Catatan Pemrograman Dan Metode Numerik - 7 (Numeric Method).pptx

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

Catatan Pemrograman Dan Metode Numerik - 7 (Numeric Method).pptx

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