catatan pemrograman dan metode numerik - 7 (numeric method).pptx
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8/17/2019 Catatan Pemrograman Dan Metode Numerik - 7 (Numeric Method).pptx
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Numerical Methods
- An Introduction -
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Chapter 1 2
Mathematical Modeling and
Engineering Problem Solving
• Requires understanding of engineeringsystems –By observation and experiment
– Theoretical analysis and generalization
•
Computers are great tools, hoever, ithoutfundamental understanding of engineeringproblems, they ill be useless!
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"
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Chapter 1 #
• $ mathematical model is represented as a functionalrelationship of the form
Dependent independent forcing
Variable %f variables, parameters, functions
• Dependent variable& Characteristic that usually re'ects thestate of the system
• Independent variables& (imensions such as time and spacealong hich the systems behavior is being determined
• Parameters& re'ect the system)s properties or composition
• Forcing functions& external in'uences acting upon thesystem
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Chapter 1 *
Newton’s 2nd law of Motion
• +tates that the time rate change ofmomentum of a body is equal to the resultingforce acting on it !-
• The model is formulated as
F = m a
F %net force acting on the body ./0m%mass of the obect .g0
a %its acceleration .m3s20
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4
D U
D
U
dv F dt m
F F F
F mg
F cv
dv mg cv
dt m
=
= +
=
= −
−=
ample, modeling of a falling parachutist&
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Chapter 1 5
• This is a di6erential equation and is rittenin terms of the di6erential rate of changedv3dt of the variable that e are interested
in predicting!• 7f the parachutist is initially at rest .v %8 at
t %80, using calculus
vm
c g
dt
dv−=
( )t mc
ec
gm
t v)/(
1)(−
−=
7ndependentvariable
(ependentvariable 9arameters:orcing
function
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;
Approximations and Round!ff Errors
• :or many engineering problems, ecannot obtain analytical solutions!
• /umerical methods yield
approximate results, results that areclose to the exact analytical solution!
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Chapter " =
• $ccuracy! >o close is a computed or
measured value to the true value• 9recision .or reproducibility 0! >o close
is a computed or measured value to
previously computed or measuredvalues!
• 7naccuracy .or bias0! $ systematic
deviation from the actual value!• 7mprecision .or uncertainty 0! ?agnitude
of scatter!
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Significant "igures
/umber of signifcant fgures indicates precision!+igni@cant digits of a number are those that can be used ith condence, e!g!, the number of certain digits plus oneestimated digit!
*",;88 >o many signi@cant @guresA
*!"; x 18# "*!";8 x 18# #*!";88 x 18# *
eros are sometimes used to locate the decimal point notsigni@cant @gures!
8!888815*" #8!88815*" #
8!8815*" #
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Error #efinitions
True Value = Approximation + Error
E t = True value – Approximation (+/-)
valuetrue
error true errorrelativefractionalTrue =
%100 valuetrue
error true error,relative percentTrue
t×=ε
True error
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•
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Chapter " 1#
• Computations are repeated until stopping criterion is
satis@ed!
7f the folloing criterion is met
you can be sure that the result is correct to at least n
signi@cant @gures!
sa ε ε 〈 9respeci@ed D tolerancebased on the noledge ofyour solution
)%10(0#$ n)-(" ×=ε
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Roundoff Errors•
/umbers such asπ, e, or cannot beexpressed by a @xed number of
signi@cant @gures!
• Computers use a base2 representation,
they cannot precisely represent certainexact base18 numbers!
• :ractional quantities are typicallyrepresented in computer using 'oatingpoint- form, e!g!,
&
em#'exponent
Base of the numbersystem used
mantissa
7ntegerpart
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Chapter "14
iure *#*
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Chapter " 15
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Chapter " 1;
:igure "!*
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Chapter "1=
1*4!5; 8!1*45;x18" in a 'oating pointbase18 system
+uppose only #decimal places to be stored
• /ormalized to remove the leading zeroes!?ultiply the mantissa by 18 and loer theexponent by 1
8!2=#1 x 181
1
1100#0
011&$#0*
1
0
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Chapter "28
Thereforefor a base18 system 8!1 EmF1
for a base2 system 8!* EmF1
• :loating point representation allos bothfractions and very large numbers to beexpressed on the computer! >oever, – :loating point numbers tae up more room!
– Tae longer to process than integer numbers!
– Roundo6 errors are introduced because mantissaholds only a @nite number of signi@cant @gures!
11m
b≤ <
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Chapter " 21
$hoppingExample%
π %"!1#1*=24*"*; to be stored on a base18 systemcarrying 5 signi@cant digits!
π %"!1#1*=2 chopping error et%8!8888884*
7f rounded
π %"!1#1*=" et%8!888888"*
• +ome machines use chopping, because rounding
adds to the computational overhead! +ince numberof signi@cant @gures is large enough, resultingchopping error is negligible!
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Truncation Grrors and the Taylor
+eries• /onelementary functions such as
trigonometric, exponential, and others are
expressed in an approximate fashion using Taylor series hen their values, derivatives,and integrals are computed!
• $ny smooth function can be approximated as
a polynomial! Taylor series provides a meansto predict the value of a function at one pointin terms of the function value and itsderivatives at another point!
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Chapter #2"
:igure #!1
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2#
Example%
To et t.e cos(x) for "mall x
f x=0#$cos(0#$) =1-0#1$+0#0001-0#00001&+
=0#2&&$2
rom t.e "upportin t.eor3, for t.i" "erie", t.e error i"
no reater t.an t.e fir"t omitte4 term#
+−+−=555
1co" x x x
x
0000001#0$#0
52
2
==∴ x for x
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Chapter # 2*
• An3 "moot. function can 'e approximate4 a" a
pol3nomial# f ( xi+1) 6 f ( xi) zero order approximation, onl3
true if xi+1 an4 xi are ver3 clo"e to eac. ot.er#
f ( xi+1) 6 f ( xi) + f 7( xi) ( xi+1- xi) first order
approximation, in form of a "trai.t line
n
n
ii
i
n
iii
iiiii
R x xn
x f
x x x f
x x x f x f x f
+−+
+−′′
+−′+≅
+
+++
)(5
)(
)(
5%
)())(()()(
1
)(
%
111
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Chapter # 24
n
n
ii
n
iiiiiii
R x xn
f
x x f
x x x f x f x f
+−+
+−′′
+−′+≅
+
+++
)(5
)(
5
))(()()(
1
)(
111
( xi+1- xi)= h step size (4efine fir"t)
)1()1(
)51(
)( ++
+
=n
n
n hn
f R
ε
• 8emin4er term, Rn, account" for all term" from
(n+1) to infinit3#
nth order approximation
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Chapter # 25
• ε i" not 9no:n exactl3, lie" "ome:.ere
'et:een xi+1;ε ; xi #
•
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Chapter #2;
• Truncation error i" 4ecrea"e4 '3 a44ition of term" to
t.e Ta3lor "erie"#
• f h i" "ufficientl3 "mall, onl3 a fe: term" ma3 'e
reBuire4 to o'tain an approximation clo"e enou. to
t.e actual value for practical purpo"e"#
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Chapter # 2=
• @verflo: An3 num'er larer t.an t.e lare"t num'er t.atcan 'e expre""e4 on a computer :ill re"ult in an overflo:#
• Cn4erflo: (?ole) An3 po"itive num'er "maller t.an t.e
"malle"t num'er t.at can 'e repre"ente4 on a computer :illre"ult an un4erflo:#
• Dta'le Alorit.m n exten4e4 calculation", it i" li9el3 t.atman3 roun4-off" :ill 'e ma4e# Eac. of t.e"e pla3" t.e role
of an input error for t.e remain4er of t.e computation,impactin t.e eventual output# Alorit.m" for :.ic. t.ecumulative effect of all "uc. error" are limite4, "o t.at au"eful re"ult i" enerate4, are calle4 "ta'leF alorit.m"#G.en accumulation i" 4eva"tatin an4 t.e "olution i"over:.elme4 '3 t.e error, "uc. alorit.m" are calle4un"ta'le#
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Chapter # "8
iure #2
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