ریاضیات عالی پیشرفته

Transcription

1 ریاضیات عالی پیشرفته Numic Mthods o Enins مدرس دکتر پدرام پیوندی ~ in Aic Equtions ~ Guss Eimintion Chpt 9

2 Sovin Systms o Equtions A in qution in n vis: n n = Fo sm (n ), in povids sv toos to sov such systms o in qutions: Gphic mthod Cm s u Mthod o imintion Nowdys, sy ccss to computs mks th soution o vy sts o in ic qutions possi Dtminnts nd Cm s Ru A [A] : coicint mti A D D : Dtminnt o A mti D D

3 Computin th Dtminnt A A 5 B Dtminnt o A D D D D D Guss Eimintion Sov A = Consists o two phss: Fowd imintion Bck sustitution Fowd Eimintion ducs A = to n upp tinu systm T = Bck sustitution cn thn sov T = o 6 Fowd Eimintion Bck Sustitution

4 Fowd Gussin Eimintion -(/) - + = 6 -(/) + + = 9 -(/7) + + = = = = = = -9 -(/7) =-(8/7) Sov usin BACK SUBSTITUTION: = =- = 7 Bck Sustitution + + = 8 + = 5 = = = 8

5 Bck Sustitution + = = = = 6 9 Bck Sustitution + = = 6 = 5

6 Bck Sustitution = 9 = 9 o i n down to do Bck Sustitution (* Psudocod *) /* ccut i */ [ i ] [ i ] / [ i, i ] /* sustitut in th qutions ov */ o j to i- do [ j ] [ j ] [ i ] [ j, i ] ndo ndo Tim Compity? O(n ) 6

7 Fowd Gussin Eimintion ji ji ii ii ji M U T I P I E R S -(/) Fowd Eimintion +6 + = 8 +5 = -(-/) 5 + = -(8/) = 7

8 Fowd Eimintion M U T I P I E R S +6 + = 8 + = -(/-) + + = 9 -(6/-) = 5 Fowd Eimintion +6 + = 8 M U T I P I E R + = + = 9?? +5 = 6 8

9 Fowd Eimintion +6 + = 8 + = + = 9 = 6 7 TOTA st coumn: n(n-) n 8 Option count in Fowd Gussin Eimintion Eimintion n n n n n n n n n TOTA# o Optionso FORWARD EIM INATION : (n-) (n-). ( n )... *() *() n i i n( n )(n ) 6 O( n ) 9

10 **hpits o Eimintion Mthods Division y zo It is possi tht duin oth imintion nd ck-sustitution phss division y zo cn occu. Fo mp: + = = - A = = 5 6 Soution: pivotin (to discussd t) 9 Pits (cont.) Round-o os Bcus computs cy ony imitd num o siniicnt ius, ound-o os wi occu nd thy wi popt om on ittion to th nt. This pom is spciy impotnt whn nums o qutions ( o mo) to sovd. Awys us dou-pcision nums/ithmtic. It is sow ut ndd o coctnss! It is so ood id to sustitut you suts ck into th oiin qutions nd chck whth sustnti o hs occud.

11 Pits (cont.) i-conditiond systms - sm chns in coicints sut in chns in th soution. Atntivy, wid n o nsws cn ppoimty stisy th qutions. (W-conditiond systms sm chns in coicints sut in sm chns in th soution) Pom: Sinc ound o os cn induc sm chns in th coicints, ths chns cn d to soution os in i-conditiond systms. Emp: + =. + =.. () (.) D () (.). + =.5 + =. D. () (.) 8 () (.5). Pits (cont.) i-conditiond systms (cont.) Supisiny, sustitution o th onous vus, =8 nd =, into th oiin qution wi not v thi incoct ntu cy: + = 8+() = (th sm!). + =..(8)+()=.8 (cos!) IMPORTANT CONCUSION: An i-conditiond systm is on with dtminnt cos to zo I dtminnt D= thn th ininity mny soutions sinu systm Scin (mutipyin th coicints with th sm vu) dos not chn th qutions ut chns th vu o th dtminnt in siniicnt wy. Howv, it dos not chn th i-conditiond stt o th qutions! DANGER! It my hid th ct tht th systm is i-conditiond!!

12 How cn w ind out whth systm is i-conditiond o not? Not sy! uckiy, most ninin systms yid w-conditiond suts! Is th systm i-conditiond? On wy to ind out: Fist sc (nomiz) ch ow such tht no coicint is thn. Thn comput th dtminnt nd chck i it is cos to zo. Anoth wy: chn th coicints sihty nd comput & comp COMPUTING THE DETERMINANT: Givn n upp tinu sys. o qutions In n, I pivotin is usd thn D = t t t nn (-) p D=t t t D=t t t nn D wh p is th num o tims th ows pivotd t t t t t t Tchniqus o Impovin Soutions Us o mo siniicnt ius dou pcision ithmtic Pivotin I pivot mnt is zo, nomiztion stp ds to division y zo. Th sm pom my is, whn th pivot mnt is cos to zo. Pom cn voidd: Pti pivotin Switchin th ows ow so tht th st mnt is th pivot mnt. Go ov th soution in: CHAP9-Pom-.doc Compt pivotin Scin Schin o th st mnt in ows nd coumns thn switchin. This is y usd cus switchin coumns chns th od o s nd dds siniicnt compity nd ovhd costy usd to duc th ound-o os nd impov ccucy

13 5 Guss-Jodn Eimintion Guss-Jodn Eimintion: Emp 8 7 Aumntd Mti: R R - (-)R R R - ( )R Scin R: R R/(-) R R - ()R R R-()R Scin R: R R/(8) 9 / R R - (7)R R R-(-5)R RESUT: =8.5, =-.89, =. Tim Compity? O(n )

14 U Dcomposition nd Mti Invsion Chpt 7 Sov A. = (systm o in qutions) Dcompos A =. U * : ow Tinu Mti U : Upp Tinu Mti 8

15 To sov [A]{}={} [][U]=[A] [][U]{}={} Consid [U]{}={d} []{d}={}. Sov []{d}={} usin owd sustitution to t {d}. Us ck sustitution to sov [U]{}={d} to t {} 9 A U A u u u u u u [ ] [ U ] 5

16 6 Guss Eimintion A U [ U ] Coicints usd duin th imintion stp A [. U ]?

17 Emp: A =. U Guss Eimintion 5 Coicints = -/-= = /-= [ ] Coicints = -/= [ U ] 5 MATRIX INVERSE A. A - = I Sov in n= mjo stps Sov ch on usin A=. U mthod.. Sov Pom.6. Soution i is vi on th w. U 7

18 Spci Mtics nd Guss-Sid Chpt 5 Ctin mtics hv pticu stuctus tht cn poitd to dvop icint soution schms (.. ndd, symmtic) A ndd mti is squ mti tht hs mnts qu to zo, with th cption o nd cntd on th min dion. Stndd Guss Eimintion is inicint in sovin ndd qutions cus unncssy spc nd tim woud pndd on th sto nd mnipution o zos. Th is no nd to sto o pocss th zos (o o th nd) 6 8

19 9 7 Sovin Tidion Systms (Thoms Aoithm) U A A tidion systm hs ndwidth o DECOMPOSITION DO k =, n k = k / k- k = k - k k- END DO Tim Compity? O(n) vs. O(n ) 8 Tidion Systms (cont.) d d d d Fowd Sustitution d = DO k =, n d k = k - k d k- END DO Bck Sustitution n = d n / n DO k = n-,, - k = (d k - k. k+ )/ k END DO { d } d d d d

20 Chosky Dcomposition (o Symmtic Positiv Dinit Mtics) [ A] [ A] A positiv dinit mti is on o which th poduct {X} T [A]{X} is t thn zo o nonzo vctos X ki ki T i j ii T ij kj o k,,, n [ A] [ A] i,,, k T kk kk T mns Tnspos o k kj j Tim Compity: O(n ) ut quis h th num o options s stndd Gussin imintion. A **h Jcoi Ittiv Mthod Ittiv mthods povid n tntiv to th imintion mthods. A D k [ D ( A D)] D ( A D) D [ ( A D) ] / k / k k / * k k k k Choos n initi uss (i.. zos) nd Itt unti th quity is stisid. No unt o convnc! Ech ittion tks O(n ) tim! k

21 Guss-Sid Th Guss-Sid mthod is commony usd ittiv mthod. It is sm s Jcoi tchniqu cpt with on impotnt dinc: A nwy computd vu (sy k ) is sustitutd in th susqunt qutions (qutions k+, k+,, n) in th sm ittion. Emp: Consid th systm ow: nw nw nw { X } { X } od od nw nw nw od od nw Fist, choos initi usss o th s. A simp wy to otin initi usss is to ssum tht thy zo. Comput nw usin th pvious ittion vus. Nw is sustitutd in th qutions to ccut nd Th pocss is ptd o,, Convnc Cition o Guss-Sid Mthod Ittions ptd unti th convnc cition is stisid: j i, i % s j i j i Fo i, wh j nd j- th cunt nd pvious ittions. As ny oth ittiv mthod, th Guss-Sid mthod hs poms: It my not conv o it convs vy sowy. I th coicint mti A is Diony Dominnt Guss-Sid is untd to conv. Fo ch qution i : Diony Dominnt Not tht this is not ncssy condition, i.. th systm my sti hv chnc to conv vn i A is not diony dominnt. n ii i j ji Tim Compity: Ech ittion tks O(n ), j