Numerical Analysis of Engineering Systems

Transcription

1 Itroductio to tri ysis Lrry Cretto echic Egieerig Numeric ysis of Egieerig Systems Ferury 7, Outie Why mtrices, sic defiitios/terms tri mutipictio Determits Iverse of tri Simuteous ier equtios: mtri from d Gussi eimitio soutio tri r d its roe i determiig type of soutios uique, ifiite, oe) Eigevues d eigevectors Why trices Simpified ottio for geer sttemets out mthemtic or egieerig systems Hve mutipe compoets tht re iterreted Eemets i mchie structure Sprigs, msses d dmpers i virtig system tri ottio provides geer retioships mog compoets tri Empe Simpe ige of two sprigs with sprig costt d dispcemets u t idividu poits Idividu ods, P, reted to idividu dispcemets, u, y mtri equtio show eow. P= P P P = + u u u = KU tri Bsics m m m m rry of umers with rows d m coums Compoets re row)coum) Size of mtri m) or y m) is umer of rows d coums Row d Coum trices trices with oy oe row or oy oe coum re ced row or coum mtrices sometimes ced row or coum vectors) The row umer is usuy dropped i row mtrices s is c c the coum umer i c c coum mtrices r r r r r r r r m m r c c c c c E Numeric ysis of Egieerig Systems

2 ore tri Bsics Two mtrices re equ e.g., = B) If oth d B hve the sme size rows d coums) If ech compoet of is the sme s the correspodig compoet of B i = i for i d ) squre mtri hs the sme umer of rows d coums digo mtri, D, hs zeros ecept for the pricip digo Digo tri Compoets of re i d i, where d i is the The digo mtri is squre mtri with ozero compoets oy o the pricip digo Kroeecer det i di i 7 8 tri Opertios C dd or sutrct mtrices if they re the sme size C = B oy vid if, B, d C hve the sme size rows d coums) Compoets of C, c i = i i utipictio y scr: C = C d hve the sme size rows d coums) Compoets of C, c i = i For scr divisio, C = /, c i = i / Nu ) d Uit I) trices For y mtri,, + = + = ; I = I = d = = The uit or idetity) mtri is squre mtri; the u mtri, which eed ot e squre, is sometimes writte m) I Trspose of tri Trspose of deoted s T Reverse rows d coums; for B = T i = i If is m), B = T is m y ) I TLB the postrophe is used to costruct the trspose: B = T tri utipictio Preview Not ituitive opertio. Loo t two coordite trsformtios s empe y z y y y ] z [ y Sustitute equtios for y i terms of ito equtios for z d z z [ ] [ ] z [ ] y E Numeric ysis of Egieerig Systems

3 tri utipictio Preview II Rerrge st set of equtios to get direct trsformtio from to z z [ z [ c c [ ] [ ] [ [ ci i ] ] ] c c ] c c c c [ [ i,;,) ] ] tri utipictio Preview III Coefficiets s mtri compoets c C c B c c C = B if ci i c C c Geerize this to y mtri size c c i,;,) utipyig trices For geer mtri mutipictio, C = B p hs rows d p coums ci B hs p rows d m coums i C hs rows d m coums i, ;, m) is eft; B is right; C is product For C = B, we get c i y ddig products of terms i row i of eft mtri) y terms i coum of B right mtri) c i = i + i + i + i + I geer, B B Geer tri utipictio For mtri mutipictio, C = B hs rows d p coums ci B hs p rows d m coums C hs rows d m coums i is eft; B is right; C is product Empe ) ) ) B ) ) ) p i B ) ) ) 7 ) ) ), ;, m) B) i is product of row i of times coum of B tri utipictio Eercise Cosider the foowig mtrices B C you fid B, B or oth? We c fid B, ecuse hs three coums d B hs three rows We cot fid B ecuse B hs four coums d hs three rows Net chrt strts the process of fidig compoets of B 7 tri utipictio Eercise II Wht is the size of C = B? B C = B hs three rows ie d four coums ie B) Wht is c? c = )) + )) + )) = 8 8 E Numeric ysis of Egieerig Systems

4 tri utipictio Eercise III Fid c, c, c, d c i C = B B c = )) + )) + )) = 8 c = )) + )) + )) = c = )-) + )) 8 + )-) = - c = )) + C )) + )) = tri utipictio Eercise IV Fid c, c, c, d c i C = B B c = )) + -)) + )) = 7 c = )) + -)) + )) = c = )-) + -)) + )-) = - 8 C 7 c = )) + -)) + )) = tri utipictio Eercise V Fid c, c, c, d c i C = B B c = )) + )) + )) = c = )) + )) + )) = c = )-) + )) + )-) = - 8 c = )) C + )) 7 + )) = tri utipictio Resuts Soutio for mtri product C = B B 8 C 7 TLB/Ece trices Hve see term-y-term opertios o rrys, +B, -B,.*B,./B d.^ tri opertios re give y foowig: +B, -B, *B, /B, \B d ^ TLB =. ^ = ^: = = 7 Ece hs mmut rry fuctio to mutipy two rrys Determits Loos ie mtri ut is t mtri squre rry of umers with rue for computig sige vue for the rry Empe t right shows ccutio of, the determit of mtri Det E Numeric ysis of Egieerig Systems

5 ore Determits Usefu i otiig geric epressios for mtri opertios, ut ot usefu for umeric computtio Equtio for y determit Det Det determit hs iors, i, otied y deetig row i d coum Cofctors, i = -) i+ i used i geer epressios for determits Geer Rue for Determits y size determit c e evuted y y of the foowig equtios Det i ) i i i ) i i C pic y row or y coum Choose row or coum with most zeros to simpify ccutios C ppy equtio recursivey; evute determit s sum of determits the get s i terms of s i i i i i i Determit Behvior determit is zero if y row or y coum cotis zeros. If oe row or oe coum of determit is mutipied y costt,, the vue of the determit is mutipied y the sme costt. Note the impictio for mtrices: if mtri is mutipied y costt,, the ech mtri eemet is mutipied y. If is mtri, =. 7 Determit Behvior II If oe row or oe coum) of determit is repced y ier comitio of tht row or coum) with other row or coum), the vue of the determit is ot chged. If two rows or two coums) of determit re iery depedet the vue of the determit is zero. 8 Determit Behvior III The determit of the product of two mtrices, d B is the product of the determits of the idividu mtrices: B) = B). The determit of trsposed mtri is the sme s the determit of the origi mtri: T ) =. TLB: det gives Ece: mdeterm; video for Ece mtri Iverse of tri For squre mtri,, iverse mtri, - my eist such tht - = - = I For the geric equtio =, = - For the mtri equtio =, = - Just s = - is ot vid if =, = - is ot vid if - does ot eist - does ot eist if Det = The iverse is importt cocept i ysis of ier systems, ut is ot used i computtio wor E Numeric ysis of Egieerig Systems

6 Formu for Iverse of tri Iverse of tri Fid the compoets of B = -, i, from determit of d its cofctors i i i If B, i ) Use to get geric equtios for compoets of iverse mtri tri computtios, if ecessry, oti compoets y tertive umeric gorithms If B ) ), i i i ) i ) ) ppy Iverse of tri i ) i B ) ), i ) ) ) ) ) ) ) TLB/Ece Iverse TLB: use B = iv to get B = - TLB c use C=B/ to get C = - B Ece fuctio miverse computes iverse Seect empty ce re sme size s mtri Eter formu =miverse<mtri ces>) Press cotro+shift+eter Origi mtri i ces :B rry formu =miverse:b) i ces C:D rry formu =mmut:b,c:d) i ces E:F Simuteous Equtio Bsics set of simuteous ier geric equtios my hve sige uique) soutio No soutio ifiite umer of soutios ier comitio of y two equtios c repce oe of the equtios d ot chge the soutio Eq I Eq II 8 8 Eq II) * Eq I) Simuteous Equtios The secod coum is equivet set of equtios tht is ier comitio of the equtios i the first coum 8 y ) No soutio E Numeric ysis of Egieerig Systems

7 Gettig to tri Form Empe of equtios uows) + 7y z = 8 y + z = y z = How c we deveop geer ottio for N equtios i N uows? C vries,, etc. C right hd side,,, etc. C top row coefficiets,,, etc. Coefficiet of i equtio i is i 7 Stdrd Form N- N- + N N = N- N- + N N = N- N- + N N = N-, + N-, N-,N N = N- N + N + N NN N = N Usu suscripts o re row,coum Row is equtio d coum is uow N c e y umer 8 Compct Stdrd Forms = N i i i,, N Equtios defied y dt: N, i, d i Summtio is sme s mtri mutipictio formu i coefficiets re mtri, Right-hd side,, d uows,, re coum vectors Empe i Stdrd Form Previous empe of equtios N = ) + 7y z = 8 y + z = y z = I stdrd form: is, y is, d z is =, = 7, = -,, = 8 =, = -, =,, = - = 8, =, = -,, = Empe i Stdrd Form Previous empe of N = equtios s = 8 + 7y z = 8 y + z = y z = 7 8 Empe i Stdrd Form utipy d s show eow = = Fi equ sig for two ) mtrices gives origi form of three simuteous equtios + 7y z = 8 y + z = y z = 8 E Numeric ysis of Egieerig Systems 7

8 E Numeric ysis of Egieerig Systems 8 Sove = i TLB/Ece TLB coud use = iv * Preferred pproch is = \ which is fster d more ccurte for sovig = Ece: Seect coum for soutio the eter rry formu: mmutmiverse< ces>), < ces>) rry formu i e:e is: =mmutmiverse:c),d:d) rry formu i f:f is: =mmut:c, e:e)-d:d Sovig = Kow the i ) d i ) Wt the uows i ) Geer System for = m m m m m m) m ) ) equtios d m uows? How c this e? We epect m = First we hve to see if the equtios re rey idepedet equtios Systems for m > hve ifiite umer of soutios Systems for > m c e soved i est squres sese Provide soutio tht hs est error i soutio: miimize i= i = m i 7 Guss Eimitio Prctic too for otiig soutios ytic too for determiig ier depedece or idepedece Bsic ide is to mipute the equtios or dt) to me them esier to sove without chgig the resuts Systemticy crete zeros i ower eft prt of the equtios or dt) 8 Upper Trigur Form UTF) Covert origi set of equtios to UTF

9 Guss Eimitio III Upper trigur form o previous side is esiy soved y c sustitutio = / - = - - )/ - -, et ceter Geer equtio for c sustitutio i i i i ii i,,,, Covetio: If ower ide is greter th upper ide i S Guss Eimitio gorithm How do we get the upper trigur form? Wor o ugmeted mtri [, ] m opertor, do ot eecute sum m m m Guss Eimitio gorithm II Repetedy repce rows y ier comitio of two rows tht produces zero i desired row/coum comitio First step: me of coum eow row zero sets r = ecept ) Secod step: me of coum eow row zero sets r = for r > ) Cotiue this id of step for rows ecept the st row Row eig sutrcted is ced pivot row Geer Guss Eimitio Use ech row from row to row - s the pivot row Digo eemet o pivot row is pivot,pivot For ech row row r) eow the pivot row utipy tht row y row,pivot / pivot,pivot Sutrct resut from row r to me row,pivot = Opertio requires sutrctio for ech coum of right of pivot coum d for Repet for ech row eow pivot row Repet for rows to - s pivot rows Repce eistig rry with resuts of ew opertios Geer Guss Eimitio II Opertios with row s pivot row Repce RowR y {RowR R / )Row} m m m m m m m m row, pivot row, coum row, coum pivot, coum pivot, pivot For pivot to For Guss Pseudocode row pivot to For coum pivot to row, coum row, coum row, pivot row row pivot pivot, pivot row, pivot pivot, pivot pivot, coum Opertios o row re the sme s opertios woud e o row,+ E Numeric ysis of Egieerig Systems

10 Soutio Detis Sove the set i) of equtios ) o the right ii 7 8 ) iii Sutrct / times i) from equtio ii) d 7/ times i) from iii) ) ) ) ) 8 ) ) Resut from first set of opertios Sutrct 7/-) times ii) from iii) Fi uppertrigur form ore Detis 8 7 Uecessry opertios 7 7 ) 7 7 ) 8) Fi uppertrigur form Bc Sustitutio We see tht = d c sustitutio gives d s foows 8 8 ) 8 ) ) ) 7 Do we hve soutio to =? swer to questio sed o the r which is defied s the umer of iery idepedet rows or coums Use Guss eimitio to determie r Use Guss eimitio to covert mtri to upper-trigur form I this form, r is umer of rows with o-zero coefficiets This is sometimes ced row-echeo form TLB hs fuctio rmtri) 8 Wht is row-echeo form? ppy Guss eimitio to get zeros eow row oe i coum oe zeros eow row two i coum two Keep this up uti you get to the fi row or uti there re o more rows with ozeros Cout umer of rows tht re ot zeros; this is the r This is wy to determie ier idepedece of set of vectors Wht is the r of ech mtri? E Numeric ysis of Egieerig Systems

11 E Numeric ysis of Egieerig Systems Fidig R Wht is r of mtri,? Wht is mimum possie vue for r? Lower right mtri is resut of ppyig Guss eimitio to Wht is its r? R = Soutios to = For system of uows If r = r [ ] = there is uique soutio If r = r [ ] < there re ifiite umer of soutios If r r [ ] there re o soutios Three Empes No soutio Origi Equtios Trigurized Set Soutios First Empe R 8 7 Origi 8 7 Rowecheo form... Here we see tht r = r [ ] = umer of uows = so we hve uique soutio Secod Empe R Origi Rowecheo form r = r [ ] = which is ess th the umer of uows ) so we hve ifiite umer of soutios Third Empe R Origi 8 Rowecheo form Here, r = r [ ] = ; therefore we hve o soutios

12 Homogeous Equtios If =, i.e., ech i =, we utomticy hve r = r[ ] so we hve soutio If this r equs the umer of uows, we hve uique soutio, = If this r is ess th the umer of uows we hve ifiite umer of soutios 7 Homogeous Equtio Empe Equtios 8 [ ] Origi [ ] mtri 8 = - mtri 8 [ ] 7 Row-echeo form 8 R = r [ ] = < uows = so there re ifiite soutios 8 Homogeous Equtio Empe II Equtios 8 Origi [ ] mtri = + mtri 8 Row-echeo form [ ] [ ] 8.8 R = r [ ] = uows = so there is uique soutio = ) R d Determits Determit r, ie mtri r, is the umer of iery idepedet rows or coums. Two equivet sttemets: determit is zero if its rows re iery depedet the size of determit is rger th its r 7 Prctic Determit Evutio Use Guss eimitio d fid the product of the eemets o the digo determit does ot chge if oe row is repced y ier comitio of tht row with other row Guss eimitio coverts determit ito upper-trigur form without chgig its vue The determit of upper-trigur rry is the product of the compoets o the pricip digo 7 Upper Trigur Determit ) ) ) 7 E Numeric ysis of Egieerig Systems

13 Determit Sig Guss eimitio uses row swppig to reduce roud-off error If two rows i determit re swpped, the determit sig chges Det = = Det = = I Guss eimitio eep cout of the row swps, Swps; fid determit from digoized rry y the formu Swps N = Homogeous Ifiite Soutios Empe 8 Row-echeo form ~ 7 α = mtri 8 ~ Det Det ))) ) 8)) ) ) ) ))) ) )) ) 8) ) 8 8 Det = => soutio of my eist Homogeous Ifiite Soutios Empe 8 Row-echeo form ~.8 ~ Det ) ).8) = + mtri 8 Det ))) ) 8)) ) ) ) ))) ) )) ) 8) ) 8 8 Det d = mes = Crmer s Rue You my ie to use this for sm systems of equtios 7 7 R d Iverses Fidig - for mtri requires the soutio of = times, where is oe coum of the uit mtri We cot sove this equtio uess r = squre mtri wi ot hve iverse uess its r equs its size tertive sttemet is tht wi ot hve iverse if = 77 R d Iverses II Rec the geer resut for the eemets, i, of B = - i = i /, where i is the cofctor of i We see tht i is ot defied if Det = - does ot eist if Det = Det = for determit shows tht R < Det d r = : two equivet coditios for ) to hve iverse 78 E Numeric ysis of Egieerig Systems

14 Why Eigevues/Eigevectors I eectric d mechic etwors, provides fudmet frequecies Shows coordite trsformtios pproprite for physic proems Provides wy to epress etwor proem s digo mtri Trsformtios sed o eigevectors used i some soutios of = Eigevues d Eigevectors Bsic defiitio squre): = is eigevector, is eigevue Bsic ide is tht eigevector is speci vector of mtri ; mutipictio of y produces mutipied y costt = => = [ I] = Homogeous equtios; requires Det [ I] = for soutio other th = 7 8 Det[ I] Det Det[ I] = Det[ I] = produces th order equtio tht hs roots for. y hve dupicte roots for eigevues. 8 Two-y-two tri Eigevues Qudrtic equtio with two roots for eigevues ) ) Eigevue soutios ) ) ) ) Det 8 Two-y-two tri Eigevues Write ) Det s dd the two soutios to get ) ) utipy the two soutios to get ) ) ) ) ) Det Det 8 Sum d Product The resuts o the previous side ppy to mtrices The sum of the eigevues is the sum of the digo eemets of the mtri, ced the trce of the mtri The product of the eigevues is the determit of the mtri Trce Det i ii i i i i 8 E Numeric ysis of Egieerig Systems

15 Two-y-two tri Eigevectors Two-y-two Eigevectors II Two eigevectors: ) = [ ) ) ] T d ) = [ ) ) ] T ) = [ ) ) ] T ) Sustitute ech eigevue soutio,, ito I) = to fid ) compoets ) ) ) ) ) ) Nottio: y i is compoet i of vector y; z ) Eigevector equtios re homogeeous, so eigevectors re determied oy withi mutipictive costt Pic ) = ) ) ) ritrry) ) ) ) ) ) ) ) is oe of vector set with compoets z )i 8 8 How y Eigevues? mtri hs distict eigevues geric mutipicity of eigevue,, is the umer of roots of Det[ I] = tht hve the sme root, Geometric mutipicity, m, of eigevue is umer of iery idepedet eigevectors for this 87 X = Digoize tri For mtri it is possie to crete mtri, X, where ech coum is oe eigevector Oe c the show tht X - X = L, where L is digo mtri whose compoets re the eigevues Net ssigmet uses TLB to do this L = N N N N N N λ λ λ N 88 Suppemet teris Items ot ped for i-css coverge tri equtios for coordite trsforms Derivtio of determit formu from geer determit formu Empe of iverse ccutio for mtri with my zeros) Empe of eigevue d eigevector ccutios rry rry Coordite trsformtios Rec previous equtios y z y y B y z y z y z 8 z y y y Defie mtrices so tht y = d z = By E Numeric ysis of Egieerig Systems

16 Coordite trsformtios II Coordite Trsformtios III Show tht mtri defiitios give trsformtio resuts y z y y y y y z z By z z y y y y y y y y y From mtri equtios y = d z = By, we hve z = B = C with C = B c c B C c c c C c c c z c z C z c c c c c c c Empe of Geer Rue Get determit of mtri y epsio og st row ) ) ) ) Empe of Geer Rue II Get determit of mtri ) ) ) Empe Proem Empe Proem Det Fid - for t right Hve the foowig formu for B = - i ) i i Geer determit formu: S i i Te sum over third row to simpify ccutio of Det Det = -) + + -) + + -) + + -) + Det ) ) Det ))) ))) ))) ))) ))) ))) E Numeric ysis of Egieerig Systems

17 Empe Proem III ppy: i = -) i+ i /Det Det = so i = -) i+ i Empes show eow ) Remove Remove for for 8 ) 7 Empe Proem Soutio We c show tht - = - = I - ) = + + -) + = - = + + -) + -) = Oy eft to chec 8 Two-y-two Empe Fid eigevues d eigevectors of Det[ I] Det[ I] ) ) )) Soutios re = d = Two-y-two Empe Cotiued Fid ) compoets for = Sove [ I] = for ) compoets ) ) ) ) ) ) ) ) Oe equtio i two uows ) ) Pic ) = the ) = from first equtio Eigevector ) is [ ] T Two-y-two Empe Cocuded Net fid ) compoets for = Sme s pproch for fidig ) ) ) ) ) ) ) ) ) Both equtios give ) = ) ) Pic ) = ) cot e determied) ) = [ ] T Chec Two-y-two Empe ) ) ) ) ) ) ) ) ) ) ) ) )) ) ) )) ) ) E Numeric ysis of Egieerig Systems 7

18 E Numeric ysis of Egieerig Systems 8 Eigevector Fctors empe showed ) = ) regrdess of choice of d This is geer resut We c pic oe eigevector compoet; typic choices re to me eigevector simpe or uit vector ) ) ) ) ) utipe Eigevue Empe I ) ) ) ) )) ) ) ) ) )) ) ) ) ) ) ) ) I Det utipe Eigevue Empe II ) ) ) ) I Det Soutios re =,, = hs geric mutipicity of Fid eigevectors) from I ) ) = ) ) ) Loo t = utipe Eigevue Empe III ) ) ) ppy Guss eimitio to these equtios ) ) ) ) ) ) ) ) ) Pic ) d ) 7 utipe Eigevue Empe IV Two iery idepedet eigevectors for = - ) ) ) Pic ) = d ) = => ) = Pic ) = d ) = => ) = - ) ) ) Pic ) d ) the 8 Cotiue Empe for = 7 ) ) ) ppy Guss eimitio to these equtios ) ) ) ) ) I ) ) ) Pic ) = => ) =-

19 Empe Resuts Eigevector Lier Depedece ) ) ) ) ) ) ) ) Eigevues = -, = -, = hve eigevectors show eow C we hve ) + ) + ) = without = = =? Homogeous equtios hve = = = if mtri hs fu r Lier Depedece II tri hs fu r it its determit is ot zero ))) )) ) Det ) ) ) )) ) ) )) )) ) Sice determit is ot zero, the oy soutio is = = =, so eigevectors re iery idepedet Quiz Three Soutios. Resuts of TLB commds >> = [ ; ; ; ] = >> B = [ ; 7 8; ] B = 7 8 >> C = [ B] Error: d B must hve the sme umer of rows >> D = [:,) B:,)] D = 8 Quiz Three Soutios II. Write TLB Commds E = E = [ - ; 7 - ] 7 7 F = F = [ 7; - ; ]. Resuts of series of two commds >> t = : >> = ogt).^ t ) >> t = :pi/:pi >> y = cost)./ ept) >> t = :: >> z = sumt) og)), og)), og)), og)), og)) cos)/e, cosp/)/e p/, cosp/)/e p/, cosp/)/e p/,cosp)/e p z = = Quiz Three Soutios III. Resuts of ) G = [E; [F F]] d ) H = [F; E ]? 7 7 G = 7 7 H = 7. G rry fter the commd G:,:) = F? 7 G = 7 E Numeric ysis of Egieerig Systems

20 First Progrmmig ssigmet Numer of studets: imum possie score: e:.7 edi: Stdrd devitio:. Grde distriutio: Commets o ssigmet Rge of methods cosidered Ce formus re simpest d quicest Use user defied fuctios UDF) whe you hve repeted ccutios d you wt to void errors i reeterig formus Rge mes etter idetify vries ot so importt for simpe ccutios, ut usefu for more compe woroos Commets o ssigmet II rry fuctio d mcro require ot of codig which is ustified oy if ccutio is repeted y sever users With rry fuctio mutipe tes c e pced o worsheet, ech drive y seprte iput ces Difficut to specify ect umer of ces y seectio cro requires recodig to use other iputs or pce resuts i other octios Uses ect specifictio of umer of ces 7 Secod Quiz Numer of studets: imum possie score: e:. edi:. Stdrd devitio:. Grde distriutio: Commets o Secod Quiz Cofusio over trspose TLB deotes trspose usuy T ) For =, T = Epressio cost).^t-) is sme s cost)).^t-) NOT cos[t).^t-) Sum ), where is D rry gives the sum of eemets i For D rry it gives sum of ech coum Secod Progrm Numer of studets: imum possie score: e:. edi: 7. Stdrd devitio: 8.8 Grde distriutio: E Numeric ysis of Egieerig Systems

21 Secod Progrm Commets Do ot geerte or do ot copy output or rge rrys Differeces etwee scripts d fuctios Fuctios receive vrie vues through rgumet ist, ust s fuctios do i other guges ie VB Vries i scripts, tht re ot set i the script, use the curret vue of the vrie i the worspce TLB vs. Ece/VB We hve focused o use of TLB from commd widow This is simir to eterig dt o the Ece spredsheet Less focus o TLB progrmmig This is compre to VB y studets prefer Ece ecuse of fmiirity Ece/VB more rediy vie, especiy i sm compies Quiz Three Soutios Iiti guesses show tht + = d - = ecuse f) > d f) < ew = f =. + f + f.88. =. f ew ) < so ew repces - ew =.. reerr = =. =.8 >.: cotiue f ew ) < so ew repces - Quiz Three Soutios II ew =.. reerr =... ew = =. =. >.: cotiue f ew ) < so ew repces - reerr = =. =. <.: fiished =. Review Lst Lecture Discussio of mtrices: row)coum) Row, coum, digo, squre, uiti), u) = B, ± B,, / scr ), T tri mutipictio: P = LR q Coums i L = rows i R = q p i = = i r P rows = L rows; P coums = R coums utipy row of L y coum of R Determits: sige umer for rectgur rry; formu depeds o rry size Iverse, - : - = - = I Review Lst Lecture II Use TLB commds det d iv for determit d iverse Use Ece formus mmut, B) for mtri mutipictio, mdeterm for determit, d miverse for iverse miverse d mmut re rry formus Simuteous ier equtios i geer mtri form, = N i = i = Vries,,, Coefficiets i mutipies i equtio i Right hd side i i equtio i E Numeric ysis of Egieerig Systems

22 Review Lst Lecture III Lier depedece: if oe equtio i system of equtios is ier comitio of oe or more other equtios, the system is sid to e iery depedet system tht is ot iery depedet is iery idepedet I mtri form equtio is row tri r = umer of iery idepedet rows = umer of iery idepedet coums TLB formu r for mtri r Review Lst Lecture IV Soutio of system of ier geric equtios, =, with uows depeds of r of d [ ] Uique soutio: r = r[ ]) = Ifiite soutios: r = r[ ]) < No soutios: r r[ ]) Gussi eimitio: process to get mtri for system of equtios i upper trigur form for simpe c sustitutio so used to determie r Wi review i deti ter 7 8 E Numeric ysis of Egieerig Systems