Numerical Methods

Part: False-Position Method of Solving a Nonlinear Equation

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Chapter 02.02: False-Position Method of Solving a Nonlinear

Equation

5

(1)

(2)

1

Introduction

6

Uxf

Uxrx

Lxf

Lx

O

xf

x

Exact root

0)( xf

0)(*)( UL xfxf

In the Bisection method

2UL

r

xxx

(3)

Figure 1 False-Position Method

7

False-Position Method

Based on two similar triangles, shown in Figure 1, one gets:

Ur

U

Lr

L

xx

xf

xx

xf

)()(

0;0)(

0;0)(

UrU

LrL

xxxf

xxxf

The signs for both sides of Eq. (4) is consistent, since:

(4)

8

LUrULr xfxxxfxx

ULrULLU xfxfxxfxxfx

From Eq. (4), one obtains

The above equation can be solved to obtain the next predicted root

UL

ULLUr xfxf

xfxxfxx

rx , as

(5)

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The above equation, UL

ULUUr xfxf

xxxfxx

LU

LU

LLr

xx

xfxf

xfxx

or

(6)

(7)

10

Step-By-Step False-Position Algorithmsas two guesses for the root such1. Choose Lx Uxand

that 0UL xfxf

2. Estimate the root, UL

ULLUm xfxf

xfxxfxx

3. Now check the following

, then the root lies between (a) If

and ; then

0mL xfxf Lx

mx LL xx and mU xx

, then the root lies between (b) If

and ; then

0mL xfxf mx

Ux mL xx and UU xx

11

, then the root is(c) If 0mL xfxf .mxStop the algorithm if this is true.

4. Find the new estimate of the root

UL

ULLUm xfxf

xfxxfxx

Find the absolute relative approximate error as

100

newm

oldm

newm

a x

xx

12

where

= estimated root from present iteration

= estimated root from previous iteration

newmxoldmx

.001.010 3 ssay5. If sa , then go to step 3,

else stop the algorithm.

Notes: The False-Position and Bisection algorithms are quite similar. The only difference is the formula used to calculate the new estimate of the root ,mx shown in steps

#2 and 4!

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Example 1The floating ball has a specific gravity of 0.6 and has a radius of 5.5cm. You are asked to find the depth to which the ball is submerged when floating in water.

The equation that gives the depth x to which the ball issubmerged under water is given by

010993.3165.0 423 xxUse the false-position method of finding roots of equations to find the depth to which the ball is submerged under water. 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 of significant digits at least correct at the converged iteration.

x

14

SolutionFrom the physics of the problem

11.00

)055.0(20

20

x

x

Rx

x

water

Figure 2 :Floating ball problem

15

Let us assume

11.0,0 UL xx

4423

4423

10662.210993.311.0165.011.011.0

10993.310993.30165.000

fxf

fxf

U

L

Hence,

010662.210993.311.00 44 ffxfxf UL

16

Iteration 1

0660.0

10662.210993.3

10662.2010993.311.044

44

UL

ULLUm xfxf

xfxxfxx

5

423

101944.3

10993.30660.0165.00660.00660.0

fxf m

00660.00 ffxfxf mL

0660.0,0 UL xx

17

Iteration 2

0611.0

101944.310993.3

101944.3010993.30660.054

54

UL

ULLUm xfxf

xfxxfxx

5

423

101320.1

10993.30611.0165.00611.00611.0

fxf m

00611.00 ffxfxf mL

0660.0,0611.0 UL xxHence,

18

%81000611.0

0660.00611.0

a

Iteration 3

0624.0

101944.310132.1

101944.30611.010132.10660.055

55

UL

ULLUm xfxf

xfxxfxx

19

7101313.1 mxf

00624.00611.0 ffxfxf mL

Hence,0624.0,0611.0 UL xx

%05.21000624.0

0611.00624.0

a

20

Iteration

1 0.0000 0.1100 0.0660 N/A -3.1944x10-

5

2 0.0000 0.0660 0.0611 8.00 1.1320x10-5

3 0.0611 0.0660 0.0624 2.05 -1.1313x10-

7

4 0.0611 0.0624 0.0632377619

0.02 -3.3471x10-

10

Lx Ux mx %a mxf

010993.3165.0 423 xxxfTable 1: Root of

for False-Position Method.

21

m

m

ma

2

2

2

1004.0

105.002.0

105.0

3,

3979.3

)3979.1(2

)04.0log(2

2)04.0log(

mSo

m

m

m

m

The number of significant digits at least correct in the estimated root of 0.062377619 at the end of 4th iteration is 3.

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References

1. S.C. Chapra, R.P. Canale, Numerical Methods for Engineers, Fourth Edition, Mc-Graw Hill.