c2_l2 - open methods

7/23/2019 C2_L2 - Open Methods

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Chapter 2Roots of Equations

- False-Position Method


2/22

Learning Outcome At the end of the lecture student should

be able to use the False-Position Method

to estimate the root of the equation


3/22

False-Position Method Let be real and continuous function

in the inter!al from to and

andha!e opposite signs

"he graph of crosses the x-a#isbet$een the !alues and and hence

a root of lies bet$een and

root

)(xf

ul xx )()( ul xfxf

0)()(


4/22

False-Position Method%o$& the points and'oined together b( a straight line "he

intersection of this line $ith the x-a#isrepresents an impro!ed estimate of theroot

root

( ))(, ll xfx ( ))(, uu xfx

ul xx

( ))(, uu xfx

)(xf

rx

( ))(, ll xfx


5/22

False-Position Method )( using the fact that the slope of the

lines connecting the points and &

and the points and are the same*

root

rl xx

ur xx

ru

u

lr

l

xx

xf

xx

xf

=

)()(

ul xx

)( uxf

)(xf

rx

)( lxf


6/22

False-Position Method +ol!ing for $e ha!e

"his is one form of the method of falseposition

rx

)1...(....................)()(

)()(

ul

ullur

xfxf

xfxxfxx

=


7/22

False-Position Method An alternati!e form can be obtained as

follo$s*

)2...(..........)()(

))((

)()(

))((

)()(

)()()()(

)()(

)(

)()(

)(

)()(

)(

)()(

)(

)()(

)()(

ul

uluu

ul

luuu

ul

uluululuu

ul

ulu

ul

luu

ul

ul

ul

lu

ul

ullur

xfxf

xxxfx

xfxf

xxxfx

xfxf

xfxxfxxfxxfxx

xfxf

xfxx

xfxf

xfxx

xfxf

xfx

xfxf

xfx

xfxf

xfxxfxx

=

+=

++=

+=

=

=


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False-Position Method As compare to Eq,. this form in!ol!es

one less function e!aluation and one

less multiplication )ecause of the input !alues $e use to

represent the quantities areappro#imate !alues& a signi/cant error

ma( arise if large arithmetic ,addition&subtraction& multiplication& di!ision.manipulations are ta0en place


9/22

False-Position Method"he !alue of computed $ith Eq(2) then replaces

$hiche!er of the t$o initial guesses& and $hich

(ields a function !alue $ith the same sign as

root

rx

ul xx

)( rxf

ul xx

)( uxf

)(xf

rx

)( lxf

)( rxf


10/22

False-Position Method 1n this $a(& the !alues of and

al$a(s brac0et the true root "he

process is repeated until the root isestimated adequatel(

"he algorithm is identical to the one forbisection $ith the e#ception that Eq(2) is

used for step 2 1n addition& the same stopping criterion

is used to terminate the computation

ul xx


11/22

E#ample "he !elocit( vof a falling parachutist is

gi!en b(

$here g34 m5s6 For a parachutist $itha drag coe7cient c80g5s& computethe mass mso that the !elocit( is

v98m5s at t3s :se the false-positionmethod to determine mto a le!el of

:se initial guesses of and

=

m

ct

ec

gm

v 1

%.1.0=s

50=lm .70=um


12/22

+olution +ubstituting the gi!en !alues in the

!elocit( equation*

"he correct mass can be determined b(/nding the root of f(m)=0

035115

8.9)(

115

8.935

135

)9)(15(

=

=

=

m

m

em

mf

em


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+olution First iteration*

"herefore& the ne$ brac0et is and

030637.1)288463929.0)(528713416.4(

)51423.60()50()()(

51423.60)085732597.4528713416.4(

)20)(085732597.4(70

)70()50(

)7050)(70(70)()(

))((


14/22

+olution +econd iteration*

"herefore& the ne$ brac0et is and

008491.0)88461.59()50()()(

%051.1%10088461.59

51423.6088461.59

88461.59

)51423.60()50(

)51423.6050)(51423.60(51423.60)()(

))((


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+olution"hird iteration*

"hus& after 9 iterations& a !alue of 834;94


16/22

+olution"his result can be !eri/ed b(

substituting into the equation for

!elocit( to gi!e

sm

ev

/00121.35

115

)84386.59(8.984386.59

)9)(15(

=

=


17/22

Pitfalls of the False-PositionMethod

Although the false-position methodseems to be the brac0eting method of

preference& there are cases $here itperforms poorl( and the bisectionmethod (ields superior results


18/22

E#ample 2 :se bisection and false-position to

locate the root of

bet$een x= and 9

1)( 10 =xxf


19/22

+olution"he true !alue of the root of f(x)=0is

"he initial guesses are and1)1(

1

01

101

10

10

==

=

=

x

x

x

0=lx 3.1=ux


20/22

+olution :sing bisection the results are summari?ed

as belo$*

After 8 iterations& the is reduced to lessthan 2>

i

= 9 =


21/22

+olution For false-position& the results are summari?ed as belo$*

After 8 iterations& the has onl( been reduced to about 83> %otethat the results ha!e undesirable feature that "his is notgood because it means that $e could stop the computation based onthe erroneous assumption that the true error is at least as good as the

appro#imate error "his is due to the slo$ con!ergence to the root

i

= 9 ==3;9= @!e 3=

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