Roots of Nonlinear Equations - Open Methods

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

Mohammad Tawfik
#WikiCourses
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

Mohammad Tawfik
#WikiCourses
Fixed Point Iterations

Mohammad Tawfik
#WikiCourses
 kk xgx 1
Fixed Point Iterations
• Solve   0xf
    0 xgxxf
• Rearrange terms:
• OR
 xgx 

Mohammad Tawfik
#WikiCourses
In some cases you do not get a
solution!

Mohammad Tawfik
#WikiCourses
Example

Mohammad Tawfik
#WikiCourses
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

Mohammad Tawfik
#WikiCourses
First trial!
• No matter how close
your initial guess is,
the solution diverges!

Mohammad Tawfik
#WikiCourses
Second trial
• The solution converges
in this case!!

Mohammad Tawfik
#WikiCourses
Condition of Convergence
• For the fixed point iteration to ensure
convergence of solution from point xk we should
ensure that
  1' kxg

Mohammad Tawfik
#WikiCourses
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

Mohammad Tawfik
#WikiCourses
Newton-Raphson Method

Mohammad Tawfik
#WikiCourses
Newton’s Method: Line Equation
 1
21
21
' xf
xx
yy
m 



The slope of the
line is given by:

Mohammad Tawfik
#WikiCourses
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

Mohammad Tawfik
#WikiCourses
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 

Mohammad Tawfik
#WikiCourses
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)

Mohammad Tawfik
#WikiCourses
Secant Method

Mohammad Tawfik
#WikiCourses
Secant Method
21
21
2
2
xx
yy
xx
yy





The line equation
is given by:
  
2
21
221 0
xx
yy
yxx




Mohammad Tawfik
#WikiCourses
Secant Method
  
2
21
221 0
xx
yy
yxx



 
21
212
2
yy
xxy
xx



  
   kk
kkk
kk
xfxf
xxxf
xx






1
1
1

Mohammad Tawfik
#WikiCourses
Secant Algorithm
1. Select x1 and x2
2. Evaluate f(x1) and f(x2)
3. Evaluate xk+1

Mohammad Tawfik
#WikiCourses
Why Secant Method?
• The most important advantage over
Newton-Raphson method is that you do
not need to evaluate the derivative!

Mohammad Tawfik
#WikiCourses
Comparing with False-Position
• Actually, false
position ensures
convergence, while
secant method does
not!!!

Mohammad Tawfik
#WikiCourses
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

Mohammad Tawfik
#WikiCourses
Roots of Nonlinear System
of Equations

Mohammad Tawfik
#WikiCourses
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

Mohammad Tawfik
#WikiCourses
 
 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



Mohammad Tawfik
#WikiCourses
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

Mohammad Tawfik
#WikiCourses
Diverging !

Mohammad Tawfik
#WikiCourses
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

Mohammad Tawfik
#WikiCourses
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

Mohammad Tawfik
#WikiCourses
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 

Mohammad Tawfik
#WikiCourses
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















Mohammad Tawfik
#WikiCourses
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

Mohammad Tawfik
#WikiCourses
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













Mohammad Tawfik
#WikiCourses
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

Mohammad Tawfik
#WikiCourses
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

Mohammad Tawfik
#WikiCourses
Homework #2
• Chapter 6, p 171, numbers:
6.1,6.2,6.3,6.16,6.17
• Homework due next week

Roots of Nonlinear Equations - Open Methods

Recommended

Recommended

More Related Content

What's hot

What's hot (20)

Viewers also liked

Viewers also liked (20)

More from Mohammad Tawfik

More from Mohammad Tawfik (20)

Recently uploaded

Recently uploaded (20)

Roots of Nonlinear Equations - Open Methods