BronsonCh2LinEq

BronsonCh2LinEq - Chapter 2 Simultaneous Linear Equatluna CDH‘EISTE-NC‘I A system all simultancnus Ijncar aquatiuns 1 3 act all

Info iconThis preview shows pages 1–12. Sign up to view the full content.

View Full Document Right Arrow Icon
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 2
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 4
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 6
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 8
Background image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 10
Background image of page 11

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 12
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Chapter 2 Simultaneous Linear Equatluna CDH‘EISTE-NC‘I' A. system all simultancnus Ijncar aquatiuns 1': 3 act all cquatiunsfl’l' thr ELI-rm "It-t1 1' la1:31 " “It-i1 * + llMir. ' bl aflxl + arng +ar1x1 4 1- aw!" =13? amt]!+flm1xl+EJ1I1+---+flm;fl-hfll [1]} Th: milking“ a}.I {ul- 1.2,. ...m'. j=1,2._.. .fl]: and U1: quunlil'laa b, {:— I.2,. .. .m] are twin-n mlfi'liflfllfi- The I: [j= 1. 2. . . . . at] an: the unkntwrm “has: 'r'a'lu-ra am: swam. A Minna” fur syslcm (2.1]. 'u. a sat at' 1mm”, nan: fat each Lining-mt. that. “Ian substitutad in Il‘Lt syslam, rcadars all its equations» talid- {Sea thlam 1L] A warm at umquItIamu lincar “luau-nus ma].- pmaasa nu- mlutlana. :xacllj' um: aululiun. {II that: than an: adutiun. Inhabit TMayatam 1l+rrtl 1l+Il-E| has an aalulia-ru. because [11th: are 11:: values [at II and I: 111:: 5.u:|'n 1a 1 amt l: samultar-aausly. 111-: ayaatm IIJIl-I Jr, rh1=2 has "be single mintiun 1"”. I:l -— Ii: and II‘IP‘“ Ex,-Ix.=1} 11:: a flalutinm. I. - 1; Eur EWIT rill": 13F 11. A act of simultancuua aquarium: is canal-slant :if it pmaaam a1 Icaal 1.1m: wlulifln: uthcrwm: il ia Incmmmt. mama: NflTA'l'IDH Systcm [2.1] is algebraically amlaaknl tn Ihr. matrix aquaticln All—H Ill") “II “I: I"In '51" x' bl a. 1: a -- a n x 5. when: Ii: -l L: a] 1 x: _i' B: _ :- I:I-IHI an? all] " 'fi-HH 1" fir! The matrix A is called thv: rwficfrnl mam-r. ham-Uta: 'III tannins like mflfifirnu fifth: unkmm. 111.2 :11. ruw ut' A. (I: I12, . . I m} mmfipanda m. the it}: Equatiun :irL ayalam {2.1 L whil: lhe flh aulumn ul' .1. [j - 1, 2. . . . _ n} mnlains all tht urn-Effician ul 1". {Int mmumtl fnl' each nquatlun. The augman mat: mam-11dng lu ayalam {2-H 1'5 IhE parlifinned matrix [AIR]. (Sal: thlam: 2.1 thluugl'l 2.1.} 12 SIMULTANEIHIE LINEAR EGLJATI'UHE [CHAR 2 TI-IEDII' {IIF EMUTJUHE 11mm Ll: The system AI - I is consistent if anti Innly if the tank at A. equals the rank at [A I I1]. Thenrem 1.1-. Denote the tank oil as it. and the number at unknowns an n. H the ayalern A]! = II is mus-talent. then the anlutitrn euntains H ~ 1' arbitrary acetate. [SEE Prflbierni 2.5 m 1?.) Syn-pm (AH is said tn- tle Frumagrnem: iEB =ll; that ia. if b, = b: = ' ' ' = it." -= I]. If B ail [i.e.. it a: [flat ene bl [.i = t. 2.. .. m]: is net earn]. the eyelet“ i5 "MW-Hm“. l-Intnnleneuu: systems. are imminent and admit the aulutinn .tr = I!" - = Jr. = 1.1. which it called the trivia-f mtutr'tm'. .a nontrivial eaiert'intr is firm: that wntttina at least L'rne rHJ-nIeI-CI value- TI'IEIJI'II'I'I 3.3: Denete the rank at a as 1:. and the rim-the: at unknown! at: n. The hamneeneeue system AI = I has a npntritli'al analtltinn if and only If rt at it. {See Problem 2-1} ELMHJF't'INI: tJPEEATIflNS Three aperalipna that attet the farm at a system nf simultaneous linear Equilimfi but de- nuI alter its setutien set are: [DI]: interchanging the anquenee pt Twp equatinne. tflI]: Multiplying an equatiun by I nunztm scalar. (Dill: Adding. to unit equation a acaJar times another equation. Applying. uperalipna fl]. ()2. and D] In: system {2.1} is equivalent tp applyu'nyI II‘IE elementary raw eperalitrns E1. E2. and E3 {see Chapter I J to the augmented matrth anufiatetf with that system. tii'tuetsian elimination i5. an algorithm Int applying these trperatinna ayuematieally, In uhtam a set at equation! that t! eaay tIJ- analyze :I'ur mnaiatcney and easy tn ant-rt: if it it. ruminant. GAUSSIAN ELD'IINA‘I'IDN fiLflGRITED'I STEP 21:.- Fu-rrn |he augmemad matrix |.HB| flfit‘ttI'iBlEfl with the git-en system of equaflnna. STEP 2.2.- Uyt eiementar'y raw upflntimta tn rranai’prrn IAI ll] II1ltJ tuw—ecthDn iLnrn {ate Chapter 1]. Dertt-te the reaulr at [El D]- .‘i TEF 2.3. Determine the ranks at t: and IE I DI. If these tanka an: equal. nominate; the ayatum is mntislent [by Then-rem 2-! ]. Ii net. amp; the uriginll ayatern haa nu eulutiun. RTEF 3:1.- CmmiIi-er the system at equatinne mrreapnnding tp |[:| [I]. din-carding any identieal'ly 1cm equatinna. [It II‘IE rank at E' is it anti the number at untnmrna is n- lhflt! wilt he a — .t such equatiem.) Suit-1: eaeh equatitm let its first! {1flWH-t indexed} uanahle having a materp enet'ficiant- STEP 2.5: Any variable not appearing an the left side at any.I equatiun is arbitrary- All other variahlte an be determined uniquely in t-Ermt- at the arhj1rary variables by haett euhtiturinn. [Sec Fruhlenls 2.5 [Ill'tllugl't 2-3.]: Either atriutrp-n prucedures are discuss-ed In Utaptfis 3. 4-. 5. and El. HVflTIHfl STRATEGIES En'tmi due to mun-dine can Mani-e a prEI-hlern in fiat-titan eliminatien. Te minim-tee the effect at mundnfl ten-tan. a armier nt' pivoting strategies have been pmpmed. earflt mnditying Step l_:t elf me eigpri'lhrn git-en in Chapter I. Pivuting strategies are merely criteria int chanting the pit-at element. CHAR 3] EJMULTAHEULIS LINEAR EGUA'I'ICIHS l3 Perrier pirating inselses searching the merit autumn ef the augmented malns fer the largest element in aheulute value app-easing in Ihe eurrenl wash: rel-r {Ir a sueeeelziing re'w. That element hemmes the new pivot. T'a use partial pirating. replace Step L3 ef the algorith [Dr Ilaniftllminfl. I man'th m raw-echelon farm Iwith the [aim-wing: STEP 1.3".- Beginning with rel- R and e-antirrujngI “trough meet-stifle rev-s. mile. the largest element in absolute value appearing in were es'llumn C. flenete II‘II': HES-1 mil in Which this element appears as new i. If I is diiferenr t'mm R. inlerEhanac was I and R {grammar}- ma- aperalien El]. Raw R sir-ill: neml.I have. in eellumn C. the largest emigre elemenl Ln abselute 1value appearing in elflumn {T of raw H flr any rea- sum-ending it. This elite-rm in new R and eelumn It." i:- ealled the primer; 1m Pdnlmln ila sraiue. (See Problems 2.9 and E. lili- Twn other printing strategies are described in Problems 2.1] and 2.12; they are successively- mare powerful but require additienal ear-tentative“. Ecinee the goal is to avoid significant mar-def! ermr, it is 11m [necessary m find the heal pis'el eiernenl at eaeh slage. but ralher ta amid. bad unes. Thus, panial pivoting is the strategy must uften implemented. Salted Frahlems ll Heterrnine whalher :l = 2. 1: - 1, and x1 -= - H is a snlut'mn get far The system 2:, 4- I: = 5 3:1 4- Eu: # I}=1 5x1+ E's-1+ J'_..- R Wtiluth: lb: ample-ed fillies far the eaten-was in!“ the lei'l slide as each equafien gi'res 2[2}+|.[1i -s stay emu Lr—1|]=1 SEEM TU}! |t-|I]=fi The I35! equalirm dunes II-L11 yrelti H- as. rammed: hence the [Imp-used values .511 nnl! :nnsts'rube .1. salutian 5ft. 1.1 Write Iha system of equatimu given in Problem 2.1 as a matrix equation. and Ihen detennirp: its asaueialed augmented matrix. 2 1 fl 1'- 5 It: 3 Ii- 1 I- I: B- l 5 ‘I' 1 I1 I!- The original system can be written as A1 I B: its augmented matrix is 2 3 11:5 ism-[s :5 iii] S'II'IIH. H SIMULTANEDUS LINEAR EQUATIDHS [CHAR 2 1.3 I|illiriti: Iht [alluring fiat-rm of aquarium :in matrix farm. and than ticlcntliru: J'tr. augmntrd matrix: 3.1, +1.12 + 11—41.] = 1 Elli-31:1 —.I:Jl -+—1 Il-m1+311-qu= '1' This ijI'EIII'I'I is equivalent IU- 1hc matrix equation II fill—nix1 | 2 511—1 In-—1 J-EIJ-EI 1 1h: umiatcd augmcnttai main: is» 3- il—lil |h|l]- 2 3 n r154 I -I5 3 -E; T Dhscrw Hill in bum A and IA. I I]. ti“ :m in thrl: rum-thud ml- :Ind third mlumn Wflflpflrfll m 111:- i-Em mEi'ficignl: nl' I." in the flew-I'll! aquarium ud' “IE ulifii ni'l 5115mm. 1.4 Writ: :11: an! of fimullancnus aquatiam Ihal Barman-and: m II'H: Businch matrix I 21'] H3- -JH.I'Jil-'1 |1l|II]= U I —21'5 1 E-l tr [1 I] [I :1] Th: mmspmtling “:1 mi aquarium is Til-E third aquatic-:1 reduce: tn the tutu-inlay III- II 311:1 n no: min-2n. Mm- du we I111: 3:11.- imam; hauLn; :- :m mfiifiml. 1.5 59hr: lhl: fit CII' :qutllimu git-an in Ficrhlnn 1.] 1:3.- lfiauflifin climzitlatifln. The :ugrncmed man-i1 In: fl'l'il swam was dawn-ruined. in mum 2.1 In tn Ilflifi [Mafia-a 1il. STIH'I- Uing 111: rend!!- at Pmblnrn 1.11, we 'lraJqufl-‘l'rn li'l‘il- rnatn: mm Iii! Imbehn harm I: 1:: n 5 5x2 [cuppa u I. ENE—mt! u L: Hi I ll [ufluws from Pushkm LII: Ila-all 1h: flunk a! [E | II] is 3. SuhmBLfil E' is :lm in :Wbl'lurl {um and has rant 2- Sin-n: II1= tank at L‘ dun-act not equal that ran]: at [1: | III. II“: urijmfl ill will cquatinru u imminent. The pmblznt is Ihl: Ian equatimi Essa-fitted unilh [Elli]. which u Elrlillzl+flII-l and which tltflflr Em. I'm mlulmn. CHAR II SIMULTAHEDUS LINEAR EflU-fi'nDNS '5 2.5 Suave ll'u". 5c: aim-Innate given in Problem 2.3 b1.- C-ausslan eliminaliun- Th: luwntfl mmri: [Dr 1-‘rI-1'I- HFHII'I w“ #flnninfll in Frnhhln 2 .1 1!: hr 2- 2 | —45 1 [Jul]:- 2 1 u —1E—L I -E: 3- —l: T Using Ihl: mulls elf Prnhlcm 1. l5. m: Inmf-mm It'll: mtrir inin thn: an-cclnlnn innn 1 21'3 11']- --=I.'3I: HEI- [cmlu u 1 4:5 L E»: u a u 1} I I] ll MIMI-s {rum Pmblem I.” Ihal L11: Tank r-EIC I [II in: I. Suhmnln'l C i:- rllmin rnv-I-rchr'lcun (am. and in all: I13 I'Bul 2. Thin. 'IJ'IE animal El. pl minim-I5. i5 :nmhfienl New. using the rnulla u! Prublr-m 1d. in: urine Il+irr+ !I_I\._i11=! 1,— i11+ 14=-I. n “I: 1:1 Ith mum-2m usincilted with [1.2.l [II]. Bulking III: firm muslin-n far 3:. and 1m.- sacnnd In: 1,. In gel II: 5—i12-111+‘:r. 1:: -I + ixi- .It. Sin: 1'. and .1‘ dc: nnt “Jpn! |-1|'I Ih: |¢E1 Lid»: Hi my :quniitm. th-nr In: Irhi1'rrlr5' Th: Minna-11 .13 i5 caught-al.3- delrl'mm-ud lI'l Harm: ud' Ih: nTb'ilnrjr Minn-«Ins. 5L1|:I:li11.|.1ing_ il imln- t'I'I-e first math-H1. 1w: mam: II- i _ il_l+ill_ll}_ ikl" III-II =I- éxl-rEL T11: mpltu: Efllfltiud'l m the migiua! m ul' tllllnll'm'fl 'n :,=!l-Lt,+2x. :3: -l iv in - Jr. with .‘Ir,I and J:- arbitrary. 1r? SDI“!- IJ'II: MID-Win: Elf: Elf hull'lflficfltm Hummus by GELIEEi-flfl :Ilminatlun: T113 +91:ll = I] 1:. + I: - J, =1] 51. +6.1: 1-in-1} fly Eun'l'cl'li-n: 1I'Ifl firs-[em III! Iflal'I-El'IMd-mfl'rifl: {arm Ind Ihrn Iranian-Mug the ma1Ti1 'm1n- m-uchflurl [mm (Step-E J..l :hrrrugh 1.3]. we gm U 1 95:] I-tlll- 2 1 -|.EIZI 5 .5 25:: I 111 -1r2:u] [Hula-[n I win H II DH! 16. H 1.9 SIMULTAHEDUE LINEAR EflUhTIDNE [Cl-LAP. 2 Th: Tani: :1 1h: mfficicnl mm A. '5 thus I. and humus: than: an “In: unkmm i11th “ti-gins! 5H nl' aquafinm. Tl'b: system has nuulnv'ut hnluliium. The 5:1. 1:! math-rm. anncLIIEfl with Eh: Inn”:an mall-i: [C I III a 1.IlFE'II- I'll-I‘D xl+h1fllfll U=fl fink-Iris In: the firm Harublt in {3:11 equatian Iwith a mmrn Enlfficicnl. In nlflain II - _i1r+ E'I'i 3:3" _?x1 mrlim. 1.. is. ubilmry. Sal-ting fat I. and x: in Itrms In! 1:,L by han: inhflutufinm we find H 1=-—u'. :,- -l[--f~'J.-,]+ EL: :1. SEQ-r: the [flllflwiflj 5c! of equations: Il+1I:— A": El Jxl+fil2+gxl- H] 21" - Ii.+ll:.=—2 The magnum-ed man-i: ahfifllal'td will: 1|“: when I5 fr I11 - 2 IrhnL'h- in Frauen] 1.13. was Inuit-{mid mm Ih: mcrlclnn fun-rm I i -|E fl [c1n1- n I fiE—ul l 3-! El 0 Hmh L' and [III- [I'] ham rant lhl'fl. in PM system is musis‘laut. Th: HI! :1! Equafm-rfl Jam—inland wilh flu":- aummwd matrix ii I 2 —I 4A|3]=[3. n I: I —'l I Jr.+1:=- :..= n x2+lfig.=—-I JIII—h] Enhing Eafli mum-an I'm- [ht flm variable with. 3 mm waif-21ml. 'l-‘E tab-tam Ih: 59mm 1" fl- lEri-Il x;- 'l 'EIL. .1_.=—1 which. can In: mind :31in hy hack :uhrlilulinn hgiurljng WiLh the fill equafiun. The mimic-n In thin. Iglum and bu- Ihu L'H'iiimI 1E1 uf equallnus In. J." - L JrJ ‘ 1.3m] I. - 'I. Salve me foaming m nl tqualiIJ-M by [1‘] mnde Gaussiu Elimumtinn and [b] fiauuian ulqmmalian will-r padial pita-ling. rounding d? rrJ-rrtpumlfans r-a four n'gmfimm figural: filllllln 1r 11:1.[fllll II'Flt-l L‘HI'I'F'. 1| 1]“ EIHULTAHEDUS LINEAR EflUAfiDHL-‘u I? {III We 'Il-TI1E1Ilt system II'I mania Elam. l'I'l-llfldi'fl]. LEIIIJI tn “Ill TIIEII In: ErIrEFuI'm 1h! Minimum-d 111mm imc. mun-gamma Inn'I'I using 1h: aIlanuIm u! Chap-1t: 1. In III: [MIMI-mg BIL-pr. IIJIIIIHII I I LIIDII I 1 E 2 —-l I mm; man] i I i 1 I I nummi Imam] —- III HEREIN—Imam I Iflflfllfli 1:13am —I-[ III 1 I I ] IN-nm: 1I1a-r m: mund “1 -I|'.'I'|]IIIIJ I'I'ict: in 111: marl-1:343“ map) 111: rfiuhmg auglI-tntcd mitrh: shims than 1h: Frucm I5 maiutm. Th-I: tquaI-nms ammttd wiIII IIIII rIIaIriI an: x.+]flflflh,-lflflflfl III! which luv: III: mluflurl I, =11 “III II = 5!- HI'HIEUIT. tuhuti'lIII'I-m iIIIn Thu: I'Irigiual :quauirrns IIIIJ'III! II'I.:I1 this is null IIII: .I-nlminn II: IIIE nr'ljhul 51.5mm. [I1] Tmflfir'llll'l'inl 111: sup-III:an .IIIaIn'II inn-n- mHEI'ILILIrI [unr- usmg pannal pimtlna gal-:15 [ULIIIII I 1 I '. 1 [Inn-I: | and I are inlcrdurlafid because my: 2 had “It hrg-m clement 'm a:de I- III: {ll-Irltfl‘t work mlumn. Rnumdhg IIZI four signifium Elam [ I 1.2] —-IZI.{IIIIJI “Ll-III- J; iii] The Sfilfll'l nf Equalinn: ambled I'ilh III: last augmqu matrix i5- cmu'umm and a .I. 4- JII':I - 1 .II:,=l II: mlulinn 'Il. Jr, - II I 1. which 'n :IIIIII- Eh: mlulinm. [n "I! uriginaJ Incl nnf equalluns. All mun-r5 mum If: a. IIIITIItIEI cf IigII'Ific-IIII figum I: 13111: dtpcnds 1:11. that marl-lint III-rim him. Than IIII :qu-atinn III the [firm In”“'&.rx,=lrlfl”“” IIII-III 35mm: resuIrI. llkc rII-il: III-f part II III'III-‘S-‘E m: Film-1mg “ring? I': uuad. II”: had Ir = I In pm i ] as :I. rule. Guiding by Intu- id'lall rI.I.IrrI:|:IEr: I'lefl lea-II In ailnlficanl mar-Lb” Efll'l-I' 111d :hnu'ld l1: a.de III-11:11 pussihlt. Sch-I: IhI: {allowing sct cl equations using Plfljal pimling: II+EI=+ 311: 13 l-I'I'l' I,- 4x,=-3fl' —5:| +311+th1= 95 'III: augmnlcd man-i: Eur 1.1m Sj'fll'flm i: I 2 .15 III 1 I «15 3-1] -5 B :1: WI IE SIMULTANEGUS LINEAR E'UUHTIGHE [lift-MP. 2 In tremicrming th'rs ITIIIIl'iI. we need In etc Stun: IJ' irrtrnetiilteiy'. with R =1 and C - I. The Its-'51:“ elemenfl in litmch value in- cuhufln | is ‘5. Inna-ring in I'd-w 1- We inlets-hinge 1h: lite: and thitsl tut-e. and the-n. emulates ill: traust'crmattcn tc row-echlten fen-n: —- —s e 1?! 915 21-45-30 - I 2 3‘. 15 —+ | —1.._s 445—19.: 2 1 -t g-au i 2 3 I 1|!- 1 -~t.e -1.e.f—tsl.2 —- u 1.2 2.3; e: 1 I 3- : Ia t —1.fi- —]_I: 49.2 c e: 2.1; :u — ii 3.5 H : 31-2 We next apply Step I.3-' With E = 2 and I: = I. COMM-Hill celjlr tune I and 1m- find that largest element in abut-tilde value in. enlutl'en 2 is 1.2. set- I- I and an m interchange is required Cumimfinl 'l'ilh the Gaussian elilti-rtatictt. we calculate t —1.e —3.e E —IsI 2 - III | til-W1: 2 u 1:. but 311 1 —1_e —s.4 g—m I] 1 awe; I —e H U l : II | —1.t. —3.e f—I'H ll 1 EMT; 2 U LI 1 r 15 The system rri equesin-ru- mated Iwith: the lee: augmenwd metrilt 1's ccrtsutent and a el- I.ftx,- 1111—-I9.2 r: +i| fleechfi 2 1,-1.5 Using haei; sul'tditurietn {beginning with IJ]. in: flhlin.‘ att Il1le stelulittn In lltiet :11 ml? equalinn: a well 15 [he tll'iflflal system. I, - Lil. 1':I ‘ '3. and 1.. '- 17.5. 1.]! Tu use snared pit-Mug. we first define. as the scale faclnr [er each new at the cuefl'teienl matrix A. the largest element in alt-suiute train-e appearing in that raw. The sale [esters are computed meet: and cult- mace and. re: E'an reference. are added cntc the augmented n'Lett-t': 1.1th as neither parlitiflned eelumtt. Then Step 1.3 at Chapter 1 is repieeed w't1h the [tallest-inn: Divide the abs-mule trislu-e at each numeral element This! is in Ih-e 'ItrurIs cchtrtm and an at heiuw the uteri; TEI'II'I' by the eeaie Fetter far its. row. The element yiclding the laTItE-t quctiem is the new Fri-um. demte its raw as raw I. it" new i' is different from the current wnrle raw {new it}. then interchange rcttlts t' and R. chr interchanges are Ihe cell): elementary raw cpetaliunts that en: pet-farmed ct] the settle liners: al] clher Hep:- tn the fiauaeten elim-t'netien are limited II-El- J. and I]. fictive thlem etc using sealed pivoting, The scale fame: in: the system cf Problem 2 I“ are t, =fl13‘lfl].2.3:l =3- s*- mul2.|.I—-1|]=-I s. - menu-SLR. JTII = I? CHM. 2; SIMULTANEDUE UHEAR EflUATtflP-«IE H 1m: add a. autumn cumming at “Its: EEIlI: factors In “I: “mum: H'IH'J'i'I: fin: III: Ifim. and rim-I lflrfil'unflin' it tn m-flhflnn firm 3:! tritium: 1. 1 3 . w J The scale-Emu: quatimla In: the I I —-I I: —‘.'.tl d riamtnts in mlumn I an: I13 = -5 3 I? i 945. J”? 0.333.. 211 =rII..‘il.'l'I. Ind 5H1 - H.194. —.. 3 | —4 I —3|:| .1 Th: Input quminr "n: fl-fiflli. In 1 1 3 . '3 _1 the pi'ml 'i: 2. which apnea-r5. in —3 a. I?! 9.5 1'? rim-1".Sinnci-h1mdfldtuhl: first and Six-rind '|'I':|'-'-|E an i11trthil1fld. —: 1 [hi -2 - [5 «I 1 1 J : i3 5 —5 E I? i ¥I 1'? 1 I15 -1 .-IS 1I .-... III 1.5 S E 3.1 .3 -fi E I'.' an 1? I “.5 -1 5- H: «I Hum Hm: rum- is 2. and tilt wart. El 1.5 3' 3-3 I mlu-mrIi-IE. Thtth-mitnlE-Bl‘l: —- TI I115. T i 21 I? I.5I'J- IJfl'III and “151'”? Il-fi-IH. | I15 —2 i— is 1' Th: inlgt'sr quoti-tam I5 LLbH-L. In I]- IIII.S ‘.|' : 1| 1'? Eh! pull-1! is. Jill-’5. which app-cars in —II 0 LE 5: 33 3 FL?!" 3.1'I'Ll'uct'lflmdthird rm Brit Int-Eirhltlgld. I “.5 -2 : -15 d -- [I 1 0.556515 1 I? III 1 fi 5 : 33 J I D El ‘1 i — IS I a I 0.51.551 i 2 1.? —- III II 1 E II .3 1 11.5 —1 5—H. t El J III-MT : 2 11' —' i 1-5 I Writing III: set at Him-rims “Fri-"Ed I.I-i|:|'in lh'fi Human-lad Mam: tip-wring the Eniumn fiscal: [aunts] and whim; lhlrrn tr].- hath auhauiutim. in: main Th: mlutim x. = 1.5, x, - —3, x. - 15.. Tu IflE twin: Nurturing. we “pint: S-tcp 1.3 at Chapter I with lhe [hunting mp5. which infill“: bin-III. MW and calm" inttrchamgfis: Ll=l the current Hurt mu.- hc R. and the mum: wart unlumn C. Scan all Ihe rltmcnts flfflhmalfifl. .1 at th: augmmcd matrix mat an: an or below. tow R and an m' tn IJII: right of column (I. In datcrmint which is largu: in alum-jut: HIDE- Dtflult Ihe raw and. wllmm in which this :lcmm: appears 3.5 Emir! and mlumn J'. H ht R. inttrchang-t nil-wi- .I1 and H; "II I! C. thlcrthang: mm- .i' and C. Baum mltll'nfl 'II'IHTEIIII'IEEE Chang: III! DTdEl' {If IhE llfltnuwnt. I hmkkccpng munhaniim far associating mltllt'lm will'l winch-I'm: mutl tn: implcnrntad. Tn do in, Hit] a new...- partit'mu-ed 1mm. rm.- u, abm‘r tilt Ultra] figummmd NIB-Iris. I11. :I=m:nt5. which are initially in the alder I. 2. - . - . .u m- dint": II'IE Euhfil'ilpls En Ill: 'III'IEIJDWEIE. will: delight“: whith uni-tnnwn is amdalcd with each column. Ell $[MULTAHEDUS LINEAR EDUATI'CIHS [Ci-IA]! I EIan 1hr syutm nI' Pnihlm 2.11] using mmplm pirating. ‘I‘I'c add me huntkupiug rm.- I] II: [he augmlmcll mam): nf Pmblc-I'I'I 1".lll‘. Thu-I1. hcgirlning 1«II-HII Inw I. In Iranslnrm III: mining fun-s inm rm-auhulnn farm. I2 Jj — H=|andf=LThclargfiI "TUE-""3- :"Tfi; chum“: I'rI :bfiuluu'. film: in 1h: g I _4 I «31] Inn-Er Lth III-Inmth is IT. in mm- —5 H 11', 9a. 3nd column 3. Wcfim inltrchunic rum 1 and 3.:I1I] than umlurl'm: l and 3. 3 .. .... ' 'IITHEI'SEII -II.2§I.I.IIII: 5mm I I I 3“ I In J _________ II.-I?II51II; nan-Ina: 5.3m -" EEHEIE ll $23523 -'-'.-H WI 1 | I IE I IJ 3 I Emmi“. -I].294IIR: MEI-I‘M III 233235 ilflllfilflE—TJII'I'IEI —' [I .T l n.53fl13fi. Lilli“:- : LEISSHE" Irnmfl=2m1d C—‘I Th: larltll Blah-ant in ahsnlum Him.- of III! Imu- 11.11.le mmidzrafiurl L5 1.35135. fnr which .I- 1 and! r 2- 5im~=1=fland1=finn iflttrc'fufllr is rEquiIfiL ".355? H ‘IJTH-J _.[-“'[I.4'I'IJSEH —II.29~IIIII: 564% II | I III-:23: : TI'I: 'Illirl'h NIH and war}: culunm -E .... "-3 ........ .J. .... .3--.-:.-_ l ISL-1115311 -l:l.29«lH-i ifi-II'I'UIEI I: I mam-I ; -’.?.' may —- III II MIL-113 - 15TH] I .... .3 ......... .3 .... _. I “AIMEE-E 4mm, 1am; I] l llJflfii‘l-I : —I.5'.|Ifl [I n I ' I.5III:I1I Thu.- firm mlumn 431’ NH: rewiring m-tchclurl matrix m-rernnds In In and 1h.- Ihi'rd :nlumn 11:- I.. In "In: uslirrcifltm “I nf aquarium is :, HIII'IIsmn—IINIIIEIIL w 5.5mm ;:+u.11Is:Iu, —- -2.5'I'H?- I = I-Hflll |. Sun-in: each equamm Eur 1hr.- fim vIIIiIIb-I: him a nor-mm :mfficinrlt. m: I'Ihliin In 5.H7D&—[llm1:+fl.29-Ill1flt, x.- -2 i-i'l-l-J-ELEEST-‘Hx, 1:: Ifll'fll which. “11m “III-.5! I33- I-uL-Ir. tubaII'IIIIIian- yklda III: salutinn I. = Lml. Jr, - 41.331}. and II - 'I'.FI|'.|IH'|'| . CHAF 2| SIMULTHHEUUS LIH EAR EEILI'AHDHS 2|. LIJ flaws-dea" Eul'r'mr'nun'm adds a slap ham-ecu Slaps. 2.3 and 2.4 ui lh‘L‘ algorith lur Gaussian n'liminaliun. Elna: :h-e mil-named man-i: has. been mind to mw-cthdun farm. ii is than maimed sliJ! furl:th Bcginning mlh 1hr. last pivot :Iun-mt and mlrlinflilrfl. ieqwfififlllf backward In th: first. :ath pi'ml :Icmcnl in used In :ranslurm all what elements in HE column :u m. UH fiamJurdm eliminaliun tu Halve Prublem M. The flu: Hm amp: Ind 111.: Gaussian chmimtiun llgn-Tilhm are usual w raducr 1h: lumnlud malru: 1n- ruw—t-rhtlun [mm a: in Pratt-m5 Hi and 1.5: I 2 -I : I'l- [I I I.‘I 1-4 I] I] I: 1 Then "1: mania. H. reduced tum-u. :5 lialhum: [I 1' '1: It] fidd-f-fimfimtlhirllmwmw I : I] I I imam] l:L1-'||'. |.'| II —] —- | I U I 5 Add 1h: thin] row m :h: firs-I ruw. [I l [I I 1’ I: II | -' —I [I1LI12 thcfimmw. —--[| [I U: 'I Add -2li.rne:th:5em11.dmwln H II li—I Th: 5H at :quauuru nun-dual lulu-Hi1. 1hr. :umnlad mall-i1 is :,-1. 1: - 2. and JrJ - — l. which 'n-II1E finlulinn 5:1 far "1: original 51mm Em hm: whatimiun u rut-quarch 1-“ Us: Gammrdfiu 'EIImil'lfiliEI-II [El sum-III: 1h: syltnm of thlcm 1.1. Th: first 11H:- fl-Epa: “I “1! flaw-slam elimi'rulinn alwri1l1m [Inn-id: 1hr augmtl'ltcd mtbclun-fmm mitrh: |~‘.:|I:I|- I] 1 9:? 1r IIIIIIIIH] 1 H2 —|..'I 4]] ar- Ir- F'mhkm 1.1 This man'h: u :Mufld furthEr by laying m: Five-1 in1htf2.flpu-Ei|:iuntfi place a ram. ju In: [1.1] pumim —- 1 El mug” Add-H2 11mm Ihrmdruwm U ] EH7: ll 1h: fill rnw. [I H El :III The 5:1 ud' Equahnrn amt-laud with IIn': :11ng 1mm: 5:. x,-'!:.,-u xl+3ix.-fl 0:“ Saul-ring In: I‘m: fi'Til variah': in: :ach tqull'iun With a mum mfficicnt. 1m.- ntllain x. I h. Ind 1,. -' ‘11-“ which :i! 1h: sululiam [at- back .nntulilutiun 'n tequirlpal] wilh .III arihiuarr. 22 SIMULTAHEDUS LINEAR EflUATFDflE ICHM‘. 2 Supplementary Problems 3-15 Which M £11} xllx,lx.-|. Eb].l:l=fl..1:==—1.A:..=fl [r3 x,1|2.rz= -3.::JL=2 [If] x,=2.:==-I.:L='i| :u'l: mlulms 1-:r1l1c synern x,+]x=+ x1=5 2r.+ Iran-15 t.+'|'l:t+511l l LIE. WIiDE I11: mime-med matrix for 1h: 51mm giutn in finbltm 2.15. 2.1? Wm: 1h: lugrnm'refl matrix fun-r system 21,-“: +h1+fix.- 4.“: 1? Eu: -3r1-4J. - 51:..- 1 2;, +31: + Jul—1:. - 11x,- III} 313 finh-E the :E1 15f Equallum. anutimtd With Each “1' the [calm-mg lugmcrned mall'iEEl: | -2 3:1'Ir I 3 21:3- [a] n I2:—3 m n U12:5 [I U 11—i- U U U IHJ 1J9 fink: rh: 55mm prim 111 thkm 2.15. 2.11} 501M: [ht- 5ys1tn1 gin-:11 i11 Pmflcmfl l?— :1 Pmnlflurru: 2.2! Ihlnugh 2.2T. m1“: In: the unknfl'kru In II'II! gin-tn Hiram. 1.1] JI+EJ;:.- hit] 13] 1:. +1.13. 4-3.:._- I Ell ' lrrrlxl-II 4xl+5|.'=rflt._=lfi 3x,+ :,+1"x1=llj T:.+Br*+lhl-=Ifi 1.1! h. - .r_. +4;.- I5 1.11 1.73513,“ +1,” . g 1:, 4 11:, +31. -r —? 2.1. ‘23:. :1 4- x1 = —.'|: -x. +1.1? =- -Li :.-3:.‘I:1--1-h- H 21:. +1I:-_. " Elm,l +41:J = —2 3,15 {11.41; 4- It, 41;,Ihl'l} |x,+ l:_.+ };.-r |l:_,--1I jJI-I- l:_.1' huh-I- I‘rfi-IE .‘x, 1 “3+ 1'x~+.'|;:.=1.1 ...
View Full Document

This note was uploaded on 04/26/2010 for the course CEG 616 taught by Professor Taylor during the Winter '10 term at Wright State.

Page1 / 12

BronsonCh2LinEq - Chapter 2 Simultaneous Linear Equatluna CDH‘EISTE-NC‘I A system all simultancnus Ijncar aquatiuns 1 3 act all

This preview shows document pages 1 - 12. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online