Matrices and Solution to Simultaneous Equations by Gaussian Elimination Method

Transcription

1 S Jose Stte Uivesity Deptmet of Mechicl d Aeospce Egieeig ME Applied Egieeig Alysis Istucto: Ti-R Hsu, Ph.D. Chpte 8 Mtices d Solutio to Simulteous Equtios y Gussi Elimitio Method

2 Chpte Outlie Mtices d Lie Alge Diffeet Foms of Mtices Tspositio of Mtices Mti Alge Mti Ivesio Solutio of Simulteous Equtios Usig Ivese Mtices Usig Gussi Elimitio Method

3 Lie Alge d Mtices Lie lge is ch of mthemtics coceed with the study of: Vectos Vecto spces (lso clled lie spces) systems of lie equtios (Souce: Wikipedi 9) Mtices e the logicl d coveiet epesettios of vectos i vecto spces, d Mti lge is fo ithmetic mipultios of mtices. It is vitl tool to solve systems of lie equtios

4 Systems of lie equtios e commo i egieeig lysis: A simple emple is the fee vitio of Mss-spig with -degee-of feedom: m k k y (t) As we postulted i sigle mss-spig systems, the two msses m d m will vite pompted y smll distuce pplied to mss m i the y diectio Followig the sme pocedues used i deivig the equtio of motio of the msses usig Newto s fist d secod lws, with the followig fee-ody of foces ctig o m d m t time t: Ieti foce: d y F () t m dt ( t) Spig foce y k : F s k [y (t)h ] () t F m d y dt ( t) F s k [y (t)h ] F s k y (t) m y y (t) Weight of Mss : W m g m y (t) Spig foce y k : F s k [y (t)-y (t)] m W m g y (t) whee W d W weights; h, h sttic deflectio of spig k d k A system of simulteous lie DEs fo mplitudes y (t) d y (t): m d y dt ( t) t ( k k ) y ( t) k y ( ) m d y dt ( t) k y t () t k y ()

5 Mtices Mtices e used to epess ys of umes, viles o dt i logicl fomt tht c e ccepted y digitl computes Mtices e mde up with ROWS d COLUMNS Mtices c epeset vecto qutities such s foce vectos, stess vectos, velocity vectos, etc. All these vecto qutities cosist of sevel compoets Huge mout of umes d dt e commo plce i mode-dy egieeig lysis, especilly i umeicl lyses such s the fiite elemet lysis (FEA) o fiite defeece lysis (FDA) Diffeet Foms of Mtices. Rectgul mtices: The totl ume of ows (m) The totl ume of colums () Aevitio of Rectgul Mtices: Mti elemets [ A] [ ] ij Colum (): m m m m Rows (m) m m m m

6 . Sque mtices: It is specil cse of ectgul mtices with: The totl ume of ows (m) totl ume of colums () Emple of sque mti: [ A] digol lie All sque mtices hve digol lie Sque mtices e most commo i computtiol egieeig lyses. Row mtices: Mtices with oly oe ow, tht is: m : { } { } A

7 . Colum mtices: Opposite to ow mtices, colum mtices hve oly oe colum ( ), ut with moe th oe ow: { A} m colum mtices epeset vecto qutities i egieeig lyses, e.g.: F A foce vecto: { F} Fy Fz i which F, F y d F z e the thee compoets log -, y- d z-is i ectgul coodite system espectively 5. Uppe tigul mtices: These mtices hve ll elemets with zeo vlue ude the digol lie [ A] digol lie

8 6. Lowe tigul mtices All elemets ove the digol lie of the mtices e zeo. 7. Digol mtices: [ A] digol lie Mtices with ll elemets ecept those o the digol lie e zeo [ A] Digol lie 8. Uity mtices [I]: It is specil cse of digol mtices, with elemets of vlue (uity vlue) [] I.... digol lie

9 Tspositio of Mtices: It is commo pocedue i the mipultio of mtices The tspositio of mti [A] is desigted y [A] T The tspositio of mti [A] is cied out y itechgig the elemets i sque mti coss the digol lie of tht mti: Digol of sque mti [ A] () Oigil mti A T digol lie () Tsposed mti

10 Mti Alge Mtices e epessios of ARRAY of umes o viles. They CANNOT e deduced to sigle vlue, s i the cse of detemit Theefoe mtices detemits Mtices c e summed, sutcted d multiplied ut cot e divided Results of the ove lgeic opetios of mtices e i the foms of mtices A mti cot e divided y othe mti, ut the sese of divisio c e ccomplished y the ivese mti techique

11 ) Additio d sutctio of mtices The ivolved mtices must hve the SAME size (i.e., ume of ows d colums): [ A ] ± [ B] [ C] with elemets cij ij ± ij ) Multiplictio with scl qutity (α): ) Multiplictio of mtices: α [C] [α c ij ] Multiplictio of two mtices is possile oly whe: The totl ume of colums i the st mti the ume of ows i the d mti: [C] [A] [B] (m p) (m ) ( p) The followig ecuece eltioship pplies: c ij i j i j i j with i,,.,m d j,,..,

12 Emple 8. Multiple the followig two mtices [ C] [ A][ B] So, we hve: [A] [B] [C] () () ()

13 Emple 8.: Multiply ectgul mti y colum mti: c c c c c c c [ C ]{} {} y ()() c Emple 8.: (A) Multiplictio of ow d colum mtices { } c c c c y y () ( scl o si gle ume) () () () (B) Multiplictio of colum mti d sque mti: { } ( sque mti) () () ()

14 Emple 8. Multiplictio of sque mti d ow mti y z y y y z z z ( colum mti) () () () NOTE: Becuse of the ule: [C] [A] [B] we hve (m p) (m ) ( p) [ A][ B] [ B][ A] Also, the followig eltioships will e useful: Distiutive lw: [A]([B] [C]) [A][B] [A][C] Associtive lw: [A]( [B][C]) [A][B]([C]) Poduct of two tsposed mtices: ([A][B]) T [B] T [A] T

15 Mti Ivesio The ivese of mti [A], epessed s [A] -, is defied s: [A][A] - [A] - [A] [I] (8.) ( UNIT mti) NOTE: the ivese of mti [A] eists ONLY if A whee A the equivlet detemit of mti [A] Followig e the geel steps i ivetig the mti [A]: Step : Evlute the equivlet detemit of the mti. Mke sue tht A Step : If the elemets of mti [A] e ij, we my detemie the elemets of co-fcto mti [C] to e: i j (8.) c ij ( ) A' i which A' is the equivlet detemit of mti [A ] tht hs ll elemets of [A] ecludig those i i th ow d j th colum. Step : Tspose the co-fcto mti, [C] to [C] T. Step : The ivese mti [A] - fo mti [A] my e estlished y the followig epessio: [ A] [ C] T (8.5) A

16 Emple 8.5 Show the ivese of () mti: 5 A Let us deive the ivese mti of [A] y followig the ove steps: Step : Evlute the equivlet detemit of [A]: A So, we my poceed to detemie the ivese of mti [A] Step : Use Equtio (8.) to fid the elemets of the co-fcto mti, [C] with its elemets evluted y the fomul: ' ) ( A c j i ij whee ' A is the equivlet detemit of mti [A ] tht hs ll elemets of [A] ecludig those i i th ow d j th colum.

17 ( ) ( )( ) ( )( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( )( ) c c c c c c c c c We thus hve the co-fcto mti, [C] i the fom: C

18 Step : Tspose the [C] mti: T T C Step : The ivese of mti [A] thus tkes the fom: A C A T Check if the esult is coect, i.e., [A] [A] - [ I ]? [] I A A

19 Solutio of Simulteous Equtios Usig Mti Alge A vitl tool fo solvig vey lge ume of simulteous equtios

20 Why huge ume of simulteous equtios i this type of lyses? Numeicl lyses, such s the fiite elemet method (FEM) d fiite diffeece method (FDM) e two effective d poweful lyticl tools fo egieeig lysis i i el ut comple situtios fo: Mechicl stess d defomtio lyses of mchies d stuctues Themofluid lyses fo tempetue distiutios i solids, d fluid flow ehvio equiig solutios i pessue dops d locl velocity, s well s fluid-iduced foces The essece of FEM d FDM is to DISCRETIZE el stuctues o flow pttes of comple cofigutios d lodig/oudy coditios ito FINITE ume of su-compoets (clled elemets) ite-coected t commo NODES Alyses e pefomed i idividul ELEMENTS isted of etie comple solids o flow pttes Emple of discetiztio of pisto i itel comustio egie d the esults i stess distiutios i pisto d coectig od: Pisto FE lysis esults Rel pisto Coectig od Discetized pisto fo FEM lysis Distiutio of stesses FEM o FDM lyses esult i oe lgeic equtio fo evey NODE i the discetized model Imgie the totl ume of (simulteous) equtios eed to e solved!! Alyses usig FEM equiig solutios of tes of thousds simulteous equtios e ot uusul.

21 Solutio of Simulteous Equtios Usig Ivese Mti Techique Let us epess the -simulteous equtios to e solved i the followig fom: m m m m (8.6) whee,,, m e costt coefficiets,,., e the ukows to e solved,,., e the esultt costts

22 The -simulteous equtios i Equtio (8.6) c e epessed i mti fom s: m m m m (8.7) o i evite fom: [A]{} {} (8.8) i which [A] Coefficiet mti with m-ows d -colums {} Ukow mti, colum mti {} Resultt mti, colum mti Now, if we let [A] - the ivese mti of [A], d multiply this [A] - o oth sides of Equtio (8.8), we will get: [A] - [A]{} [A] - {} Ledig to: [ I ] {} [A] - {},i which [ I ] uity mti The ukow mti, d thus the vlues of the ukow qutities,,,, my e otied y the followig eltio: {} [A] - {} (8.9)

23 Emple 8.6 Solve the followig simulteous equtio usig mti ivesio techique; - Let us epess the ove equtios i mti fom: [A] {} {} whee A {} {} d d Followig the pocedue peseted i Sectio 8.5, we my deive the ivese mti [A] - to e: 9 A Thus, y usig Equtio (8.9), we hve: {} {} 8 ) )( ( A fom which we solve fo d

24 Solutio of Simulteous Equtios Usig Gussi Elimitio Method Joh Cl Fiedich Guss ( ) A Gem stoome (plet oitig), Physicist (molecul od theoy, mgetic theoy, etc.), d Mthemtici (diffeetil geomety, Gussio distiutio i sttistics Gussio elimitio method, etc.) Gussi elimitio method d its deivtives, e.g., Gussi-Jod elimitio method d Gussi-Siedel itetio method e widely used i solvig lge ume of simulteous equtios s equied i my mode-dy umeicl lyses, such s FEM d FDM s metioed elie. The picipl eso fo Gussi elimitio method eig popul i this type of pplictios is the fomultios i the solutio pocedue c e edily pogmmed usig cocuet pogmmig lguges such s FORTRAN fo digitl computes with high computtiol efficiecies

25 The essece of Gussi elimitio method: ) To covet the sque coefficiet mti [A] of set of simulteous equtios ito the fom of Uppe tigul mti i Equtio (8.5) usig elimitio pocedue [ A ] pocess Vi elimitio A uppe ' ' '' ) The lst ukow qutity i the coveted uppe tigul mti i the simulteous equtios ecomes immeditely ville. ' ' '' ' '' / ) The secod lst ukow qutity my e otied y sustitutig the ewly foud umeicl vlue of the lst ukow qutity ito the secod lst equtio: ' ' ' ' ' ' ) The emiig ukow qutities my e otied y the simil pocedue, which is temed s ck sustitutio

26 The Gussi Elimitio Pocess: We will demostte this pocess y the solutio of -simulteous equtios: We will epess Equtio (8.) i mti fom: (8.,,c) o i simple fom: [ A ]{ } { } 8.) We my epess the ukow i Equtio (8.) i tems of d s follows:

27 Now, if we sustitute i Equtio (8. d c) y we will tu Equtio (8.) fom: (8.) You do ot see i the ew Equtio ( d c) ymoe So, is elimited i these equtios fte Step elimitio The ew mti fom of the simulteous equtios hs the fom: (8.) The ide umes ( ) idictes elimitio step i the ove epessios

28 Step elimitio ivolve the epessio of i Equtio (8.) i tem of : fom to (8.) d sumitted it ito Equtio (8.c), esultig i elimite i tht equtio. The mti fom of the oigil simulteous equtios ow tkes the fom: (8.) We otice the coefficiet mti [A] ow hs ecome uppe tigul mti, fom which we hve the solutio The othe two ukows d my e otied y the ck sustitutio pocess fom Equtio (8.),such s:

29 Recuece eltios fo Gussi elimitio pocess: Give geel fom of -simulteous equtios: m m m The followig ecuece eltios c e used i Gussi elimitio pocess: Fo elimitio: i > d j> Fo ck sustitutio j ij ij i i i i i i ij j j i m with i,,..., ii (8.6) (8.5) (8.5) (8.6)

30 Emple Solve the followig simulteous equtios usig Gussi elimitio method: z y z y z Epess the ove equtios i mti fom: () y z If we compe Equtio () with the followig typicl mti epessio of -simulteous equtios: we will hve the followig: () d

31 Let us use the ecuece eltioships fo the elimitio pocess i Equtio (8.5): j ij ij i Step, so i, d j, Fo i, j d : i i i with I > d j> i, j : i, j : i : Fo i, j d : i, j : i, j : o i :

32 So, the oigil simulteous equtios fte Step elimitio hve the fom: We ow hve:

33 Step, so i d j (i >, j > ) i d j : ( ) ( ) The coefficiet mti [A] hs ow ee tigulized, d the oigil simulteous equtios hs ee tsfomed ito the fom: We get fom the lst equtio with () (), fom which we solve fo. The othe two ukows d c e otied y ck sustitutio of usig Equtio (8.6): ( ) ( )( ) / / / j j j d ( ) ( ) ( ) { } / / / j j j We thus hve the solutio: ; y - d z