1. Open Methods Chapter 6 The Islamic University of Gaza
Faculty of Engineering
Civil Engineering Department
Numerical Analysis
ECIV 3306
Chapter 6
Open Methods

2. Open Methods
Bracketing methods are based on assuming an interval of the function which brackets the root.
The bracketing methods always converge to the root.
Open methods are based on formulas that require only a single starting value of x or two starting values that do not necessarily bracket the root.
These method sometimes diverge from the true root.

3. 1. Simple Fixed-Point Iteration
Rearrange the function so that x is on the left side of the equation:
Bracketing methods are “convergent”.
Fixed-point methods may sometime “diverge”, depending on the stating point (initial guess) and how the function behaves.

4. Simple Fixed-Point Iteration
Examples: 1.
f(x) = x 2-2x+3  x = g(x)=(x2+3)/2
f(x) = sin x  x = g(x)= sin x + x
f(x) = e-x- x  x = g(x)= e-x

5. Simple Fixed-Point Iteration Convergence
x = g(x) can be expressed as a pair of equations:
y1= x
y2= g(x)…. (component equations)
Plot them separately.

6. Simple Fixed-Point Iteration Convergence
Fixed-point iteration converges if :
When the method converges, the error is roughly proportional to or less than the error of the previous step, therefore it is called “linearly convergent.”

7. Simple Fixed-Point Iteration-Convergence

8. Steps of Simple Fixed Pint Iteration
1. Rearrange the equation f(x) = 0 so that x is on the left hand side and g(x) is on the right hand side.
e.g f(x) = x2-2x-1 = 0  x= (x2-1)/2
g(x) = (x2-1)/2
2. Set xi at an initial guess xo.
3. Evaluate g(xi)
4. Let xi+1 = g(xi), and set xi at xi+1.
5. Find a=(Xi+1 – xi)/Xi+1
6. Repeat steps 3 through 5 until |a|<= a

9. Example: Simple Fixed-Point Iteration
f(x) = e-x - x
f(x)
f(x)=e-x - x
1. f(x) is manipulated so that we get x=g(x) g(x) = e-x 2. Thus, the formula predicting the new value of x is: xi+1 = e-xi 3. Guess xo = 0 4. The iterations continues till the approx. error reaches a certain limiting value
Root x
f(x)
f1(x) = x
g(x) = e-x x

10. Example: Simple Fixed-Point Iteration
i xi g(xi) ea% et%

11. Example: Simple Fixed-Point Iteration
i xi g(xi) ea% et%

12. Flow Chart – Fixed Point
Start
Input: xo , s, maxi
i=0
a=1.1s 1

13. 1 False Stop True while a< s & i >maxi
xn=0
x0=xn
Print: xo, f(xo) ,a , i
False
True