Relating to, connected or concerned with, quality or qualities. Now usually in implied or expressed opposition to quantitative.

Transcription

1 . Mathematical bacgou I you chose poessio, it will be ecessay to mae egieeig esig ecisios. Whe it comes to pogammig, you will ote have a selectio o possible algoithms o ata stuctues; howeve, whe you compae two algoithms, it is ot possible to simply say that algoithm A is aste tha algoithm B. Such a compaiso qualitatively compaes two values, e.g., a > b o b > a. Fom the Oxo Eglish Dictioay (OED): qualitative, a. Relatig to, coecte o cocee with, quality o qualities. Now usually i implie o expesse oppositio to quatitative. Egieeig esig ecisios must be mae by choosig a metic, measuig the possible alteatives usig that metic, a compaig the two measuemets, o quatities. Thus, we equie a quatitative compaiso: quatitative, a. Relatig to, cocee with, quatity o its measuemet; ascetaiig o expessig quatity. We will be able to quatitatively compae algoithms a ata stuctues by looig at elatively ates o gowth i time equiemets a memoy equiemets as the poblem size iceases. To o this, we will use mathematics a elemetay calculus; howeve, we will also equie a ew othe tools... Floo a ceilig uctios The loo uctio maps ay eal umbe x oto the geatest itege less tha o equal to x. Fo example, You ca itepet this as ouig towa egative iiity. The ceilig uctio maps x oto the least itege geate tha o equal to x. Fo example, The ceilig uctio ca be itepete as ouig towa positive iiity. The cmath libay implemets these two uctios as ouble loo( ouble ); ouble ceil( ouble ); The justiicatio o the etu value beig ouble a ot a log is that a ouble-pecisio loatigpoit umbe ca be as lage as 4 while the lagest umbe that ca be epesete by log is 63. Page o

2 .. L Hôpital s ule I you ae attemptig to etemie the limit lim g but both eivatives lim a lim g, the we ca etemie the limit by taig the limit o the lim lim. g g This ca be epeate as ecessay. Note, that i this couse, the th eivative will always be witte as () (x), eve i =. Thus, we will pee () (x) a () (x) to (x) a (x)...3 Logaithms Recall that i = e m, we eie m = l(). It is also always tue that e l() = ; howeve, o l(e ) =, it must also be tue that is eal. Because expoetials gow aste tha ay polyomial, that is, e lim o ay >, it ollows that the ivese, the logaithm, must gow moe slowly tha ay polyomial, which we ca see by applyig l Hopital s ule oce: l / lim lim lim. Thus, the logaithm l() gows moe slowly tha o eve.. Aothe popety you may have see but i ot appeciate is that all logaithms ae scala multiples o each othe. Recall that log b b l, l a, theeoe, lg() = log () gows at a acto o /l().447 ( 3/9) aste tha l(). Fo example, Figue shows a plot o log () = lg(), l(), a log (). Page o

3 A ew otes: Figue. A plot o lg(), l(), a log ().. The base- logaithm, log (), is witte as lg(),. I most libaies, the atual logaithm, l(), has the sigatue ouble log( ouble );, a 3. The commo logaithm, log (), has the sigatue ouble log( ouble );. A popety o the logaithm that we will epeately use is: log b (m) log () b = m Fially, it is a useul popety that = 4 is close to : a theeoe lg( ) = lg( ) = lg( 3 ) lg( 6 ) lg( 9 ) 3 lg( ) 4 ilo mega giga tea Page 3 o

4 ..4 Aithmetic seies A aithmetic seies iceases by a costat (usually ) a vey ote we will use Poo : A the seies twice: ( + ) + ( + ) + ( + ) ( + ) + ( + ) + ( + ) = ( + ) Havig oe so, we ivie the esult by. Poo : Usig iuctio, we obseve the statemet is tue o = : a. Next, we assume that the statemet is tue o a abitay. Usig this assumptio, we must ow show that the statemet is also tue o + : 3 Theeoe, the statemet is tue o = a the tuth o the statemet o implies the tuth o the statemet o +. Theeoe, by the pocess o mathematical iuctio, this statemet must be tue o all. Page 4 o

5 ..5 Othe polyomial seies It is possible to epeat the above pocess to pove that the omulas, 6 3 4, a ae all tue; howeve, such pecisio is ote uecessay. Istea, we ca usually ely o easoable appoximatios: I geeal, we ca say that appoximatig the sum by the coespoig itegal:. This is actually a vey goo appoximatio: we i it by x x x. x The actual sum is the aea ue the blue piecewise costat uctio i Figue while the itegal is the aea o the e egio. Figue. The appoximatio o by x x. Page 5 o

6 The accumulatig eo is show i Figue 3. Figue 3. The accumulatig eo i appoximatig a sum by a itegal. Fotuately, otice that the eo ca be shite ito a sigle colum o with a height, as is show i Figue 4. Figue 4. Shitig the eos i Figue 3. The aea o the etie ectagle i Figue 4 is a the pat coloue blue is the eo. Now, by ispectio, the blue egio is hal o moe o the etie ectagle, a theeoe, we have the iequality. Page 6 o

7 Page 7 o Thus, the eo gows o aste tha but the magitue o ou appoximatio is a theeoe the elative eo must go to zeo as becomes lage. Asie: you may ote om the eo aalysis that it might be easoable to use the ollowig bette appoximatio:. This is a sigiicatly bette appoximatio (egaless o the value o ); howeve, such accuacy will ot be ecessay o this class. You ae expecte to memoize, i othig else om this sectio, the omula a appoximatio a...6 Geometic seies The value o a iite geometic seies ca be evaluate usig the omula. I < (whethe is eal o complex), it is also tue that. Poo : Multiply both sies by :.

8 a oig a chage-o-iex, we get: Poo : By iuctio, we ote the omula is coect o = :. Assume that the omula is coect o a abitay. I that case, we have Theeoe, the statemet is tue o = a the tuth o the statemet o implies the tuth o the statemet o +. Theeoe, by the pocess o mathematical iuctio, this statemet must be tue o all. Commo geometic seies that we will see i this class ae whe = ½ a = : Page 8 o

9 ..7 Recuece elatios I the tems i a sequece ae give by a explicit omula, it is possible to calculate ay tem i the sequece. Fo example, i x the x 4 =.5. How much time, howeve, is it to peom a biay seach o a sote list? The algoithm is:. Chec the mile ety o a aay i that is the object we e looig o, etu tue,. I the object is less tha the mile ety, o a biay seach o the let hal, othewise 3. The object must be geate tha the mile ety, so o a biay seach o the ight hal. I the hal that we ae seachig is eve empty, we etu alse. This is a ecusive esciptio o the biay seach algoithm. We ca escibe a ecusive algoithm usig a ecuece elatio; o example, the ollowig ae all ecuece elatios the last beig the ecuece elatio escibig the umbe o calls to a biay seach: x x x x x x x x x x x x x / Beoe we go o, we will itouce a alteative otatio. I geeal, we will use ecuece elatios to escibe the u times o memoy usage o ata stuctues a algoithms. Cosequetly, we usually eote these quatities as T() a Mem(). Theeoe, ou ecuece elatios will usually be witte usig uctioal otatio: / We woul lie to tae these a i explicit omulas. I the ist two cases, this is elatively easy: a, with the help o Maple, x. Page 9 o

10 The last two ecuece elatios ae slightly moe complex. Beyo the scope o this couse, it is possible to show that 3 ( ) whee.68 is the gole atio. I the last case, thee is o close-om solutio, but it ca be show that ( ) l. We will ivestigate such ecuece elatios i the topic o ivie-a-coque algoithms...8 Weighte aveages Give objects, x, x,..., x, the aveage o these umbes is x x x x 3 I, howeve, we have a sequece o coeiciets c, c,..., c such that c + c + + c =, we ca also calculate the weighte aveage c x + c x + + c x. Fo example, we ca appoximate a itegal om a to c by taig a weighte aveage o the values o the uctios at a, b, a c whee b is the mipoit, as is show i Figue X.. Figue X. The itegal x x. c a Page o

11 We coul tae the aveage o these thee poits a multiply the esult by the with c a, 3 a b c c a ; howeve, a bette solutio is to use the weighte aveage a b cc a Fo example, cos( x) x si().993. Appoximatig this with the weighte aveage yiels cos cos cos while the simple aveage is sigiicatly less accuate:..9 Combiatios Give istict items, oe may as cos cos cos o, equivaletly, How may ways ca you choose items? How may ways ca you combie items om items? Fo example, give the set {,, 3, 4, 5, 6}, you ca choose thee items i ieet ways: {,, 3}, {,, 4}, {,, 5}, {,, 6}, {, 3, 4}, {, 3, 5}, {, 3, 6}, {, 4, 5}, {, 4, 6}, {, 5, 6}, {, 3, 4}, {, 3, 5}, {, 3, 6}, {, 4, 5}, {, 4, 6}, {, 5, 6}, {3, 4, 5}, {3, 4, 6}, {3, 5, 6}, {4, 5, 6} The geeal omula o this is!!! a this is ea choose. Usually, we will be iteeste i the speciic case whee = :!.!! Fo example, give the set {,, 3, 4, 5, 6}, you ca choose two items i 5 ieet ways: {, }, {, 3}, {, 4}, {, 5}, {, 6} {, 3}, {, 4}, {, 5}, {, 6} {3, 4}, {3, 5}, {3, 6} {4, 5}, {4, 6} {5, 6} Page o

12 You may have see these umbes i polyomial expasios: o example, x y x y ; x y x y y xy x y x y x y 4xy 6x y 4x y x You may also have see these umbes i Pascal s tiagle: Page o