ENEM602 Spring 2007

Dr. Eng. Mohammad Tawfik

Roots of Nonlinear Equations

Open Methods

ENEM602 Spring 2007

Dr. Eng. Mohammad Tawfik

Objectives

• Be able to use the fixed point method to find a root of an equation

• 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

ENEM602 Spring 2007

Dr. Eng. Mohammad Tawfik

Fixed Point Iterations

ENEM602 Spring 2007

Dr. Eng. Mohammad Tawfik

kk xgx 1

Fixed Point Iterations

• Solve 0xf

0 xgxxf

• Rearrange terms:

• OR

xgx

ENEM602 Spring 2007

Dr. Eng. Mohammad Tawfik

In some cases you do not get a solution!

ENEM602 Spring 2007

Dr. Eng. Mohammad Tawfik

Example

ENEM602 Spring 2007

Dr. Eng. Mohammad Tawfik

Example

22 xxxf Which has the solutions -1 & 2

To get a fixed-point form, we may use:

22 xxg

xxg 21 2 xxg

12

22

x

xxg

ENEM602 Spring 2007

Dr. Eng. Mohammad Tawfik

First trial!

• No matter how close your initial guess is, the solution diverges!

ENEM602 Spring 2007

Dr. Eng. Mohammad Tawfik

Second trial

• The solution converges in this case!!

ENEM602 Spring 2007

Dr. Eng. Mohammad Tawfik

Condition of Convergence

• For the fixed point iteration to ensure convergence of solution from point xk we should ensure that

1' kxg

ENEM602 Spring 2007

Dr. Eng. Mohammad Tawfik

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< s; end

6. Let xk=xk+1; goto step 4

ENEM602 Spring 2007

Dr. Eng. Mohammad Tawfik

Newton-Raphson Method

ENEM602 Spring 2007

Dr. Eng. Mohammad Tawfik

Newton’s Method: Line Equation

121

21 ' xfxx

yym

The slope of the line is given by:

ENEM602 Spring 2007

Dr. Eng. Mohammad Tawfik

Newton’s Method: Line equation

121

1 ' xfxx

xf

11

12 ' xf

xfxx

kk

kk xf

xfxx

'1

Newton-RaphsonIterative method

ENEM602 Spring 2007

Dr. Eng. Mohammad Tawfik

Newton’s Method: Taylor’s Series

1121 ' xfxxxf 11

12 ' xf

xfxx

kk

kk xf

xfxx

'1

Newton-RaphsonIterative method

11212 ' xfxxxfxf

ENEM602 Spring 2007

Dr. Eng. Mohammad Tawfik

Example

ENEM602 Spring 2007

Dr. Eng. Mohammad Tawfik

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< s; end

5. Let xk=xk+1; goto step 4

ENEM602 Spring 2007

Dr. Eng. Mohammad Tawfik

Convergence condition!

• Try to derive a convergence conditions similar to that of the fixed point iteration!

ENEM602 Spring 2007

Dr. Eng. Mohammad Tawfik

Secant Method

ENEM602 Spring 2007

Dr. Eng. Mohammad Tawfik

Secant Method

21

21

2

2

xx

yy

xx

yy

The line equation is given by:

2

21

221 0xx

yy

yxx

ENEM602 Spring 2007

Dr. Eng. Mohammad Tawfik

Secant Method

2

21

221 0xx

yy

yxx

21

2122 yy

xxyxx

kk

kkkkk xfxf

xxxfxx

1

11

ENEM602 Spring 2007

Dr. Eng. Mohammad Tawfik

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< s; end

5. Let xk=xk+1; goto step 3

ENEM602 Spring 2007

Dr. Eng. Mohammad Tawfik

Why Secant Method?

• The most important advantage over Newton-Raphson method is that you do not need to evaluate the derivative!

ENEM602 Spring 2007

Dr. Eng. Mohammad Tawfik

Comparing with False-Position

• Actually, false position ensures convergence, while secant method does not!!!

ENEM602 Spring 2007

Dr. Eng. Mohammad Tawfik

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

ENEM602 Spring 2007

Dr. Eng. Mohammad Tawfik

Homework #2

• Chapter 6, p 157, numbers:6.1,6.2,6.3

• Homework due next week