ࡱ> `!)zqr+??NJ XJxcdd``> @c112BYL%bpu;vv0o8L+KRsnePdhB 0Cbd`+XNI`!hh 'a V2  &XJxcdd``db``baV d,FYzP1n:&&^! KA?H1Zx ㆪaM,,He`c&,eBܤ9~31ܘ!+(|-oI3d*F\u m#dn? J"y ;!2]J{Iq Nd6c ebAu錧ĺq7D sm`aS 㗱p,8Ly;#俐br< 7=W@|C8U ?3 9 *Q{ |.h@pp@;g+KRs A`PdhB a Հ`!?C4͹@q8m`PH xcdd`` @c112BYL%bpuRS.i @2RxڕRJ@}ID<"ŋţ^$t$D$*Zlo*m:Q I"38'lY<*Oq,ZpQŽӌxyκ}7l+VX(}?vV!~h~fAhy<\_9K+WMYbx~*oS^}И"ACVs;hBVu4ԫPWݷBZjR143p1ܣN]`!(@9_`<# RXJxcdd``> @c112BYL%bpugB8b/# , #lpIEM&sp8>B7,'޽2 \ =8 BBӞ;+KRsePdhB ~`م`!\ZN=>o^ `x *xڝ=hAߛٯy*X"jJ8HR9Dq$V6BP++  STlDQD:þ`1̼1W+#XDB3NO-xAp@y?^NωHkb<-9jU}AU{_zd~7]?_spcxz_~ϡQӶ\'-d}< zըe|O๱.p"' B}[WM5 /$/ +vm]^[p]a^`!JCY\^ ޟ!ڹ!xڥkPg&MM]hl-<]H馮UZAxaA^< EdK)Tl|o^^-HMxa>7o(Rjbb!qHٙΡ-{:70'!bpwJv)b7+ A~(40(8G}3UV,M~O8C pٿ6;K=xtQ n5'Z#ӯOяA)(nnH`$y1=\2E(?dT_t}7M8(O{h!/NGP=ɫ_Xyk q^-OlQ[+:ǵ~=W7%y9ŧ(e_g=?|L"ɕ( ɵiDs9`}uԼ)Otc$_=h=d{|1Q;y? Jn oN^\8$:;Z^ 0#q68@~z_C V`!@~_Bm3q2@`0xڥT;,CQқT0\!BfHDTP$,BX,ubi"bQ,E-$zssns b/حt e7ZyaI}l&S1Mcmx7/dL\ZΚkY98uW?;TAz%ыw\"G?ԃ{c=%6q}^UozT}UH}cym8rx$>"nkxڥ/A߼VmiAXAW6z!/Q(¡w"E$5fvS&nf>;73 Z' :@<.>h[ȘZE kU|:4n639uG)0㇩hn%Rĸx7f2w &fnw4-K u`$eXޜM')_a/yl^ޒ9k=B~<.y=''xU'U!%l>YGp*^ e=ϿՃԫNJIwqlpx}UIRW%sRz `$do߄YB͸EDU͸Lc=M4~Md^O'(xV=ؿ2D>3g^\=J\7M36,Z@tN!vX.]Cb: AuO4 `!'Uُ\B`"f`!xcdd``$d@9`,&FF(`Tic A?d=bAm@ UXRY`,@1[)!Ar؅,L ! ~ Ay 7. _=ӖCdeYQuՌ`S!v0y{ddbR ,.Ie`ԡ)En&0h3X?O=`![ qhq%@ PVZxcdd``f 2 ĜL0##0KQ* WفSRcgbR vx@f0@0&dT0pE1At 2Bab`RfFv;35v! @Hfnj_jBP~nbÍB4u`tUB=l|d~>*.X>f^q;dBr%AI9 \I*τpb.KYAF\$ / 8MNLLJ% w0u(2t58b@33X?Nq+`!zC3(| aghxcdd`` @c112BYL%bputI9@B\_> N>pguL mb ')n{1W̓l ;9 yCsLLJ% SA0u(2tA4!t?3e7zu`!\Q+WQ0n^hv` hZxcdd``b 2 ĜL0##0KQ* W!d3H1)fYˀX(T UXRYPd,@1[)fRcűX@n{# &br<??bCak@A*D`03M!kdL@(\R˻NCaڳ ; Ű",`nO_.pnv0o8021)W2ԡ"b@#3X?ʀ`!(.BjρBvݒ@|xcdd``> @c112BYL%bpu @c112BYL%bpu;vv0o8L+KRsePdhB 0Cbd`L`!4qkZ8cXsBJ`h0xcdd`` @c112BYL%bpu @c112BYL%bpuVdw ⅑x 1JyP͉͟* NJ^b> L I90Vy&Dd1*ӭ ܞt 矔mf1T PmA|]8;y9POgGߎ ķ 깠'SH0y{e #RpeqIj.(:@ !4 ߿3`!5b_چJ:r`h0xcdd`` @c112BYL%bpuFnoJ.!o `p121)W2<ԡ-υ`~ /Q]`! %7̾~XA;_Frxڥ+DQ=1oyYHϏD~,d&xel$ 6,l% eCYXXHɏf)@yndǛNϽs=y@@@=E4BgHt]g[ K]Ab4D\&vhlZ;T4K͏'l`  4Z]y3RZ[O{ABg gbڊ @c112BYL%bpu @c112BYL%bpu;vv0o8L+KRsePdhB 0Cbd`lM`!)FJe 䢡 U2@H|xcdd``> @c112BYL%bpu @c112BYL%bpuɼ{pv>ae]:&;bAL'I>3}5+PS#?tEu:(Qg=T,%Hl>^Ϫ&)U~~*w1eCxcB9Iq*5 1"gp B|,{}`!CKD D5|~H?Hxcdd`` @c112BYL%bpu ~E20DIOإ\28' F&&\' @ ] @2EJ|p`!/ GhDLr^R$ jXu#8xcdd`` @c112BYL%bpu^PL f%$MDfd, l8$\Dl9[IkRi8O~/8Zqޛw=FWm gaީnm!:7 =sRϘ(34tKK){k˻aPW_O@_>,YpX3゗s>c:d9~#gq9] uo]"E֘Ώt9DvSօk9ҩR\o!Epܞl`/^܈x(  % FEquation Equation.30,Microsoft Equation 3.00FEquation Equation.30,Microsoft Equation 3.00u(FEquation Equation.30,Microsoft Equation 3.00+FEquation Equation.30,Microsoft Equation 3.002FEquation Equation.30,Microsoft Equation 3.003FEquation Equation.30,Microsoft Equation 3.007FEquation Equation.30,Microsoft Equation 3.009FEquation Equation.30,Microsoft Equation 3.00BFEquation Equation.30,Microsoft Equation 3.00DFEquation Equation.30,Microsoft Equation 3.00HFEquation Equation.30,Microsoft Equation 3.00LFEquation Equation.30,Microsoft Equation 3.00NFEquation Equation.30,Microsoft Equation 3.00PFEquation Equation.30,Microsoft Equation 3.00QFEquation Equation.30,Microsoft Equation 3.00SFEquation Equation.30,Microsoft Equation 3.00 UFEquation Equation.30,Microsoft Equation 3.00 ZFEquation Equation.30,Microsoft Equation 3.00[FEquation Equation.30,Microsoft Equation 3.00]FEquation Equation.30,Microsoft Equation 3.00`FEquation Equation.30,Microsoft Equation 3.00cFEquation Equation.30,Microsoft Equation 3.00dFEquation Equation.30,Microsoft Equation 3.00eFEquation Equation.30,Microsoft Equation 3.00fFEquation Equation.30,Microsoft Equation 3.00 gFEquation Equation.30,Microsoft Equation 3.00!hFEquation Equation.30,Microsoft Equation 3.00$iFEquation Equation.30,Microsoft Equation 3.00%kFEquation Equation.30,Microsoft Equation 3.00&lFEquation Equation.30,Microsoft Equation 3.00'mFEquation Equation.30,Microsoft Equation 3.00(nFEquation Equation.30,Microsoft Equation 3.00+oFEquation Equation.30,Microsoft Equation 3.006uFEquation Equation.30,Microsoft Equation 3.007vFEquation Equation.30,Microsoft Equation 3.00<wFEquation Equation.30,Microsoft Equation 3.00OFEquation Equation.30,Microsoft Equation 3.00WFEquation Equation.30,Microsoft Equation 3.00ZFEquation Equation.30,Microsoft Equation 3.00]FEquation Equation.30,Microsoft Equation 3.00^FEquation Equation.30,Microsoft Equation 3.00bFEquation Equation.30,Microsoft Equation 3.00dFEquation Equation.30,Microsoft Equation 3.00gFEquation Equation.30,Microsoft Equation 3.00hFEquation Equation.30,Microsoft Equation 3.00oFEquation Equation.30,Microsoft Equation 3.00rFEquation Equation.30,Microsoft Equation 3.00uFEquation Equation.30,Microsoft Equation 3.00xFEquation Equation.30,Microsoft Equation 3.00{FEquation Equation.30,Microsoft Equation 3.00FEquation Equation.30,Microsoft Equation 3.00oFEquation Equation.30,Microsoft Equation 3.00pFEquation Equation.30,Microsoft Equation 3.00qFEquation Equation.30,Microsoft Equation 3.00rFEquation Equation.30,Microsoft Equation 3.00FEquation Equation.30,Microsoft Equation 3.00vFEquation Equation.30,Microsoft Equation 3.00wFEquation Equation.30,Microsoft Equation 3.00xFEquation Equation.30,Microsoft Equation 3.00FEquation Equation.30,Microsoft Equation 3.0/ 0DTimes New RomanF0Wo 0DArialNew RomanF0Wo 0" DWingdingsRomanF0Wo 00DPalatino Linotype0Wo 0@DMonotype Corsiva0Wo 0BPDSymbole Corsiva0Wo 0c(.2  @n?" dd@  @@``    .!                   $ ,      '        !            , 3'        )  ,h/ %+   O  g+   FJ '    !?(  2$zqr+??NJ12$hh 'a V12$C4͹@q8mG2$=f Cn"{F2$>RS.i`2$@9_`<#02$Z=!e珓2$ZN=>od 2$CY\^ ޟ!ڹR+ 2$@~_Bm3q2}2$oZ.4Ix2$Uُ\B`"/2$[ qhq%2$zC3(| agK2$\Q+WQ0n^hv"2$.BjρBv02$ 9Iպ6a12$qkZ8cXsBJ<2$Jdow<S2$s 5gqU=2$&f d$ۡޗ4 02$kpEÙfm;ߥ=SD2$b_چJ:r=@"2$%7̾~XA;_F}#2$&U1| OV%2$7G A@i=DA'2$Kˍ7M0$~0)2$|3N[Vd *2$Dyg= 21+2$FJe 䢡 U21-2$0 [/:K-1.2$i'pu'ٯe^/2$Int4Sb Q*02$KD D5|~H?K22$"s bj@X2_j32$MNdIɣ5eP52$?$G+FE*i62$M%oFCn82$Edj92$ EEȲB ;2$Gbv ekfi:<2$PcRkn?=2$ GhDLr^R$ 7*?$2$uh= AQ#aAc $0e0e     A@ A8c 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E|| @8 g4KdKd0 ~ppp@ G4BdBd0s ʚ;f"2ʚ;<4!d!dЁ 0<4ddddЁ 0<4BdBdЁ 00___PPT10 n___PPT9P/ 04=?D$ 0Essential Math for GamesO =W,B %Marq Singer (marq@essentialmath.com) & T              The Problem   3What are they Why do we care What are they good for4  / The Basics   Constraint  something that keeps an entity in the system from moving freely For our purposes, we will treat each discreet entity as one particle in a system Particles can be doors on hinges, bones in a skeleton, points on a piece of cloth, etc.Z   Box Constraints3Simplest case Movement constrained within a 2D areaF;Box Constraints -Box Constraints (cont)Restrict P to extents of the box Recover from violations in position (last valid, rebound, wrap around) Simple, yet the basis for the rest of thisBead on a Wire\The Problem: Restrict bead to path Solutions: Explicit (parametric) method Implicit method L  /  /W<Parametric Constraints DBead on a WireFrom Baraff, Witkin N = gradient f = force fc = constraint force f  = f + fc@     P .Implicit Representation3 legal position legal velocity legal acceleration43AImplicit Representation1Constraint force = gradient vector times scalar l,20Spring ConstraintsuSeems like a reasonable choice for soft body dynamics (cloth) In practice, not very useful Unstable, quickly explodesStiff ConstraintsA special spring case does work Ball and Stick/Tinkertoy Particles stay a fixed distance apart Basically an infinitely stiff spring Simple Not as prone to explode/ k!Cloth SimulationNUse stiff springs Solving constraints by relaxation Solve with a linear system0Cloth SimulationZ $2Cloth SimulationForces on our clothX=Cloth SimulationRelaxation is simple Infinitely rigid springs are stable Predetermine Ci distance between particles Apply forces (once per timestep) Calculate D for two particles If D != 0, move each particle half the distance If n = 2, you re done!9" 9  G),F3g$Relaxation MethodsZ Y>Relaxation MethodsZ %Cloth SimulationWhen n > 2, each particle s movement influenced by multiple particles Satisfying one constraint can invalidate another Multiple iterations stabilize system converging to approximate constraints Forces applied before iterations Fixed timestep (critical)&Z  "More Cloth SimulationUse less rigid constraints Vary the constraints in each direction (i.e. horizontal stronger than vertical) Warp and weft constraints#Still More Cloth Simulation Sheer Springsp@Still More Cloth Simulation Flex Springs &Articulated BodiesPin Joints Hinges Angular ConstraintsHRestrict the angle between particles Results in a cone-shaped constraintEAngular ConstraintsEUnilateral distance constraint Only apply constraint in one direction'Angular Constraints7Dot product constraint Recovery is a bit more involved!1 Stick ManUses points and hinges Angular (not shown) allow realistic orientations Graphic example of why I m an engineer and not an artist(Using A Linear SystemdCan sum up forces and constraints Represent as system of linear equations Solve using matrix methods, Basic Stuff}Systems of linear equations Where: A = matrix of coefficients x = column vector of variables b = column vector of solutionsT~%14 Basic StuffdPopulating matricies is a bit tricky, see [Boxerman] for a good example Isolating the ith equation::IZZW > $ *3Jacobi Iteration 3Solve for xi (assume other entries in x unchanged):(4 ( (55Jacobi Iteration In matrix form:CJacobi Iteration JDefinition (diagonal, strictly lower, strictly upper): A = D - L - UHKZ (*Gauss-Seidel Iteration+Uses previous results as they are available,,;6Gauss-Seidel IterationIn matrix form:<7Gauss-Seidel Iteration =8 Successive Over Relaxation (SOR)  eGauss-Seidel has convergence problems SOR is a modification of Gauss-Seidel Add a parameter to w G-S 0f_A9 Successive Over Relaxation (SOR)  T = a Gauss-Seidel iterate 0 < w < 2 If w = 1, simplifies to plain old Gauss-SeidelNU*G:Gauss-Seidel IterationIn matrix form: +!Lots More Math (not covered here)6I highly recommend [Shewchuk 1994] Steepest Descent Conjugate Gradient Newton s Method (in some cases) Hessian Newton variants (Discreet, Quasi, Truncated)$y  References   Boxerman, Eddy and Ascher, Uri, Decomposing Cloth, Eurographics/ACM SIGGRAPH Symposium on Computer Animation (2004) Eberly, David, Game Physics, Morgan Kaufmann, 2003. Jakobsen, Thomas, Advanced Character Physics, Gamasutra Game Physics Resource Guide Mathews, John H. and Fink, Kurtis K., Numerical Methods Using Matlab, 4th Edition, Prentice-Hall 2004 Shewchuk, Jonathan Richard, An Introduction to the Conjugate Gradient Method Without the Agonizing Pain, August 1994. http://www-2.cs.cmu.edu/~jrs/jrspapers.html Witken, Andrew, David Baraff, Michael Kass, SIGGRAPH Course Notes, Physically Based Modeling, SIGGRAPH 2002. Yu, David, The Physics That Brought Cel Damage to Life: A Case Study, GDC 2002 Z R +O>K9             5              &     8        !     |                                  g    )    ` f33` 3f3` ___>?" dd@,?" dd@  " @ `"  n?" dd@   @@``@n?" dd@  @@``PR    @ ` ` p>>D$ 0Essential Math for Games r(    Ngֳgֳ ?P  X Click to edit Master title style!!  2  Htgֳgֳ ?  RClick to edit Master text styles Second level Third level Fourth level Fifth level!    S    Z!1 ?``  b* 2    Z(1 ?`   X*(2    Z+1 ?`   Z*(2  N  6޽h? ? 33 Default DesignE D$ 0Essential Math for Games @ 3(     Njgֳgֳ ?p  X Click to edit Master title style!!    HXmgֳgֳ ? p  [#Click to edit Master subtitle style$$    ZX1 ?``  d* 2    Z\]1 ?`   \*(2    Ze1 ?`   \*(2  N  6޽h? ? 333 0 ( ~@K@   Br 1?P%  r X* 2     Br 1? % r Z*(2   p  0 ?*P  r,  Br 1? @2 r RClick to edit Master text styles Second level Third level Fourth level Fifth level!    S    H$lr 1?P  r X* 2     Hor 1?  r Z*(2   H  0h0 ? ̙3380___PPT10.QA9 XP( \@d@   Bxr 1?P%  r X* 2     B|r 1? % r Z*(2     HԀr 1?P  r X* 2     H܄r 1?  r Z*(2   H  0h0 ? ̙3380___PPT10.Qp {G VN0 (     NXtgֳgֳ ?0<  Constrained Dynamics,    Thugֳgֳ ?  J  H  0޽h ? 3380___PPT10.#\z  P@$(  @r @ S .P   r @ S   H @ 0޽h ? 33  ` $(   r  S  P   r  S D   H  0޽h ? 33  p8 $(  8 r 8 S  P   r 8 S +  H 8 0޽h ? 33    )(   x  c $EwP  w x  c $Fw  w v  N jJ?vh v2  NjJ? vB  NDjJ?  v2  NjJ?  ; @  HBw jJ?  7P 2   s *A ??, 8x d  Hw jJ? V  9100 2  Hw jJ?   70 2H  0޽h ? 33   0(   x  c $D # T 2  ?D # 4 ~B @D  NDo?vv~B AD  NDo?ss~B BD NDo?'s~B CD NDo? 2 ~B DD NDo?  2 ~B ED NDo?  2 ~B FD NDo? k 2 ~B GD NDo?n Q 2 ~B HD NDo?T 62 ~B ID NDo?:p2 JD  TjJ?}:2 KD  TjJ? } :T   LD # T 2  MD # 4 ~B ND  NDo?vv~B OD  NDo?ss~B PD NDo?'s~B QD NDo? 2 ~B RD NDo?  2 ~B SD NDo?  2 ~B TD NDo? k 2 ~B UD NDo?n Q 2 ~B VD NDo?T 62 ~B WD NDo?:p2 XD  TjJ?}:2 YD  TjJ? } :L  D # ZVT   D # T 2  D # 4 ~B D  NDo?vv~B D  NDo?ss~B D NDo?'s~B D NDo? 2 ~B D NDo?  2 ~B D NDo?  2 ~B D NDo? k 2 ~B D NDo?n Q 2 ~B D NDo?T 62 ~B D NDo?:p2 D  TjJ?}:2 D  TjJ? } :T   D # T 2  D # 4 ~B D  NDo?vv~B D  NDo?ss~B D NDo?'s~B D NDo? 2 ~B D NDo?  2 ~B D NDo?  2 ~B D NDo? k 2 ~B D NDo?n Q 2 ~B D NDo?T 62 ~B D NDo?:p2 D  TjJ?}:2 D  TjJ? } :T   D # T 2  D # 4 ~B D  NDo?vv~B D  NDo?ss~B D NDo?'s~B D NDo? 2 ~B D NDo?  2 ~B D NDo?  2 ~B D NDo? k 2 ~B D NDo?n Q 2 ~B D NDo?T 62 ~B D NDo?:p2 D  TjJ?}:2 D  TjJ? } :L  D # Z!T   D # T 2  D # 4 ~B D  NDo?vv~B D  NDo?ss~B D NDo?'s~B D NDo? 2 ~B D NDo?  2 ~B D NDo?  2 ~B D NDo? k 2 ~B D NDo?n Q 2 ~B D NDo?T 62 ~B D NDo?:p2 D  TjJ?}:2 D  TjJ? } :T   D # T 2  D # 4 ~B D  NDo?vv~B D  NDo?ss~B D NDo?'s~B D NDo? 2 ~B D NDo?  2 ~B D NDo?  2 ~B D NDo? k 2 ~B D NDo?n Q 2 ~B D NDo?T 62 ~B D NDo?:p2 D  TjJ?}:2 D  TjJ? } :T   D # T 2  D # 4 ~B D  NDo?vv~B D  NDo?ss~B D NDo?'s~B D NDo? 2 ~B D NDo?  2 ~B D NDo?  2 ~B D NDo? k 2 ~B D NDo?n Q 2 ~B D NDo?T 62 ~B D NDo?:p2 D  TjJ?}:2 D  TjJ? } :L  D # ZT   D # T 2  D # 4 ~B D  NDo?vv~B D  NDo?ss~B D NDo?'s~B D NDo? 2 ~B D NDo?  2 ~B D NDo?  2 ~B D NDo? k 2 ~B D NDo?n Q 2 ~B D NDo?T 62 ~B D NDo?:p2 D  TjJ?}:2 D  TjJ? } :T   D # T 2  D # 4 ~B D  NDo?vv~B D  NDo?ss~B D NDo?'s~B D NDo? 2 ~B D NDo?  2 ~B D NDo?  2 ~B D NDo? k 2 ~B D NDo?n Q 2 ~B D NDo?T 62 ~B D NDo?:p2 D  TjJ?}:2 D  TjJ? } :T   D # T 2  D # 4 ~B D  NDo?vv~B D  NDo?ss~B D NDo?'s~B D NDo? 2 ~B D NDo?  2 ~B D NDo?  2 ~B E NDo? k 2 ~B E NDo?n Q 2 ~B E NDo?T 62 ~B E NDo?:p2 E  TjJ?}:2 E  TjJ? } :F  E   W T   E # T 2  E # 4 ~B  E  NDo?vv~B  E  NDo?ss~B  E NDo?'s~B  E NDo? 2 ~B  E NDo?  2 ~B E NDo?  2 ~B E NDo? k 2 ~B E NDo?n Q 2 ~B E NDo?T 62 ~B E NDo?:p2 E  TjJ?}:2 E  TjJ? } :T   E # T 2  E # 4 ~B E  NDo?vv~B E  NDo?ss~B E NDo?'s~B E NDo? 2 ~B E NDo?  2 ~B E NDo?  2 ~B E NDo? k 2 ~B E NDo?n Q 2 ~B E NDo?T 62 ~B  E NDo?:p2 !E  TjJ?}:2 "E  TjJ? } :T   #E # T 2  $E # 4 ~B %E  NDo?vv~B &E  NDo?ss~B 'E NDo?'s~B (E NDo? 2 ~B )E NDo?  2 ~B *E NDo?  2 ~B +E NDo? k 2 ~B ,E NDo?n Q 2 ~B -E NDo?T 62 ~B .E NDo?:p2 /E  TjJ?}:2 0E  TjJ? } :F  1E  J W T   2E # T 2  3E # 4 ~B 4E  NDo?vv~B 5E  NDo?ss~B 6E NDo?'s~B 7E NDo? 2 ~B 8E NDo?  2 ~B 9E NDo?  2 ~B :E NDo? k 2 ~B ;E NDo?n Q 2 ~B E  TjJ?}:2 ?E  TjJ? } :T   @E # T 2  AE # 4 ~B BE  NDo?vv~B CE  NDo?ss~B DE NDo?'s~B EE NDo? 2 ~B FE NDo?  2 ~B GE NDo?  2 ~B HE NDo? k 2 ~B IE NDo?n Q 2 ~B JE NDo?T 62 ~B KE NDo?:p2 LE  TjJ?}:2 ME  TjJ? } :T   NE # T 2  OE # 4 ~B PE  NDo?vv~B QE  NDo?ss~B RE NDo?'s~B SE NDo? 2 ~B TE NDo?  2 ~B UE NDo?  2 ~B VE NDo? k 2 ~B WE NDo?n Q 2 ~B XE NDo?T 62 ~B YE NDo?:p2 ZE  TjJ?}:2 [E  TjJ? } :F  \E  VT   ]E # T 2  ^E # 4 ~B _E  NDo?vv~B `E  NDo?ss~B aE NDo?'s~B bE NDo? 2 ~B cE NDo?  2 ~B dE NDo?  2 ~B eE NDo? k 2 ~B fE NDo?n Q 2 ~B gE NDo?T 62 ~B hE NDo?:p2 iE  TjJ?}:2 jE  TjJ? } :T   kE # T 2  lE # 4 ~B mE  NDo?vv~B nE  NDo?ss~B oE NDo?'s~B pE NDo? 2 ~B qE NDo?  2 ~B rE NDo?  2 ~B sE NDo? k 2 ~B tE NDo?n Q 2 ~B uE NDo?T 62 ~B vE NDo?:p2 wE  TjJ?}:2 xE  TjJ? } :T   yE # T 2  zE # 4 ~B {E  NDo?vv~B |E  NDo?ss~B }E NDo?'s~B ~E NDo? 2 ~B E NDo?  2 ~B E NDo?  2 ~B E NDo? k 2 ~B E NDo?n Q 2 ~B E NDo?T 62 ~B E NDo?:p2 E  TjJ?}:2 E  TjJ? } :H D 0޽h ? 33  @ $(  @ r @ S HVwP  w r @ S Ww w H @ 0޽h ? 33  L $(  L r L S @\wP  w r L S \w w H L 0޽h ? 33   8E 0(   8 r  > r6@ aj   j    T 8c?aj    T 8c?    T 8c? `j `@        T 8c?^g   T 8c?\f nN      _   T 8c?^g   T 8c?\f nN      i    T 8c?^g   T 8c?\f nN      ^h    T 8c?^g   T 8c?\f @ r2 % r2~2   N8c?r2~2 !  N8c? 2~2 "  N8c? a 2~2 #  N8c?u  2~2 $  N8c?2N r2 &  rD~2 '  N8c?r2~2 (  N8c? 2~2 )  N8c? a 2~2 *  N8c?u  2~2 +  N8c?2N r2 ,  s~2 -  N8c?r2~2 .  N8c? 2~2 /  N8c? a 2~2 0  N8c?u  2~2 1  N8c?2N r2 2  s 3 ~2 3  N8c?r2~2 4  N8c? 2~2 5  N8c? a 2~2 6  N8c?u  2~2 7  N8c?2N r2 8  sG  ~2 9  N8c?r2~2 :  N8c? 2~2 ;  N8c? a 2~2 <  N8c?u  2~2 =  N8c?2x  c $rP  r x  c $@r r  @  s *A ??x d A  s *A ??x d B  s *A ?? x d C  s *A ??,x d D  s *A ?? x d E  s *A  ??9zz x  dH  0޽h ? 33  zr0  (   x  c $HrP  r x  c $r r    s *A u??T x udH  0޽h ? 33  @ F(   x  c $$rP  r   c $|*r r "p`PpH  0޽h ? 33  0(P*X  (  X x X c $rP  r r X S r r 8 D X |<B X   `DԔ?x2 X  H8c?Tx2 X  H8c?D X  s *A  ??RHx  d X  s *A ??K.x d X  c $A ??{ x d X  s *A ??|zx dF Dl X   I B  X   `DԔ?((~2 !X  N8c?Sl~2 "X  N8c?Dlp2 #X H8c? PI p2 $X H8c? eI vB %X ND8c?7 7 vB &X @ ND8c?7 D7  'X  s *A %??c qw x %d (X  s *A &?? x &d )X  s *A '??+ x 'd *X  c $A (?? q x (dH X 0޽h ? 33  0(` (   x  c $hrP  r x  c $$r r F Dl   DB    `DԔ?((~2   N8c?Sl~2   N8c?Dlp2  H8c?Pp2  H8c?evB  ND8c?qqvB  @ ND8c?qDqF Dl    D B    `DԔ?((~2   N8c?Sl~2   N8c?Dlp2  H8c? X  p2  H8c? u vB  ND8c? 5 vB  @ ND8c?      s *A ??6x d   s *A ??+x d   c $A ??C x d   c $A ?? ; x d   s *A  ??n  x  d   s *A !??v B % x !d   s *A $??x $d   s *A +??Z x +dH  0޽h ? 33  p\ F(  \ x \ c $rP  r  \ c $rP r "p`PpH \ 0޽h ? 33  P $(  P r P S ArP  r r P S Br r H P 0޽h ? 330  //ffT V/(  T F !   ZT  V [ VN  [T  xB \T  HD8c?xB ]T B HD8c?VN  ^T   !  xB _T  HD8c?xB `T B HD8c?VN  aT   ! xB bT  HD8c?xB cT B HD8c?VN  dT   xB eT  HD8c?xB fT B HD8c?F !   MT  V  VN  NT  xB OT  HD8c?xB PT B HD8c?VN  QT   !  xB RT  HD8c?xB ST B HD8c?VN  TT   ! xB UT  HD8c?xB VT B HD8c?VN  WT   xB XT  HD8c?xB YT B HD8c?F !   @T  [  VN  AT  xB BT  HD8c?xB CT B HD8c?VN  DT   !  xB ET  HD8c?xB FT B HD8c?VN  GT   ! xB HT  HD8c?xB IT B HD8c?VN  JT   xB KT  HD8c?xB LT B HD8c?8 !   ?T !  H@  5T xB 3T  HD8c?xB 4T B HD8c?VN  6T   !  xB 7T  HD8c?xB 8T B HD8c?VN  9T   ! xB :T  HD8c?xB ;T B HD8c?VN  T B HD8c?x T c $Tr  r x T c $Urx  r F r  T  io N aj  T  j ~ T  N 8c?aj ~ T  N 8c? ~ T  N 8c? `j nN    T     T  T 8c?^g  T  T 8c?\f nN    T   _  T  T 8c?^g T  T 8c?\f nN   T   i  T  T 8c?^g T  T 8c?\f nN   T   ^h  T  T 8c?^g T  T 8c?\f N r2 T  r2~2 T  N8c?r2~2 T  N8c? 2~2 T  N8c? a 2~2 T  N8c?u  2~2 T  N8c?2N r2 T  rD~2 T  N8c?r2~2 T  N8c? 2~2 T  N8c? a 2~2 T  N8c?u  2~2  T  N8c?2N r2 !T  s~2 "T  N8c?r2~2 #T  N8c? 2~2 $T  N8c? a 2~2 %T  N8c?u  2~2 &T  N8c?2N r2 'T  s 3 ~2 (T  N8c?r2~2 )T  N8c? 2~2 *T  N8c? a 2~2 +T  N8c?u  2~2 ,T  N8c?2N r2 -T  sG  ~2 .T  N8c?r2~2 /T  N8c? 2~2 0T  N8c? a 2~2 1T  N8c?u  2~2 2T  N8c?2H T 0޽h ? 33v<  &<<<> ;(   x  c $(r  r x  c $rx  r F r    iPN aj    j ~   N 8c?aj ~   N 8c? ~   N 8c? `j nN         T 8c?^g   T 8c?\f nN      _   T 8c?^g   T 8c?\f nN      i    T 8c?^g   T 8c?\f nN      ^h    T 8c?^g   T 8c?\f N r2   r2~2   N8c?r2~2   N8c? 2~2   N8c? a 2~2   N8c?u  2~2   N8c?2N r2   rD~2   N8c?r2~2   N8c? 2~2   N8c? a 2~2   N8c?u  2~2   N8c?2N r2 !  s~2 "  N8c?r2~2 #  N8c? 2~2 $  N8c? a 2~2 %  N8c?u  2~2 &  N8c?2N r2 '  s 3 ~2 (  N8c?r2~2 )  N8c? 2~2 *  N8c? a 2~2 +  N8c?u  2~2 ,  N8c?2N r2 -  sG  ~2 .  N8c?r2~2 /  N8c? 2~2 0  N8c? a 2~2 1  N8c?u  2~2 2  N8c?2| 5 S T0e0e    BqCDEF A@  8c 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||qJhzi@   6 @ c Z0e0e    BqCDEF @  8c 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||qJhzi@   7 s `0e0e    BqCDEF @  8c 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||qJhzi@  [tD 8 s `0e0e    BqCDEF @  8c 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||qJhzi@  ktT 9 @ c Z0e0e    BqCDEF @  8c 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||qJhzi@  w `  : @ c Z0e0e    BqCDEF @  8c 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||qJhzi@    ; @ c Z0e0e    BqCDEF @  8c 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||qJhzi@  ? (  < s `0e0e    BqCDEF @  8c 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||qJhzi@   t  = s `0e0e    BqCDEF @  8c 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||qJhzi@  + t  > s `0e0e    BqCDEF @  8c 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||qJhzi@  tH  0޽h ? 33  ` H(  ` x ` c $prP  r x ` c $,r r 8 6Z ` 6Zf@ R     ` F Zr `   `G 8c?R 4 ~B ` ND8c?   hN R    `  J 6Jr `  ZG 8c?R 4 xB ` HD8c?   ~2 `  N8c? * d8  H]  `  P ] R `  T 8c?  / R ` B T 8c?  H/ ~2 `  N8c?j \ ~2 `  N8c?l ] H ` 0޽h ? 33   ` X  H  (  H r H S [rP  r r H S \\r r 2 H C x8c?ZD 8 0 | H S T0e0e    BCDEF A@  8c 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||o`P1C @  @ x ( pB H @ HDԔ?0  2 H # CENG6H?TJ&TQ 8c? `T2t`T2t&T2t&T  R  H c Z0e0e    BCDEF @  8c 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||o`P1C @   r " v2 H N8c? X0 H H 0޽h ?H  33   >(    F C~    # ^ bN ]d    C~ E ~B  ND8c?]` ~B   ND8c?$ d 2   T8c?p l 2   T8c?u ! Y    s *A v?? p x vd   s *A w?? ~ x wdB   TD8c? x  c $frP  r x  c $fr r    6A x?? x xdH  0޽h ? 33>     d r (  d x d c $LP  r x d c $   d  6A ??T b@x dT 8 X^  d v| a hT ]d  d # # F s ~B d ND8c?]` ~B d  ND8c?$ d 2  d  Z8c?P 4 l 2  d  Z8c?U !   d  0A 6?? P x 6d  d  0A 7?? ^ x 7dB  d  ZD8c?p p  d  s *A <??X!  0 x <d2 d  T8c? E  d   x0e0e    B%C\DEF A@  A1 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||\M#?$0%!@   | H d 0޽h ? 33  ^V 0I P(   x  c $`aP   x  c $b  B H Z 8c?hR B @  `G 8c?  R C  `GQ) 8c? R O (  D A3   2 E  T8c?O w 2 F  T8c? ( ~B G  ND8c?q  R <  `G 8c? V R =  `GQ) 8c? V ~R  Qm S  4 A3 H R 5 B ZG 8c? Qm R R 6 ZG H 8c? R e S R  @ m d  7 A3 >2 8 B T8c?E @ m d 2 9 B T8c? @ d ~B : B ND8c? R K R B   `l 8c?A Q 2  Z8c?: V B  Z 8c? o6B  Z 8c?6obB  Z 8c?bo 2  Z8c?S&Dq2  Z8c?%&DC|B  TD8c?-<2 ! Z8c?&D |B # @ TD8c? 3 |B $ TD8c? 2 2 % Z8c?6T 2 & Z8c?  2 ' Z8c?  2 ( Z8c? &D R  Qm S  3 3 " R * B  `G 8c? Qm R R +  `G H 8c? R e S  R  @ m d  2 3 ,2 , B Z8c?E @ m d 2 - B Z8c? @ d B . B TD8c? R K R B / @  ` `d 8c?3  2 0 @ Z8c?& A 2 1 Z8c?3 R O (  > 3  `% 2 ?  T8c?O w 2 @  T8c? ( ~B A  ND8c?q  H  0޽h ? 33  0h 0(  h x h c $,P   x h c $  H h 0޽h ? 33  nf@x (  x r x S P   r x S h   x  s *A ??Hv x dH x 0޽h ? 33  NFP (   x  c $P   x  c $D     s *A ??  x d  H4gֳgֳ ?P ng  T  H  0޽h ? 33  xp` (   x  c $P   x  c $     s *A ??T' x d  H4gֳgֳ ?  ~2(Which is basically what we did a few slides back)3 3 H  0޽h ? 33B  p (   x  c $P   x  c $Ro     s *A! ??x dp  Htgֳgֳ ? fD, -L, -U are subparts of A D = diagonal -L = strictly lower triangular -U = strictly upper triangular`g        H  0޽h ? 33  zr  (   x  c $P$P   x  c $ %Ro     s *A- ??q x dH  0޽h ? 33   p :(  p x p c $*P    p T5gֳgֳ ?)9    p  s *A ??' x dH p 0޽h ? 33  zr  (   x  c $h4P   x  c $6"     s *A ??%w x dH  0޽h ? 33"   b(   x  c $=P     HJgֳgֳ ? 2Components depend on previously computed components Cannot solve simultaneously (unlike Jacobi) Order dependant If order changes the components of new iterates change8    XIH  0޽h ? 33    `(   x  c $HNP     TOgֳgֳ ?r  H  0޽h ? 33   >(   x  c $lTP     T(Ugֳgֳ ?\     s *A ??lgIx d   TA ? 8c?@ x dH  0޽h ? 33  zr  (   x  c $ZP   x  c $`"     s *A+ ??' x dH  0޽h ? 33  t 0(  t x t c $*P   x t c $D0  H t 0޽h ? 33   0 0(  0 x 0 c $;P   x 0 c $0< 0  H 0 0޽h ? 33ExWMLQv[R"4kM  m9C%㹐jh %ENzG 0QD/&:f[jTf37=fxݽņU(@.mITt~v@vo%ǹkuC I|@lIu1.KiG~]f ̞r!VM;iL' !^\}ڧH>9OK01HC@<Ω^`fTc^ʐˑ=@1@"W"W!W8({)驓]//hF~ҩ1<SI5 RTĚe mag. F] r !0mn*BZYyΡ2dQ%,hy }{Œ=^fzC亄5ʽf(ǧ&MUkO?`Ud]>>-̨ɐv oWig7DܾRIEz糭pzwRxx's R^x|twW:=blF#YS+#Œ8n8 =D,czw*M:[MrВjQ\]N&,;wQ G`Ț. dC2Ո ࡨYƹ?Mgb :Ìo'H)|"|xQ$ě}RꬃkXaLa#v!/%<5>|L/[[mm>Rax=(t~8Iѹ{ TvKFxVkAf6fhhU!B ౉$d&5M( /P$G zߛ kD(|d{3y߼yogWomG` bg)Df̭>>S{_SpF%z/Y ~ s| }>vwՔS( &v?JH`g~r84|=Fuyu!o}kvR ׸_Ui:鞽C?3KNUDy9];%{2C6H$>Yy<敚{B7d&Nf1KB %EŨr],aL\&5XWYH c&]=7X"Sܟ䅦;5يY.a&5X !9/ gg6U?(fp}ȜduSGcBQHH?ըTE/W}l|҂%$[ 靽Zx#J60)8:43[y@H͟{DIViNΌ_Wwc;p9xWMlE~3c&nӖ`A4TDr)($BM8R:51DOHFBNJ*H[[E8pH . ff75**Y_vyoߛyfvr|;^y4Yb}LCJ׼l6R'"04ҼqǶ.2pa9 ֣_e3VL~3J=weLGV̸8Y|>Ia'q%qL1>4~R~|`PV=Tij7Z1@"D܆@9*=@5E v|!Dܥ SqCÞ/ <@F( &9r)U{{j`oly<#3Xrw Rdyd`'sL zQCXB*.3.{ XaJCN'TY.BVgvRo6@s`KD6[^?x`u@=u7@A|o֖ 6bvhW|C0š`Ts rɟoN6ẗe6Y*U C⣚X,32ӾuLOS#EWR=fG n_ g#~:e/3e6[~\KQWq+a@91Ǟ4C` Y'j A sAyJ'#gN\_fMWXK?={lc{?}$r@Wykز<WB )/NZ3֖'ʐ>HЋ7'ȶ`L0'צ#_+C?[BVReVQన}X]ӏxB5S)In oEOcU]\}"I^Ybi7[9_xVMhA~3M6dZJU7Kk⡢ VH4?mR "ZĻzfx!A AEAi{aTQ ;of̼f^?o_ R<}LD+X*R H~T`RdHwrhňqeQ=//8çsu_ /2*d8cI:p{ߏ' vYO~Qq@|Mՠl+vQ74"@jZcf N]H- wo 9Ʒ .t'*sLj4kOFV*ivuP:5\2]qI/N^I; W80$h #YahH.EnMMsR!4b%32RhUq]vΝ"6\lǞarUz4ѧ֫ɽT[[~Tr6{W;ڻA8+wU頲  cq~p8@rkOT `sDNFTF_n[e[0&~G[`mЍ4qx\3z>z$3φNF>Vq(,%QU_2w{IճVuԪoc; TxMOQpۥ– -4r `@@UડZhG 0GBDxb%:f`4* v|vݽGA2.S( +>J>)@t=?K>h>q0?cG/;ƲE_.(Y>g1.L}ܑO;=~ ;CtRNN~2V>CP#Pwb%Tfښو9@5p#"z"恸 T5%;H.k٧h ] źfp EͺJk,8O~us%P6W4$N֎q#BJ-JN1Q AUNr<(U3G0P!||j|T]EV]j P'5_cO2dž$(.َ-_AfM4g-?']n_s#ܰ%φh.{af!_~awD *z}RϪ2@*V,bxVoSG!GRUm"A %vq{ㄘ*п '$npAH=B/H8B@ ffw)U)cvfwfgg?~};PG`AVMh@+j5lвĭz mҸǶB>"J8L.p,>R7m=SWEUrF ߔO@aΧ~K/ői.3Ԩ2Ali݄%W^1ĉk vP9 黁x#&ݶBkj~s 9^(ْ_mo0jxǥ_gԾqh"uT)$\1 %YHCHOm.v+6lvB+ӴvjǰMeK2Scy ET=@ǽE;pY,82B! / Qӣ;Ğ>wSH_8 ލVQ m~N3ָ">m7snJ B_>8B^Ǵma fPze4ԒQy՟wj~VbUa\~KNr^~?r]Uܨ~*l!~B_p}v:S8,dg^Y':c-/K?\g͉ń7{ZL edkzX[r>%eDx~(n3x_ ƅ'DuW>$g݃S|r|3?woshҔ,}0' K/?ALkmC2ּO|fkZnPW]0蒲%%Μ !NUxܙa3d.IHلn}ٔUML9P@"s~ R LT7N81}h%FoexWOA3[ LVB $* &F01j(i i@$x7(\L ě4Zjl}붠M ߼a}ͻt~ (;n#qPs|^ɮ/Lsخ=;2q|RAw 2[e?fSWyz#0} i(1c}B@;Bpd~=d}*?Cc]]6fcc'6JlU@\XNB1^ ,/"k DOœє79LE1Ro饃N7i|+n01WS3YmZDﵑ:94,]?lwC9V +[6(` $pKpt6',S6 H4m0 3[4LcJDMQɆڜM~R^P^k']Ʉ^gˡWX+Y;© .Uz!0ń1[U9JwJFu֨chMt';%K Pv3> )rĮ䑝߸zxzÑЈJb(,L-%1bvh.A749m>yn{K{frNߩg8fN8qYF̓q=5_׾F`}WsPk-D'u?,Ǽ{]ss$\®v<^n$^=|O훰$m6@Rw@܏Ź8ٝ"GzW-; !8|'yJ93-Oo(_*Rc3زXp< ˢ%`zwP߁:իD$ {a&YB6lhl~֌eܚBKDa+MX$.LB\͛~iqxWMLA~3Zʏc5"A=AcňL bX x41QOOEpAbDo^yds^(1U\.g6俒/X`)R8m[?/!OB #RcEETmOyZ4OgJ C N?<1C@-??\ #4ǽ*֌Gֳj?-O /P|XJa) Ti5PNԩ&?yu1\M͐)O_'gѱ`|$YRG&M{`iQwKWۋɗƄ3Y{Lr&w1)Dժ6կ] e"?ԫ=G;akXf0gucO)t'ϞHC.{Of!pb0Mx&/OcfN&n#f e]^]~!ܡp$|MA串_˰r>O(s0dlzK,8 agV [,,ک%jSS9쟿ޅٝ^fy"ݮ u>evgؾrb> ϖ8ǿ~9փM0ގ 1v=fN+~cNxx}Ƴ vX^̩_8O8v9w[:[vVva'i8B-Su./+3q17h҅?ʘj~G%9lDdۙƘ ]g,nZƧ!>|jYƧ[A6unWktPkxKU0|E$!FILS`UR5$I@& _N1Ǜ ރG? EW>ᆳ_3!n xVOkQv'&Ak)֢ŋakAZAHchLkb#MiRx*H~GE,*ę.鶕o{ȻFvAޖ5D h6rw u<*`l<*` 5()yV_k ;3*,B-VYk>~w096iRN~?tOB,qǰ_obפP4%t(!C(?u3kk=oş]ss9D)]Jk#6=.'^*ed_8 ,xi+YThIc/!VkA/A싢6;#”5j(װu51LeaCnKpk =9s ٲV);(2ru4xƨkz(q"˒ v`>1pN[} 6ﶝ}t`m s㸮j.W|osBqlC=qTg!$wXSZ-@xh(xsd[0%X ,w@RY|M,y⇓=hO>9yVv?0$t1:coX!`t} 8 ~xVkAlҚMi o(TzmSL4ۆEÐ($8^3w>È~wwGr{R[#yY+ó;c!ë)q P~'?װ브)5P_ίxh!GB˵хY df9A]د kH# tc?\jըVV5"C狽>Y >z /yeoL_E~N'fg>Ic!ݶ,)uk'q'LȣM=Xn%p= ©2-{mei~xVkA٤5@`k V Q4( iRECP/=xY$ήiEQZ[^vͼf<~EÀӉa($uxݝ>,Q}vwX@&,dQR["缬P[ldHjUD1g7L'W븆)5P_/yh`B=1k} Y dd9~]8 kD# tc7\ZQ+5l},תtf !κdܳz0~+97:kc+HwըZn 6Z}q^B\(ŭCHKJe }f9 7U.rˊtn,$ʕb[BRý.e}kҼ 99eОvvm8= eԾ}nЬMz:J.WRԢh+ve8{g\_b\䋴ַ>~{D "LJ"IܝRjdQ: |w@Nk~w= )3-{mdnxVkQm&4ZKԐ'/E<4Q4԰/كj`=I7{vI#E[&;~ofgΛ&ЇypQDzHBvݮ7H%aHߝ!P 9T5/x- wJ̓SȐE8g/L'Nϲy7p{`$( /ח.8E4VWpLhC˽Fտc9/ ?+s:pga_'g|s֬ZV^hkUk.F./c9_j㗼2CpJCmeg)5+3=*<ͭbG(-Ez,毋#`R\Bz6kX(nunruUXN7U+JimkV)Te}i/&Jyrkrܥj۳*{Ʒg]@|PJE9k;:.WB๊ 1l7[ "򹲛3$<\.?6h崶w΍au{LA&%eab$6wg 62X/: |w@^[wGᔕ1k>A727`pxVkQm&4hO5$ɃE<4J[/FI0b~5aoP<(^ěس<9EILF RͷLv潗1 zpo=,A{a UEk[O1WWB?uMޜ|< inO9( p35E\?JwE y| $r;w{s_g CxUsęaN !Y,]Z6Ŧ盥jŚO$"lFb;7%RsX˲Ir3x TH#I=Eq֎0k.&fI^b5Q,曗W7",]wTb+ra -kZWާ6e}nuBU1rj2mnKߴڷ=3N_P|mSBE9c;:WL 1l72Vse79gd]oGwȝʝVo4|VOLãJ7 2)~#c!>!:֑瓿`rںT8_. grxVkQmMiKO5-E$F(mFI0jRފP<œ={Pz[/ILF 62ٙ^{3;3o߽yl>,@Fg(NuC(|g wՄ *|_ ~q@j3}wmj3< inOU Aqi:n g~;~օGR?0n@c,#,aa9>X3}qf{kFش2|TXs#7?p{4ܕIFg^Mi{8V_PAdIiĝiy`,G Ա<%>+m_'~8ee,ٶaPs~xVkAlҚM۠5 o(TzmSL4[CRЃx]DԃcYEfv6dEiɷ켙ͼf7^o{N@׍a(Xzu]bG;KTaDyMC;̣9`)-&弬PwOɐהsr(~Oaqˡ/kt'(_m/ΗBV=GB˵օY dea]ح {`# bo0\jը8vVr="Cg>Y > /ye&6?Ou1 G01GF40z{1vX Ckau~HPڣ8 U=ݰS_ߒ˫)װR a4bq'_@_~5l_/9Ǵ?_)=sn^Zj!,=ׅ r=82,ۍ?? kf\cT5M!CW}),ϿhW*CpJÌ=<$5ƑlrwFHfj`i̋ ):ILŕzr c91,[U.rCHt˕j)\W5|~H5(ggɳ&ukQ {nB6arum^=!!N-NI|&>v WOͼIwt"ȦIBĽ JqC(_{yu+KzSeƹ?[kxVkQm&4hڃ( F_Mj[Pœ"OAڣzJA*X &{vI#Eio{/[o;&1y zpo=,A{aUEk[o1WB?w/2_! )P0!;?Q}\ō/{t/\_Ժ_@J]1 /yW7e/=a0dG}ac| G`㟅ҵjZlZzYVd Q!¶k$sv_|+=,$8wNYajNK?BiIco]Tv*4,xm%Do^_kjtSU-ʅ+j9_[,u&VHNSfޫm~ϞUvv]#Oo*ޙYpbwTH%,ёϕ$zw>p{4ܥ)F/(E&ndRv!Bb#qkb# `!:֑瓿`rںT8?. sxVKkAl<6GPE#(f I"nd7dCH@xh@I ADAA47qPqתqf2+fzLU=YVy 4!cj p\ΞiEW u|(U8}+ߧ.Ha`X1^t\ߑ}n9UɺLC~"`)$$ &8sw*a)D?zDh@ɯ}S`x/u)++0 fE>-'U^|Co0?~`vt=Rŧ˼xċEw-8>n+՛> 0d:3?;i&lpz79᝖u "IB%zq^md)"?6g:wv/!?g,O=sEbwo+]-\ Ƙ j 2>>|:F7 NG-h\ݒw6m!5qN+3{}c Pt~}R4l,dV`J~ZI\;" a~xVkAlR&mZcՃ7CjA"14^%դKxR]DԃcYEfv6dK%f&7ޛ]}3Qj8 ]7>Ң`B]unwm,Q}5 uC6ldQRW#yYkz3sw|NT UBqÄ~o|3 ^E\ ͟d~Y;An w2 $\[]Ȼ@],Yb{< ;]`O đa^ >\Z嶝m.ەz͞Ngࡳ^Jz_N%udd)s[YGl}yip[T߅ˆ5_ZvԱb } ;J^N36xV= eԾ}]e7z&t0<.KpiKU,HZ­g${\?;;cRԳ_{0A&%oa$nri KE(_{9] t+ۼ$̌ ˿m{xVkAlR&$X1 xESڂ(_Mj-E<g=y c< 'i٘F]%f&7ޛ>i1 >2L6g2^KӂDU~3^ ~s^NC_˫)pN Q4븂̯jr/_2ma/;hb^?1kWkh yߜe/Kj'uak|d# vco<bjY+dVlkUM#C}),6O?kW*CpJÌ=ˬ'5ƑlrwNHfj`Y΋ )iLŕzr a91,[V:u.r]Hwu+Ӕy-jbl{,R~kIʿI{zLǵ{(=uhK nc0Qp6/ǖ;͖SAVU( ?}I[E^sN'f^|c[3mYdSV4!JvHȡ5/=Xn%h=é2ԟ-{mdkxVkQm&4hڃ( F̯&5-IE/ك ^գzJA*X fF RͷLv潗W/ǻMma 0s svFU0ソP 9T{5/x!wLC! )pš8aBP?AC'_@NP~u?ϑ/cR_u[7~o^^ {,Q¿ r2qf{jZeVVSiy?۞F0~+9>SVlL6@;feGU,5hRuq*:"t%׬=`9k#,1&BҊ]&WW_@b\)6|m-**^bORWwܚ3V7c{CeӶy|4~y;RQڦ΄"˕PxbFB,V|&Gp> 7;OM7y6q?6t ׳01ik`,rh`>|V; NeOpXmvSs+xVkQmR&mK kjHڋ^#xh@$i~5aoA=O ^<ٳGUALF RͷLv潗W/':M`0E07FZ,@h _Y4Ÿ2|5a# 7 bae?/N_k%fޅ>'CS.`8aBP?Aw/CǙ_@NP~uȑ/# /yW7e/=AdG}_G`៥jZlfYVd 1:!ʶg${j_|+=;,$87NYinK?BiIJ豨.&D[Y;>6)]s,yKTa)߼Ըa:0$`ʅ+j9_[;,u*v'ȭyCo~ӎg?To+;|GCMIcjL,b\c3bvi4 etse79[gqx\ !w&w~plZYm}>ID1NJA\J?`E_s9m] t*~SVm{krT}xVkAlҚM۠5V i Q4( iE2I&/ci,J1`5\C&2g&?.dlE.\Vܸ pDE#EAKϹwCjߔ63ys=ܟ→~~ۆ%XE=`@&(.EA`5 7L4Y~wG;i3Vk)>׽uex#f [)GJ'8'0qɒ[@SP?x?O t_pWbm KͿ<ل- crl1l*U(5 sZ Tu6`ۊQm55LY{>ٙxMd3NQJ}^(/tnOO~C CɾEdXoR(w9cMիrO/Ɵ6UG*l'wr cH'GFs8) ߹*v&m'88q5Zn_ev})s,!>2Ny{(ӯ_/a_/M4ˑ8Hxn:]qBYw9|>L>~ˏEtn4w:2v僋eh I7kt~]0ua%82N&psN.бUϰj~B_g'º~ רWc;)OQDVɰVQ}עn=~Yן%t\Rcݛt+e~'nti`:uAއ+<)t_1ag&J"n@?Zf X G00/ |ePh.Ы'e/SiZ9$7˵J f2%:; \`J9|<ߊ9j9Nߥ-4W;EqwiC%9Jj^{LoB/xVMHTQ>qtX󣩉KDTfrUjIUi+E j"hA-$h岥HˠZT49x3&<{9ݝ7GO[LAB^GS*$.EX#%Pj =Qv?U#+B `o&k.k.ڷe%;:kNdDDENс3g{6x)= J1Y֨3Y(D/e[b)ZT~o]?e|7zĸzSf7/\w/s7򫰤Հ`Cp*JXb٘Ȥixa|Zި9s2uV7FM }pR#Kn.b}o^ n³xXqKnlha疗S;cgrfj:3+aީ)zmxM)l|^d􈩫LAU,-(i"?f.HA;6. H+/(%˿*6Q/ |>MӘڦ^_-DntAxu?㷣!z+ }|ȪBFa!'$x)Tf;'n8ivT(ܖ&~xVkAlR&4h7Cڂh CSQiEY,> /yeDYt[g7+^!yaƞrUVHB;GUGrXH JӬń=Jr4R&kn=a9φYfN n\As}!JZj|#b nv,R~MʿSi{L׵x(=uhG nk0Qp6/Ǘv1~&>v OͼN gt"ȦiB JCk(_{yu+KzSeƙ?[?lxVkQm&mZKԐ%F(mFI0b~5aoA=O ^<ٳS W1q۷KQ6(J52ٙ^{3;3o_}8,@Eo(Xvyýv; {/i?%TjFl?~~ ^5؝>ӿgdHz9\Aa(~Oǟa˨>. =P_/j]hs h {֍ߛWȞe/K2ƣc|G`ŵjZlzYVd q:!ƶg$}b_|+=,$87NYQinK?BiIc1]\Hv$2)]s,'ym%ıo^ZqjjuRUVJBZR-:n;ҿc١jidz({ַ甝r\v㣡Hۦr1w&t1\.K 142:򹲛M3<\.;;;86h錶W?ǽ؄ "LJ\#Hl Bl_Ȣu/9X:m~? )+c׶nfoslxVOlA3mj-TCT\4%E+qPb+OuiV !q "qq$"7nJD$.!Įf| ѷoޛ޼yKWk$ZdsEp1\.gwfhZl>(˱ПM_,~AfQCŠmys?.ܺ ga]B4/n23|ZRؓ03?*a0 mj}yXϪus:hy [[ Rle (@Ȝ jes6~kaW}3׸FVdc)koާ۹/9L[=CұdZ&Jeenŵ!o/ճq(h|$J68URq8фZY+v|貨ʰɟ d5! lHޜIE)3>q9TK]Ɂo WW8{@懳r)qmţVotڔ$ŷ#OYH \ڞx;o' 7;M.o{„glBC%So m>en|X_e%/UIP-;o\Wq$X,^v0q98^˯T8POm%Ҡn\.q_obu L|Gr/Ĭkz> (Rm^*]?6˭`.,#LB]L'ti`:uAއyS8ct4k(t΃bu 1`j]ݻ;"z`bJիO2 }xty|n9DnRx"Y*di#<~6J1||{Өr7Tr)H^"S8j)޸,|xVAkAlR&$XA`h,^K[9J4Mj[xxQzVAz1bٸ (JKeL7o7mwqj#8 A>2%l0d1Q,j^Eg漪ϝ޹ܱ/R֔KR N/leXu\ƵIW5P_/0/ΗĬ~c 977U-P5^n96!Gh\jY/dVlU59E- eUs<_x+3; %5ƑlrwFHdj`i΋ )щILŕz2φ^D͜r ͽ%;: CJԔR[.֫>`kY\P'''O=io3^ڣ@H.]v;Y % +kT qdl((Zg$ܵ;\_?bFhE菾MaskJ-"lJ\5IA)v5h` E+|; oneɻ'~8UfeoCPm~xVAkAlR&$h`hSK[jSL4z*xOPA?»˫)gp @!8_<'_@_~5h?;_@q(Z[S;ܼW@m,Y"P{< [^ ?Ȱl7s̔.,WbCfk+FZIx|Q=%=O+^!ya^pYVnHC;GUGrXH JӸń)ɭ#H+uz5sb7..8K\f57!MY/ʅrZWк"D;HnNf6}fZdzG޷=ktZ]ulz&J.W\b߼Soh)ZgIxkw+ w,wߩĄ>ѩ')n[Dٔ k%q-$RlidQ |w@h}ʒwTqϖA6636oe~xVAkAlR&$h`H-^h FL4Ei}nIAdI+0F I7fRlk2|Ww@IkWC~= )3-{mlex.nXxVkQmR&bXAАړDJ?( FLiRCC@?@Q={՛[E('uy?6AdRiɷ̾y?{3gSXȈBa #(dI3L66}XO. q`)#"yYƻzzWdQM[( ϳUpsh~e >F+%4WO|LCK[Bs_YŨvƷ=;#IJu !~-T t bܹ\CQ'7K^ RvϤ4ɹ#Hwլ4R#PZ&X.#%c1er.+f5K?t%mdPlXnp+h (ք@vR-5\rj6X\[+3s}.)omXldha8n4KU'+Qܧh~r9H.FٗL <1t"^aQ@k(_{<ӻVvIND;`X cLxVOkAlR6`\Ra =yTbJB#MiR JO~Ðoc{CO 6( VKES\jw' zhXB>N/Lp{G֟"KX&;p=~(_/hkP"!V${^ ziD b cG0u<;eac3[TBNWVsS DiBÓ՞ħÊ?BT]I;1?=krUdz)UłbDH*F5o/aXD:1]s15QOi9LUl8.SKHݟZSYxW~N8bRjgu{)x}c}mg¦of{3/z6:p "0Uh4ݍԷma&= $Povg)%o弨kWN=aך5":8k'X";pͳ5PVW8y%Pcc9Cs\Q$k9*"=țłcDH.U_k^oh&bX\XxM: ΜsBHf{\LWꎅk:g \HjƼY̖6a|"MqmqkEYp..Ӏj]<&Cܻrfh]b/u6uk`Ĉܰɭ!^M`kiLgS7#YU/ $P2t,= Z]Y^O<@t47xVkQ6fu]Z~xPQbu[tM5 xcAzs ԓz QОy]2ͼ73o_`RK BKO?# Wr(C ЎY_G~V;AlSU4 şGB3[cNGZj(q`{ 0@?k|`4(??\neMbWdBgINOk/vȹ1N9d$wwϲHS%E$Crl.A+:c`DƼFDư5 Ț΍E.ŹҥpNE(hŊUjRۖY?77Q cyL/2N#A٫)w}h!hך1uz}'3?%!zIGoNzG{lv&N\WAF9Bpk3%܆S`jww>sԡ'G,..Q~`òm0y[!D!*rһ3@ 5Xo ރUޭ@I(g~m)xVOkAlR6`\RaĞ1/l=I;@=4҈@Ơ7`;yw1 T]gxe٩:jElOҹl'ӫ=O^!v b';&(woS8Tjpg_7Xð:6cbkX%s֘'\ma]&?զp]~N8bsRjgu{)xyc}mԧ¦S5yʼ̶x5D'uK"ղe>Yxcd>xn8L Seqy:)Q1tdX3'n/aS](n3sBFɜ\?2`ɭ=b_``&nצoi}4*XAn"݃eu_M?:E']O~xVkAl6j¶HocC4RI0dVPxP?A=ԀWт' of'q٠(c{o7o g "c$WfjwWFW{S?i%X` }+K/1'E=/r/̷LZ\ r}tCgnwmqUc! w˴9Ta;F> [o9$GZ jDA5BD0s1ss1Mwb;+YRVf$kZ*sQSb'ë݈eo^>8y%PcS9O\Q$k 9*"=ۜłcDH.FUPg^oiG0r.,&nhdNƈ9!${)uM{:g \HƒU̖6aw1)5մG" p'?;.C%\>t䲎ܨܝ ɌQdgi[WCE5D!|*mWk"bBn۹֞{AZKR{GL iFr_I^cLc:G79< 2֐@`)պT6zw=03{ouxVkQ6fhmKHaC%FQ$4d5 xQP=P zhAēS1qmld2ͼ73o߽=mUP݉C7FphT۝N p%*0"ۇlYq4Tno)#F yVb{M6x5&@1*P'{OKp'42w%٦[EhɟGB3[cNWZj1(q`{ 0@A@5G0U^o|a7clW٤CZ;$gaʫz0~9"rn簓l#Yݝ,k,f-BQo%ɨKPI911lMwkXLνKs*BA˗+Ŧ)])Tom.Y̿KDlR--Vc]]I}ozIH;^՛䧓^{N9r~SU*(Gn6<#~ɜtvQ3w>)}b1g/&Mo~3OA 9y\:Bo2ރU@I(=8?#HxVKTQ3OefRr"H\0 !)?FpfӷhѲ]-"hHl "uνM0w޹?s;Y{ն~Ap M5c6 mA8bG;1cv0"=exH@Y@\Kǟڵɷ#υ"yQ5m'vdӤ7?@6'*?_&gQBR?i5ϺϤ]gs-^j`-v>a-zE?AѠ{~sWfdKeʹbIΏVtx>ly8y9c0ys/(S4Q4ϣ*?.WEc=F$ux/KlW+q~彋v&nBg'ڗCXiвB GKi~²v609d=XiۻV{$ ''oWEG;uXxVkQmҚM4XA`HۓDJ?( FL4[CC@?@Q={7=ZEԓARɷ̾y?{3fӛiX"cdD!  v~$&LcF9,EԹ]CglHs^Z^>$$k%Dy((lwckXe;95P_]/ΗBA=1eO /mm o-Y dgكQ=Agp#ֱ7Xh5*Slj ;y|=_xN@畲}"-HΥ8B;feD, 4euѣX;6&k^tCwYfM%Jg\Aq]_[x_&+Z嶳ԨxK{q3ʂ霜E۱ s ͑U9 vt^v~L*"Ȧ7Y(u=X݊u+;|z3-oc7bxVKTQ3OefRr"H4W- r p̑&uA"Zl@-EZRk -h^{4Xo q;sΝWoPjLئ헃 ](|0f.PI=Wr|)S3=Pvwυ"yA5 &'dSn?@6'*?_%gPD~u3IZ5/}>\[uA9&js fMم&0\v:{yt4~xν`NDYE tӳ 2L$$Nf+ꋻ$s6l5Iֻd i`t}ğʰg8JVœ"!˒C$mXorlV.k cl.SR9oKާ jy.<1nsGwo\ʯcГscIm3y뽻 Զ|MjG^wKTjz}zʁH%D_H1Kƙ+ ~僋v&nRSBgڗXilղB G)ȥ^’p+6$1Y=)ۻV{$ ''oWEG;v xVMLQ7 J  AMdSb -BoDccF]$~P3 v'( S^yhR*NLeٹH&}2N;m2Fp{4뎟y9=JҪOɸGl ˑǢ5ԂM58qCEZ uq&lv-:.Tca4o\`j]Լ+F-ϣ;>+0O&5vCO%v=/1>{;ir^Q>/:xÓ󰽰evPFPo-[@H'26bx6ayn < 31v~A\YH۫p*uCCZ ={A>nY}c*^ZODr]RǕR괖R2W~ rqѿ[gr?5y6OBo)y`|h˚@^;d%A;zM('(my#I\zE;;ܵ7`tgt~d``&,gaaۓx|툡E=X]t+[d}k=¹dކn6v27 \xVkAlR&ŸhA`HIhxT4٦Z p?.lR[伬PaKSo{j9̡g7L'v!?70~Z(c/״.tyh'ԅ忬ocَA=A5ľfxzsSsbk֝=/ ]˝6َ$>}܊/yyN +w,H߃ wLʓfHJӈnKwe* C;0s.+f텴x[,i=&{7Ee?d lk:s!|p7?.;õ5a s9C9m35 #rx|'![mr>Y96Ekx嫅=ug(,MEZщgھa7V9D$Vj6E u?newLmlemaqxVOQtAHЬċ5!J%P6]-B=bx&z0<9GQ^$μ}[Kt}fޛo^{q4(P* PǤl]n`W;_] OtB9(Ԗ"JS=௸_W<%SP?sj%'jY?nY? sh \̯!?@-?OIKGq`AxܣCO$rZ@5 ҄cf=({|7{k| a({FDδT^fy="΍WDx9~rI?R<+a{=NcnHqG4q \?T5ԎZSe*jk(Ӹk#QɗJw甏CxH'-=\t<>/lׅcZj;W>hPR'c:CۢK8bP?PD c Sh9IЊ ,Co~ ~d뚫[YGzV)/N^w3E,hE%\5)9xVMHUA>3w|Ң\ZDMi>?^"BUmaE6ED䢠]KVP#Jzsf^3<|3sΙs̹u'`\@MRkr^wnVf= @U7)|@=t܅Tz_q?/=;ϙh~./fl,OrOB .~ S $O;4ϵG0qjOeاz߂A!0Pc+[)_wWieSq>3.ƞ% '\o [³_}TGکhM'RjJu~L/TڇdKfvѝ\'v) 3}<O5Ɛ7r2==NbGbt_?CM*IE)hXS ZgZ,rQNZ526ɶTrL(/ǔߠED}.0`1aa<zh~x0d0ݰ} O ߒHxiJ%[}p |/L\70qeeQG>2-˔x_`"oL̔f0;,L< J}Tlv<98I8Fqj7cw,NwT 69>s7Wr W bCu}w K6,6q b^y56b?/v17b1.܎ J914);__^QE0b̋J/n6jt.[փ۴FYxVkQmR&b\Ra X0Q40J4&!7L!̹E/kXcTw;#IJu ^~W.ԛER<2tzZ 4> SSb?F'0#--jū {0 u]!MR8 ~p*Kn^⡥XX*1ݖp.:9r:"j֨\֨ f+BgD,7xW> BӶzOOשvΪf*'[RYL$(aq~b: yXϿZPL~1ۡtt=rEs97s63ɬm&bW&VQF`t^[٦ݏIND3`P 6`?yxVkAl&4hւE*Cci{iA$*w={ԃ-xԃY4쪴[^{oѵIXqȈF u]vV5g1-i8ȣ ?"h/U= ;{dהr(~O?װv\iW5P_ͯ]gyhaB?1fW5kU TŲ%úSA5r@# bo0Lh5*7WNQS $y| nx ƯxU 3/'O[J{'mOߚc1owٺWl]YCE <{8?RV+a >gy{\/||>کĄ<n s_Ǥ11o4ҁ_KQBߤ{MeջV6h$ Ȍf6?p:xVMLSAh@yD $r"c(*m)-6bT &C<]O!&\Ջ#fufw_i!-1&vgo~иU9 `&rp~cQ>Աw)>{KC~R'@M>\E.֭U!;;5ef TB2B{ hf4ܪXMaseU$G|S_@?a]na'P pc;tؼի>Rx}e\ (ZoR%9b; #@3d~+Dsd<T$z{D( Dx9Bw/lN"J[ַpb 'z)If*/ NsF!f)=d8!n֦j|]ȝr5@9ʄIBEO51 # {6D7Cpw1yv~B'(r݄+ :D$JZPG1X[Jo,;z“L|8?i_lQ*GB9Ngb>l;=̖P.v|vt:>h2N7w[/EE 9oϣ"y7k$wM6 P$D}x#q9n'S(L@ n0f NK,6*]<ݰξmƘ f.Ak`P\ 1;z|0|#_Iwz-^XO t',,*Ns/$w/ڏ-U tw?`6|xVAkAlR&$XA`h,^K[9J4Mj[xxQzVAz1bٸ (JKeL7o7mwqj#8 A>2%l0d1Q,j^Eg漪ϝ޹ܱ/R֔KR N/leXu\ƵIW5P_/0/ΗĬ~c 977U-P5^n96!Gh\jY/dVlU59E- eUs<_x+3; %5ƑlrwFHdj`i΋ )щILŕz2φ^D͜r ͽ%;: CJԔR[.֫>`kY\P'''O=io3^ڣ@H.]v;Y % +kT qdl((Zg$ܵ;\_?bFhE菾MaskJ-"lJ\5IA)v5h` E+|; oneɻ'~8UfeoCPm}xVAkAlR&$hA`hSK[j#&ii<' A=9""&M[^v}f{o$:1cE06GF40zyݽ~̻4¿W*CpJÌ!5ƑlrwJHfj`Iϋ )Q[Ƒ29W^s!6%jĐ+8[K\4w!MY/+ņrV)TѾ"D;LnNf}fZg#ov謶ڲm(a\q]'zOH Sj?$")}3RF|ݍ ݶ,)yc8!JfHȢ/=XѮ%`=é2ȟ-{mlen> xVMLQ7-eAۮH0Y9pɘb0Jb1%ЖO1hx3ğy?[JUlfwf{y;3σokل rZ٘bATz.;|*{;UrJ!IW,= [ ~JM)< }]/=s H>;=L@,b2Yq<&|Q?@'4rO*~ |5ZGح x!?c:zzVrBG] {n+Strrqy> JC}Dch#g^$K Ʌ:Y<^${ТjW:Iw-*vC$su8;IMnR>!v+Fu6\,~tp,M[5GMb4_8Xe/=ByD6y π8.;GCXrnиw*{s={ߊgp ~ PuD${<YۆL4kcpC,M(Q'gZZ1(}]#<511i䕙 s1\f )}cL@Xi3KO뛊3GSW}zO57pGc? ?~w{m8rŲeAn^uџ|&1pv!OfF_wiih2d݈ fpF ODieT58L/hkC*g_8]̡#$׍j.P Ua7B{M;TP:A/MZszd\{(QnpNr!2ίZּ%&DΈ% x&OÛƊR9hiEKWtP7Odzx,D!&\>ڧr\{{~26cK)^/YҟkrgD:.-B\RәYl~xSx?Eڝ:Zlo/VhEޔxM1wXec)vr.*؜w#pI;~%Fq3MbK;@~ N/4SUOA`Q/Vv\WrD0x 0zR'{9 } 1aM]t@Uv@/߸f~!Z$(~* |10y57 9*;0A ӒDQF0>ٖMLANP IS_UWPӟi ]`cpauk`IS u0_k h "~6 "%(#+%.ۄ1-469;>\AC[EFJOh+'0 hp   $Math for ProgrammersPJim Van Verthamim Jim Van Vertham255Microsoft PowerPointP@pϥ@htq@! :(Gg  S%& &&#TNPPD2OMi & TNPP &&TNPP    3--- !-----X--j}w@ j}ww0- @"Arialw@ }ww0- .%2 SConstrained Dynamics+! ! ! !+!!1 .---- @"Arialw@ s}ww0- .2 )Marq   . .2 )pSinger  .@"Arialw@ }ww0- . 2 )( . . 2 )marq . . 2 ):@!. .2 )[ essentialmath  . .2 ) .com)  .--"System 0-&TNPP &՜.+,D՜.+,h$    On-screen ShowRed Storm Entertainment* 2Times New RomanArial WingdingsPalatino LinotypeMonotype CorsivaSymbolDefault DesignMicrosoft Equation 3.0PowerPoint Presentation The Problem The BasicsBox ConstraintsBox ConstraintsBox Constraints (cont)Bead on a WireParametric ConstraintsBead on a WireImplicit RepresentationImplicit RepresentationSpring ConstraintsStiff ConstraintsCloth SimulationCloth SimulationCloth SimulationCloth SimulationRelaxation MethodsRelaxation MethodsCloth SimulationMore Cloth SimulationStill More Cloth SimulationStill More Cloth SimulationArticulated BodiesAngular ConstraintsAngular ConstraintsAngular Constraints Stick ManUsing A Linear System Basic Stuff Basic StuffJacobi IterationJacobi IterationJacobi IterationGauss-Seidel IterationGauss-Seidel IterationGauss-Seidel Iteration!Successive Over Relaxation (SOR)!Successive Over Relaxation (SOR)Gauss-Seidel Iteration"Lots More Math (not covered here) References  Fonts UsedDesign TemplateEmbedded OLE Servers Slide Titles*8_AdHocReviewCycleID_EmailSubject _AuthorEmail_AuthorEmailDisplayName6ePPointE lmb@ndl.comLars M. Bishopp%_NL rJim Van VerthJim Van Verth  !#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~Root EntrydO)PicturesCCurrent UserSummaryInformation(PowerPoint Document("rLDocumentSummaryInformation8