RROR ANLYSTS
’Sources of Error
I. Input Enor
2. Nlgorihinie Ener
3. Computtional Eror
(1)Arpoximade Numbers.
There aue too types nunbers exct and cproimte.
Exct nmbe aro 2,4,4,6. 46... elc but there are
numbers such that(,355.) 2 .414213.) and
(T 3.lAIS92.. ) ohi h cannot be erprtssed by number
These may be approximatd by number
§ intinite digits. -The
1.333,1414 and 2.14lG respetively
numbetr which represent the given number to
Such cre caled aPproximate
a certin deqree ccuraey
numbes
(2) Signikcant Digits are callecd
Ihe digik wed to exprec a number
significant cigte
The digits 12: 3S, 4, 5, Gt,8, q ere
cigniticar digik.
decd to
o' is alo significant digit except, cohen it iscnknoon or
ix he demal poin or to i he placel d
cliscarded digit.
A. The aignihcant hgure in a number in positional notction
Censists ot
non-Zero digits
cüs Zero igik wich
Ca) ie betoeen aignititant digits lot:s9.
Cb) ie to the right of deeimal poin t cnd at he
Same time, to the right o a non-zero igt
'are SpeciBically indiated to be signicat.
B. The signiticant iqure in a nunber wrten in selentiic
notatiorn Ceq. Mx|O) censiste 4 al the digik expuy
in M.
Number Signiicantdigits:; No. o siqnitkant
digits
3464 3,9,6,9
3
3060 3,0
3q00
34.64 3, 4, 6, 4
04
O. 3469 3,4, 6, 4
3, 9, O, o o4
34,0O
O.O0034 3,4
O. 00 B40 3, 4,;
3. O064. 3, oo, G,4
3,9 x IO6. i 3,4
3, , o, 4 O4
6X 102 0
Suestion
243oO
6234SO 23o0 S,R,3,,5, 0,2)3 O
t o023 1o,0, 2,3 o S
20.82 sO 2, 0, 3, 2,S,oO6
O . 0 7 0 3 5 0 ,o,3,S O O5
O. o00o20 2,O ’ o2
(3) Rounding at
There are number oith large no otigits e q i .3.)423541
In pratice, it is desirable to imut such humbes to a
mana geable no. of gi t such a 314 er 3.143 his
Pproce s ot cropping unwanted ougits is called roucig-o.
Numbes re rourded -o4 accoreing to following rule :
To round-ott a number to n signi ticant digits di scard all dgits
do the ight 4 rth digit and 4.this dis caded nnber s
Ú les: than 5 in (ntD" place , leowe the nth i'gt
unatered . eg .843 lo .84
(0 greater than 5 in (n+1ym place ihrease the nh
igit by uty eg 6.3456 to G.346
un eraclg Sin Cn+4)th place, nreose the nth,
igit b uity i} it is odel oiberoise leave it
unchanged 12.6& bdo)
l2. 646
12.688 12. 68-oNe n
rounded- of i's Soud to be rcorret do
The number hus
signiicant iguresls it easiy.
to grasp
ALH is being provided
Rounded - ot4 to
Numbers four digits Fie digits
Three
GO,543
digi O0.5432 O0.E4324
O0.54 324 29.52 34.5 26
39. 5
39.82B5 64.416
29 64.42
69,4155
O0. 66&ln.O0. 66t
O0. 664646
to four sígni&cant
numbers, corret
Round-o the dolowing
digits 3.28425, ’ 3.264 8S8,
’ 35.44 o.
35 4635
448 55 64 i4486OO0 35
O.f0035 ’ o,
’ o.
O. o OO322
1. 6683 ’I.668
O. B5348:
3. 141Sq3.1429 it,
too deannal pces
Round -oft the
48.2|416
folouring 48:2 (i1))tort
52.at5’ E2.28A
2 3 4 5 , 2.33
2. 3 S ’ 238
81. 255’ 81. 26
GERRORS AND tsypes
ErRDR-True value - pproxinate value
(1)Inherent emors in the stdement ot a
Enors ohich are already presentcalled inberent errOrS.
solution ue
probern before.its taking beter
be minimized by
Inherent erors can computing ods.
precision
laia or by ang hijh
Sue shion: DiHerence betuoeen Accurccy
and Recisíon:
of
curd precison are toO measures
4) Accuracy
obseraional enors measurement
cdose a Hven sets o
how
(2) AcCuracy is howvue, whch preu'slon ís
re to their rue are to e a h
other.
measurmen t
Aose the
number of sigifcant digits
reters to the R accurate to 5 sigiieant
C3) Aceuracy reers
53.46S
ina vaue 'eg.
digits nunber o decímal postion or
Treci'sion reters to the the volue
he lat digit in
order ot magriBude ot lo-3
precision fs
e.g. in 55.465 ,
’ Iruncahon Enrors
by wing appoxímate result or on replocig
’ They are caued
process by a Hiníte one ,
ininite
wirg adecimal computer havnga tked oord
rourding ot 13.658 is 13.Z6
lengn o 4 digit,
truuncation tis oould, be 13.6S
where au n
Alo
’ Truncadion Eror ís a type g alqoithm enor,
4T ... (say îs
turcated to 4ta+42 e x(say) then uhcahon
error 2 X *
Aird he truncatiorn Enor or e at 2
Example is rs turee terms cre relahed in expansíon.
Eror True value - Approinate value
hen, (+ ().(t
24 120
is Absolute Eror bekueen 4h
Absslute eror is numerital di#erence
cnd its approimate value
true l u e o a
vaue o a quantity and a is ite
t e is the trie
’
ue l |s called ihe abso lkete
then
approímate
eor nd Is denoteo by Easea
Ea -|a-a
i) Relatve Enor Absolute et or
|x-x|
Er > er True value
) Percentage Eror
eror
Remask Correct to m deeimal placeA then
t a number s -n
CoTrect to 4 leimal poce
nunber3,1416 is
Ex he
then ernor equal
(upto 5 deimal
cor ect to 4 sigrihcant digis
. 8. 634662
the
Suppose 1.414-is wed
as an a pproination to 2.find
eror?
asolute and relative.
> Lo2 =1.4|4213562 |. 4|42136
Irue value
Appror vaue x I.414
O.O0oa36
eax-x l.414 2136 - .414
9An approxímate value 4 7 is gíven by 3.142854 | and
t ue vaue is 3.l415926 Find absolute and relatie error.
eo lx-x| "3.4|5424- 3.142 854 | -o. o1265|
O.o0266
O,O01265
er
O.|4|5q26
And he relative erTOr & thenumber 8.6 e both ß} 1H
digite r e Coreet.
places .
8.6 correct to one decimal
20.0o58|4
8.6
enor i# 625.483 is a pproximnatee
rd the percentage
to three signiticant iqure 1,
625
Eror
G26.483 -G25) > 0.483O00
ea a
Round of he nunbers 865 260 and 31.46235 tor4
signiicant qure ord compute
ea.er.ep.
x 0625O
' 86E200
la 80B25o - 86S200 > gO
86525O
