Issuu

Instructor’s Solutions Manual for Prepared by

Youngstown State University

Richard

L. Burden

Youngstown State University Australia • Brazil • Japan • Korea • Mexico • Singapore • Spain • United Kingdom • United States Numerical Analysis 9th EDITION

J. Douglas Faires

Youngstown

Richard L. Burden

State University

Youngstown

Numerical Analysis 9th Edition Burden Solutions Manual Full Download: http://testbanktip.com/download/numerical-analysis-9th-edition-burden-solutions-manual/ Download all pages and all chapters at: TestBankTip.com

J. Douglas Faires

State University

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Preface vii MathematicalPreliminaries 1 ExerciseSet1.1..........................................1 ExerciseSet1.2..........................................6 ExerciseSet1.3..........................................12 SolutionsofEquationsofOneVariable 19 ExerciseSet2.1..........................................19 ExerciseSet2.2..........................................22 ExerciseSet2.3..........................................26 ExerciseSet2.4..........................................31 ExerciseSet2.5..........................................34 ExerciseSet2.6..........................................36 InterpolationandPolynomialApproximation 41 ExerciseSet3.1..........................................41 ExerciseSet3.2..........................................47 ExerciseSet3.3..........................................49 ExerciseSet3.4..........................................52 ExerciseSet3.5..........................................56 ExerciseSet3.6..........................................68 NumericalDiﬀerentiationandIntegration 71 ExerciseSet4.1..........................................71 ExerciseSet4.2..........................................78 ExerciseSet4.3..........................................82 ExerciseSet4.4..........................................87 ExerciseSet4.5..........................................91 ExerciseSet4.6..........................................94 ExerciseSet4.7..........................................96 ExerciseSet4.8..........................................99 ExerciseSet4.9..........................................101 iii

TableofContents

iv Initial-ValueProblemsforOrdinaryDiﬀerentialEquations 105 ExerciseSet5.1..........................................105 ExerciseSet5.2..........................................108 ExerciseSet5.3..........................................113 ExerciseSet5.4..........................................119 ExerciseSet5.5..........................................129 ExerciseSet5.6..........................................133 ExerciseSet5.7..........................................144 ExerciseSet5.8..........................................149 ExerciseSet5.9..........................................152 ExerciseSet5.10.........................................159 DirectMethodsforSolvingLinearSystems 169 ExerciseSet6.1..........................................169 ExerciseSet6.2..........................................174 ExerciseSet6.3..........................................180 ExerciseSet6.4..........................................188 ExerciseSet6.1..........................................190 ExerciseSet6.6..........................................194 IterativeTechniquesinMatrixAlgebra 201 ExerciseSet7.1..........................................201 ExerciseSet7.2..........................................206 ExerciseSet7.3..........................................209 ExerciseSet7.4..........................................215 ExerciseSet7.5..........................................219 ExerciseSet7.6..........................................221 ApproximationTheory 231 ExerciseSet8.1..........................................231 ExerciseSet8.2..........................................232 ExerciseSet8.3..........................................236 ExerciseSet8.4..........................................238 ExerciseSet8.5..........................................244 ExerciseSet8.6..........................................247 ApproximatingEigenvalues 251 ExerciseSet9.1..........................................251 ExerciseSet9.2..........................................257 ExerciseSet9.3..........................................262 ExerciseSet9.4..........................................266 ExerciseSet9.5..........................................267 ExerciseSet9.6..........................................273

v NumericalSolutionsofNonlinearSystemsofEquations 281 ExerciseSet10.1.........................................281 ExerciseSet10.2.........................................284 ExerciseSet10.3.........................................287 ExerciseSet10.4.........................................290 ExerciseSet10.5.........................................291 Boundary-ValueProblemsforOrdinaryDiﬀerentialEquations 295 ExerciseSet11.1.........................................295 ExerciseSet11.2.........................................299 ExerciseSet11.3.........................................303 ExerciseSet11.4.........................................308 ExerciseSet11.5.........................................313 NumericalSolutionstoPartialDiﬀerentialEquations 319 ExerciseSet12.1.........................................319 ExerciseSet12.2.........................................323 ExerciseSet12.3.........................................333 ExerciseSet12.4.........................................336

vi Preface

Preface

This Instructor’sManualfortheNinthEditionofNumericalAnalysisbyBurdenand Fairescontainssolutionstoalltheexercisesinthebook.Althoughtheanswersto theoddexercisesarealsointhebackofthetext,wehavefoundthatusersofthe bookappreciatehavingallthesolutionsinonesource.Inaddition,theresultslisted inthisInstructor’sManualoftengobeyondthosegiveninthebackofthebook.For example,wedonotplacethelongsolutionstotheoreticalandappliedexercisesin thebook.Youwillﬁndthemhere.

AStudentStudyGuidefortheNinthEditionofNumericalAnalysisisalsoavailableandthesolutionsgivenintheGuidearegenerallymoredetailedthanthoseinthe Instructor’sManual.Inordertomakeitconvenientforinstructors,wehaveplaced anasterisk(*)inthisManualinfrontofeachexercisewhosesolutionisgiveninthe StudentStudyGuide.Hopefullythiswillhelpwithyourhomeworkassignmentsand testproblems.

WehaveaddedanumberofexercisestothetextthatinvolvetheuseofaComputer AlgebraSystem.WehavechosenMapleasourstandard,becausetheir NumericalAnalysis packageparallelsthealgorithmsinthisbook,butanyofthesesystemscan beused.Inourrecentteachingofthecoursewefoundthatstudentsunderstoodthe conceptsbetterwhentheyworkedthroughthealgorithmsstep-by-step,butletthe computeralgebrasystemdothetediouscomputation.

IthasbeenourpracticetoincludestructuredalgorithmsinourNumericalAnalysis bookforallthetechniquesdiscussedinthetext.Thealgorithmsaregiveninaform thatcanbecodedinanyappropriateprogramminglanguage,bystudentswitheven aminimalamountofprogrammingexpertise.

Atourwebsiteforthebook, http://www.math.ysu.edu/∼faires/Numerical-Analysis/

youwillﬁndallthealgorithmswrittenintheprogramminglanguagesFORTRAN, Pascal,C,Java,aswellasintheComputerAlgebraSystems,Maple,MATLAB, andMathematica.FortheNinthEdition,wehaveaddednewMapleprogramsto reﬂectthechangesintheirsystemandtoincludeportionsoftheir NumericalAnalysis package.

vii

Thewebsitealsocontainsadditionalinformationaboutthebookandwillbeupdatedregularlytoreﬂectanymodiﬁcationsthatmightbemade.Forexample,we willplacethereanyresponsestoquestionsfromusersofthebookconcerninginterpretationsoftheexercisesandappropriateapplicationsofthetechniques.

Wewillhaveasetofpresentationmaterialreadysoonformanyofthemethods inthebook.ThesearebeingconstructedbyProfessorJohnCarrollofDublinCity UniversityusingtheBeamerpackageofLATEX,andwillbeplacedonthewebsite. TheBeamerpackagecreatesPDFﬁlesthataresimilartoPowerPointpresentations butincorporatesmathematicalelementseasilyandcorrectly.Wearequiteexcited aboutthismaterialandexpecttohavemanyofthepresentationsreadybeforethey wouldnormallybecoveredintheFalltermof2010.Ifyousendusane-mailwewill keepyoupostedonourprogress.

WehopeoursupplementpackageprovidesﬂexibilityforinstructorsteachingNumericalAnalysis.Ifyouhaveanysuggestionsforimprovementsthatcanbeincorporatedintofutureeditionsofthebookorthesupplements,wewouldbemostgrateful toreceiveyourcomments.Wecanbemosteasilycontactedbyelectronicmailatthe addresseslistedbelow.

viii Preface

YoungstownStateUniversity RichardL.Burden burden@math.ysu.edu August20,2010 J.DouglasFaires faires@math.ysu.edu

MathematicalPreliminaries

Note:Anasterisk(*)beforeanexerciseindicatesthatthereisasolutionintheStudent StudyGuide.

ExerciseSet1.1,page14

*1.Foreachpart, f ∈ C[a,b]onthegiveninterval.Since f (a)and f (b)areofoppositesign,the IntermediateValueTheoremimpliesthatanumber c existswith f (c)=0.

2.(a)[0, 1]

(b)[0, 1], [4, 5], [ 1, 0]

*(c)[ 2, 2/3], [0, 1], [2, 4]

(d)[ 3, 2], [ 1, 0.5],and[ 0.5, 0]

3.Foreachpart, f ∈ C[a,b], f ′ existson(a,b)and f (a)= f (b)=0.Rolle’sTheoremimplies thatanumber c existsin(a,b)with f ′(c)=0.Forpart(d),wecanuse[a,b]=[ 1, 0]or [a,b]=[0, 2].

4.Themaximumvaluefor |f (x)| isgivenbelow.

*(a)(2ln2)/3 ≈ 0 4620981

(b)0.8

(d)1.582572

*5.For x< 0, f (x) < 2x + k< 0,providedthat x< 1 2 k.Similarly,for x> 0, f (x) > 2x + k> 0, providedthat x> 1 2 k.By Theorem1.11,thereexistsanumber c with f (c)=0.If f (c)=0 and f (c′)=0forsome c′ = c,thenbyTheorem1.7,thereexistsanumber p between c and c′ with f ′(p)=0.However, f ′(x)=3x2 +2 > 0forall x.

6.Suppose p and q arein[a,b]with p = q and f (p)= f (q)=0.BytheMeanValueTheorem, thereexists ξ ∈ (a,b)with f (p) f (q)= f ′(ξ)(p q)

But, f (p) f (q)=0and p = q. So f ′(ξ)=0, contradictingthehypothesis.

7.(a) P2 (x)=0

(b) R2(0 5)=0 125;actualerror=0 125 1

135372on[0, 1]

(c)

(d)Anerrorboundis7

(a) P3 (0 5)=0 312500,f (0 5)=0 346574.Anerrorboundis0 2916,andtheactualerror is0 034074.

(b) |f (x) P3 (x)|≤ 0 2916on[0

(d)Anerrorboundis0 0583,andtheactualerroris4 687 × 10

2 ExerciseSet1.1 (c) P2 (x)=1+3(x 1)+3(x 1)2 (d) R2(0 5)= 0 125;actualerror= 0 125 8. P3(x)=1+ 1 2 x 1 8 x2 + 1 16 x3 x 0 5 0 751 251 5 P3 (x)1.22656251.33105471.55175781.6796875 √x +11 22474491 32287571 51 5811388 |√x +1 P3(x)| 0.00181760.00817900.05175780.0985487 *9.Since P2 (x)= 1 + x and R2(x)= 2eξ(sin ξ +cos ξ) 6 x 3 for some ξ between x and0,wehavethefollowing: (a) P2 (0 5)=1 5and |f (0 5) P2 (0 5)|≤ 0 0932; (b) |f (x) P2 (x)|≤ 1 252; (c) 1 0 f (x) dx ≈ 1 5; (d) | 1 0 f (x) dx 1 0 P2 (x) dx|≤ 1 0 |R2 (x)| dx ≤ 0 313,andtheactualerroris0 122. 10. P2(x)=1 461930+0 617884 x π 6 0 844046 x π 6 2 and R2(x) = 1 3 eξ(sin ξ+cos ξ) x π 6 3 for some ξ between x and π 6 . (a) P2 (0.5)=1.446879and f (0.5)=1.446889.Anerrorboundis1.01 × 10 5,andtheactual erroris1.0 × 10 5 .

|f (x) P2 (x)|≤ 0.

(b)

1 0 P2(x)

1 0 f (x) dx =1

378025

dx =1.376542and

2 1

403 × 10

,andtheactualerroris1 483 × 10 3 11. P3(x)=(x 1)

2 (x 1)3

1 5 0 5 (x 1) ln x

088020

12.(a)

4+6

2 4x3; P3(0 4)= 2

P3 (x)=

x x

016

4) P3(0 4)

4+6x x2 4x3; P4(0 4)= 2 016

(b) |R3 (0 4)|≤ 0 05849; |f (0

013365367 (c) P4 (x)=

4) P4(0 4)

(d) |R4 (0 4)|≤ 0 01366; |f (0

013365367

*14.Firstweneedtoconvertthedegreemeasureforthesinefunctiontoradians.Wehave180

MathematicalPreliminaries 3 13 P4(x) = x + x3 (a) |f (x) P4 (x)|≤ 0 012405 (b) 0 4 0 P4 (x) dx =0 0864, 0 4 0 xex 2 dx =0 086755 (c)8 27 × 10 4 (d) P ′ 4 (0 2)=1 12,f ′(0 2)=1 124076.Theactualerroris4 076 × 10 3

◦ = π radians,so1◦ = π 180 radians.Since, f (x)=sin x,f ′(x)=cos x,f ′′(x)= sin x, and f ′′′(x)= cos x, wehave f (0)=0, f ′(0)=1, and f ′′(0)=0. Theapproximationsin x ≈ x isgivenby f (x) ≈ P2 (x)= x, and R2(x)= cos ξ 3! x3 . Ifwe usethebound | cos ξ|≤ 1,then sin π 180 π 180 = R2 π 180 = cos ξ 3! π 180 3 ≤ 8 86 × 10 7 15.Since42◦ =7π/30radians,use x0 = π/4.Then Rn 7π 30 ≤ π 4 7π 30 n+1 (n +1)! < (0 053)n+1 (n +1)! For |Rn( 7π 30 )| < 10 6 , it suﬃcestotake n =3.To7digits, cos42◦ =0 7431448and P3(42◦)= P3( 7π 30 )= 0 7431446, sotheactualerroris2 × 10 7 *16.(a) P3 (x)= 1 3 x + 1 6 x 2 + 23 648 x3 (b) Wehave f (4)(x)= 119 1296 ex/2 sin x 3 + 5 54 ex/2 cos x 3 , so f (4)(x) ≤ f (4)(0 60473891) ≤ 0 09787176, for0 ≤ x ≤ 1, and |f (x) P3 (x)|≤ f (4)(ξ) 4! |x|4 ≤ 0 09787176 24 (1)4 = 0.004077990.

arethesameat x0

(b) P2 (x)=3+4(x 1)+3(x 1)2

23.(a)Theassumptionisthat f (xi)=0foreach i =0, 1,...,n.ApplyingRolle’sTheorem oneachontheintervals[xi,xi+1 ]impliesthatforeach i =0, 1,...,n 1thereexistsa number zi with f ′(zi)=0.Inaddition,wehave

<z0 <x1 <z1 < <zn 1 <xn ≤ b.

(b)Applythelogicinpart(a)tothefunction g(x)= f ′(x)withthenumberofzerosof g in [a,b]reducedby1.Thisimpliesthatnumbers wi,for i =0, 1,...,n 2existwith

g ′(wi)= f ′′(wi)=0, and a<z0 <w0 <z1 <w1 < wn 2 <zn 1 <b.

(c)Continuingbyinductionfollowingthelogicinparts(a)and(b)provides n+1 j distinct zerosof f (j) in[a,b].

(d)Theconclusionofthetheoremfollowsfrompart(c)when j = n,forinthiscasethere willbe(atleast)(n +1) n =1zeroin[a,b].

*24.Firstobservethatfor f (x)= x sin x wehave f ′(x)=1 cos x ≥ 0,because 1 ≤ cos x ≤ 1 forallvaluesof x.Also,thestatementclearlyholdswhen |x|≥ π,because | sin x|≤ 1.

(a)Theobservationimpliesthat f (x)isnon-decreasingforallvaluesof x,andinparticular that f (x) >f (0)=0when x> 0.Hencefor x ≥ 0,wehave x ≥ sin x,andwhen 0 ≤ x ≤ π, | sin x| =sin x ≤ x = |x|

25.Since R2 (1)= 1 6 eξ,forsome ξ in(0, 1),wehave |

andobtaintheresult.

4 ExerciseSet1.1 17.(a) P3 (x)=ln(3)+ 2 3 (x 1) + 1 9 (x 1)2 10 81 (x 1)3 (b)max0≤x≤1 |f (x) P3 (x)| = |f (0) P3 (0)| =0.02663366 (c) ˜ P3 (x)=ln(2)+ 1 2 x2 (d) max0≤x≤1 |f (x) P3 (x)| = |f (1) P3 (1)| =0 09453489 (e) P3 (0)approximates f (0)betterthan ˜ P3 (1)approximates f (1). 18. Pn(x)= n k=0 xk,n ≥ 19 19. Pn(x)= n k=0 1 k! x k ,n ≥ 7 20.For n odd, Pn(x)= x 1 3 x3 + 1 5 x5 + ·· · + 1 n ( 1)(n 1)/2xn.For n even, Pn(x)= Pn 1(x). 21.Aboundforthemaximumerroris0 0026. 22.(a) P (k) n (x0 )= f (k)(x0 )for k =0, 1,...,n. Theshapesof Pn and f

a ≤ x0

| = 1 6 |1 eξ| ≤ 1 6 (e 1)

e t2 = ∞ k=0 ( 1)kt2k k! tointegrate 2 √π x 0 e t2 dt,

E R

(1)

26.(a)Usetheseries

(d)0

MathematicalPreliminaries 5

Wehave 2 √π e x 2 ∞ k=0 2kx2k+1 1 3 (2k + 1) = 2 √π 1 x 2 + 1 2 x 4 1 6 x7 + 1 24 x8 + ·· · x + 2 3 x3 + 4 15 x5 + 8 105 x7 + 16 945 x9 + = 2 √π x 1 3 x 3 + 1 10 x 5 1 42 x 7 + 1 216 x 9 + ·· · = erf(x)

8427008

(b)

8427069

n andpositive x wehave thebound erf(x) 2 √π n k=0 ( 1)kx2k+1 (2k +1)k! < x2n+3 (2n +3)(n + 1)! . Wehavenosuchboundforthepositivetermseriesinpart(b). 27.(a)Let x0 beanynumberin[a,b] Given ǫ> 0, let δ = ǫ/L. If |x x0 | <δ and a ≤ x ≤ b, then |f (x) f (x0 )|≤ L|x x0| <ǫ. (b)UsingtheMeanValueTheorem,wehave |f (x2) f (x1)| = |f ′(ξ)||x2 x1 |, forsome ξ between x1 and x2 ,so |f (x2 ) f (x1 )|≤ L|x2 x1 | (c)Oneexampleis f (x)= x1/3 on[0, 1]. *28.(a)Thenumber 1 2 (f (x1 )+ f (x2 )) istheaverageof f (x1 )and f (x2 ),soitliesbetweenthese twovaluesof

f (ξ)= 1 2 (f (x1 )+ f (x2 )) = 1 2 f (x1)+ 1 2 f (x2 )

m =min

,f (x2 )} and M =max{f (x1),f (x2 )} Then m ≤ f (x1) ≤ M and m ≤ f

≤ M, so c1m ≤ c1f (x1 ) ≤ c1 M and c2m ≤ c2f (x2 ) ≤ c2M. Thus (c1 + c2)m ≤ c1f (x1 )+ c2f (x2) ≤ (c1 + c2 )M and m ≤ c1f (x1 )+ c2f (x2) c1 + c2 ≤ M.

theIntermediateValueTheorem1.11appliedtotheintervalwithendpoints

ξ between x1

x2 forwhich f (ξ)= c1f (x1)+ c2f (x2 ) c1 + c2 .

(e)Theseriesinpart(a)isalternating,soforanypositiveinteger

f .BytheIntermediateValueTheorem1.11thereexistanumber ξ between x1 and x2 with

(b) Let

{f (

)

(x2)

and

,thereexistsanumber

and

(c)Let f (x)= x2 +1, x1 =0, x2 =1, c1 =2,and c2 = 1.Thenforallvaluesof x,

)

29.(a)Since f iscontinuousat p and f (p) =0,thereexistsa δ> 0with

f (x) f (p)| < |f (p)| 2 , for |x p| <δ and a<x<b.Werestrict δ sothat[p δ,p + δ]isasubsetof[a,b].

Thus,for x ∈ [p δ,p + δ],wehave x ∈ [a,b].So

If f (p) > 0, then

If f (p) < 0, then |f (p)| = f (p),and

Ineithercase, f (x) =0,for x ∈ [p δ,p + δ].

(b)Since f iscontinuousat p and f (p)=0,thereexistsa δ> 0with

f (x) f (p)| <k, for |x p| <δ and a<x<b.

Werestrict δ sothat[p δ,p + δ]isasubsetof[a,b].Thus,for x ∈ [p δ,p + δ],wehave |f (x)| = |f (x) f (p)| <k.

ExerciseSet1.2,page28

1.Wehave

6 ExerciseSet1.2

f (x

> 0but c1f (x1)+ c2f (x2 )

1 + c2 = 2(1) 1(2) 2 1 =0

|f (p)| 2 <f (x) f (p) < |f (p)| 2 and f (p) |f (p)|

<f (x) < f (p) + |f (p)|

2 > 0, so f (x) > f (p

(p) |f (p)|

= f (p)

) |f (p)| 2 > 0.

f (p)

2 = f (p) f (p) 2 = f

p) 2 < 0.

(x) <f (p)+ |

(

AbsoluteerrorRelativeerror (a)0.0012644 025 × 10 4 (b)7 346 × 10 6 2 338 × 10 6 (c)2 818 × 10 4 1 037 × 10 4 (d)2.136 × 10 4 1.510 × 10 4 (e)2.647 × 101 1.202 × 10 3 (f)1.454 × 101 1.050 × 10 2 (g)4201 042 × 10 2 (h)3 343 × 103 9 213 × 10 3

2. Thelargestintervalsare:

(a)(3 1412784, 3 1419068)

(b)(2 7180100, 2 7185536)

*(c)(1 4140721, 1 4143549)

(d)(1 9127398, 1 9131224)

3.Thelargestintervalsare

(a)(149.85,150.15)

(b)(899.1,900.9)

(c)(1498.5,1501.5)

(d)(89.91,90.09)

4.Thecalculationsandtheirerrorsare:

MathematicalPreliminaries 7

× 10 3 (c)(i)139/660(ii)0.211(iii)0.210(iv)2

3,3 × 10 3 (d)(i)301/660(ii)0.455(iii)0.456(iv)2 × 10 3,1 × 10 4

ApproximationAbsoluteerrorRelativeerror (a)134 0.0795 90 × 10 4 (b)133 0.4993 77 × 10 3 (c)2.00 0.327 0.195 (d)1.67 0.0031 79 × 10 3 *(e)1.80 0.154 0.0786 (f) 15.10.05463.60 × 10 3 (g)0.2862.86 × 10 4 10 3 (h)0.00 0.0215 1.00 6.Wehave ApproximationAbsoluteerrorRelativeerror (a)133.9 0.0211.568 × 10 4 (b)132.5 0.0017 55 × 10 6 (c)1.700 0.0270.01614 (d)1.673 0 0 (e)1.9860.032460.01662 (f) 15 160.0053773 548 × 10 4 (g)0.28571 429 × 10 5 5 × 10 5 (h) 0.017000.00450.2092

(a)(i)17/15(ii)1.13(iii)1.13(iv)both3 × 10 3 (b)(i)4/15(ii)0.266(iii)0.266(iv)both2 5

× 10

5.Wehave

8 ExerciseSet1.2

ApproximationAbsoluteerrorRelativeerror (a)133 0.9216.88 × 10 3 (b)132 0.5013.78 × 10 3 (c)1.00 0.673 0.402 (d)1.67 0.0031 79 × 10 3 *(e)3.55 1.60 0.817 (f) 15 20.04540.00299 (g)0.2840.001710.00600 (h)0 0.02150 1

ApproximationAbsoluteerrorRelativeerror (a)133.9 0.0211.568 × 10 4 (b)132.5 0.0017 55 × 10 6 (c)1.600 0.0730.04363 (d)1.673 0 0 (e)1.9830.029450.01508 (f) 15 150.0046223 050 × 10 4 (g)0.28552 143 × 10 4 7 5 × 10 4 (h) 0 017000.00450.2092

ApproximationAbsoluteerrorRelativeerror *(a)3.145576133.983 × 10 3 1.268 × 10 3 (b)3.141621032.838 × 10 5 9.032 × 10 6 10.Wehave ApproximationAbsoluteerrorRelativeerror (a)2.71666670.00161525 9418 × 10 4 (b)2.7182818012.73 ×10 8 1 00 × 10 8

7.Wehave

8.Wehave

9.Wehave

MathematicalPreliminaries 9 11 (a)Wehave lim x→0 x cos x sin x x sin x = lim x→0 x sin x 1 cos x = lim x→0 sin x x cos x sin x =lim x→0 2 cos x + x sin x cos x = 2 (b) f (0.1) ≈−1.941 (c) x(1 1 2 x2 ) (x 1 6 x3) x (x 1 6 x3 ) = 2 (d) Therelativeerrorinpart(b)is0 029.Therelativeerrorinpart(c)is0 00050. 12.(a)lim x→0 ex e x x =lim x→0 ex + e x 1 =2 (b) f (0.1) ≈ 2.05 (c) 1 x 1 + x + 1 2 x 2 + 1 6 x3 1 x + 1 2 x 2 1 6 x3 = 1 x 2x + 1 3 x3 =2 + 1 3 x 2 ; usingthree-digitroundingarithmeticand x =0 1,weobtain2 00. (d)Therelativeerrorinpart(b)is=0 0233.Therelativeerrorinpart(c)is=0 00166. 13.Wehave x1 AbsoluteerrorRelativeerror x2 AbsoluteerrorRelativeerror (a)92.260.015421 672 × 10 4 0.0054196 273 × 10 7 1 157 × 10 4 (b)0.0054211 264 × 10 6 2 333 × 10 4 92 264 580 × 10 3 4 965 × 10 5 (c)10.986 875 × 10 3 6 257 × 10 4 0.0011497 566 × 10 8 6 584 × 10 5 (d) 0 0011497 566 × 10 8 6 584 × 10 5 10 986 875 × 10 3 6 257 × 10 4 14.Wehave Approximationfor x1 AbsoluteerrorRelativeerror (a)92.24 0.0045804.965 × 10 5 (b)0.0054172.736 × 10 6 5.048 × 10 4 (c)10.98 6 875 × 10 3 6 257 × 10 4 (d) 0 0011497 566 × 10 8 6 584 × 10 5 Approximationfor x2 AbsoluteerrorRelativeerror (a)0 0054182 373 × 10 6 4 377 × 10 4 (b) 92 255 420 × 10 3 5 875 × 10 5 (c)0 0011497 566 × 10 8 6 584 × 10 5 (d) 10 986 875 × 10 3 6 257 × 10 4

15.Themachinenumbersareequivalentto

(a)3224

(b) 3224

*(c)1.32421875

(d)1.3242187500000002220446049250313080847263336181640625

16.(a)NextLargest:3224 00000000000045474735088646411895751953125; NextSmallest:3223.99999999999954525264911353588104248046875

(b)NextLargest: 3224.00000000000045474735088646411895751953125; NextSmallest: 3223 99999999999954525264911353588104248046875

*(c)NextLargest:1.3242187500000002220446049250313080847263336181640625; NextSmallest:1 3242187499999997779553950749686919152736663818359375

(d)NextLargest:1 324218750000000444089209850062616169452667236328125; NextSmallest:1 32421875

17.(b)Theﬁrstformulagives 0 00658,andthesecondformulagives 0 0100.Thetruethreedigitvalueis 0 0116

18.(a) 1 82

(b)7.09 × 10 3

(c)Theformulain(b)ismoreaccuratesincesubtractionisnotinvolved.

19.Theapproximatesolutionstothesystemsare

10 ExerciseSet1.2

(a) x =2 451, y = 1 635 (b) x =507 7, y =82 00 20.(a) x =2.460 y = 1.634 (b) x =477.0 y =76.93

f (x)=(((1 01ex 4 62)ex 3 11)ex +12 2)ex 1 99 (b) 6 79 (c) 7 07

|− 7 61 ( 6 71)| =0 82and |− 7 61 ( 7 07)| =0 54 Nestingissigniﬁcantlybettersincetherelativeerrorsare 0 82 7 61 = 0.108and 0 54 7 61 = 0.071,

375 ≤ Volume ≤ 86 625and71 5 ≤ SurfaceArea ≤ 119 5.

n =77

*21.(a)Innestedform,wehave

(d)Theabsoluteerrorsare

22.Wehave39

23.(a)

When P isdoubledand V ishalved,1 99 ≤ P ≤ 2 01and0 0497 ≤ V ≤ 0 0503sothat 286.61◦ ≤ T ≤ 293.72◦ . Notethat19◦C=292.16K.Thelaboratoryﬁguresarewithinan acceptablerange.

MathematicalPreliminaries 11 (b) n =35 *24.When dk+1 < 5, y fl(y) y = 0.dk+1 × 10n k 0.d1 × 10n ≤ 0 5 × 10 k 0 1 = 0.5 × 10 k+1 . When dk+1 > 5, y fl(y) y = (1 0.dk+1 ) × 10n k 0.d1 × 10n < (1 0 5) × 10 k 0 1 = 0.5 × 10 k+1 . 25.(a) m =17 (b)Wehave m k = m! k!(m k)! = m(m 1) · ·· (m k 1)(m k)! k!(m k)! = m k m 1 k 1 m k 1 1 (c) m = 181707 (d)2,597,000;actualerror1960;relativeerror7 541 × 10 4 26.(a)Theactualerroris |f ′(ξ)ǫ|,andtherelativeerroris |f ′(ξ)ǫ|·|f (x0 )| 1,wherethenumber ξ isbetween x0 and x0 + ǫ (b)(i)1 4 × 10 5;5 1 × 10 6 (ii)2 7 × 10 6;3 2 × 10 6 (c)(i)1 2;5 1 × 10 5 (ii)4 2 × 10 5;7 8 × 10 5

(e)0.0065

(h)

*28.Since0

≤ P ≤ 1

≤ V ≤ 0 1005,0 082055 ≤ R ≤ 0 082065,and0

N ≤

Notethat15

C=288

27.(a)124.03 (b)124.03 (c) 124.03 (d) 124.03

(f)0.0065 (g) 0.0065

0 0065

995

005,0 0995

004195 ≤

0 004205,wehave287 61◦ ≤ T ≤ 293 42◦

◦

16K.

Theactualvalueis1.549.Significantround-offerroroccursmuchearlierinthefirst method.

(b)Thefollowingalgorithmwillsumtheseries

INPUT N ; x1,x2 ,...,xN OUTPUT SUM

STEP1 Set SUM =0 STEP2 For j =1,...,N set i

STEP3 OUTPUT(SUM ); STOP.

2.Wehave

ExerciseSet1.3

1 1 + 1 4 .. . + 1 100 =1.53; 1 100 + 1 81 + .. . + 1 1 =1.54.

ExerciseSet1.3,page39 1.(a)

N i=1 xi inthereverseorder.

N j

SUM = SUM + xi

ApproximationAbsoluteErrorRelativeError (a)2.7153 282 × 10 3 1 207 × 10 3 (b)2.7162 282 × 10 3 8 394 × 10 4 (c)2.7162 282 × 10 3 8 394 × 10 4 (d)2.7182 818 × 10 4 1 037 × 10 4

(b)

4.4terms *5.3terms 6.(a) O 1 n (b) O 1 n2 (c) O 1 n2 (d) O 1 n

Theratesofconvergenceare: (a) O(h2 ) (b) O(h) (c) O(h2 ) (d) O(h)

n(n

n +2)(n 1)

2additions. Numerical Analysis 9th Edition Burden Solutions Manual Full Download: http://testbanktip.com/download/numerical-analysis-9th-edition-burden-solutions-manual/ Download all pages and all chapters at: TestBankTip.com

*3.(a)2000terms

20,000,000,000terms

*8.(a)

+1)/2multiplications;(