for where min and max are the minimum and maximum eigenvalue of M, respectively. Since k q i k 1 applek q i k 2 we also have that

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Audrey Stewart
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1 All s Well That Ends Well: Guaanteed Resolution o Siultaneous Riid Body Iact : All s Well That Ends Well: Suleentay Poos This docuent coleents the ae All s Well That Ends Well: Guaanteed Resolution o Siultaneous Riid Body Iact ovides detailed oos o seveal clais theein: that aiwise GaussSeidellike aloiths Genealized Reections when odied accodin to the telate shown in Aloith 3 satisy all the inexact iact oeato desideata hence ae uaanteed to teate ust as ae thei exact aithetic counteats A DETAILED INEXACT ARITHETIC PROOFS Hee we will ove the clais in 7: that both the inexact aiwise GaussSeidel ethod descibed in Aloith 3 as well as the Sith et al s Genealized Reections aloith [] satisy the inexact iact oeato axios ( NOR) ( OD) We will assue the ollowin coutation odel: eal nubes ae aoxiated usin oatinoint aithetic with achine esilon < iu eesentable anitude < We assue that no inteediate calculation oveows; we then have an associated oundin oeato [x] so that evey exact uantity x x x ale [x] ale x x Fo calculations we will ake use o the weake oe convenient bound x x ale [x] ale x x Aithetic oeations suae oots ae assued to take lace in innite ecision then ounded; we will wite [E] to denote that evey oeation in the exession E is eed in this way e [x ] = [[x] [ ]] Finally we will assue that i sall intee constants ae eesented exactly but that N ust be ounded I is too lae the oeties ( NOR) ( DRIFT) ( OD) cannot be uaanteed We will ove that both aiwise GaussSeidel Genealized Reections satisy these oeties suciently sall ive a constuctive bound in tes o the anitudes o inut uantities like N etc Fo both aloiths we will st look at dit constuct a C which is used in the denition o ( DRIFT) as a ceticate that eney cannot ow unbounded ove the couse o seveal iteations The oo o no dit will aleady iose a bound on ; intuitively i the achine ecision is too lae the enoalization o the velocity ate evey iteation in Aloiths 3 4 itsel intoduces so uch eo into the coutation o i that desite the enoalization its anitude cannot be bounded Once we have constucted a C we also need an We will show that ( NOR) ioses a lowe bound o that this lowe bound deceases to zeo as deceases We end by ovin ( OD) hold ovided that is not too lae The ue bound is constant the lowe bound shinks as shinks so that it is always ossible to nd an i is suciently sall A Paiwise GaussSeidel In this section we deive an C which the odied aiwise GS aloith descibed in section 7 satises the six citeia ( NOR) ( OD) Thee o these ( KIN) (ONE) ( VIO) ae obvious o the constuction o the aloith We st ove ( DRIFT) by induction on the iteation i: suose it holds the st i iteations o Aloith 3 Then k i k ale k k C k i k alek k C k i k ale axk k C k i k ale C = s = ax k k ax ae the iu axiu eienvalue o esectively Since k i k alek i k we also have that k i k ale C We now bound = [ i h i ni n] n is soe constaint adient selected by Aloith 3 The ollowin act will be useul: a seuence o nubes x x d it can be shown by induction on d that dx 3 dx x i [x i ] ale 6 4= * dx d [x i ] / ( ) d = = We now oceed to bound Fist [ i n ] i n ale i [n ] n denotes the th coodinates o the vecto n We can wite these bounds as [ i n ] i n ale C Since i n su ove ives = (knk ( ) ) = (knk ( ) ) alek i k knk aleknk ( [h i ni] h i ni ale 3 3 C 3 = ( knk )d( ) d 3 = (knk )d( ) d AC Tansactions on Gahics Vol 36 No 4 Aticle Publication date: July 7

2 : E Voua et al Switchin eas ale k so that k ale n n ale (knk ) ale k n k n ale 4 4 = 7 knk 4 4knk 3 Su aain ove we can bound ale n k n k ale 5 Now since 5 = knk 4 ( ) d ale n k we have that ale h i ni n ale d knk 5 h i ni n ale ( = ( 3 )d knk 5 dknk = 3 d knk 5 d knk 5 3 we have ade libeal use o the act that < to siliy the above exessions Then ale h i ni n h i ni n ale ( = 4 6 d knk 7 = 4 6 d knk Finally we bound in tes o = i h i ni n We have that ale ( 8 8 (4) 8 = 7 d knk 8 = 7 d knk Next we need to bound the no [k k ] in the denoato o the coecient o the velocity udate ste We can use the act that to et alekk ale kk = k i k ale k k C k k ale ( = ( 3k k ) * 9 = ( 3k k ) 8 The suation ula then ives 8 k k! ( ) k () k ale ( = d 9 * k k dk k = d 9 dk k Next cobinin the last seveal bounds 9 d( ) d 9! d( ) d ( ) () ale ( C = dk k k k 8 ( 8 d k k ) k k k k = dk k 8 8 ( 8 dk k ) k k ( 8 d k k ) = d k k 8 ( 8 d k k ) We aly the suation ula a second tie to et the suaed no T T ale C = d ( dk k d )d( ) d = d (d d )d( ) d = d d ( ) d We can ewite this bound in oe convenient by coletin the suae in anticiation o takin the suae oot: T T ale! C * Finally we have a bound on the no o : 4 [k k ] kk ale ( 3 3 (5) AC Tansactions on Gahics Vol 36 No 4 Aticle Publication date: July 7

3 All s Well That Ends Well: Guaanteed Resolution o Siultaneous Riid Body Iact * 3 = 3 = v t / 4 Notice that since k k > so the denoatos in 3 ae bounded well away o zeo The last iece we need coutin i is the no o the initial velocity To bein with k k 4 4 Since yields k d k k k k alyin the suation ula 5 Then = k k k k k k ( ) ( ) = d 4 ( d k k k k ( ) ( ) 7 Finally [ ] with 6 Let 8 k k = 9 = 8 C kk ( 3 3 C * d ax ( 3 3 kk 4 axd 8 axd = 4 ax d 4 axd = C= (6) Cobinin euations (4) (6) ives ( 9 9 k = 4 ax d 8 = k k 7 P Take d ( )d 8 k k L A I < 3 < 4 ( ) ( 3 ) 4 4( 3 ) then aiwise GaussSeidel satis es ( DRIFT) Notice that these conditions ae satis ed i is su ciently sall 6 = d 6 k k ax d )d = d k k k k k k 5 Alyin the suation ula a second tie ives 7 6 = 3 9 k i k 5 4 ) d ( = 3 9 Theee : Now we ae at last eaed to bound the next velocity iteate " # i = k k Suose that kk > ( 3 3 Then by the evious bound euation (5) C i (7) kk kk ( 3 3 * ( 3 ) 4( 3 ) 4 ( 4 ) // k k Since < 3 C ( 3 3 hence the bound in euation (7) is valid oeove we can substitute to et this ineuality into the bound on k i k k i k k ( C kk ( 3 3 Then i satis es ( DRIFT) wheneve ( )C in aticula wheneve 4 ( C 4 ) AC Tansactions on Gahics Vol 36 No 4 Aticle Publication date: July 7

4 :4 E Voua et al We now ove the eainin oeties which ae elatively staihtwad Fist we have that 3: 4: 5: C (k Then aiwise GaussSeidel satis es ( NOR) Notice that both ihth sides vanish as deceases ax d P Let C be as in the evious lea By constuction o the aloith ( VIO) we know that the value o is = h i ni > k i k ( the last ineuality ollows o ( DRIFT) Fo the bound (7) on the coonents o c we have that C kck ax d < P At evey iteation a constaint with adient n is violated k i i k = k n ck ( > kck ) A 6: 7: 8: 9: : : : unction R I A ( ) N A C G () i := do NV V N( i ) // Ti NV < k i k i NV = ; then etun i end i a k NV i k st i NV k i NV k i end end unction the couted solution We assue that aoxiately satis es the KKT conditions o the QP NVT NV NVT i this is less than when C ( ax d Lastly since k i k C we have that kck k i k wheneve C ax d (k L A3 Paiwise GaussSeidel satis es ( OD) when : : L A Let C be as in the evious lea suose s ax d ( > k k(k k Aloith 4 Inexact Genealized Relections Genealized Relections The enealized e ection oeato o Sith et al [] ioves on aiwise GaussSeidel by uaanteein esevation o syeties oe accuately odelin shock oaations at the cost o an R that is oe exensive to coute Aloith 4 shows how to odiy it so that it satis es all the inexact desideata euied uaanteed teation Notice that these odi cations io those o GaussSeidel: constaints whose violation does not exceed a theshold ae uned o consideation evey tie a e ection is alied the velocity is enoalized evey ste to event eney dit Coutin at each iteation o Aloith 4 euies solvin a uadatic oa (QP) Let be the exact solution to this QP the coesondin ositivity constaint Laane ultilies AC Tansactions on Gahics Vol 36 No 4 Aticle Publication date: July 7 k i k? is an accuacy aaete indeendent o i ; notice that this condition is a stad elative eo teation citeion in nueical QP codes The oal now will be to bound the inteediate ste = i NV in tes o the tue ste = i NV ; the oo o ( DRIFT) will then ollow diectly o identical calculations to that in aiwise GaussSeidel Once we have a value o C we will ove that inexact GR satis es ( NOR) ( OD) As in the case o GaussSeidel ( KIN) (ONE) ( VIO) all hold by constuction o Alotih 4 Let N A NV be the set o constaints that ae active in the inexact QP solution A the coesondin ats o The atix NVT NV has ones alon the diaonal o diaonal enties o anitude at ost one; theee by the Geshoin Cicle Theoe its axiu eienvalue is at ost the nube o total constaints in N Then we have the ollowin useul bound on : T N k kn A A A k k = k A k T N k d kn A A A d T (kn A i k k i k )! d k i k k i k ax µ C

5 All s Well That Ends Well: Guaanteed Resolution o Siultaneous Riid Body Iact with = µ = * d ax * d ax = d µ 7 = d d µ 7 = d µ 7 Coletin the suae ives k NV NV k k NV NV k ax ( 8 µ 8 as usual we have used < to siliy exessions Now n the th ow o NV we have the bound nk k nk k ( 3 µ 3 with = 3kN k µ 3 = 3kN k µ µ 3 k NV ( 5 µ 5 k NV 5 = (5k k ) 4 (k k )kn k µ 5 = (5k k )µ 4 (k k )kn k µ so that alyin the suation ula ives NV NV ( 6 µ 6 6 Theee d µ 4 = (µ 3 kn k µ ) d ( )d Then * µ 7 8 = ax 7 7 µ8 = so alyin the suation ula ives NV NV ( 4 µ 4 4 = ( 3 kn k ) d ( ) :5 = d 5 ( d k k kn k 5 )d ( )d µ 6 = dµ 5 (µ 5 d k k kn k µ )d ( )d Bee we can bound we need to elate the iulse usin the aoxiate ultilies to that usin the exact ultilies We can do so by akin use o the act that the QP s KKT conditions ae nealy satis ed : k NV NV V k )T (N T NV =( NVT NV ) V k d k i k h i h i k k d k i k k (k k k k ) d ( ( µ d ( ( 7 µ 7 C 7 v t ax NV NV 7 µ 7 / 4 7 ( 9 µ 9 sily 9 = 6 8 µ 9 = µ 6 µ 8 We then have ( µ = 9 d k k knv k µ = µ 9 d k k knv k µ The oo o ( DRIFT) now ollows identically the auents aiwise GaussSeidel with µ takin the lace o 8 8 As in the aiwise GS case constuction o a C cetiyin ( DRIFT) euies that be su ciently sall We now ove that GR satis es the eainin oeties ( NOR) ( OD) L A4 Let C be as in the oo o ( DRIFT) suose that a a b a= ( C b = 4 ax d C ax d (k Then Genealized Re ections satis es ( NOR) Notice that both ihth sides vanish as deceases P Since at least one constaint ust be violated by ( VIO) kn T NV k k i k k i k V ( AC Tansactions on Gahics Vol 36 No 4 Aticle Publication date: July 7

6 :6 E Voua et al we have used ( DRIFT) aain the act that the laest eienvalue o NVT NV is at ost Fo the bound (7) on the coonents o c we have that C kck ax d this is less than k k when C d ( ax d Reaanin ives C ( ax d the st ineuality above Lastly since k i k C we have that kck k i k wheneve C ax d (k as in the case o aiwise GaussSeidel At last we end with L A5 I < 4 then Genealized Re ections satis es ( OD) P At evey iteation a constaint is violated k i i k = k NV ck k N k kck k k k i k kck k k k i k k k k i k / k k k i k k k k i k / The ihth side is ositive when < 4 AC Tansactions on Gahics Vol 36 No 4 Aticle Publication date: July 7

CMSC 425: Lecture 5 More on Geometry and Geometric Programming

CMSC 425: Lecture 5 More on Geometry and Geometric Programming CMSC 425: Lectue 5 Moe on Geomety and Geometic Pogamming Moe Geometic Pogamming: In this lectue we continue the discussion of basic geometic ogamming fom the eious lectue. We will discuss coodinate systems