COMPUTER SCIENCE 349A SAMPLE EXAM QUESTIONS WITH SOLUTIONS PARTS 1, 2

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1 COMPUTE SCIENCE 49A SAMPLE EXAM QUESTIONS WITH SOLUTIONS PATS, PAT.. a Dene he erm ll-ondoned problem. b Gve an eample o a polynomal ha has ll-ondoned zeros.. Consder evaluaon o anh, where e e anh. e e I s o be evaluaed n loang-pon arhme e.g., k 4 demal dg, dealzed, roundng loang-pon, or eah o he ollowng ranges o values o, spey wheher he ompued loang-pon resul wll be aurae or naurae. a s large and posve or eample, > 4 k 4 b s lose o or eample,. k 4 s large and negave or eample, < 4 k 4. Consder sn h sn g h, h h where he argumens or sn are n radans. When h s lose o, evaluaon o g h s naurae n loang-pon arhme. In a and d below, use 4 demal dg, dealzed, roundng loang-pon arhme. I s a loang-pon number, assume ha l sn s deermned by roundng he ea value o sn o 4 sgnan dgs. a Evaluae l g h or h. 5. Noe ha sn..8488l, sn l and sn.8447l. b Taylor's Theorem an be epressed n wo equvalen orms: gven any ed value,

2 L!! or, usng a hange o varable replang by ndependen varable, h, so ha h s he h h h h L.!! Usng he laer orm o Taylor's Theorem whou he remander erm, deermne he quadra n h Taylor polynomal appromaon o sn h. Noe: leave your answer n erms o os and sn; do no evaluae hese numerally. Use he Taylor polynomal appromaon rom b o oban a polynomal appromaon, say p h, o g h. d Show ha p h s muh beer han g h or loang-pon evaluaon when h s lose o by evaluang l p.5. Noe ha sn.8447l and os.54l..4 I a, b,, d, e, have known values, hen a b d y e s a sysem o lnear equaons n he unknowns and y. I ad b, hen he soluon s de b and ad b a e y. ad b Consder he lnear sysem y.8 Show ha he problem o ompung he soluon s ll-ondoned. y

3 .5 a For wha values o he real varable, where >, s he ollowng epresson subje o subrave anellaon ha wll produe a very naurae resul n erms o relave error usng loang-pon arhme?, where >. b How should be evaluaed n loang-pon arhme n order o avod he subrave anellaon n a?.6 Le sn e,. Noe: s n radans or he sne unon. a In he ollowng, use 4 demal dg, dealzed, hoppng loang-pon arhme. I w s a loang-pon number, ompue l e w and lsn w by hoppng w he ea value o e and sn w, respevely, o 4 sgnan dgs. For he sne unon, w s n radans. Evaluae l.. Noe ha sn..69l and. e.88l. b To 4 sgnan dgs, he ea value o. s. 546, so he ompuaon n a s naurae. In order o oban a beer ormula or appromang when s lose o, use he Taylor polynomal appromaons or e and sn boh epanded abou n order o oban a quadra polynomal appromaon or. Noe: you know hese requred Taylor epansons, s no neessary o show her dervaons. Use he polynomal appromaon or rom b whh s aurae when s lose o o show ha he ompuaon o l. n a s unsable. Noe: onsder he perurbed problem wh ˆ. ε, where ε. s small..7 a Use 4 demal dg, dealzed, hoppng loang-pon arhme n he 4 ollowng. I w s a loang-pon number, appromae l w / by hoppng he ea / 4 value o w o 4 sgnan demal dgs. The evaluaon o

4 / 4 g s naurae n loang-pon arhme when s appromaely equal o. Very hs 4 by evaluang l g.5. Noe ha he ea value o.5 / s.79l. Usng real arhme, he ea value o g.5 s.48659l. b Deermne he seond order n Taylor polynomal appromaon or / 4 epanded abou. Inlude he remander erm. Leave hs polynomal n erms o epressons nvolvng powers o. Do no mulply ou hese powers o. Subsue he polynomal appromaon rom b, whou he remander erm, no he ormula or g, and smply n order o oban a polynomal appromaon or g. Ths polynomal appromaon s aurae usng loang-pon arhme when s lose o. Noe: leave hs polynomal n erms o epressons nvolvng. d Deermne a good upper bound or he runaon error o he Taylor polynomal appromaon n b when by boundng he remander erm. Gve a leas 4 orre sgnan dgs..8 Usng dealzed, roundng loang-pon arhme base, preson k 4, he evaluaon o l l w* l y * z or w y. z gves a resul o., whereas he ea value s.9. The relave error o hs ompued resul s 4%. Usng he denon o sably gven n lass, show by usng only a perurbaon o y ha he above loang-pon ompuaon s sable. PAT.. Le denoe any posve number. a Apply Newon s mehod o n order o deermne an erave ormula or ompung /.

5 b For arbrary >, le be he nal appromaon o {,, L} /, le, be he sequene o ompued appromaons o / usng he erave ormula rom a, and le e, or,,,, K Show usng algebra ha e e. e Noe: rom hs, ollows ha lm lm e ormula n a s quadraally onvergen., provng ha he erave. a I Newon s mehod s used o ompue an appromaon o a zero o 5 4 P usng he nal appromaon, onvergene s obaned o he zero o P. I hs ompuaon s arred ou, wha s he order o onvergene? Jusy your answer. b Gve or MATLAB saemens ha ould be used o ompue all o he zeros o he polynomal usng he MATLAB unon roos. 5 4 P a Le denoe any posve number. Apply Newon s mehod o n order o deermne an erave ormula or ompung. Smply he ormula so ha s n he orm

6 g h where g and h are smple polynomals n. b Consder he ase. Gven some nal value, he erave ormula n a onverges o, wha wll be he order o onvergene? Very brely jusy your answer, reerrng o any resuls rom your lass noes or he ebook..4 a Wh regard o an algorhm or ompung a roo o, wha s he denon o order o onvergene? b The ollowng sequene o values s onvergng o a roo a Wha s he order o onvergene? Could he ompued appromaons n b have been ompued usng Newon s mehod? Jusy your answer..5 a Show how o evaluae usng nesed mulplaon. P a a a L n a n b Gve pseudoode or he ompuaon n a.

7 .6 A good appromaon o one o he zeros o 4 P s. 64. I s used as an appromaon o a zero o P, use synhe dvson ha s, Horner's algorhm o deermne he assoaed delaed polynomal. Show all o your alulaons. Noe: do no do any ompuaons wh Newon s mehod..7 a Fll n he 7 blanks n he ollowng MATLAB ode so ha he unon M- les.m and sean.m ould be used o ompue one zero o sn e usng he Sean mehod. The unon M-le sean.m has he ollowng npu parameers: nal appromaons and mamum number o eraons N error olerane ol ess relave error and prns eah suessve ompued appromaon o a zero o. I he unon doesn onverge whn N eraons, hen an error message s prned. The M-le.m : unon y y ; The M-le sean.m : unon roo sean,, N, ol ; q ; q ; whle < N roo ; prn' %g',,prn' appromaon %8.\n',roo < ol reurn end ; ; q ; ; q ; end prn'mehod aled o onverge n %g',n,prn' eraons\n' b I he above MATLAB M-les.m and sean.m are used o ompue one zero o

8 sn e 6 wh nal appromaons and, N and ol, hen a ompued appromaon o roo s obaned. Wha s he order o onvergene or hs ompuaon o hs zero o? Brely jusy your answer usng resuls gven n lass..8 Use Taylor s Theorem o derve Newon s mehod or ompung a roo o..9 The volume o lqud n a spheral ank o radus lled o a deph h s V h / π h. Gve one MATLAB saemen ha uses he MATLAB bul-n unon zero o ompue he deph o whh a ank o radus mus be lled so ha he volume s. In hs saemen, spey an approprae nerval ha onans he answer and ha an be used wh zero. SOLUTIONS PAT.. a A problem s ll-ondoned s ea soluon an hange grealy wh a small hange n he daa denng he problem. 5 b P. Polynomal roos wh mulply greaer han one are llondoned.. a naurae, sne anh s appromaely equal o. b aurae, sne anh s appromaely equal o alhough l anh may no be oo aurae aurae, sne anh s appromaely equal o.. a l h lsn h lsn l.5 l.4 lsn h sn l.8447l.845 lsn h sn / h l.5.4 l.8465l.846 l l./.5 or. or l.5989l.598.

9 Noe: ea value o g h s and he relave error n he above.598 ompued appromaon s. or % b d h sn h sn hos sn h sn hos sn sn p h h h os sn los l.54l.54 l h / l.5/.755 or.755 lsn l.8447l.845 l h / sn l l los h / sn l l whh has all 4 sgnan dgs orre. or Almos any perurbaon o he 6 onsans n he daa, wll do: or eample, ˆ yˆ.8 has ea soluon..96 whereas he gven sysem has soluon..5 a For suenly large and posve. Noe ha here s no problem when sne hen l, whh s aurae. b

10 .6 a lsn l e lsn e lsn e l l l l.6. l.74.7 l.7.7 or.7 l.59.5 l l.7 /.5 l b 4 sn, e gven problem ompued soluon..469 perurbed problem sn ˆ e ˆ ˆ ˆ. ε ˆ And ˆ ˆ ˆ ˆ s lose o 4. ε. ε ε O ε ε.546 or all ε suh ha. s small..7 a Sne hs s no lose o. 469 or all small ε, he ompuaon s unsable.

11 l l / 4 / 4 l l g l.79l. l... or. l.5..5 or.5 l./.5. or. b Thus d / 4 4 / / 4 / 4 / 4 /6! 7 8 ξ! 4 [ ξ ] / / 4 4 g [ ξ ] / ma [ ξ ] ξ.6.95 / 4 / 4.6, ma.95.6 whh s appro..6.8 To show sably, nd a value ε or whh he ea value o ε s appro. equal o. By onnuy, any suh value mus be appro. equal o he value ε suh ha ε Solvng or ε gves ε

12 Thus, or eample, y. s perurbed o y ˆ y ε.. 5 or you ould use ε. 49, hen he ea value o w yˆ z.5, ε whh s very lose o.. And sne. s sable. PAT..5. s small, he ompuaon n a. a / / / or b e / / rom a / / e n. a 4 P 5 8 8, so P Thus, he zero a p has mulply m, whh mples ha Newon s mehod has lnear onvergene ha s, he order o onvergene s. b or roos [ ] p [ ] ; roos p

13 . a b s a smple zero, so a he roo, / ha s, he mulply o he roo s and hus Newon s mehod onverges quadraally he order o onvergene s α.4 a The order o onvergene s α here es onsans > λ and α suh ha α λ lm b By nspeon, he order o onvergene s lnear onvergene. Yes, hese appromaons ould have been ompued wh Newon s mehod he roo has mulply m, sne n hs ase, he order o onvergene o Newon s mehod s only..5 a a n a n a a a P L b b a b n n k a b k k k n n,,,, or K

14 .6 b b 4 b a b b a b b a b a b The delaed polynomal s a y sn ep- ; roo q * /q q ; abs - / roo < ol ; q q ; roo ; q roo ; b Order o onvergene s.68 sne hs s he order o onvergene o he Sean mehod or a smple zero mulply. The mulply s beause roo, roo sne learly os roo > and e >..8 The lnear n Taylor Theorem epanson or epanded abou s ξ,! or some value ξ beween and. I we ake where s a roo o he equaon, hen rom above ξ!, sne s a zero o. On droppng he remander erm, hs gves,

15 whh mples ha. Ths suggess ompung, whh s he rs sep o Newon's mehod, and hen onnue erang wh..9 zero p * ^ * , [ ]