1. EE:211 Computational Techniques in Electrical Engineering Lecture#2
Dr. Mubashir Alam
King Saud University

2. Outline Chapter#3 Rootfinding Bisection Method Newton's Algorithm
Secant Method

3. Introduction

4. The Bisection Method

6. Example: 3.1.1

8. Error Bounds
α is the true root and cn is its estimate
Error: |α-cn|

10. Newton’s Algorithm

14. If x1 is the root of p(x) then p(x1)=0
Repeat the next with x1 as the initial estimate and determine x2
And so on

15. Newton’s Method

16. Example: 3.2.1

17. Stopping criteria

18. Example n xn f(xn) xn-xn-1 1.5 8.890625 ------------------ 1
1.5 1 2 3 4
x 10-3
x 10-3
Stopping criteria, ε = 1 x 10-2

19. Example: 3.2.1

21. Example: 3.2.2

24. Secant Method
From this perspective, other straight-line approximation to y=f(x) would also lead to methods for approximating a root of f(x).
One such straight-line approximation leads to SECANT METHOD

25. The two pints (x0,f(x0)) and (x1,f(x1)) , on the graph of y=f(x), determine
a straight line , called a secant line.
This line is an approximation to the graph of y=f(x), and its root x2 is an
approximation to true root α.

26. Equation of lines
Equation of line with slope = m and passing through a point (x1,y1):
y-y1 = m(x-x1)
y = y1+ m(x-x1)
Slope of the line between points (x1,y1) and (x2,y2)
Slope = m = (y2- y1) / (x2- x1)

27. Secant Method Find the equation of the line and then its root x2
Having found x2, we can drop x0, and use x1,x2 as a new set of approximate values for α.
This will lead to an improved value x3.
Continue this process …..

28. Secant Method General Formula:
Two point method, since two approximate values are needed to obtain the next improved value.

29. Example: 3.3.1

35. Matlab Function: fzero
This function uses ideas involved in the bisection and the secant method.
Use: root=fzero(f_name, [a,b])
Produces a root within [a,b], assume f(a)f(b) <= 0
Use: root=fzero(f_name, xo)
Find a root near x0.

36. Matlab Function: fzero
Define function for: f(x)=x6-x-1
function f = myfun(x)
f = x.^6-x-1; x=