i error analysis
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I ERROR ANALYSIS
Text: Ch 4
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Objective:
Understand:
◦ Measures of Error used in numerical
computation◦ Causes of Error in Numerical
Computation
◦ Some measure to reduce errors incomputation
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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
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*+* 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.
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• !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.
-.
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• 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
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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
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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
$
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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$.0035.0ˆ"
44#4.354#.35ˆ"
32222.0222.0ˆ"
====
==
p pc
p pb
digitst significan p pa
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Relative rror -9. #!n indicator ofthe correct number of si%ni'cant
di%its
2
4
3
102$.002144.0
0213$.002144.0"
10$5.04#4.23
4#.234#4.23"
10#.02222.0
222.02222.0"
−
−
−
×=−
=
×=−=
×=−
=
ε
ε
ε
c
b
a
Exp of *< indicates the ? accurac" of thesi%ni'cant di%its
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*+/ 5inite !rithmetic
=ef+: – Precision# indicates ho& repetitive
– !cc"rac# # ho& near to the actual)true value
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*+/+* Causes of Error: Numbers"stems used n Computers
• @imited ,an%e of Numbers -Max andMin. –$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
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*+/+/ 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.
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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
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*+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
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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 +−+≅
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*+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#
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Example: Subtractivecancellation
0;11
1"'
lnln"'
≈−+
−
−
x x
c
xb
x(a)
oncancellatiesubtractivtosensitivenotisthatform
aobtaintosexpressiontheReorder
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*+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;*
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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
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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
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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."'
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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'"!'
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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#.$1<usin< enm/s0.002<is<itintrouceerrorte@9 1"
m 5 m 5
m 6 m 6
7m
−− ×=∆×=×=∆×=
=∆=
=∆=
= =∆
2
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*+8Condition
=ef+
Sensitivit" of the =ependent variable toChan%es in the ndependent variable
Cond+ ?;*2 relative error is identical to the
relative error in xCond+ ?K*2 relative error is ma%ni'ed
/Cond+ ?L*2 relative error is attenuated
"'
"'7
x f
x f x 6umber Condition =
2
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*+GError Control
• Not ,ossile to etermine e+act error
• Control trate<ies
– A8oi sutracti8e cancellation
• earran<e/ e9ormulate te ,rolem• Bse e+tene ,recision
• A smallest numer 9irst
–Carry out numerical e+,eriments
– or critical numerical com,utation soul e one y 2 or more ine,enent <rou,s
2$