C H A P T E R
1 Mathematical Preliminaries
andErrorAnalysis
1.1 Introduction
This book examines problems thatcanbesolved bymethods ofapproximation ,techniques
called numerical methods .Webegin byconsidering some ofthemathematical andcom-
putational topics thatarise when approximating asolution toaproblem .Nearly allthe
problems whose solutions canbeapproximated involve continuous functions ,socalculus
istheprincipal tooltouseforderiving numerical methods andverifying thatthey solve the
problems .Thecalculus definitions andresults included inthenextsection provide ahandy
reference when these concepts areneeded later inthebook.
There aretwothings toconsider when applying anumerical technique .The firstand
most obvious istoobtain theapproximation .The equally important second objective is
todetermine asafety factor fortheapproximation :some assurance ,oratleast asense ,of
theaccuracy oftheapproximation .Sections 1.3and1.4deal with astandard difficulty that
occurs when applying techniques toapproximate thesolution toaproblem :
 Where andwhy iscomputational error produced andhow canitbecontrolled ?
Thefinal section inthischapter describes various types andsources ofmathematical software
forimplementing numerical methods .
1.2 Review ofCalculus
Limits andContinuity
The limit ofafunction ataspecific number tells,inessence ,what thefunction values
approach asthenumbers inthedomain approach thespecific number .Thelimit concept is
basic tocalculus ,andthemajor developments ofcalculus were discovered inthelatter part
oftheseventeenth century ,primarily byIsaac Newton andGottfried Leibnitz .However ,it
wasnotuntil 200years later thatAugustus Cauchy ,based onwork ofKarl Weierstrass ,first
expressed thelimit concept intheform wenow use.
Wesaythatafunction /defined onasetXofrealnumbers hasthelimit Lat*o,written
lim^^^,/(x)=L,if,given anyrealnumber e>0,there exists arealnumber 8>0such
that\f(x)— L\<ewhenever 0<\x— *ol< <$ This definition ensures thatvalues ofthe
function willbeclose toLwhenever xissufficicndy close toJCQ.(SeeFigure 1.1.)
1
O.'pyrijj'tt20I2Cenjjayc Lcirrii /AIRjihuReserved May notbecopied Manned orduplicated towhole ortopan.Doctoelectronic rifhu.**ncthird party content nu>besupprcsied riom theeBook and/oreCh<xemi .Editorial review h*>
deemed thatanyxipprecxd content dee *,not i*uxtia:'.>affect theoverall Icamir .actpcncncc Ceityape Leaning roervev theright toremose additional contort atanytime ifvubvcijjcni nphre restrictions require it.2 C H A P T E R 1 Mathematical Preliminaries andError Analysis
Figure 1.1
y
-1
L+'-ss
L!L /i/i/i
i. i
i
i
i
i!
x0-Sx o X g+6 x
Afunction issaid tobecontinuous atanumber initsdomain when thelimit atthe
number agrees with thevalue ofthefunction atthenumber .Soafunction /iscontinuous
atX Qiflim,^f(x)=f(x0).
Afunction /iscontinuous onthesetXifitiscontinuous ateach number inX.We
useC(X)todenote thesetofallfunctions thatarecontinuous onX.When Xisaninterval
oftherealline,theparentheses inthisnotation areomitted .Forexample ,thesetofall
functions thatarecontinuous ontheclosed interval [a,b]isdenoted C[a,b].
Thelimit ofasequence ofrealorcomplex numbers isdefined inasimilar manner .An
infinite sequence {x„ }jjijconverges toanumber *if,given anye>0,there exists apositive
integer N(e)such that \x„— x\<ewhenever n>N(e).Thenotation lim^ocx„=x,or
x n-xasn-oo,means thatthesequence {x,,}^converges tox.
Continuity andSequence Convergence
If/isafunction defined onasetXofrealnumbers andxoeX,then thefollowing are
equivalent :
a./iscontinuous atXQ.
b.If isanysequence inXconverging toxo,then
lim/(*„ )=f(x0).n-*oo
Allthefunctions weconsider when discussing numerical methods arecontinuous
because thisisaminimal requirement forpredictable behavior .Functions thatarenot
continuous canskipover points ofinterest ,which cancause difficulties when weattempt
toapproximate asolution toaproblem .
More sophisticated assumptions about afunction generally lead tobetter approxima -
tion results .Forexample ,afunction with asmooth graph would normally behave more
predictably than would onewith numerous jagged features .Smoothness relies onthecon-
cept ofthederivative .
Copyright 2012 Cengagc Learn in*.AIRights Reserved May r*xbecopied .scanned .orimplicated ,inwhole orinpar.Doc tocjectronie rights .some third puny content may besupplied rican theeBook and/orcChapccmi .Editorial review h*>
deemed Cutanysuppressed content does notmaterialy alTcct theoverall learning experience .('engage Learning reserves theright 10remove additional conceal atanytime ifsubsequent nghts restrictions require It.1.2 Review ofCalculus 3
Differentiability
If/isafunction defined inanopen interval containing xo,then/isdifferentiable atx0
when
/'(*o)=lim/(*)-/(xo)
X— Xo
exists .The number f'(xo)iscalled thederivative of/atxo.The derivative of/atxois
theslope ofthetangent linetothegraph of/at(XQ,/(xo)),asshown inFigure 1.2.
Figure 1.2
yi
The tangent linehasslope/'(XQ)
f(xo)/(XQ)) y— y(*)
Xo x
Afunction thathasaderivative ateach number inasetXisdifferentiable onX.
Differentiability isastronger condition onafunction than continuity inthefollowing
sense.
Differentiability Implies Continuity
Ifthefunction /isdifferentiable atxo,then/iscontinuous atXQ.
Thesetofallfunctions thathave ncontinuous derivatives onXisdenoted C"(X),and
thesetoffunctions thathave derivatives ofallorders onXisdenoted C00(X).Polynomial ,
rational ,trigonometric ,exponential ,and logarithmic functions areinC^CX),where X
consists ofallnumbers atwhich thefunction isdefined .
Thenextresults areoffundamental importance inderiving methods forerror estimation .
Theproofs ofmost ofthese canbefound inanystandard calculus text.
Mean Value Theorem
If/ C[a,b]and/isdifferentiable on(a,b),then anumber cin(a,b)exists such
that(seeFigure 1.3)
f\c)m-m
b— a
Copyright 2012 Cenfajc Learn in*.AIRights Reversed Mayr*xbecopied.scanned .ocimplicated ,inwhole orinpar.Doctocjectronie rtghtv.some third pur.ycontent may besupplied rican theeBook and/orcChaptcnM .Editorial review h*>
deemed Cutanysuppressed content does notnuxtlaly alTcct theoverall learning experience .Ccagagc [.camonreserves theright 10remove additional eonteatatanytime ifsubsequent rights restrictions require It4 C H A P T E R 1 Mathematical Preliminaries andError Analysis
Figure 1.3
y
Slope/'(c)
Slopeb-aParallel lines
y=m
m-m
a c b x
Thefollowing result isfrequently used todetermine bounds forerror formulas .
Extreme Value Theorem
If/ C[a,b],thenc\andciin[a,b]exist with f(c\)<f(x)<f(ci)forallxin
[a,b].If,inaddition ,/isdifferentiable on(a,b),then thenumbers c\andcioccur
either atendpoints of[a,b]orwhere/'iszero.
Thevalues where acontinuous function hasitsderivative 0orwhere thederivative does
notexist arecalled critical points ofthefunction .SotheExtreme Value Theorem states
thatamaximum orminimum value ofacontinuously differentiable function onaclosed
interval canoccur only atthecritical points ortheendpoints .
Ourfirstexample gives some illustrations ofapplications oftheExtreme Value Theorem
andMATLAB .
Example 1UseMATLAB tofindtheabsolute minimum andabsolute maximum values of
f(x)=5cos2x-2xsin2x
ontheintervals (a)[1,2],and(b)[0.5,1].
Solution The solution tothis problem isone thatiscommonly needed incalculus .It
provides agood example forillustrating some commonly used commands inMATLAB
andtheresponse tothecommands thatMATLAB gives.Inourpresentations ofMATLAB
material ,input statements appear left-justified using atypewriter -like font.Toadd
emphasis totheresponses from MATLAB ,these appear centered andincyan type.
Forbetter readability ,wewilldelete the»symbols needed forinput statements as
wellastheblank lines from MATLAB responses .Other than these changes ,thestatements
willagree with thatofMATLAB .
Thefollowing command defines f(x)=5cos2x— 2xsin2xasafunction ofx.
f*inline (,5*cos(2*x)-2*x*sin(2*x)*,*x*)
Copyright 2012 Cenfajc Learn in*.AIR.(huRocncd Mayr*xbecopied.v.'aitned.orimplicated ,inwhole ormpart.Doctocjectronie rifhu.*wcthird pur.ycontent may besupplied rican theeBook and/orcChaptcnM .Editorial roiew h*>
deemed CutanyMpprcucd content does nottnaxrUXy alTcct theoverall Icarmr .itexperience .Ceagage [.camonroenn theright K>rcnxyve additional conteatatanytime ifsubvoyjem nghtc mtrictinna require It1.2 Review ofCalculus 5
andMATLAB responds with (actually ,theresponse isontwoseparate lines,butwewill
compress theMATLAB responses ,here andthroughout )
Inline function :f(x)=5*cos(2*x)— 2*x*sin(2*x)
Wehave now defined ourbase function /(*).Thexinthecommand indicates thatxisthe
argument ofthefunction /.
Tofindtheabsolute minimum andmaximum values off(x)onthegiven intervals ,we
also need itsderivative /'(*),which is
f'(x)=— \2sm2x— 4xcos2x.
Then wedefine thefunction fp(x)=f'(x)inMATLAB torepresent thederivative with
theinline command
fp=inline(*-12*sin(2*x)-4*x*cos(2*x)*,’x’)
Bydefault ,MATLAB displays only afive-digit result ,asillustrated bythefollowing com-
mand which computes /(0.5):
f(0.5)
Theresult from MATLAB is
ans=1.8600
Wecanincrease thenumber ofdigits ofdisplay with thecommand
formatlong
Then thecommand
f(0.5)
produces
ans=1.860040544532602
Wewill usethisextended precision version ofMATLAB output intheremainder ofthe
text.
(a)The absolute minimum andmaximum ofthecontinuously differentiable function /
occur only attheendpoints oftheinterval [1,2]oratacritical point within thisinterval .
Weobtain thevalues attheendpoints with
f(1),f(2)
andMATLAB responds with
ans=-3.899329036387075 ,ans=-0.241008123086347
Todetermine critical points ofthefunction /,weneed tofindzeros off'(x).Forthiswe
usethefzero command inMATLAB :
p=fzero (fp,[1,2])
Copyright 2012 Cc«£»fcLearn in*.AIR.(huReversed Mayr*xbecopied ,canned ,o*duplicated.»whole o tmpan .Doctoelectronic rifhu.vomc third pony content may besupposed ftem theeBook and/orcCh<xcn»l.Editorial roiew h*>
deemed Cutanyvuppicwed content dee>notimxttaly alTect theoverall Icamir .itexperience .Ccitit apeLearn xiprexxvev therljtlu lorenxyve additional conceal atanytime i ivutoeqjrni nghtv rotrictionv require It.6 C H A P T E R 1 Mathematical Preliminaries andError Analysis
andMATLAB responds with
p=1.358229873843064
Evaluating /atthissingle critical point with
f(p)
gives
ans— — 5.675301337592883
Insummary ,theabsolute minimum andabsolute maximum values off(x)ontheinterval
[1,2]areapproximately
/(1.358229873843064 )=-5.675301337592883 and/(2)=-0.241008123086347 .
(b)When theinterval is[0.5,1]wehave thevalues attheendpoints given by
/(0.5)=5cos1-1sin1=1.860040544532602 and
/(1)=5cos2-2sin2=-3.899329036387075 .
However ,when weattempt todetermine critical points intheinterval [0.5,1]with the
command
pi=fzero (fp,[0.51])
MATLAB returns theresponse
???Error using==>fzero at293
This indicates thatMATLAB could notfindasolution tothisequation ,which isthecorrect
response because /isstrictly decreasing on[0.5,1]andnosolution exists.Hence the
approximate absolute minimum andabsolute maximum values ontheinterval [0.5,1]are
/(1)=-3.899329036387075 and/(0.5)=1.860040544532602 .
Thefollowing fivecommands plot thefunction ontheinterval [0.5,2]with titles for
thegraph andaxes onagrid.
fplot (f,[0.5 2])
title ('Plot off(x)*)
xlabel ('Values ofx')
ylabelC 'Values off(x)')
grid
Figure 1.4shows thescreen thatresults from these commands .They confirm theresults
weobtained inExample 1.Thegraph isdisplayed inawindow thatcanbesaved inavariety
offorms foruseintechnical presentations .
Copyright 2012 Cengagc Learn in*.AIRights Reserved May r*xbecopied.scanned .ocimplicated ,inwhole orinpar.Doctocjectronie rights.some third pur.ycontent may besupplied rican theeBook and/orcChapccmi .Editorial review h*>
deemed Cutanysuppressed content does notmaterialy alTcct theoverall learning experience .('engage l.cammu reserves theright 10remove additional conceal atanytime ifsubsequent rights restrictions require It.1.2 Review ofCalculus 7
Figure 1.4
Plotof/(x)
2
I
0
iS-l
"-2s.U-zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA3-3>
-4
-5
1 1.5
Values ofx2
Thenext result istheIntermediate Value Theorem .Although itsstatement isnotdiffi-
cult,theproof isbeyond thescope oftheusual calculus course .
Intermediate Value Theorem
If/6C[a,b]and Kisanynumber between /(a)and f(b),then there exists anumber
cin(a,b)forwhich/(c)=K.(Figure 1.5shows oneofthethree possibilities forthis
function andinterval .)
Figure 1.5
yi
(«./(«))
f(a)-
Kii iAiiiL
m1 \i v
(bjm
— i— !1
a c bx
Example 2Show thatx5— 2x3+3x2— 1=0hasasolution intheinterval [0,1].
Solution Consider thefunction defined by/(x)=x5— 2x3+3x2— 1.Thefunction /is
continuous on[0,1].Inaddition ,
/(0)=-1<0 and 0<1=/(1).
Copyright 2012 Cc«£»fcLearn in*.AIRifhb Rocncd May rotbecopied ,canned ,ocdaplicated.»whole oempan .Doc 10electronic rifhu.*wtethird pony content may besuppressed rrom theeBook and/orcCh<xcn»l.Editorial toiew h*>
deemed Cutanysuppressed content dees notimxttaly alTcct theoverall Icamir*experience .C'cnpape[.camon reserves thety-ht10remote additional conceal atanytimeiisubvoyjem nRhts restrictions require It8 C H A P T E R 1 Mathematical Preliminaries andError Analysis
TheIntermediate Value Theorem implies thatanumber xexists in(0,1)withx5— 2x3+
3x2— 1=0. u
Asseen inExample 2,theIntermediate Value Theorem isused tohelpdetermine when
solutions tocertain problems exist.Itdoes not,however ,giveanefficient means forfinding
these solutions .This topic isconsidered inChapter 2.
Integration
The integral istheother basic concept ofcalculus .TheRiemann integral ofthefunction
/ontheinterval [a,b]isthefollowing limit ,provided itexists :
[f(x)d x= limVf(n)AxitJa maxAx,-+0^where thenumbers XQ,XJ, xnsatisfy a=xo<x\<    <xn=band where
Ax,=x,— x,_
i,foreach i=1,2 n,andz,isarbitrarily chosen intheinterval
[**'— 1>%i] 
Afunction /thatiscontinuous onaninterval [a,b]isalso Riemann intcgrablc on
[a,b\.This permits ustochoose ,forcomputational convenience ,thepoints x,tobeequally
spaced in[a,b]andforeach i=1,2,...,n,tochoose z,=x,.Inthiscase,
fJ a/(*)dx=nb— alim V/(*.),
oon'i=i
where thenumbers shown inFigure 1.6asx,arcx,=a+(i(b-a)/n).
Figure 1.6
y,
/7N>
7-/w/
a=xox,x2...x,.,x,... x„ _
ib— xn x
Two more basic results areneeded inourstudy ofnumerical methods .The firstisa
generalization oftheusual Mean Value Theorem forIntegrals .
Mean Value Theorem forIntegrals
If/eC[a,b],gisintegrable on[a,b]tandg(x)does notchange sign on[a,b]tthen
there exists anumber cin(a,b)with
[f(x)g(x)d x=/(c)fg(x)dx.
J a J a
Copyright 20I2C««cLearn in*.AIRights Reversed May notbecopied ,canned ,o*dedicated .inwhole orinpar.Doc toelectronic tights .some third pur.ycontent may besuppreved rrom sheeBook and/orcChapcerOri .Editorial review h*>
deemed Cutanysuppressed content does notmaterial yalTcct theioera'.llearning experience .('engage [.cammereserves theright K>remove additional conteat atanytime ifsubsequent rights restrictions require IL1.2 Review ofCalculus 9
When g(x)=1,thisresult reduces totheusual Mean Value Theorem forIntegrals .It
gives theaverage value ofthefunction /over theinterval [a,b]as
1 [b
/(c)=7/f(x)dx.b-aJa
(SeeFigure 1.7.)
Figure 1.7
y-
=/Mj/(c)-
1
1
1
1
1
c C £ X
Taylor Polynomials andSeries
Thefinal result inthisreview from calculus describes thedevelopment oftheTaylor polyno -
mials .Theimportance oftheTaylor polynomials tothestudy ofnumerical analysis cannot
beoveremphasized ,andthefollowing result isused repeatedly .
Taylor 'sTheorem
Suppose / Cn[a,b]and/("+1)exists on[a,b].Letxobeanumber in[a,b].For
every xin[a,b],there exists anumber £(x)between xoandxwith
/(x)=P„ (x)+Rn(x),
where
F„ (x)=/(x0)+/'(*0)(x-*o)+ ~xo)2+ **+*-fo\x-x0)"2! n!I
= -*<>)*k\
and
f^Hm ), ,„ +1RAX )~(n+1)!(X X0)
Here Pn(x)iscalled thenthTaylor polynomial for/about xo,and Rn(x)iscalled
thetruncation error (orremainder term )associated with Pn(x).Thenumber £(x)inthe
truncation error R„(x)depends onthevalue ofxatwhich thepolynomial Pn(x)isbeing
evaluated ,soitisactually afunction ofthevariable x.However ,weshould notexpect to
Copyright 2012 Cc«£»fcLearn in*.AIRighb Rocncd May notbecopied .N.anncil .ocdaplicated.»whole oempan .Doc 10electronic itfhu .*wtethird pony content may besuppreved Tiom theeBook and/wcCh<xcn»l.Editorial roiew h*>
deemed*»aiany Mjppic-cdcement dee>nottmxtlaly alTect theoverall lcamir«experience .Cent!ape[.camon roervei theright loremote additional conceal atanytime ifsubvoyjeM nght »rotrictionc requite IL10 C H A P T E R 1 Mathematical Preliminaries andError Analysis
Brook Taylor (1685-1731 )
described thisseries in1715 in
thepaper Methodus
incrementorum directa etinverse .
Special cases oftheresult ,and
likely theresult itself,hadbeen
previously known toIsaac
Newton ,James Gregory ,and
others .b eable toexplicitly determine thefunction £(*).Taylor ’sTheorem simply ensures that
such afunction exists ,andthatitsvalue liesbetween xand XQ.Infact,oneofthecommon
problems innumerical methods istotrytodetermine arealistic bound forthevalue of
y(#»+1>(£(*))forvalues ofxwithin some specified interval .
The infinite series obtained bytaking thelimit ofPn{x)asn-ociscalled theTaylor
series for/about XQ.The term truncation error intheTaylor polynomial refers tothe
error involved inusing atruncated (thatis,finite )summation toapproximate thesumofan
infinite series .
Inthecase x0=0,theTaylor polynomial isoften called aMaclaurin polynomial ,
andtheTaylor series iscalled aMaclaurin series .
Example 3Letf(x)=cos*and X Q=0.Determine
(a)thesecond Taylor polynomial for/about xo;and
(b)thethird Taylor polynomial for/about JCO-
Solution Since/C°°(R),Taylor ’sTheorem canbeapplied foranyn>0.Also ,
f\x)=— sin*,/"(*)=— cos*,/"'(*)=sinx ,and/(4)(x)=COSJC ,
so
/(0)=1,/'(0)=0,/"(0)=-1,and/"'(0)=0.
(a)Forn=2and X Q=0,wehave
cos*=/(0)+/'(0)*+@*2+0*^3
2! 3!
1,1,._
=1-r*+z xsm£(x),2 o
where f(x)issome (generally unknown )number between 0andx.(SeeFigure 1.8.)
Figure 1.8
y
-
I
^y=cosx
7Tyf 71
2/y 2
— zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA1T // \\ 7T X
y=P2(x)=1-£*2
When x=0.01 ,thisbecomes
cos0.01=1-
^(0.01)2+*(0.01 )3sin£(0.01 )2 60.99995 + sin£(0.01).6
Copyright 2012 Cc«£»fcLearn in*.AIRights Reserved .May r*xbecopied ,scanned ,ordaplicated.»whole ormpan .Doc 10electronic rights .some third pony content may besuppressed riom sheeBook and/orcClupccmi .Editorial review has
deemed Cutanysuppressed content decs notimxtlaly alTcct theiAera'.llearning experience .Ceng ageLearning reserves theright »oremove additional conceal atanytimeiisubsequent rights restrictions require IL1.2 Review ofCalculus 11
Theapproximation tocos0.01 given bytheTaylor polynomial istherefore 0.99995 .The
truncation error,orremainder term,associated with thisapproximation is
sinf (O.Ol)=0.16 x10-6sin|(0.01),6
where thebarover the6in0.16 isused toindicate that thisdigit repeats indefinitely .
Although wehave noway ofdetermining sin§(0.01 ),weknow thatallvalues ofthesine
lieintheinterval [-1,1],soabound fortheerror occurring ifweusetheapproximation
0.99995 forthevalue ofcos0.01 is
|cos(0.01)-0.999951 =0.16 x10~6|sin$(0.01)|<0.16 x10~6.
Hence theapproximation 0.99995 matches atleast thefirstfivedigits ofcos0.01,and
0.9999483 <0.99995 -1.6x10“ 6<cos0.01
<0.99995 +1.6x1(T6<0.9999517 .
Theerror bound ismuch larger than theactual error.This isdueinparttothepoor
bound weused for|sin§(*)|.Itisshown inExercise 16thatforallvalues ofx,wehave|sinx|<|x|.Since 0<§<0.01 ,wecould have used thefactthat|sin§(x)|<0.01 inthe
error formula ,producing thebound 0.16 x10“ 8.
(b)Since/'"(0)=0,thethird Taylor polynomial with remainder term about xo=0has
nor3term.Itis
cos*=1— 
^x2+^*4cosf (*),
where 0<§(*)<0.01.The approximating polynomial remains thesame ,andtheap-
proximation isstill 0.99995 ,butwenow have much better accuracy assurance .Since|COS§(JC)|<1forallx,wehave
^*4cosf (*)<-
^(0.01)4(1)*4.2 X1(T10.
So
|cos0.01-0.999951 <4.2x10-,°,
and
0.99994999958 =0.99995 -4.2x10"10
<cos0.01 <0.99995 +4.2x10'10=0.99995000042 .
Example 3illustrates thetwobasic objectives ofnumerical methods :
 Find anapproximation tothesolution ofagiven problem .
 Determine abound fortheaccuracy oftheapproximation .
Thesecond andthird Taylor polynomials gave thesame result forthefirstobjective ,butthe
third Taylor polynomial gave amuch better result forthesecond objective .
Copyright 2012 Cc«£»fcLearn in*.AIR.phtsReserved .May notbecopied ,scanned ,ordaplicated.»whole ormpan .Doe 10electronic rights ,some third pony content may besuppressed riom sheeBook and/orcClupccrtsl .Editorial tesiew h*>
deemed Cutanysuppressed cement decs notimxttaly alTect theoverall lcamir«experience .C'cnitapc Learn opreserves thert|>httoremove additional conceal atanytimeiisubsequent rights restrictions require IL12 C H A P T E R 1 Mathematical Preliminaries andError Analysis
Illustration Wecanalsousethethird Taylor polynomial anditsremainder term found inExample 3to
approximate J*001cos*d x.Wehave
={‘-ix,}yhrx ‘mHx )i‘
l lf0-1
=0.1--(0.1)3+— *4cosf (jr)dx.
Therefore
1/ U.l
/cosxdx*0.1-2(0.1)3=0.09983 .
Jo O
Abound fortheerror inthisapproximation isdetermined from theintegral oftheTaylor
remainder term andthefactthat|cosf (x)|<1forallx:
cosf (*)d x x4|cos£(x)|</x
<-F'x^J x-(0'1)5
— 8T-;10-8-24JoX X120X
The truevalue ofthisintegral is
fcosxdx=sinxo.i
0sin0.1 %0.099833416647 ,
sotheactual error forthisapproximation is8.3314 x108,which iswithin theerror
bound .
MATLAB canbeused toobtain these results byfirst defining f(x)=cos*andthe
second Taylor polynomial T2(x)m7*2(x)=1— \x2with
f=inline(*cos(x)*,’x*)
T2=inline(,1-0.5.x.''2Vx’)
Thenextcommands evaluate /at0.01,72at0.01 andcompute theerror inapproximating
cos(0.01 )with the72(0.01 ).
yl-f(0.01),y2=T2(0.01),errl=abs(yl-y2)
giving
yl=0.999950000416665 ,y2=0.999950000000000 ,
errl=4.166652578518892*-010
Toobtain agraph similar toFigure 1.8requires creating anM-filewhich canload more
than onecommand atatime.Weneed thisinorder todefine both f(x)andT2(x)ifwe
want toplotthem both onthesame graph.
Copyright 2012 Cengagc Learning .AIRights Reserved .May notbecopied ,scanned ,ctduplicated .inwhole orinp«.".Doctocjectronie rights.some third pur.ycontent may besuppressed rrom theeBook and/oreChaptcnnl .Editorial review h*>
deemed Cutanysuppressed content dees nottnaxrUXy alTcet theoverall Icamir .itexperience .Catfage [.camonreserves theright 10remove additional conteatatanytime ifsubsequent rights restrictions require It.1.2 Review ofCalculus 13
An M-fileiscreated byselecting File ontheMATLAB toolbar .Then select New
andScript .The three statements areentered and theresult issaved asafilenamed
ourfunctionl .The M-fileconsists ofthefollowing commands :
functionY=ourfunctionl(x)
Y(:,l)=cos(x(:));
Y(:,2)=l-0.5.*x(:).~2;
From theworksheet ,weneed toenter thefollowing command tocreate areference to
thefunction M-file:
fh=Qourfunctionl
Theresponse from MATLAB is
fh=@ourfunctionl
Thegraph shown inFigure 1.9canthen becreated with thecommands
fplot(fh,[-pi pi])
grid
Figure 1.9
l
0.5
0
0.5
-1
1.5
-2
2.5
-3
3.5
-40-3-2 2 3 -1 1
Wecanalso compute theintegrals of/(x)andtheTaylor polynomial Ti{x)onthe
interval [0,0.1]using MATLAB .Weusethecommands
ql=quad(’cos(x)*,0,0.1)
q2=quadC’1-0.5*x.~2\0,0.1)
andMATLAB produces
q\=0.099833416646828 ,ql=0.099833333333333
Copyright 2012 Cengagc Learn in*.AIRights Reversed Mayr*xbecopied.scanned .o*implicated ,inwhole orinpar.Doctoelectronic rights.some third pur.ycontent may besupplied ftem theeBook and/orcChaptcnsl .Editorial review h*>
deemed Cutanysuppressed content does nottnaxriaXy alTcct theioera .1learning experience .Ccagagc [.cammureserves theright toremove additional eonteatatanytime ifvutoeqjmi nghts restrictions require It14 C H A P T E R 1 Mathematical Preliminaries andError Analysis
E X E R C I S E S E T 12
1. Show thatthefollowing equations have atleast onesolution inthegiven intervals .
a.xcos*-2x2+3x-1=0,[0.2,0.3]and[1.2,1.3)
b.(x— 2)2— Inx=0,[1.2]and [e,4]
c.lxcos(2x)— (x— 2)2=0,[2,3]and[3,4]
d.x-(lnx)*=0,[4,5]
2. Find intervals containing solutions tothefollowing equations .
a.x-3~x=0
b.4x:— e*=0
c. X2-lx2-Ax+3=0
d.x2+4.001 x2+4.002*+1.101=0
3. Show thatthefirstderivatives ofthefollowing functions arezero atleast once inthegiven intervals .
a. f(x)=1-e*+(e-1)sin((;r/2)x),[0,1]
b.f(x)=(x-l)tanx+xsin 7rx,[0,1]
c.fix)=xsin7T X— (x— 2)lnx,[1,2]
d.fix)=(x—2)sinx ln(x+2),[-1,3]
4. Find maxu <^<fc|/(x)|forthefollowing functions andintervals .
a.fix)=i2— ex4-2x)/3,[0,1]
b.fix)=iAx-3)/(x2-2x),10.5,1]
c.fix)=2xcos(2x)-ix-2)2,[2,4]
d./(x)=l+tf-co^-1>t[1,2]
5. Letfix)=x2.
a.Find thesecond Taylor polynomial P2ix)about xo=0.
b.Find R2(0.5)andtheactual error when using P2(0.5)toapproximate /(0.5).
c.Repeat (a)with x0=1.
d.Repeat (b)forthepolynomial found in(c).
6. Letfix)=-Jx+1.
a.Find thethird Taylor polynomial P3(x)about xo=0.
b.UsePiix )toapproximate >/53,y/0.75,Vl-25,andVT3.
c.Determine theactual error oftheapproximations in(b).
7. Find thesecond Taylor polynomial P2{x)forthefunction fix)=e*cosx about x0=0.
a.UseF2(05)toapproximate /(0.5).Find anupper bound forerror \f(0.5)— /^(O.S)!using the
error formula ,andcompare ittotheactual error.
b.Find abound fortheerror|f i x )-Pzix )\inusing Piix )toapproximate /(x)ontheinterval
[0.1].
c.Approximate fQlf(x)dxusing f01P2ix)dx.
d.Find anupper bound fortheerror in(c)using fQ1|R2(x)dx|,andcompare thebound totheactual
error.
8. Find thethird Taylor polynomial P3(x)forthefunction /(x)=(x-1)lnxabout x0=1.
a.Use/M0.5)toapproximate /(0.5).Find anupper bound forerror \f(0.5)— /M0.5)|using the
error formula ,andcompare ittotheactual error.
b.Find abound fortheerror|/(x)-P3(x)|inusing P3(x)toapproximate f i x )ontheinterval
[0.5,1.5].
c.Approximate /0‘55f i x)d xusing/0'55P3(x)dx.
d.Find anupper bound fortheerror in(c)using/Jj3|/?3(x)dx\%andcompare thebound tothe
actual error.
Copyrifht 2012 Cc«£»fcLcarniit*.AIRights Reversed May notbecopied ,canned ,orduplicated .»whole oempar.Doctoelectronic rtghtv.vonte third pur.ycontent may bevupprcstcd riom theeBook amtar eOncxcnul .Editorial roiew h*>
deemed Cutanysuppressed content does notmaxrUXy alTect theoverall learning experience .Ccagagc [.cammu reserves theRIGHTK >renxyve additional contort atanytimeiisubseqjcnt rights restrictions require It1.3 Round -OffError andComputer Arithmetic 15
9. Use theerror term ofaTaylor polynomial toestimate theerror involved inusing sin*%xto
approximate sin1°.
10. UseaTaylor polynomial about 7r/4toapproximate cos42toanaccuracy of106.
11. Let/(x)=e4/2sin(x/3).UseMATLAB todetermine thefollowing .
a.The third Maclaurin polynomial /*3(x).
b.Abound fortheerror|f i x)-P3(x)|on[0,1J.
12. Let/(x)=ln(x2-I-2).UseMATLAB todetermine thefollowing .
a.TheTaylor polynomial P*(x)for/expanded about xo=1.
b.Themaximum error|f(x)— /*3(x)|for0<x<1.
c.TheMaclaurin polynomial P3(x)for/.
d.Themaximum error|/(JC)— P3(x)|for0<x<1.
e.Does P3(0)approximate /(0)belter thanP3(l)approximates /(1)?
13. Thepolynomial P2(x)=1-\x2istobeused toapproximate f(x)=cosx inl-*,J].Find abound
forthemaximum error.
14. ThenthTaylor polynomial forafunction /atx0issometimes referred toasthepolynomial ofdegree
atmost nthat“ best” approximates /near XQ.
a.Explain why thisdescription isaccurate .
b.Find thequadratic polynomial thatbestapproximates afunction /nearxo=1ifthetangent
lineatx0=1hasequation y=4x—1,andif/"(1)=6.
15. Theerror function defined by
gives theprobability thatanyoneofaseries oftrials willliewithin xunits ofthemean ,assuming that
thetrials have anormal distribution with mean 0andstandard deviation J2/2.This integral cannot
beevaluated interms ofelementary functions ,soanapproximating technique must beused ,
a.Integrate theMaclaurin series fore~‘~toshow that
erf(x)?oc
k=0(2k+1)*!’
b.Theerror function canalso beexpressed intheform
erf(x)=A 2kx2k*1
1 3 . 5 -..(2*+l)’
Verify thatthetwoseries agree fork=1,2,3,and4.[Hint:UsetheMaclaurin series fore~x‘.)
c.Usetheseries in(a)toapproximate erf(l)towithin 101.
d.Usethesame number ofterms used in(c)toapproximate erf(l)with theseries in(b).
e.Explain why difficulties occur using theseries in(b)toapproximate erf(x).
16. InExample 3itisstated thatxwehave|sinx|<|x|.Usethefollowing toverify thisstatement .
a.Show thatforallx>0wehave/(x)=x-sinxisnon-decreasing ,which implies thatsinx<x
with equality only when x=0.
b.Reach theconclusion byusing thefactthatforallvalues ofx,sin(-x)=— sinx.
1.3 Round -OffError andComputer Arithmetic
Thearithmetic performed byacalculator orcomputer isdifferent from thearithmetic that
weuseinouralgebra andcalculus courses .From your pastexperience ,youmight expect
thatwealways have astruestatements such things as2+2=4,4 8=32,and(V^3)2=3.
Copyright 20I2C««cLearning .AIRight*Reversed May notbecopied ,canned ,orduplicated.»whole ormpar.Doctoelectronic right*.vome third pur.yconcent may bevupprcv *driom theeBook and/orcCh<xcn»l.Editorial roiew h*>
deemed Cutanysuppressed content dec*notmaterial yalTcct theoverall learning experience .C'cngagc Learn aigreserves theright loremove additional conteatatanytime iisubsequent right*restrictions require It.16 C H A P T E R 1 Mathematical Preliminaries andError Analysis
Error duetorounding should be
expected whenever computations
areperformed using numbers that
arenotpowers of2.Keeping this
error under control isextremely
important when thenumber of
calculations islarge.
IllustrationInstandard computational arithmetic weexpect exact results for2+2=4and4 8=32,
butwewillnothave precisely (V3)2=3.Tounderstand why thisistruewemust explore
theworld offinite-digit arithmetic .
Inourtraditional mathematical world wepermit numbers with aninfinite number of
digits.Thearithmetic weuseinthisworld defines \/3asthatunique positive number which
when multiplied byitself produces theinteger 3.Inthecomputational world ,however ,each
representable number hasonly afixed andfinite number ofdigits.This means ,forexample ,
thatonly rational numbers — andnoteven allthese— canberepresented exactly .Since >/3is
notrational ,itisgiven anapproximate representation within themachine ,arepresentation
whose square will notbeprecisely 3,although itwilllikely besufficiently close to3tobe
acceptable inmost situations .Inmost cases ,then,thismachine representation andarithmetic
issatisfactory andpasses without notice orconcern ,butattimes problems arise because of
thisdiscrepancy .
Theenor thatisproduced when acalculator orcomputer isused toperform real-
number calculations iscalled round -offerror .Itoccurs because thearithmetic performed
inamachine involves numbers with only afinite number ofdigits ,with theresult that
calculations areperformed with only approximate representations oftheactual numbers .
Inatypical computer ,only arelatively small subset oftherealnumber system isused for
therepresentation ofalltherealnumbers .This subset contains only rational numbers ,both
positive andnegative ,andstores thefractional part,together with anexponential part.
Binary Machine Numbers
In1985,theIEEE (Institute forElectrical andElectronic Engineers )published areport called
Binary Floating Point Arithmetic Standard 754-1985.Anupdated version waspublished
in2008 asIEEE 754-2008.This provides standards forbinary anddecimal floating point
numbers ,formats fordata interchange ,algorithms forrounding arithmetic operations ,and
forthehandling ofexceptions .Formats arespecified forsingle ,double ,andextended
precisions ,andthese standards aregenerally followed byallmicrocomputer manufacturers
using floating -point hardware .
Forexample ,double precision realnumbers require a64-bit(binary digit )representa -
tion.The first bitisasign indicator ,denoted s.This isfollowed byan11-bitexponent ,c,
called thecharacteristic ,anda52-bitbinary fraction ,/,called themantissa .Thebase for
theexponent is2.
Thenormalized form forthenonzero double precision numbers has0<c<2n— 1=
2047.Since cispositive ,abias of1023 issubtracted from ctogive anactual exponent
intheinterval (— 1023 ,1024 ).This permits adequate representation ofnumbers with both
large andsmall magnitude .The first bitofthefractional partofanumber isassumed to
be1andisnotstored inorder togive oneadditional bitofprecision totherepresentation ,
Since 53binary digits correspond tobetween 15and16decimal digits ,wecanassume that
anumber represented using thissystem hasatleast 15decimal digits ofprecision .Thus,
numbers represented innormalized double precision have theform
(— 1)'2c_
1023(1+/).
Consider themachine number
010000000011 1011100100010000000000000000000000000000000000000000 .
The leftmost bitiss=0,which indicates thatthenumber ispositive .The next 11bits,
10000000011 ,give thecharacteristic andareequivalent tothedecimal number
c=l-210+0-29+---+0-22+l-2‘+l-2°=1024+2+1=1027.
Copyright 2012 Ccngagc Learn in*.AIR.phtsReserved Mayr*xbecopied ,canned ,o*duplicated.»whole o tmpan.Doc loelectronic right*.some third pony content may besuppressed ftem sheeBook and/orcCh<xcn»l.Editorial review h*>
deemed that artysuppressed content dees notmaterial yalTcct theoverall learning experience .('engage [.cannon reserves theright Mremove additional conceal atanytime i!subsequent right*restriction*require It.1.3 Round -OffError andComputer Arithmetic 17
Theexponential partofthenumber is,therefore ,210271023=24.Thefinal 52bitsspecify
thatthemantissa is
Asaconsequence ,thismachine number precisely represents thedecimal number
(— 1)S2c-1023(1+/)=(-1)° 21027-1023(1+G+§+i+^+256+4i6))
=27.56640625 .
However ,thenext smallest machine number is
01 111011100100001111111111111111111111111111111111111111 ,
andthenext largest machine number is
01 l l l l l l l 111011100100010000000000000000000000000000000000000001 .
This means thatouroriginal machine number represents notonly 27.56640625 ,butalsohalf
oftherealnumbers thatarebetween 27.56640625 andthenext smallest machine number ,
aswellashalfthenumbers between 27.56640625 andthenext largest machine number .To
beprecise ,itrepresents anyrealnumber intheinterval
[27.5664062499999982236431605997495353221893310546875 ,
27.5664062500000017763568394002504646778106689453125 ).
Thesmallest normalized positive number thatcanberepresented hass=0,c=1,
and/=0,andisequivalent tothedecimal number
2-IK2.(1+o)«0.225 x10-307.
The largest normalized positive number thatcanberepresented hass=0,c=2046 ,and
/=1— 2“ 52,andisequivalent tothedecimal number
2'023 (1+(1-2-52))0.17977 x10309.
Numbers occurring incalculations thathave toosmall amagnitude toberepresented result
inunderflow ,andarcgenerally setto0with computations continuing .However ,numbers
occurring incalculations thathave toolarge amagnitude toberepresented result inoverflow
andtypically cause thecomputations tostop.Note thatthere arctworepresentations for
thenumber zero;apositive 0when s=0,c=0,and/=0andanegative 0when 5=1,
c=0,and/=0.
Decimal Machine Numbers
Theuseofbinary digits tends tocomplicate thecomputational problems thatoccur when a
finite collection ofmachine numbers isused torepresent alltherealnumbers .Toexamine
Copyright 2012 Cengagc Learn in*.AIR.phtvReversed May notbecopied ,canned ,o*duplicated.»whole otmpan .Doc 10electronic right*.vonte third pony content may besuppressed ri«nsheeBook and/orcCh<xcn»l.Editorial roiew h*>
deemed Cut artyMipprcucd content dcc>notmaterial'.yafTcet theoverall learnire experience .Ccnitapc [.cannon roetve*theright Mremove additional conceal atanytime i!vuhvcyjcw nphtv rotrictions require it18 C H A P T E R 1 Mathematical Preliminaries andError Analysis
these problems ,wenowassume ,forsimplicity ,thatmachine numbers arerepresented in
thenormalized decimal form
± 0.did2...4x10",1<<4<9,0<4<9
foreach i=2,...,k.Wecallnumbers ofthisform k-digit decimal machine numbers .
Any positive realnumber within numerical range ofthemachine canbenormalized to
achieve theform
Theerror thatresults from
replacing anumber with its
floating -point form iscalled
round-offerror regardless of
whether therounding or
chopping method isused.y=0.d\d2... <4+2...x10” .
Thefloating -point form ofy,denoted by//(y),isobtained byterminating themantissa
ofyatkdecimal digits.There aretwoways ofperforming thetermination .One method ,
called chopping ,istosimply chop offthedigits <4+I<4+2   toobtain
//(y)=0.<4<4...<4x10".
Theother method ofterminating themantissa ofyatkdecimal points iscalled round -
ing.Ifthek+1stdigit issmaller than5,then theresult isthesame aschopping .Ifthek-I-1st
digit is5orgreater ,then 1isadded tothek\hdigit andtheresulting number ischopped .
Asaconsequence ,rounding canbeaccomplished bysimply adding 5x10"-(*+l)toyand
then chopping theresult toobtain//(y).Note thatwhen rounding up,theexponent ncould
increase by1.Insummary ,when rounding weaddoneto <4toobtain//(y)whenever
<4+15,thatis,weround up;when <4+i<5,wechop offallbutthefirstkdigits ,sowe
round down.
The next examples illustrate floating -point arithmetic when thenumber ofdigits being
retained isquite small .Although thefloating -point arithmetic thatisperformed onacal-
culator orcomputer will retain many more digits ,theproblems thisarithmetic cancause
arcessentially thesame regardless ofthenumber ofdigits .Retaining more digits simply
postpones theawareness ofthesituation .
Example 1Determine thefive-digit (a)chopping and(b)rounding values oftheirrational number TT.
Solution Thenumber nhasaninfinite decimal expansion oftheformn=3.14159265—Written innormalized decimal form ,wehave
?r=0.314159265 ...x10'.
(a)Thefloating -point form ofnusing five-digit chopping is
=0.31415 x101=3.1415 .
(b)Thesixth digit ofthedecimal expansion ofnisa9,sothefloating -point form ofn
using five-digit rounding is
Therelative error isgenerally a
better measure ofaccuracy than
theabsolute error because ittakes
intoconsideration thesizeofthe
number being approximated .fl(n)=(0.31415 +0.00001 )x101=3.1416 .
There arctwocommon methods formeasuring approximation errors .
Theapproximation p*tophasabsolute error \p— p*\andrelative error|p— p*|/|p|,
provided that p^0.
Example 2Determine theabsolute andrelative errors when approximating pbyp*when
(a)p=3.000 and p*=3.100 ;
(b)p=0.003000 and p*=0.003100 ;
(c)p=3000 andp*=3100.
Copyright 2012 Cengagc Learning .AIRights Reserved .May rotbecopied ,scanned ,orduplicated ,inwhole orinpar.Doctocjectronie lights ,some third party content may besuppressed rrom theeBook aml/ofcCh<xcnsl .Editorial review h*>
deemed thatanysuppressed content decs notmaterial yafleet theoverall learning experience ,('engage Learning reserves (heright Mremove additional conceal atanytime ifsuhscqjcnt rights restrictions require it1.3 Round -OffError andComputer Arithmetic 19
Solution First weneed towrite these numbers instandard floating -point form:
(a)Forp=0.3000 x101andp*=0.3100 x101theabsolute error is0.1,andtherelative
error is0.3333 x10_
1.
(b)Forp=0.3000 x10-3and p*=0.3100 x10-3theabsolute error is0.1x10-4,
andtherelative error is0.3333 x10_
1.
(c)Forp=0.3000 x104and p*=0.3100 x104,theabsolute error is0.1x103,and
therelative error isagain 0.3333 x10-1.
Wcoften cannot findanaccurate
value forthetrueerror inan
approximation .Instead wcfinda
bound fortheerror,which gives
usa“ worst-case” error.This example shows thatthesame relative error ,0.3333 x10-1,occurs forwidely varying
absolute errors .Asameasure ofaccuracy ,theabsolute error canbemisleading andthe
relative error more meaningful ,because therelative error takes intoconsideration thesize
ofthevalue.
Finite -Digit Arithmetic
Thearithmetic operations ofaddition ,subtraction ,multiplication ,anddivision performed
byacomputer onfloating -point numbers also introduce error.These arithmetic operations
involve manipulating binary digits byvarious shifting andlogical operations ,buttheactual
mechanics ofthearithmetic arenotpertinent toourdiscussion .Toillustrate theprob-
lems thatcanoccur ,wesimulate thisfinite -digit arithmetic byfirst performing ,ateach
stage inacalculation ,theappropriate operation using exact arithmetic onthefloating -point
representations ofthenumbers .Wethen convert theresult todecimal machine -number
representation .The most common round -ofTerror producing arithmetic operation involves
thesubtraction ofnearly equal numbers .
Example 3Use four-digit decimal chopping arithmetic tosimulate theproblem ofperforming the
computer operation n-j.
Solution Thefloating -point representations ofthese numbers are
0.3142 x10*. //(* )=0.3141 x10’and //
Performing theexact arithmetic onthefloating -point numbers gives
//(jr)-fl
^=-0.0001 x10',
which converts tothefloating -point approximation ofthiscalculation :
p'=fl
^flOi )-fl(y))=-0.1000 x10~2.
Although therelative errors using thefloating -point representations for JTandyaresmall ,
7T-//(TT)
71<0.0002 and ¥-//(¥)
22
7<0.0003 ,
therelative error produced bysubtracting thenearly equal numbers nand^isabout 700
times aslarge:
(w-f)-P*
(*-?)«0.2092 .
Copyright 2012 Cc«£»fcLcarni #*.AIRighb Rcscncd May ncabecopied ,canned ,ordaplicatod .»whole oempar.Doc 10electronic tlfht».*wcthird pur.ycontent may besuppressed riom theeBook and/orcCh<xcn»l.Editorial toiew h*>
deemed Cutanysuppressed content decs notmaterial yalTcct theioera .1Icamir*experience .C'cnitasc Learn xii:reserves thert|>httoremose additional conteatatanytime iisubsequent rights restrictions require IL20 C H A P T E R 1 Mathematical Preliminaries andError Analysis
Rounding arithmetic inMATLAB involves theuseofthefunction round .This function
rounds tothenearest whole number and thefunction fix chops offthefractional part.
Suppose wedefine thenumbers x\,x2,x3,andx4with theMATLAB command
xl—123.4 ,x2=-123.5 ,x3=123.4 ,x4=123.5
MATLAB responds with,
xl=-1.234000000000000^+002
x2=-1.235000000000000*+002
x3=1.234000000000000*+002
x4=1.235000000000000*+002
Invoking thecommands round andfixgives ,forround ,
ans=— 123,ans=—124,ans=123,ans=124
and,forf i x,
ans=— 123,ans=— 123,ans=123,ans=123
Exercise 12illustrates how these functions canbeused toperform rounding andchopping
arithmetic toaspecific number ofdigits .
E X E R C I S E S E T 1 . 3
l.
2.
3.Compute theabsolute error andrelative error inapproximations ofpbyp*.
a. p=7T,p*= b.p=7T,p‘=3.1416
c. p=e,p*=2.718 d.p=\/2,p*=1.414
e. p=ei0,p'=22000 f.p=10T,p*=1400
g.p=8!,p"=39900 h.p_9|p-_,/ig^(9/e)»
Perform thefollowing computations (i)exactly ,(ii)using three-digit chopping arithmetic ,and
(iii)using three-digit rounding arithmetic ,(iv)Compute therelative errors in(ii)and(iii).
a.
c.4 1
5+3/1_3\ 3_
\3liy+20b.
d.41
5'3
G+n)3_
20
Usethree-digit rounding arithmetic toperform thefollowing calculations .Compute theabsolute error
andrelative error with theexact value determined toatleast fivedigits.
a.
c.
e.
8-133+0.921
(121-0.327 )-119
13_6
14 7
2e-5.4
G) (!)b.133-0.499
d.(121-119)-0.327
f.— 10JT+6e— 62
Copyright 2012 Cengagc Learning .AIR.ghtvReversed May notbecopied ,canned ,ordaplica '.cd.»whole oempar.Doctoelectronic rtghtv.vontc third pur.ycontent may besuppreved riom theeBook and/orcCh<xcn»l.Editorial roiew h*>
deemed Cutanyvuppicvvcd content dee*nottmxtlaly alTect theoverall learning experience .Ceng age[.cammgreserve*theright loremote additional contort atanytime iivubveqjcni right*rotrictionv reyuirc It.1.3 Round -OffError andComputer Arithmetic 21
4. Repeat Exercise 3using three-digit chopping arithmetic .
5. Repeat Exercise 3using four-digit rounding arithmetic .
6.
7.Repeat Exercise 3using four-digit chopping arithmetic .
The firstthree nonzero terms oftheMaclaurin series forthearctanx arex— *x3+jx5.Compute the
absolute error andrelative error inthefollowing approximations ofnusing thepolynomial inplace
ofthearctanx :
a.4arctanG)+-~G)]b.16arctanG)_4arc,an(i)
8. The two-by-twolinear system
ax+by=e,
cx+dy=/,
where a,b,c,d,e,/aregiven ,canbesolved forxandyasfollows :
csetm= provided a/0;a
d\=d— mb\
f\=f-m e\
x_(g~b y)
a
Solve thefollowing linear systems using four-digit rounding arithmetic .
a.1.130 x-6.990 y=14.20 b. 1.013 x-6.099 y=14.22
8.1lOx+12.20 y=-0.1370 -18.1lx-I-112.2 y=-0.1376
9. Suppose thepoints (x0,y0)and(x!,yi)areonastraight linewith y\/y0.TWo formulas areavailable
tofindthex-intercept oftheline:
xxoyi-xiyo . (*i-x0)yo— — and x=x0 — yi-y o y\-y o
10.a.Show thatboth formulas arealgebraically correct .
b.Usethedata (x0,yo)=(1.31*3.24 )and(x1(yj)=(1.93,4.76 )andthree-digit rounding arith-
metic tocompute thex-intercept both ways .Which method isbetter ,andwhy?
TheTaylor polynomial ofdegree nfor/(x)=e1is0x'//!.UsetheTaylor polynomial ofdegree
nine andthree-digit chopping arithmetic tofindanapproximation toe~5byeach ofthefollowing
methods .
a.e-59(— 5y
i!9
=£(-1)'5'
i!b.e-=-U'E’=o5-/i!
Anapproximate value ofe~5correct tothree digits is6.74 x103.Which formula ,(a)or(b),gives
themost accuracy ,andwhy?
11. Arectangular parallelepiped hassides 3cm,4cm,and5cm,measured tothenearest centimeter .
a.What arethebestupper andlower bounds forthevolume ofthisparallelepiped ?
b.What arethebestupper andlower bounds forthesurface area?
Copyright 2012 Cc«£»fcLearn in*.AIR.(huReversed May notbecopied ,canned ,ordaplicated .inwhole ormpan .Doctoelectronic ilfhu.vonte third pony content may besuppressed rrom theeBook and/orcClupccnM .Editorial roiew h*>
deemed Cutanysuppressed content dees notimxttaly alTect theoverall leamir*experience .C'crtitape[.camon roentt thertpltt 10renxyve additional conceal atanytimeiisubsequent rights restrictions require It2 2 C H A P T E R Mathematical Preliminaries andError Analysis
12. Thefollowing MATLAB M-filerounds orchops anumber xtotdigits where md=1forrounding
andmd=0forchopping .
function [res] CHIP (rnd,t,x)
'/.This program isused toround orchop anumber xtoaspecific number t
Xofdigits ,
ifx-0
w-0;
else
ee=fix(loglO (abs(x) ));
ifabs(x)>1
ee-ee+1;
end;
ifrnd ==1
v-round (x*10~(t-ee) )*10~(ee-t);
else
v=fix(x*10“ (t-ee))*10“ (ee-t);
end;
end;
res =w;
Verify thattheprocedure works forthefollowing values .
a.x— 124.031 ,t=5 b.x=124.036 ,/=5
c.x=— 0.00653 ,t— 2 d.x=-0.00656 ,t=2
Thebinomial coefficient
/m\_ m!\k) k\(m— k)\
describes thenumber ofways ofchoosing asubset ofkobjects from asetofmelements .
a.Suppose decimal machine numbers areoftheform
±0.d|*/2^3^4x10",with1<d i<9,0<d,<9,ifi=2,3,4and|n|<15.
What isthelargest value ofmforwhich thebinomial coefficient (T1)canbecomputed forallk
bythedefinition without causing overflow ?
b.Show that(")canalsobecomputed by
(m\_f m\fm— \\ fm— k+\\[k j-\k)OnT /Ai/
c.What isthelargest value ofmforwhich thebinomial coefficient (")canbecomputed bythe
formula in(b)without causing overflow ?
d.Usetheequation in(b)andfour-digit chopping arithmetic tocompute thenumber ofpossible
5-card hands ina52-card deck.Compute theactual andrelative errors.
1.4 Errors inScientific Computation
Intheprevious section wesawhow computational devices represent andmanipulate num-
bers using finite -digit arithmetic .Wenow examine how theproblems with thisarithmetic
cancompound andlook atways toarrange arithmetic calculations toreduce thisinaccuracy .
Copyright 2012 Cengagc Learn in*.AIRights Reversed Mayr*xbecopied.scanned .ocimplicated ,inwhole orinpar.Doctocjectronie rlghtv.some third pur.ycontent may besupplied ftem theeBook and/orcChaptcnM .Editorial review h*>
deemed Cutany suppic-cdcontent does notmamlaXy alTcct theoverall learning experience .('engage [.camonreserves theright 10remove additional eonteatatanytime ifsubsequent nghts restrictions require It1.4 Errors inScientific Computation 23
Thelossofaccuracy duetoround -offerror canoften beavoided byacareful sequencing
ofoperations orareformulation oftheproblem .This ismost easily described byconsidering
acommon computational problem .
Illustration The quadratic formula states thattheroots ofax2+bx+c=0,when a/0,are
*i— b+ y/b2— 4ac
2aand*2— b— y/b2— 4ac
2a(1.1)
Consider this formula applied totheequation x2+62.10*+1=0,whose roots are
approximately
*i=-0.01610723 and x2=-62.08390 .
Wewillagain usefour-digit rounding arithmetic inthecalculations todetermine theroot.In
thisequation ,b2ismuch larger than 4ac,sothenumerator inthecalculation for X]involves
thesubtraction ofnearly equal numbers .Because
vV-4ac=>/(62.10 )2— (4.000 )(1.000 )(1.000 )
=-v/3856 ."
— "4.000 =V385Z=62.06 ,
wehave
//(*1)=-62.10 +62.06
~l/666-0.04000
~
2.000=-0.02000 ,
apoor approximation tox\=-0.01611 ,with thelarge relative error
1-0.01611 +0.020001
1-0.016111«2.4x10_1.
Ontheother hand ,thecalculation forx2involves theaddition ofthenearly equal numbers-band— y/b2— 4ac.This presents noproblem because
//(*2)=-62.10-62.06
2.000-124.2
2.000-62.10
hasthesmall relative error
1-62.08 +62.101
|— 62.08 |»3.2x10“ 4.
Theroots x\andxiofageneral
quadratic equation arerelated to
thecoefficients bythefactthat
b cx\+xi=— and X\X2— a a
This isaspecial case ofVifcta 's
formula forthecoefficients of
polynomials .Toobtain amore accurate four -digit rounding approximation forx\,wechange theform of
thequadratic formula byrationalizing thenumerator :
*i=— b+y/b2— 4ac(— b— y/b2-4ac
2ab2— (b2— 4ac)
— b— y/b2— 4acI 2a(— b— y/b2— 4ac)’
which simplifies toanalternate quadratic formula
-2cx\=
Using Eq.(1.2)gives
f K x i)=b+y/b2— 4ac
-2.000 -2.000(1.2)
62.10 +62.06 124.2-0.01610 ,
which hasthesmall relative error 6.2x104.
Copyright 2012 Cengagc Learn in*.AIRights Reversed May notbecopied ,canned ,ocimplicated ,inwhole orinpar.Doctocjectronie rlghtv.some third pur.yCOMCK may besuppressed rican theeBook and/orcChapccriM .Editorial review h*>
deemed Cutanysuppre-edcontent docs notmaterial yalTcct theoverall Icamir .itexperience .('engage [.cammereserves theright K>remove additional eonteatatanytime ifvubveqjeni nghtv restrictions require It24 C H A P T E R 1 Mathematical Preliminaries andError Analysis
The rationalization technique canalso beapplied togive thefollowing alternative
quadratic formula forx2:
*2 — 2c
b— >/b2— 4ac(1.3)
This istheform touseifbisanegative number .IntheIllustration ,however ,themistaken use
ofthisformula forx2would result innotonly thesubtraction ofnearly equal numbers ,but
alsothedivision bythesmall result ofthissubtraction .Theinaccuracy thatthiscombination
produces ,
fl(x2) — 2c
b— V*2— 4ac-2.000
62.10-62.06-2.000
0.04000-50.00 ,
hasthelarge relative error 1.9x10"1.
Nested Arithmetic
Example 1Evaluate /(x)=x3— 6.lx2+3.2x+1.5atx=4.71 using three-digit arithmetic .
Solution Table 1.1gives theintermediate results inthecalculations .
Table 1.1X x2x36.1x23.2x
Exact 4.71 22.1841 104.487111 135.32301 15.072
Three -digit (chopping ) 4.71 22.1 104. 134. 15.0
Three -digit (rounding ) 4.71 22.2 105. 135. 15.1
Remember thatchopping (or
rounding )isperformed after
each calculation .Toillustrate thecalculations ,letusfirstlook atthose involved with finding x3using
three -digit rounding arithmetic .
First wefind
x2=4.712=22.1841 ,which rounds to22.2.
Then weusethisvalue ofx2tofind
x3=x2 x=22.2 -4.71=104.562 ,which rounds to105.
Also ,
6.lx2=6.1(22.2 )=135.42 ,which rounds to135,
and
3.2x=3.2(4.71)=15.072 ,which rounds to15.1.
Using finite -digit arithmetic ,theway inwhich weaddtheresults canaffect thefinal
result.Suppose thatweaddlefttoright.Then forchopping arithmetic wehave
Three -digit (chopping ):/(4.71)=((104.-134.)+15.0 )-I-1.5=-13.5,
andforrounding arithmetic wehave
Three -digit (rounding ):/(4.71)=((105.— 135.)+15.1)+1.5=-13.4.
Copyright 2012 Cc«£»fcLearn in*.AIR-phivReversed Mayr*xbecopied ,canned ,o*duplicated.»whole o tmpan .Doctoelectronic rifhu.*wcthird pony contcac may besoppre ^tedftem theeBook and/orcCh<xcn»l.Editorial roiew h*>
deemed CutanyMpprcucd content dee>notnuxtiily alTca theoverall Icamir .itexperience .C'crtitape l.camzip roentt thertpht 10renxyve additional conceal atanytime i itutoeqjcni npht »rotrictionc require It1.4 Errors inScientific Computation 25
Illustration(Youshould carefully verify these results tobesure thatyour notion offinite-digit arithmetic
iscorrect .)Note thatthethree-digit chopping values simply retain theleading three digits ,
with norounding involved ,anddiffer from thethree-digit rounding values.Theexact result
oftheevaluation is
Exact :/(4.71)=104.487111 -135.32301 +15.072 +1.5=-14.263899 ,
sotherelative errors forthethree -digit methods are
Chopping :-14.263899 +13.5~
14.263899%0.05 ,and
Rounding :-14.263899 +13.4-14.263899%0.06.
Asanalternative approach ,thepolynomial /(x)inExample 1canbewritten inanested
manner as
/(*)=X1-6.IJC2+3.2X+1.5=((*-6.1)JC+3.2)x+1.5.
Using three -digit chopping arithmetic now produces
/(4.71)=((4.71-6.1)4.71+3.2)4.71+1.5=((-1.39)(4.71 )+3.2)4.71+1.5
=(-6.54+3.2)4.71+1.5=(-3.34 )4.71+1.5=-15.7+1.5=-14.2.
Inasimilar manner ,wenowobtain athree-digit rounding answer of— 14.3.Thenewrelative
errors are
Three -digit (chopping ):-14.263899 +14.2^14.263899%0.0045 ;
Three -digit (rounding ):-14.263899 +14.3-14.263899%0.0025 .
Nesting hasreduced therelative error forthechopping approximation tolessthan 10%
ofthatobtained initially .Fortherounding approximation ,theimprovement hasbeen even
more dramatic ;theerror inthiscase hasbeen reduced bymore than95%.
Characterizing Algorithms
Wewill beconsidering avariety ofapproximation problems throughout thetext,andin
each case weneed todetermine methods thatproduce dependably accurate results fora
wide class ofproblems .Because ofthediffering ways inwhich theapproximation methods
arederived ,weneed avariety ofconditions tocategorize their accuracy .Notallofthese
conditions willbeappropriate foranyparticular problem .
One criterion wewill impose ,whenever possible ,isthat ofstability .Amethod is
called stable ifsmall changes intheinitial data produce correspondingly small changes
inthefinal results .When itispossible tohave small changes intheinitial data producing
large changes inthefinal results ,themethod isunstable .Some methods arestable only for
certain choices ofinitial data.These methods arecalled conditionally stable .Weattempt
tocharacterize stability properties whenever possible .
Oneofthemost important topics affecting thestability ofamethod isthewayinwhich
theround-offerror grows asthemethod issuccessively applied .Suppose anerror with
Copyright 2012 Cengagc Learn in*.AIRights Reserved Mayr*xbecopied.scanned .orimplicated ,inwhole orinpar.Doctoelectronic rights.some third pur.ycontent may besupplied ftem theeBook and/orcChapccmi .Editorial review h*>
deemed Cutanysuppressed content dees notmaterial yalTcct theoverall learnire experience .Cette jgcLearning rese-tvestheright Mremove additional conceal atanytime i!suhvojjcni right*restrictions require It.26 C H A P T E R 1 Mathematical Preliminaries andError Analysis
magnitude Eo>0isintroduced atsome stage inthecalculations andthatthemagnitude of
theerror after nsubsequent operations isE„.There aretwodistinct cases thatoften arise
inpractice .
 Ifaconstant Cexists independent of«,with En%CnEQ ,thegrowth oferror islinear .
 Ifaconstant C>1exists independent ofn,with En%C"£o»thegrowth oferror is
exponential .
Itwould beunlikely tohave En^CnE 0,with C<1,because thisimplies thattheerror
tends tozero.
Linear growth oferror isusually unavoidable and,when CandEoaresmall ,the
results aregenerally acceptable .Methods having exponential growth oferror should be
avoided because theterm Cnbecomes large foreven relatively small values ofnandEo.
Consequently ,amethod thatexhibits linear error growth isstable ,while oneexhibiting
exponential error growth isunstable .(SeeFigure 1.10.)
Figure 1.10
 
Unstable exponential error growth.En=CnE 0
 
Eo 
#  Stable linear error growth
 . * En=CnE0. *
1 2 3 4 5 6 7 8n
Rates ofConvergence
Iterative techniques often involve sequences ,andthesection concludes with abrief dis-
cussion ofsome terminology used todescribe therateatwhich sequences converge when
employing anumerical technique .Ingeneral ,wewould liketochoose techniques thatcon-
verge asrapidly aspossible .Thefollowing definition isused tocompare theconvergence
rates ofvarious methods .
Suppose that {(*„ }£!,isasequence thatconverges toanumber aasnbecomes laige.
Ifpositive constants pand Kexist with
%|a— an\<— ,foralllarge values ofn,
nP
then wesaythat{an}converges toawith rate,ororder ,ofconvergence 0{\/n p)(read
“ bigohof1/np").This isindicated bywriting an=a+0(l/n p)andstated as“ a„ -a
Copyright 2012 Cc«£»fcLearn in*.AIR.(huReversed Mayr*xbecopied ,canned ,o*duplicated.»whole ormpan.Doctoelectronic rifhu.vomc third pony content may bevupprcv «dftem theeBook and/orcCh<xcn»l.Editorial roiew h*>
deemed Cutanyvuppicwed content dee>notnuxtiily alTect theoverall leamir*experience .C'cnitapc l.cammu roentt thert|>htKVrenxyve additional conceal atanytime i ivutoeqjrni nghtv rotrictionv require It1.4 Errors inScientific Computation 27
There arcnumerous other ways
ofdescribing thegrowth of
sequences andfunctions ,some of
which require bounds both above
andbelow thesequence or
function under consideration .
Any good book thatanalyzes
algorithms ,forexample [CLRS ],
willinclude thisinformation .with rateofconvergence \/np.” Wearegenerally interested inthelargest value ofpfor
which an=a4-0(1/np).
Wealsousethe“ bigoh” notation todescribe howsome divergent sequences grow as
nbecomes large.Ifpositive constants pand Kexist with
|or„ l<Knp,foralllarge values ofn,
then wesaythat{a„ }goes tooowith rate,ororder ,0(np).Inthecase ofasequence
thatbecomes infinite ,weareinterested inthesmallest value ofpforwhich anis0(np).
The “ bigoh” definition forsequences canbeextended toincorporate more general
sequences ,butthedefinition aspresented hereissufficient forourpurposes .
Example 2Suppose that,forn>1,
»+l .A"+3an=— — and &n=— — .n2 n'
Both ofthesequences converge to0,butthesequence {&„ }converges much faster than the
sequence {<*„ }.Using five-digit rounding arithmetic ,wehave thevalues shown inTable 1.2.
Determine rates ofconvergence forthese twosequences .
Table Mn 1 2 3 4 5 6 7
C*n 2.00000 0.75000 0.44444 0.31250 0.24000 0.19444 0.16327
4.00000 0.62500 0.22222 0.10938 0.064000 0.041667 0.029155
Solution Define thesequences pn=l/nand =1/n2.Then
l«n-0|=n+1 n+n
n2 n2=2--=2fi„n
and
A1n+3 n+3n 1h -01=— 3-<J— =4-r=4p„.rtJ n-5n2
Hence therateofconvergence of{(*„ }to0issimilar totheconvergence of{l/n}to0,
whereas {&„ }converges to0atarate similar tothemore rapidly convergent sequence
{l/n2}.Weexpress thisbywriting
a,i=O+0G)and an=0+0&) 
The “ bigoh” notation isalso used todescribe therateofconvergence offunctions ,
particularly when theindependent variable approaches 0.
Suppose that Fisafunction thatconverges toanumber Lashgoes to0.Ifpositive
constants pandKexist with
\F(h)-L\<Khp, as/i-0,
then F(h)converges toLwith rate,ororder ,ofconvergence 0(hp).This iswritten as
F(h)=L+0(hp)andstated as“ F(h)—Lwith rateofconvergence
Wearegenerally interested inthelargest value ofpforwhich F(h)=L+0(hp).
Copyright 2012 Cerijaxc Leirnin*.AIR.(huRocrscd May ncabecopied ,canned ,ortiaplicated .inwhole orinpar.Doctoelectronic itfht».xvtic third pur.yconunc may be>upptcvcd riom*ceBook amMr cCh<xcn»l.Editorial roiew h*.
deemed CutanyMpprccscd content does nottnaxrUXy alTea theoverall leamir*experience .Ceagace [.cammere\me»theright*>rerrxyve additional conteatatanytime iisutoeqjcoi nghts rotrictionv require It28 C H A P T E R 1 Mathematical Preliminaries andError Analysis
EXERCISE SET 1.4
l.
2.
3.(i)Usefour-digit rounding arithmetic andEqs.(1.2)and(1.3)tofindthemost accurate approximations
totheroots ofthefollowing quadratic equations ,(ii)Compute theabsolute errors andrelative errors
forthese approximations .
1,123 18*3*’T X+6"°
c.1.002 x2-ll.Olx+0.01265 =012123 1„ b.-X2+— x--=03 4 6
d.1.002 x2+ll.Olx+0.01265 =0
Repeat Exercise 1using four-digit chopping arithmetic .
Letf(x)=1.013 x5-5.262 x3-0.01732 x2+0.8389 x-1.912 .
a.Evaluate /(2.279 )byfirstcalculating (2.279 )2,(2.279 )3,(2.279 )4,and(2.279 )5using four-digit
rounding arithmetic .
b.Evaluate /(2.279 )using theformula
fix)=(((1.013 x2-5.262 )x-0.01732 )x+0.8389 )x-1.912
andfour-digit rounding arithmetic ,
c.Compute theabsolute andrelative errors in(a)and(b).
4. Repeat Exercise 3using four-digit chopping arithmetic .
5. Thefifth Maclaurin polynomials fore2*ande_2jare
*w"((((£*+f)*+s)*+2)*+2)x+1
ft(l)'(((("B'+5)'"0x+2)Jt_
2)x+1
a.Approximate e-098using Ps(0.49)andfour-digit rounding arithmetic .
b.Compute theabsolute andrelative error fortheapproximation in(a).
c.Approximate e“ °9Kusing l//>5(0.49)andfour-digit rounding arithmetic .
d.Compute theabsolute andrelative errors fortheapproximation in(c).
6. a.Show thatthepolynomial nesting technique described intheIllustration onpage 25canalso be
applied totheevaluation of
fix)=l.Ole4*— 4.62e3*-3.1U2*+12.2e'-1.99.
b.Usethree-digit rounding arithmetic ,theassumption thate153=4.62 ,andthefactthate"(,-53)=
iel53)ntoevaluate /(1.53)asgiven in(a).
c.Redo thecalculation in(b)byfirstnesting thecalculations .
d.Compare theapproximations in(b)and(c)tothetruethree-digit result/(1.53)=— 7.61.
7. Usethree-digit chopping arithmetic tocompute thesum2,'=!I/*2first by}+\4 4-^and
then by^^H h}.Which method ismore accurate ,andwhy?
8. TheMaclaurin series forthearctangent function converges for— 1<x<1andisgiven by
arctanx =lim P„ix)
IXn-limyV-l)m '.(2i-1)
a.Use thefactthat tan 7r/4=1todetermine thenumber ofterms oftheseries thatneed tobe
summed toensure that|4P„ (1)— n\<10“ 3.
b.TheCprogramming language requires thevalue ofTTtobewithin 10_,°.How many terms of
theseries would weneed tosum toobtain thisdegree ofaccuracy ?
Copyright 2012 Cengagc Learn in*.AIRights Reserved May notbecopied.scanned.orduplicated .inwhole orinpar.Doctoelectronic rights.some third pur.ycontent may besuppressed rrom theeBook aml/oreChaptcnnl .Editorial review h*>
deemed Cutanysuppressed content does nottnaxrUXy alTcct theioera .1learning experience .('engage [.camon reserves theright 10remove additional conceal atanytime ifsubvciyjem rights restrictions require It1.5 Computer Software 29
9. The number eisdefined bye= 1/**!»where n!=n(n— 1)   2 1,forn^0and0!=1.
(i)Usefour-digit chopping arithmetic tocompute thefollowing approximations toe.(ii)Compute
absolute andrelative errors forthese approximations .
a.51^n\
/>=05E
7=o1
(5-7)!
c.101
n=010E
7=01
(10-7)!
10. Find therates ofconvergence ofthefollowing sequences asn-oo.
a.limsin(-]=0
n-*oc\nJb.limsinf-z|=0
n-*oc\n*J
c- =0 d.lim[ln(n+1)— ln(n)J=0
(l-*90
11. Find therates ofconvergence ofthefollowing functions ash— 0.
sinh-hcosha-hm =0h\-ehb.lim— — =-1
h->oh
..sin/ic.hm— — =1oh1-coshd.hm =0
h-+o h
12. a.How many multiplications andadditions arerequired todetermine asum oftheform
EEfl-v
i=l7=1
b.Modify thesum in(a)toanequivalent form thatreduces thenumber ofcomputations .
13. The sequence {/%,}described byF0=1,F\=1,and Fn+2=Fn4-Fn+Uifn>0,iscalled
aFibonacci sequence .Itsterms occur naturally inmany botanical species ,particularly those with
petals orscales arranged intheform ofalogarithmic spiral .Consider thesequence {*„ },where
xn=F„+i/F„.Assuming thatlim„ ¥0cxn=xexists ,show thatxisthegolden ratio (1+V5)/2.
1.5 Computer Software
Computer software packages forapproximating thenumerical solutions toproblems are
available inmany forms .Onourwebsite forthebook
http://www.math.ysu.edu/~faires/Numerical -Methods /
wehave provided programs written inC,FORTRAN ,Maple ,Mathematica ,MATLAB ,
andPascal ,aswellasJAVA applets .These canbeused tosolve theproblems given inthe
examples andexercises ,andwillgive satisfactory results formost problems thatyoumay
need tosolve.However ,theyarewhat wecallspecial -purpose programs .Weusethisterm
todistinguish these programs from those available inthestandard mathematical subroutine
libraries .Theprograms inthese packages willbecalled general purpose .
General Purpose Algorithms
The programs ingeneral -purpose software packages differ intheir intent from thealgo-
rithms andprograms provided with thisbook.General -purpose software packages consider
Copyright 2012 Cc«£»fcLearn in*.AIR.(huReversed Mayr*xbecopied ,canned ,o*duplicated.»whole otmpan .Doctoelectronic rifhu.vomc third pony content may besupposed ftem theeBook and/orcCh<xcn»l.Editorial roiew h*>
deemed Cutanyvuppicwed content dee>notimxttaly alTect theoverall Icamir .itexperience .C'criitapeLearn oprexxvev therljtlu Kvrenxyve additional conceal atanytime i ivutoeqjrni nphtv rotrictionv require It