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The document discusses numerical computations, focusing on exact and approximate numbers, significant digits, and errors in calculations. It explains concepts such as rounding, truncation, and types of errors, including absolute, relative, and percentage errors. Additionally, it covers floating-point representation and the limitations of numerical precision in computing.
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0% found this document useful (0 votes)
9 views23 pagesM3
The document discusses numerical computations, focusing on exact and approximate numbers, significant digits, and errors in calculations. It explains concepts such as rounding, truncation, and types of errors, including absolute, relative, and percentage errors. Additionally, it covers floating-point representation and the limitations of numerical precision in computing.
Uploaded by
Bhavesh Kumar 1019Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
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/ 23
Att DA Nisha Cufita
a DWximoation aol Enos mm rumenrical Compubations |
Q)
Alstroximase rueblas: There Wu two type of Mumbtns exack
aol ofspropimate . Exact mumbers are 21 4/9, 13 T]a, 6245 -e
Buk thu ant mumbos such as 4/3 (= 13333-), Wa (= hulya3--)
nol TW (=B.NIS9Q--) WHICH cammot be expressed by a finite
Wht, of digits . These moup be approximates! by qumbers
13353, 1 4Iya onal 3.1WG Auspectively Such ruumbers wich
Aaphtsted the givin numbers bo a certain clea rte of accursey-
ant calles! ofproximate rurmbts,
beginning with the Lofomast nomero cligit Anal ending with
th alghtmast covred digit, including Final zeros thd Oe
exach,
Bt
(2) Significant Dights of Prrcision: Significand oligits ant digits
BUS, 3.589, O-47S§ Contains Four significant oligits
Ex: 00386, 0.000587, 000203 contains ond thrre significa
cugits.
(sy Accunauy anol Phocision; Accurate to» dlacimal places means
that you can trust n digits tothe night of the checrnal place .
Accurate ton Significant cligit means that you ead
of n cligits as being meamaful beginning woith-the Laplmast noe,
wy Rounding off The, are MUmbers Lolth, number of digits
eqs 3.2/7 = 3.14985 7143, Tn practice , It Is desinable to Limit
SUCH Mtumbers to O rraseno.gecub fa rumber of digits Such as 51
os 3.1US, This process of clroffng Unluarteel oLigits is called
Acumoling a t t i Da. Nis hy Gupte
yy
A ruumber is houndesl t position» by the Follcring Aude.
CY Discard all obgits tothe Aug ht of the orth digit
(i) Tp this cliscarelecl robes is
(a Lussthan b half Q unit inthe nth place,
Umchangcol;
(h) Qa eat than heety a unt mthe nth place , mcrcase the
{eave th nith ligt
nth cligit by unity >
Co exactly half q unit inthe nth place, mcdcase tha bt
oligid by unity if Hd iS oclel othimense Lease tt ue charged.
For erample Gy gy 767 Actmalid to thrce sirafleant fine = $4600
Gi) 36567 Acuncledl to -thace Signsficat figures 3.57
CW 8.73500 Apunsled to two diumal plots = §74
dy 724500 Aol to two divmasplacr= 784
wW —-(11-34576523 Actuncled to -five clecimal fis ur 1134577
Eprors: In any. numetal Computation ie Come across the
Following type of Crdors:
CO Trhured Onions: Eptons whith ane alatacly Preset inthe
stattmert of a problun before its solution are callicl mbersd
CrAOAS . Such ertors arbse eilhrv due to the Given data being
approrimade or dut to tie Drartafien of ynithimahical table,
colculaters or the cligital Comput. Tritrd COs Cas be
ourunazed by labung better clala oy by tong, Nigh fredsion
Computer Cudls. Dar Nish Gubty ay
(a) Rewwoling ennons; These Uvioks anise from the process of
Acunding off the amass during the. Computation. Such wors
ane unavoidable in mast of the calculations lie to the Limitaftons
Of the computing acl, Reuncling enrors Caer, however be Aecluced |
ch by changing the, calculation procedunt 50 ato avoid subtraction
of nearly Lea) Numbers W divisiow bya mall ruber 5
ud by Achainirg at Least one mare signaticach Fagus ad each step
than that given inthe dade anol Attunding off at the Last step,
(3) Truncagjon earons: These ennnas LL cassecl by uring approximate
posulls ox on acplacing du Infinite Process by a Finite ont.
For example, we Consider-the Teuylor serits expansion f F)
about x= e, C€[arb]- Lf we actain-the First » teams, We
Geb the afypro ximahe~
Fla) FCO) + (1-9 F ela eeepc) + = + eee)
al -i)I
anol Lhe totuncttion canon (TE) is given bye
TiE= Cxcell p@) thse fits bebe Cans x
3
TEL
<. Absolute enor = [X-X*] = 0-0008aaxlo-~
= 0+8.38x lo-> Da. Nishe Gupta
(7)
Cd Giver X = 000545828 i
= o.syseaexlo .
After rcuncled off +o tps leomal Place) its approxing
Valu x*= 0.546 x io
+ Absolute Enron = 1X-x*]
en
= [0.545838 xlo—0.54 6000 X Io
= 0-000I7&x la”
= altaxlo”
Buus Finodthe relative canon if the number
X= 0-004997 is
W truncated tothe olecimal places
cb Aounolesl off to thrte olecimal oligats
Sof We have X = 0.004997 = 04997 x10"
Uy After -trumeadeol to three olecimal Place, its approximate
valu X* = 0-499 x10
Relative Bd = [X=x* |
ix!
= [0.4997 x10 —0-499x lo
= ON = fan
ae = 000140 = 0-l4ax
o49a7 aa
Ud Aften roumoleal off to thrte oltcrmal Places, Hs
Approximate Value x*=0-005 = 0.500 x lo perl DA Nisha bast,
Robadtive error, = @®
“TT
= [0.4997 x lo — 0-5000 x 10 |
10.4997xlo]
00003 g.gpogeo = 0-booxlos AL
o4agT
Que Using Teuglth serite expansion of @-% abaut C= 0: Iyginaiho.
WD maximum error for x E11 thin the Fiast four toms
ane used inthe abproximahion, Afso —find athe Maxine
nr0h Whim X =03,
Gd the east rember of terms paguiras in the Ofsproximosion
such that Jensoal 0)
3 e*r2
= gxz dn
a
MS LIng=0-23l0e-
ied hn fs e
Similarly XO ANAK <—O85IOS at most 5 git Ishnd. DAs Nishy Gupta ®
Homer thr sete, (2) showlol be tseol fior 1X1 co-aslos Quel
For Ix > 023105, Fromula (1) cay be used.
Saray Rockurction
Another cause of Loss of significant Fiqurts isthe evaluation
of variou Library Functions with Lange apmects , Fer exe
A basic paoputy of the-funchin sinx, is fis povodicry;
sinxe sin (xtanm) for al Ava! valu of %
anol -for oll indegen valu of,
Because of this Aelahoshif, we need fo know oly -tte velues
of sin x in Some -fineol interval of funath giv to Compude,
sinx Fox anbitnany x. This “property Con be Useaf in the
Computer. Cyaluation of Sinx auol ts Callecl Allinge Aaduction,
Qua Porsinx, how many binary. bits of SlontPicanee Ont
Dost im Range AeckucHon to the intewat (0/21) 7
Sol Given Qn et X 7 27, we Final Qe Irdeger » that
salisfity 0