How to find the roots of Nonlinear Equations? Newton-Raphson method is not the only way! How about a system of nonlinear equations? #WikiCourses https://wikicourses.wikispaces.com/Topic+Roots+of+Nonlinear+Equations

1. Roots of Non-Linear Equations
Mohammad Tawfik
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http://WikiCourses.WikiSpaces.com
Roots of Nonlinear Equations
Open Methods

2. Roots of Non-Linear Equations
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Objectives
• Be able to use the Newton Raphson
method to find a root of an equations
• Be able to use the Secant method to find a
root of an equations
• Write down an algorithm to outline the
method being used

3. Roots of Non-Linear Equations
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Fixed Point Iterations

4. Roots of Non-Linear Equations
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
 kk xgx 1
Fixed Point Iterations
• Solve   0xf
    0 xgxxf
• Rearrange terms:
• OR
 xgx 

5. Roots of Non-Linear Equations
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
In some cases you do not get a
solution!

6. Roots of Non-Linear Equations
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Example

7. Roots of Non-Linear Equations
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Example
  22
 xxxf Which has the solutions -1 & 2
To get a fixed-point form, we may use:
  22
 xxg
  x
xg 21
  2 xxg
 
12
22



x
x
xg

8. Roots of Non-Linear Equations
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
First trial!
• No matter how close
your initial guess is,
the solution diverges!

9. Roots of Non-Linear Equations
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Second trial
• The solution converges
in this case!!

10. Roots of Non-Linear Equations
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Condition of Convergence
• For the fixed point iteration to ensure
convergence of solution from point xk we should
ensure that
  1' kxg

11. Roots of Non-Linear Equations
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Fixed Point Algorithm
1. Rearrange f(x) to get f(x)=x-g(x)
2. Start with a reasonable initial guess x0
3. If |g’(x0)|>=1, goto step 2
4. Evaluate xk+1=g(xk)
5. If (xk+1-xk)/xk+1< es; end
6. Let xk=xk+1; goto step 4

12. Roots of Non-Linear Equations
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Newton-Raphson Method

13. Roots of Non-Linear Equations
Mohammad Tawfik
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http://WikiCourses.WikiSpaces.com
Newton’s Method: Line Equation
 1
21
21
' xf
xx
yy
m 



The slope of the
line is given by:

14. Roots of Non-Linear Equations
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Newton’s Method: Line equation
   1
21
1
' xf
xx
xf


 
 1
1
12
' xf
xf
xx 
 
 k
k
kk
xf
xf
xx
'
1 
Newton-Raphson
Iterative method

15. Roots of Non-Linear Equations
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Newton’s Method: Taylor’s Series
     1121 ' xfxxxf 
 
 1
1
12
' xf
xf
xx 
 
 k
k
kk
xf
xf
xx
'
1 
Newton-Raphson
Iterative method
       11212 ' xfxxxfxf 

16. Roots of Non-Linear Equations
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Newton-Raphson Algorithm
1. From f(x) get f’(x)
2. Start with a reasonable initial guess x0
3. Evaluate xk+1=xk-f(xk)/f’(xk)
4. If (xk+1-xk)/xk+1< es; end
5. Let xk=xk+1; goto step 4

17. Roots of Non-Linear Equations
Mohammad Tawfik
#WikiCourses
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Secant Method

18. Roots of Non-Linear Equations
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Secant Method
21
21
2
2
xx
yy
xx
yy





The line equation
is given by:
  
2
21
221 0
xx
yy
yxx




19. Roots of Non-Linear Equations
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Secant Method
  
2
21
221 0
xx
yy
yxx



 
21
212
2
yy
xxy
xx



  
   kk
kkk
kk
xfxf
xxxf
xx






1
1
1

20. Roots of Non-Linear Equations
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Secant Algorithm
1. Select x1 and x2
2. Evaluate f(x1) and f(x2)
3. Evaluate xk+1
4. If (xk+1-xk)/xk+1< es; end
5. Let xk=xk+1; goto step 3

21. Roots of Non-Linear Equations
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Why Secant Method?
• The most important advantage over
Newton-Raphson method is that you do
not need to evaluate the derivative!

22. Roots of Non-Linear Equations
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Comparing with False-Position
• Actually, false
position ensures
convergence, while
secant method does
not!!!

23. Roots of Non-Linear Equations
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Conclusion
• The fixed point iteration, Newton-Raphson
method, and the secant method in general
converge faster than bisection and false position
methods
• On the other hand, these methods do not ensure
convergence!
• The secant method, in many cases, becomes
more practical than Newton-Raphson as
derivatives do not need to be evaluated

24. Roots of Non-Linear Equations
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Roots of Nonlinear System
of Equations

25. Roots of Non-Linear Equations
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Objectives
• Be able to use the fixed point method to
find a root of a set of equations
• Be able to use the Newton Raphson
method to find a root of a set equations

26. Roots of Non-Linear Equations
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Fixed Point Iterations

27. Roots of Non-Linear Equations
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
 kk xgx 1
Fixed Point Iterations
• Solve   0xf
    0 xgxxf
• Rearrange terms:
• OR
 xgx 

28. Roots of Non-Linear Equations
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
 
 kkk
kkk
yxgy
yxgx
,
,
21
11




Fixed Point Iterations (cont’d)
• Solve  
  0,
0,
2
1


yxf
yxf
   
    0,,
0,,
22
11


yxgyyxf
yxgxyxf• Rearrange terms:
• OR
 
 yxgy
yxgx
,
,
2
1



29. Roots of Non-Linear Equations
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Example
 
  0573
010
2
2
2
1


xyyxf
xyxxf
Which has a solution x=2 & y=3
To get a fixed-point form, we may use:
With initial values: x=1.5 and y=3.5
kkk
k
k
k
yxy
y
x
x
2
1
2
1
357
10





38.245.3*5.1*357
214.2
5.3
5.110
2
1
2
1




y
x
7.42938.24*214.2*357
209.0
38.24
214.210
2
2
2
2





y
x

30. Roots of Non-Linear Equations
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Diverging !

31. Roots of Non-Linear Equations
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Another trial
k
k
k
kkk
x
y
y
yxx
3
57
10
1
1





861.2
5.1*3
5.357
179.25.3*5.110
1
1




y
x
05.3
179.2*3
861.257
941.1861.2*179.210
2
2




y
x

32. Roots of Non-Linear Equations
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Condition of Convergence
• For the fixed point iteration to ensure
convergence of solution from point xk and yk we
should ensure that
1
1
21
21












y
g
y
g
and
x
g
x
g

33. Roots of Non-Linear Equations
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Newton-Raphson Method

34. Roots of Non-Linear Equations
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Newton’s Method: Taylor’s Series
   1
21
1
' xf
xx
xf


 
 1
1
12
' xf
xf
xx 
 
 k
k
kk
xf
xf
xx
'
1 
Newton-Raphson
Iterative method
     112 ' xxfxfxf 

35. Roots of Non-Linear Equations
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Taylor’s series in multiple variables
   
   
y
f
y
x
f
xyxfyxf
y
f
y
x
f
xyxfyxf












22
112222
11
111221
,,
,,
 
 112
22
111
11
,
,
yxf
y
f
y
x
f
x
yxf
y
f
y
x
f
x















36. Roots of Non-Linear Equations
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Manipulating the equations
 
 




































112
111
22
11
,
,
yxf
yxf
y
x
y
f
x
f
y
f
x
f
Solve for x and y then evaluate:






















y
x
y
x
y
x
1
1
2
2
Repeat until convergence

37. Roots of Non-Linear Equations
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Example
 
  0573
010
2
2
2
1


xyyxf
xyxxf
Which has a solution x=2 & y=3
With initial values: x=1.5 and y=3.5
Get the derivatives
xy
y
f
y
x
f
x
y
f
yx
x
f
613
2
222
11













38. Roots of Non-Linear Equations
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Example
Using initial values: x=1.5 and y=3.5
Get the derivatives
5.3275.36
5.15.6
22
11












y
f
x
f
y
f
x
f






















625.1
5.2
5.3275.36
5.15.6
y
x
















656.0
536.0
y
x













844.2
036.2
2
2
y
x

39. Roots of Non-Linear Equations
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Conclusion
• The fixed point iteration and Newton-
Raphson methods were used to find a
solution for a system of nonlinear
equations in a manner similar to that used
in single-variable problems

40. Roots of Non-Linear Equations
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Homework #2
• Chapter 6, p 171, numbers:
6.1,6.2,6.3,6.16,6.17
• Homework due next week