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Chapter 03.05Secant Method of Solving a Nonlinear Equation More ExamplesCivi l Engineering

Example 1

You are making a bookshelf to carry books that range from 8" to 11" in height and wouldtake up 29" of space along the length. The material is wood having a Youngs Modulus of

, thickness of 3/8" and width of 12". You want to find the maximum verticaldeflection of the bookshelf. The vertical deflection of the shelf is given by

Msi667.3

xxxxxv 018507.01066722.01013533.01042493.0)( 465834

where x is the position along the length of the beam. Hence to find the maximum deflection

we need to find where 0)( dx

dvxf and conduct the second derivative test.

x

Books

Bookshelf

Figure 1 A loaded bookshelf.

The equation that gives the position x where the deflection is maximum is given by

0018507.01012748.01026689.01067665.0 233548 xxx

Use the secant method of finding roots of equations to find the position x where the

deflection is maximum. Conduct three iterations to estimate the root of the above equation.

Find the absolute relative approximate error at the end of each iteration and the number ofsignificant digits at least correct at the end of each iteration.

03.05.1

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03.05.2 Chapter 03.05

Solution

Let us take the initial guesses of the root of 0xf as 101 x and .x 150

Iteration 1

The estimate of the root is1x

10

1000

xfxf

xxxfx

018507.01012748.01026689.01067665.0 2033

0

54

0

8

0 xxxxf

018507.0151012748.0151026689.0151067665.0 233548 4102591.8

018507.01012748.01026689.01067665.0 2133

1

54

1

8

1

xxxxf

018507.0101012748.0101026689.0101067665.0 233548 3104956.8

344

1104956.8102591.8

1015102591.815

x

557.14

The absolute relative approximate error a at the end of Iteration 1 is

1001

01

x

xxa

100557.14

15557.14

%0433.3

The number of significant digits at least correct is 1, because the absolute relativeapproximate error is less than .%5

Iteration 2

The estimate of the root is

01

011

12xfxf

xxxfxx

018507.01012748.01026689.01067665.0 2133

1

54

1

8

1

xxxxf

018507.0557.141012748.0

557.141026689.0557.141067665.0

23

3548

5

109870.2

45

5

2102591.8109870.2

15557.14109870.215

x

572.14

The absolute relative approximate error a at the end of Iteration 2 is

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Secant Method-More Examples: Civil Engineering 03.05.3

1002

12

x

xxa

100572.14

557.14572.14

%10611.0

The number of significant digits at least correct is 2, because the absolute relative

approximate error is less than .%5.0

Iteration 3

The estimate of the root is

12

12223

xfxf

xxxfxx

018507.01012748.01026689.01067665.0 2233

2

54

2

8

2 xxxxf

018507.0572.141012748.0

572.141026689.0572.141067665.0

23

3548

9100676.6

59

9

3109870.2100676.6

557.14572.14100676.6572.14

x

572.14

The absolute relative approximate error a at the end of Iteration 3 is

100

3

23

x

xxa

100572.14

572.14572.14

%101559.2 5

The number of significant digits at least correct is 6, because the absolute relative

approximate error is less than .%00005.0

NONLINEAR EQUATIONS

Topic Secant Method-More Examples

Summary Examples of Secant MethodMajor Civil Engineering

Authors Autar Kaw

Date August 7, 2009

Web Site http://numericalmethods.eng.usf.edu

http://c/Documents%20and%20Settings/sgarapat/http://c/Documents%20and%20Settings/sgarapat/http://c/Documents%20and%20Settings/sgarapat/