i error analysis

8/10/2019 I Error Analysis

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I ERROR ANALYSIS

Text: Ch 4

1



Objective:

Understand:

◦ Measures of Error used in numerical

computation◦ Causes of Error in Numerical

Computation

◦ Some measure to reduce errors incomputation

2



ntroduction

• Numerical !nal"sis:# – nexact mathematics – E$ective to%ether &ith computer

• Source of Error – Machine number representation – !rithmetic Error – Mathematical !pproximation

• Challen%es of N! – denti'cation of Error – (uanti'cation of Error – Control)limit &ithin pre#speci'ed error

3



*+* Error !nal"sis: Measuresof Error

• Measures of True Error – True Error

– !bsolute Error

– True ,elative Error

.ˆ;

;ˆ

Approx pvalueTrue p

p p E t

−−−=

;ˆ p p E t

−=

p

p pt

ˆ−=ε

-*.

-/.

-0.

4



• !pproximate Error – True value - p.# not available in real problems

– - p.# available onl" &hen dealin% anal"ticall"solvable function

– f - p. is not 1no&n2 use the best approx+available

• !pproximate ,elative error3 for 1 step process

• !pprox+ Error3 for Iterative Process%100ˆ

ˆˆ1

i

ii

a

p

p p−

−=ε

%100ion Approximat

Error ion Approximat a=ε -4.

-.

5



• Error Control/Limitation in Numerical Analysis

desired number t significann

valueTolerable specified e

OthershScarboroug

Criteria

s

n

s

n

s

sa

−

−−=

×=

<

−

−

/Pr

2/10%105.0

2

ε

ε

ε

ε ε



Example *:

!000015.0ˆ!00002.0"

!###ˆ!10000"

14.3ˆ!142$5.3"

==

====

p pc

p pb

p pa5ind the Error 6 ,elative Error

%25000005.0"

%1044"

%100#.#002$5.0"

2

2

==

×==

×==

−

−

t t

t t

t t

E c

E b

E a

ε

ε

ε



Example /: 5unction!pproximation

!pproximate e x for x=0.5 correct to 0si%ni'cant di%its2 p= e0.5 =1.648721

5unction !pprox+

Error Criteria&&3&2

132

n

x x x xe

n

x

+++=

%05.0%105.0 32 =×= −

sε

$



Si%ni'cant =i%its

Def.: The number of leadin% di%its of anapproximation that is correct to thecorrespondin% di%its in the true value

countin% ri%ht &ard from the *st non#>erodi%it

• ndicates the level of Condence

Example:

2035$.0035.0ˆ"

44#4.354#.35ˆ"

32222.0222.0ˆ"

====

==

p pc

p pb

digitst significan p pa

10



Relative rror -9. #!n indicator ofthe correct number of si%ni'cant

di%its

2

4

3

102$.002144.0

0213$.002144.0"

10$5.04#4.23

4#.234#4.23"

10#.02222.0

222.02222.0"

−

−

−

×=−

=

×=−=

×=−

=

ε

ε

ε

c

b

a

Exp of *< indicates the ? accurac" of thesi%ni'cant di%its

11



*+/ 5inite !rithmetic

=ef+: – Precision# indicates ho& repetitive

– !cc"rac# # ho& near to the actual)true value

12



*+/+* Causes of Error: Numbers"stems used n Computers

• @imited ,an%e of Numbers -Max andMin. –$ver t%e realmax# overow -NaN.

–&ess t%an realmin#underow -<.• 5inite number can be ,epresented

-step b)n numbers $x. 6 ncrease &iththe number x% $x

Example -%#pot%etical.&1' &1 &1"

• Error E t proportional to x

13



*+/+/ Errors: =ue to Choppin%6 ,oundin%!ll numbers C!NNOT be represented exactl"Error is introduced in numerical computation!pproximation made b" Ro"ndin' (C%oppin'

C%oppin'# ? stored to the lo&er end ofinterval Example# p*& stored as &1' +&"&2

$x*&&&2 )ax rror ;# $x -BiasedAA.

Ro"ndin'#? stored to the @o&er or Upper Example# p*& stored as &1' +&"&2

$x*&&&2 )ax rror *, $x-' -UnbiasedAA.

14



Example #Smart)5e&er calculationsusin% 0#di%its roundin% !rithmetic

2$5125.52#.5"25.1'

4*4

"""5'4'2'3"'

"'

24.5"25.1'

4*10

5423"'

"'

432

==

+−++−+=

=

+−+−=

valueTrue.

additionstionmultiplica

x x x x x.

b /orm

0

additionstionmultiplica

x x x x x 0

a /orm

1



*+0+* Truncation Error: ,eplacin%a complex function b" a simpler

function pdxe x f

x

=== ∫ 4544#$104$.0"'

25.0

0

1

4101.0 −×=t ε

Example:

&&4&3&21"'

2$4

22

n x x x x x x 0 e

SeriesTalor

n

n x +++++=≅

;&3&2

1"'4

2

x x x x 0 +++≈

544#$.0"'&3"5'&23

5.0

0

%535.0

0 =

+++=∫ x x x xdx 0



Order of !pproximation O-hn.

"'&4&2

1"cos'

"'&3&2

1

42

432

hOhh

h

hOhhheh

++−=

++++=

"cos'heh +

1$

"cos'heh

Example:

!pprox+ of Sums3

!pprox+ of Broducts3

"'&3

2

"'"'&4&3

2

43

443

hOh

h

hOhOhh

h

+++≅

+++++≅

"'&3

1 43

hOh

h +−+≅



*+0+/ @oss of Si%ni'cance)Subtractive Cancellation

xa,ple: Evaluate the function usin% 8di%it roundin% for x;<<

%22.0"'

%10#5.$"'

14.11*

14$.111

"'"

1500.11"1'"'"

4

factor lossahas x f

error lessinvolves x g

baof valueTruedigits

x x

x x g b

x x x x f a

−×

=++

==−+=

1#



Example: Subtractivecancellation

0;11

1"'

lnln"'

≈−+

−

−

x x

c

xb

x(a)

oncancellatiesubtractivtosensitivenotisthatform

aobtaintosexpressiontheReorder

20



*+4Error Bropa%ation,esults of calculations made &ith numberscontainin% E,,O,

• Errors due to Ordinar" !rithmeticOperations

x x x

x x

x

E E E x E x E E x x

tion 2ultiplica

E E x E E x x

Addition

E x x E Errors

valuese Approximat xvaluesTrue x

×+++×=+×+=×

+++=+++=+

=−=−

ˆˆ"ˆˆ'"ˆ'"ˆ'

"'"ˆˆ'"ˆ'"ˆ'

ˆ;ˆ

ˆ*ˆ;*

21



E+am,le- Aritmetic ,eration Error

Fiven: x;*<)*03 ";<+G8H/<*3 >;HIG8+H3&;<+<<<******2 determine the absolute and therelative error of the operation usin% di%itarithmetic

,eration E+act A,,ro+. As. Error el. Error

a '+)y" 2.##+10)5

'+)y"/ 2.#2+10)1

c '+)y" 3.0141+10)10

y6 #.$#+104

22



Bropa%ation Error

@ar%e Number computation – Series Computation

– !ddin% small and lar%e numbers

+++=∑ 21 x x x n

$3...44.1"2/'1"1/'1/1/1

$...44.1#/14/11/1

22210

1

10

1

=+−+−+=

=+++=

∑

∑=

=

=

=

nnn x

x

n

n

n

n

n

n

23



Error: Sin%le Jariable 5unctionEvaluation

Error x f x f

3alue /unction

x x f x x x f x f x f x f Estimate Error

x

x f

±=

∆•=−=−=∆

−−

"ˆ'"'

ˆ"ˆ'7"ˆ"'ˆ'7"ˆ'"'"ˆ'

8aluea,,ro+.anˆ

8arialeine,.te9or 8alue9unc."'

24



Error: More than one Jariable5unction Evaluation

t t

f 4 4 f

f x

x f t 4 x f

f x x

f x f Estimate Error

f x x

x

f x f x f

SeriesTalor

∆∂∂+∆

∂∂+∆

∂∂+∆

∂∂=∆

∆∂∂+∆

∂∂=∆=

−∂∂

+−∂∂

+=

ˆˆˆ"!ˆ!ˆ!ˆ'

:eneral

ˆˆ"ˆ!ˆ'

"ˆ'"ˆ'"ˆ!ˆ'"!'

25



Examples

4$4$

211211

2

2

1001.010.

/1005.0E/101.2E

1m0.05.1L

N1 N500

9olloin<=

<i8en tene9electioon teerrorteEstimateti,.at te

loa ,ointatosu>ecte eamcantile8er o9 ?e9lection2"

N.1in< oy ei<aon

'm"masson teerrorteEstimate.m/s#.$1<usin< enm/s0.002<is<itintrouceerrorte@9 1"

m 5 m 5

m 6 m 6

7m

−− ×=∆×=×=∆×=

=∆=

=∆=

= =∆

2



*+8Condition

=ef+

Sensitivit" of the =ependent variable toChan%es in the ndependent variable

Cond+ ?;*2 relative error is identical to the

relative error in xCond+ ?K*2 relative error is ma%ni'ed

/Cond+ ?L*2 relative error is attenuated

"'

"'7

x f

x f x 6umber Condition =

2



*+GError Control

• Not ,ossile to etermine e+act error

• Control trate<ies

– A8oi sutracti8e cancellation

• earran<e/ e9ormulate te ,rolem• Bse e+tene ,recision

• A smallest numer 9irst

–Carry out numerical e+,eriments

– or critical numerical com,utation soul e one y 2 or more ine,enent <rou,s

2$

i error analysis

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