Numerical Methods

Part: False-Position Method of Solving a Nonlinear Equation

http://numericalmethods.eng.usf.edu

For more details on this topic

Go to http://numericalmethods.eng.usf.edu

Click on Keyword Click on False-Position Method of

Solving a Nonlinear Equation

You are free to Share – to copy, distribute,

display and perform the work to Remix – to make derivative works

Under the following conditions Attribution — You must attribute the

work in the manner specified by the author or licensor (but not in any way that suggests that they endorse you or your use of the work).

Noncommercial — You may not use this work for commercial purposes.

Share Alike — If you alter, transform, or build upon this work, you may distribute the resulting work only under the same or similar license to this one.

Chapter 03.06: False-Position Method of Solving a Nonlinear

EquationMajor: All Engineering Majors

Authors: Duc Nguyen

http://numericalmethods.eng.usf.edu

Numerical Methods for STEM undergraduateshttp://

numericalmethods.eng.usf.edu 504/22/23

Lecture # 1

(1)

(2)

1

Introduction

6

http://numericalmethods.eng.usf.edu

Uxf

Uxrx

Lxf

LxO

xf

x

Exact root

0)( xf

0)(*)( UL xfxf

In the Bisection method

2UL

rxxx

(3)

Figure 1 False-Position Method

http://numericalmethods.eng.usf.edu7

False-Position MethodBased on two similar triangles, shown in Figure 1, one gets:

Ur

U

Lr

L

xxxf

xxxf

)()(

0;0)(0;0)(

UrU

LrL

xxxfxxxf

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

(4)

http://numericalmethods.eng.usf.edu8

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)

http://numericalmethods.eng.usf.edu9

The above equation, UL

ULUUr xfxf

xxxfxx

LU

LU

LLr

xxxfxf

xfxx

or

(6)

(7)

http://numericalmethods.eng.usf.edu10

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 Lxmx LL xx and mU xx

, then the root lies between (b) Ifand ; then

0mL xfxf mxUx mL xx and UU xx

http://numericalmethods.eng.usf.edu11

, 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 xxx

http://numericalmethods.eng.usf.edu12

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!

http://numericalmethods.eng.usf.edu13

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

http://numericalmethods.eng.usf.edu14

SolutionFrom the physics of the problem

11.00)055.0(20

20

xx

Rx

xwater

Figure 2 :Floating ball problem

http://numericalmethods.eng.usf.edu15

Let us assume11.0,0 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

http://numericalmethods.eng.usf.edu16

Iteration 1

0660.010662.210993.3

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

http://numericalmethods.eng.usf.edu17

Iteration 2

0611.0101944.310993.3

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,

http://numericalmethods.eng.usf.edu18

%81000611.0

0660.00611.0

a

Iteration 3

0624.0101944.310132.1

101944.30611.010132.10660.055

55

UL

ULLUm xfxf

xfxxfxx

http://numericalmethods.eng.usf.edu19

7101313.1 mxf

00624.00611.0 ffxfxf mL

Hence,0624.0,0611.0 UL xx

%05.21000624.0

0611.00624.0

a

http://numericalmethods.eng.usf.edu20

Iteration1 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 offor False-Position Method.

http://numericalmethods.eng.usf.edu21

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(

mSommm

m

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

http://numericalmethods.eng.usf.edu22

References1. S.C. Chapra, R.P. Canale, Numerical Methods for Engineers, Fourth Edition, Mc-Graw Hill.

THE ENDhttp://numericalmethods.eng.usf.edu

This instructional power point brought to you byNumerical Methods for STEM undergraduatehttp://numericalmethods.eng.usf.eduCommitted to bringing numerical methods to the undergraduate

Acknowledgement

For instructional videos on other topics, go to

http://numericalmethods.eng.usf.edu/videos/

This material is based upon work supported by the National Science Foundation under Grant # 0717624. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

The End - Really