c2_l2 - open methods
TRANSCRIPT
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Chapter 2Roots of Equations
- False-Position Method
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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
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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)()(
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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
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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
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False-Position Method +ol!ing for $e ha!e
"his is one form of the method of falseposition
rx
)1...(....................)()(
)()(
ul
ullur
xfxf
xfxxfxx
=
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False-Position Method An alternati!e form can be obtained as
follo$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
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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
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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
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E#ample "he !elocit( vof a falling parachutist is
gi!en b(
$here g34 m5s6 For a parachutist $itha drag coe7cient c80g5s& computethe mass mso that the !elocit( is
v98m5s at t3s :se the false-positionmethod to determine mto a le!el of
:se initial guesses of and
=
m
ct
ec
gm
v 1
%.1.0=s
50=lm .70=um
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+olution +ubstituting the gi!en !alues in the
!elocit( equation*
"he correct mass can be determined b(/nding the root of f(m)=0
035115
8.9)(
115
8.935
135
)9)(15(
=
=
=
m
m
em
mf
em
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+olution First iteration*
"herefore& the ne$ 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)()(
))((
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+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
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+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(
=
=
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Pitfalls of the False-PositionMethod
Although the false-position methodseems to be the brac0eting method of
preference& there are cases $here itperforms poorl( and the bisectionmethod (ields superior results
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E#ample 2 :se bisection and false-position to
locate the root of
bet$een x= and 9
1)( 10 =xxf
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+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
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+olution :sing bisection the results are summari?ed
as belo$*
After 8 iterations& the is reduced to lessthan 2>
i
= 9 =
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+olution For false-position& the results are summari?ed as belo$*
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=