 2.740

12. Method of Iteration

13. METHOD OF ITERATION can be applied to find a real root of the equation f (x) = 0 by rewriting the same in the form,
(2.3)
Example,
f (x) = cos x – 2x + 3 = 0.
It can be
rewritten as

14. Also, f ’(x) and f ’’(x) do not vanish in (0, 1) and f (x) and f ’’(x) will have the same sign at x = 1.
Therefore, we take the first approximation x0 = 1, and using N-R method, we get

15. The second approximation is The required root is 0.853.

16. Example
Find a real root of the equation x3 – x – 1 = 0 using Newton - Raphson method, correct to four decimal places.

17. Numerical Analysis
Lecture 5