1. Numerical Analysis
Lecture 7

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 Secant Method Muller’s Method Graeffe’s Root Squaring Method

4. Secant Method

5. We choose x0, x1 close to the root ‘a’ of f (x) =0 such that f (x0) f(x1)
As a next approximation x2 is obtained as the point of intersection of y = 0 and the chord passing through the points
(x0, f(x0 )), (x1 f(x1 )).

6. Putting y = 0 we get
Generally
This sequence converges to the root ‘b’ of f (x) = 0 i.e. f( b ) = 0.

7. Convergence of Secant Method
Here the sequence is generated by the rule
starting with x0 and x1 as
It will converge to ‘a’ ,that is f ( a ) = 0

8. The Secant method converges faster than linear and slower than Newton’s quadratic.

9. Example
Do three iterations of secant method to find the root of
taking

13. Though X5 is not the true root, yet this is good approximation to the root and convergence is faster than bisection.

14. Comparison:
In secant method, we do not check whether the root lies in between two successive approximates Xn-1, and Xn.
This checking was imposed after each iteration, in Regula –Falsi method.

15. Muller’s Method

16. In Muller’s method, f (x) = 0 is approximated by a second degree polynomial; that is by a quadratic equation that fits through three points in the vicinity of a root. The roots of this quadratic equation are then approximated to the roots of the equation f (x) = 0.

17. This method is iterative in nature and does not require the evaluation of derivatives as in Newton-Raphson method. This method can also be used to determine both real and complex roots of f (x) = 0.

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

19. 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

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

21. Fig: Quadratic polynomial.

22. Eliminating a, b, c, we obtain

23. which can be written as

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

25. The above equation can be written as

26. We further define

27. With these substitution we get a simplified Equation as

28. Or

29. 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:

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

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

32. Example
Find the root of the equation x3 – x–1 = 0 using Muller’s method.

33. Numerical Analysis
Lecture 7