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(  L@:RS|HS0-8B .8?2j82FCAUpؘ*4@CAAACBA@CB@AC@@B@@@ABAAAaYaY!2C!2!-!# !<!   2L@:BNuc0@1#[pb-#@:DaA#:<@@BC@BCA& BJ"D@`6 @2Ajl2 @2@2DjD2D" $DXT> PKkxHC<<Index/MasterSlide-4.iwa8 J#?*B@@ABBCB@ @"31B/ Transitionnone?)`0X傧*B2@: R Title - TopBA@BZتxCLorem Ipsum Dolor$B!*BA   >  B B ZD*C%l?%@A-?0(8$Ah*b.=T   _:*C 0*C >D$B 4H*BA" ) Text* 4A2 :BPr @60 ABN0MC0060V0$BB50A BB)1*BA"* A2 : BPr ( T*+C  >  B@C @QDB%n#,@A-?0(8 Ch*b.=T    _:B 0B >D+."R*bJ% TThe QR algorithm: 56bJ6JZ6()(*A+ .8BM%8$ZDC8A.8=f8"=8C28V8+.:<*KB%:5$BJ$KZ$>Z C+="C%")ZZ^Z+. }.Z,EZsignumY^L (Pfrang, Deift, M.)rL  en )y(7(*C2uC#%b B C" 58J(Zbj b xy :R (6hB:hC 6DA! hB.h=fha :R0++L*@,ijuo@JoBJ,Zz, ,  N1U1(CI1>D1 5M1  ZA1A1~159f :fRf:1 +AN BA1****6V+.y<*AM/U*ȺF.At%"KD,BAǪ=jS_9 ,:,R(++uQW*Jź<Hamiltonian = ,* Jfb (JtuC DE_ݡtC 5mC  ZaCaC~C59V C>F Ev+.?S. TodaY( PKkxH ~Index/Slide-192.iwa^71*%  B @ā B"31B/ Transitionnone?)?0X*2: :ā :@Cā  /  2 *  ?  ?9CB @ 4D%!<,@A-?0(8(h*b?((BT  E _ a6,B0,B >D   hi*J"Moser's solu!x formula (1975)* 2 <: JPr en ! )T4*B  1V C 6D V B=U>fUB2e :U^U5W$M*@B)Y*1'@J<BJ1].H==  -  !XAU *m =TB@!T !IDE!f m=T >j@T `Q :T^T 5R *B́ )QDBThus, L(t) may be recovered by Stieltjes inverse spectral mapping.* Ḿ R.V 3,*C2 ! Z6Cv CE 1Dp)BC" 68J8Z b b4>a1 J4J %%Cv CR 6 $Z%b%z @.Nd6A 2!MZDCA}9>f`C.J u;<*BĉBJ2y/B`2`B%'<)` >`j`  N2_Then!!6a^a D)*a*a*6aVa6aAUaAJa^a 0* n >*AA pNpA.2āFɁ]aTEHDF >1f1 Ɂ 12HThe Toda lattice 8@solved explicitly@i>6\theory. Suppose we know\ data4 initiaqtrix. If5denote* }}PKkxH, Index/Slide-191.iwa 7XR*F@ֿ̿׿Bʿ B"31B/ Transitionnone?) p0X*2ֿ: ::@׿        b )*C2 ?  )Dc;D CC'A%!t%@A-?0(8C"68J CpBZbT  ] CB@  >D!  8*m   B Bn ZD*C4mh*b.>b ` *C.N  @<*AB"* A2 :BPr A8  S,* }-'<D_϶C A%%+Cr 4*..? >-k )6B>Bai *mYfA A 09,B%Az.2J,Bb1 `U ,2JU *JE,Phase space and QR iterates for 3x3 matrices* XJVJ.HAj,en   ]cl>VMAei6Vs6 Km%sumJDK.D2)׿:)m>)C 6Ai.>bȡ!U :BZ)UM*@Ͽm51@J1BJQϿZ=  $  /0* @ DE~Aa*A\ﵪBPjA & [Bz  ( (! / K.( 5D - (- P! \ s (Y6B||>:^ >."74C5!u"?:    :'6'rA~&4nB%p~:' >$f' " p8 ^ ɿp&ų>1Deift, Nanda, Tomei (SIAM J. Numer. Anal., 1983).* J6+ :ZC " @A !Z9#!Zi z|  C* >HjZρb9Z:^ *   )[>>Convention: The eigenvalues are labelled in decreasing order:5h h  =s%*6 Da>*6\=C0_F \6\ 0* n >*mBeH"% NV jBf>f@    ^ f.<*  >> A^ B.02Z D6!(%4Zp~6? (<* x&"/LB  :DqD/"60FDZbj v5& FD6 FD*  :* ƨ f!N:  B B  ^ F }4 " C\!" Ҙ ?   6 PKkxHIndex/Slide-101.iwaT4.*"B@ B"31B/ Transitionnone?)|?0X*2::@= d  \*A   ?  BMC ZDC%%@A-?0(8Ah*b.>T   `6C 0>D  @<*BB"* B2 :BPr ! S,*C2 -'wMD.!' ZCgC'C" 6p8J %MC|DZ b ^;0! -+@  J;A?6 B6 C 6D!% B. >f AuY B.N U M*@I U@JBJ RZ  Mp$  ar  $*C u]`K lB /D%aJuv]C=T >(BbT 2C`-D ,:T,^T 5Rx*Je\.From random matrix theory to a sorting network*1[J|J.pi en}  m:sm9sTܬAla|$sm=s>jfs2s :CZ 5s*1t lInterest!Y. *1ڲ6jAVs a*LA=>b "jɃ*6J8 6sAgbAJ4^s<0* n >*AAepNpA.V}K&::.C$( 4DM%pC~ 2 >j:n ` :^ }:*q:`8J. Moser -- Finitely man,ss-points on, line (1976)uJ :m2sҧ2mH~jm =m *JET With probability 1, each choice of initial condidetermin:?ډPKkxH,_66Index/Slide-136.iwa2X1+*B@ B"31B/ Transitionnone?)t?0X*2: :@7 Q ˨*C  ?  pBC gD%! d@A-?0(8Ch*b>0RCT  DP CFC >D  \*J"Tracy and Widom provide formulas for the joint law of k-top eigenvalues, but,largest gap one must simultaneously consider the entire pdpprocess. An explicit descripA(6l rxis not known. * 2 !: !JPr en A+:2 jBb ]2L|Theorem [M., 2015]:Fhnspectrum1\stochastic Airy operator! Z4finite a.s. *9 X |ڹaMH* u >*AAe*QAJ{A.4])$ ,*A CA+MA+,ZDC%a~CAC>j+`S:CCZC=+<*BB6)BJ)BZ)$ )*6{C98l{!8Vc>{r{ }c6{Proofalvia a new inverse scatteringacory[:Z<, based on treat..`.J>.B6.C 6D.B..>f.ȁ. 2N Y M*@Ӊ1_@JBJbBB$  " 6V B!V***6J$ 6A-S5NN^PKkxH=RXJIndex/Slide-190.iwaB"#u*((((B(((@ B"31B/ Transitionnone?)`0X*(2(:x(@(((&,((*+C  >  MC:C @IDBC%#,@A-?0(8$Ch*b.=T    _:BC 0BC >D0++ *J"For each algorithm, the rescaled (zero mean, unit variance) empirical distributions collapse onto a single curve that depends only on the , (i.e. not o tensemble, matrix size or deflau toler$. *  2 !v: !`JB  f sPr+*!8en U m s  8(I?(*B+ YB,C 6D - BB.BIfBB2R B6BBZB$++UD L*@,BIA*1W@JlBJ ,.c=6, M,  >6DC-B!L.DBe|!誎=fL "y :L^L++5Jy*J )I' EuL0universality :U JnJ.cME 2v( I( -<*+ ++ >*BA5AJB.|:(A:(B ) ZD*C(A.(=j_=*:j*^j+.&<*Am&5$Ⱥ$:$ +A6L)$MŎ$$6$V+:$KĉJ5$BJHKZHPKkxHmgIndex/MasterSlide-3.iwaL%@ *BBBB@B?AA@ B"31B/ Transitionnone?)t0X*@2@:?::RPhoto - VerticalA Media?BBbӪA Lorem IpsumDonec quis nunc?@(@B%8*AA   >  @DHB C"D%!"l@%@A-?0(8$Ah*b.=T H _:"D 0"D >D<AA @<*BA"* (A2 :BPr B  Q@63A@ 23B:3C3@F3C3C 1S%3CZ3AA=3H!3C%3, Title Text5?C? ?)@0v*CA2QcVC50@sD*"cC"A 5*A8JAA0DZbj (A  UY.?bA  - aB(C 8D O @D!} @B6C 6D%E;%a~B2 =ja%}B 2BZ@a} @BEU@R=$B:$eC%$6$ =j$".$6V$BB5$B)%@NBody Level One Two2hre2 Four.!Five*A !8  / ?2(r /?PKkxHrY}}Index/Slide-189.iwayT4ĉ.*"ԉ݉Bȉ։މ܉ʼn@ B"31B/ Transitionnone?)h0X*܉2݉:@ԉȉ=    6  ژ*ЉB   ?  C 6D %!%@A-?0(8Bh*b.>T . BY6BB >D Љ `M*@B"* @2 :@BJ Pr#    g  E1!6gA6gMC!$ZDCgA.g>fg"5gC2gVg5g<*KB%d5_BJ_KZSA -:,*C2QCv!' kBhA'C" 68JAXBpBZbT>^5!9;@  J;ȉM20*ʉ ʉ=N >*Bm6PmJPBbP ։FP -PnCu!P CpaBP6-P D$ZbzPC6 @6PaR -Q *CPCdݷB T3D,BC}W>bW "Q, ,2NuUd*JEtThe largest gap conjecture* J#J.# enx6AۉN !P7***6Jl ۉ=R=*B %R56 J6bY ʼn>y6yB8aYI$ĉn72y>Hfy `q56)V)Ay.'sAy %'$)Denote t:in(spectrum by5R JR:CyC.!] :DB]>H~]f] u2^0Our guess: aobserved "universal distribution" for%z,Toda flow isTedge-scaling limit of .xQR and signum are more subtle.*HPKkxHqH.Index/Slide-188.iwa B"#u*%%B%%%%@% B"31B/ Transitionnone?)`0X*%2%:x%@%%%&(%%*A(   >  B B ZD*C%l#%@A-?0(8$Ah*b.=T   _:*C 0*C >D<(( @<*BA"* (A2 :BPr % R*(C9"cCc!",T3D,B%~"C9"=b" "5" ,B6",BZ"((5 w*J%!H8Empirical deflation time statistics: unshifted QR on GOE* 2 >\JZ\ %)(*C2Q@&AC 9%]DC"58JADCCZbj b59v 6J:c (B6cC 6DaIAB]A=bA!OUA :A^A$((UC L*B@+ID uj@RjJqu+Z+ mu@  % M1 :^ARMABb:)CN)$(: BK%\5WBJWKZL% -K*(m (.j >*AB6lͲ-PKkxHIIIndex/Slide-134.iwaED$#*@''''''B'' B"20B. Transitionnone?)\0XօI*'2':x'@'''+,''*C)  >  CXB D%(#,@A-?0(8$Ch*b=((BT  C$  2k:,B +>D0)) Bn*J"$From deflation to global dynamics...* 2 : cJPr en 0' )S'  <*) )) >*AB*rAJB.|> )C9 >B~f5D%n! B%pC=b "5 :B V).p*J%!1Deift, Nanda, Tomei (SIAM J. Numer. Anal., 1983).5JU6=U')(*B) 9WC 6)WA &B}&MjWB.6 :W^W$))U{ L*B@+)Z U@J+BJQ+Z+ M,  :N )A6NMCA$ZDCNA.N=fN"]C2tV).<*BK%L5GBJGKZ<'C(*C2ֱ#,?!#DFD#C"} 58J(Zbj T>l5C}FD e FDN<:A:UI {)***6V): AB6ȲPKkxH1j#Index/MasterSlide-12.iwaJ#@*C@ABBBACB A"31B/ Transitionnone?)`0X޹*@2C:RBulletsACBbتlBMaecenas aliquam mHligula nostra, accumsan taciti. Sociis mauris in integer, a dolor non dui [`et feugiat augue imperdiet laoreet Sagittis felis sodales, dolor sociis mauris El eu libero cras \rdum at eget habitasse elementum est, pede class Wisi mattis leo suscipit nec amet, nisl ferm B$tempor ac ! nbifend!$venenatis,( vestibulum Din hamenaeos A$C Ap @ *AA   >  B B ZD*C%Akl@%@A-?0(8$Ah*b.=T   _:*C 0*C >D<AA pH*BA" Title Text* 4A2 :BPr B  ^B6@A>@C%@ D@=f@ "9@ D -`)@DZ@AA=@H@6$iIHe)H=f$ "9$ H$@@5$[B)%@NBody Level One Two2hre2 Four.!Five* i!8  / ?2(q /?}^PKkxH{Index/Slide-187.iwaN. (*   @B B"31B/ Transitionnone?)H0X* 2 :::@   1,  0*    >*Bm"* m2 : BPr   pN* A   ?  BMC ZDC% %@A-?0(8 Ah*b.>T   `6C 0>D  =+<*KB6)BJ)KZ)! -~ *C9'<CD <)6D!v'C=' >Bj'C`1:'CZ' UP *J%'REOverlay of statistics of 1/(largest gap) and observed deflation time.* JoJ.oIen Aac6Z BYC*C**6|V 5~=*B %}56 J6bJ:.CȺs!)(T3D,B2>b "u ,BQeV) ='xA%'.The ]x conjecture: numerical evidence5WJW!0B 60C 6 dB >bd B2  :dBZd U$M*B @)h a@J< BJ m .} n    K  ,*C2͕[ClAׄCpϒC" 68JTzC`CZbb 0U `:`RPKkxHFIndex/Slide-1.iwaT4.*"ƁB@ B"31B/ Transitionnone?)|?0X*2:Ɓ:@=Ɓ  ۨ*mÁ  ?  7BAC UD%!d@A-?0(8mh*b>0HBT  DP CFC >DÁ  Bn*J"$Asymptotics: Sobolev, Sirazhdinov. * 2 : JPr en !BX9XB EC D%EX >(fXT3 `-H ,B )XVX =X_*1XStatis!W( of saddles5IIFI.IOBM!MjfI U2HTheorem [Moser/Tomei]: Each equilibrium is a permutation. The dimension!Pthe unstable manifold9given by"number0 acents in2\ 

The/-4 with index k,B_Eulerian9A(n,k).Uf,*C2))BC6wA,JIDB%ArfC" 6p8J DRBZ)b)b0mqj@ Jz0* UF >*AA*AJxA.™ a)O,*A B ZD*CR<A>j`5> *6J> =+<*AB6)N)BZ)_  My6:V:MA ::6:V:6:B::BJc^:vF)q%mC]!'یC$Z6iCRaaz0aq@iBN; 6 B6CC 6DNaCB.C>fCayC 2N nM*@mFt@J BJ V bWaW   PKkxH ]]Index/Slide-195.iwaY^71*%B@ B"31B/ Transitionnone?)t0X*2:@C      *C2 ?  CC Cp%B%!*t%@A-?0(8C"68JLZbT d  t> p%B 0>D!  0*   >*Bm"* m2 : BPr !  p8*m  1RB BZD*CR mh*b.>f>`9N*C.>N> =+<*AB6)A):)m6)C 6D!%r{!)2)>f)!9) 2gZ)UT$M*@B%,UZ@R1J Zf  M  Aa   *mY}BC!T f}}; `Mm :T^Tށq,*JƱ )THA modern inverse spectral theorem [Bloch, Flaschka, Ratiu] reveals that. iso>Hmanifold is a convex polytope --the permutahedron. 

The iterates of )LQR algorithm live in rnterior )isb.*Q J+XJB  xƱ .4 enB&.& BbV5&>bz "uz & U& *E&I# :Assume L)ʰJacobi matrix with distinct eigenvalues, say*~opFddCF!o7Cpd6d Zdzd0$m Jd>m6Mńfw `YQ C.J 6 KmEQNK.2xBx6xB1+!'uxxZx2*}{1 ar6f.cesn  .rV   f B.< K,!A\,>b " ,2B s*J|)@dynamical system:Ňpha:acexRxnPKkxH<޽Index/Slide-184.iwa)rJD*8B B"-+B) Transitionnone?) X0*2:hi          K)Gp*A h ?  0BC jDA%! !%@A-?0(8A  @ 0A33>L  @<*AB"* OA2 Q:BPr !  <*  H)gDh*b ?T !* ":D 0D >D 5+ >*)- ,=.Z/Ac )0AJU*E5=!=)=A%=), @ `eAW@ & x3AU@ F@rA ( A (/]Ak Ǡ@ D@( X7D \ (xD \  AB.(*C Y!I%@TABeAnC.>f "Y B6BZqŁqq*BbEHamiltonian system of n particles with unit mass on the line. Each interacts with its neighbors through an exponential potential.*Yb}@bn$* 2$?*Bu2r m,*C2 uAAUUaȒC" 68J D0AZ b j f5Y5 A65AN5::[AFJ6CBK!,څ >Lj "]P p6P,^a2jaeDThe Toda lattice.du 8  en* J yfCa!`Bu6iD Zaaz 59m :mRm :A.B*A-aaF2 :AA]Ei i A . !>j J u@Ya) y6iDaaz 5y ]J D&AuBi6  Zz 5 :ZeN$ TVT TC RTCf VTC6CN_ " =>S涡 >- >-A 7E% za2@ Ab- "5 : R 5- N*B-0"(Z5JjU $  :Y>!WYYYYY>PKkxHm++Index/Slide-156.iwa'L%# *B''''''''@ B"31B/ Transitionnone?)l0X*'2':x'@'''+(''*B*   >  C 6D %l#%@A-?0(8$Bh*b.=T . B_6BB >D<** \L*B@+"* /@2 :BJ +Pr" +  d@  ' *C*9L8,ChB  ,BLC=L=bL "1L ,:L,^L**5J`*J)I!Toda phase space for 4x4 matrices* 25;5hJZ]:A:A(MCA4$ZDCA=G=bG "9G C2V$**5I<*BK%HUBJKZ$$'e(*C2qA^4" .D@aD#C"58JaDDFDZbj f5U FD2<J<>L *CyL>B~f5n B%pC~2L=f " o6^L*.Lp*qL1Deift, Nanda, Tomei (SIAM J. Numer. Anal., 1983).*q\J8b\' M8' 0** *. >*AB6AJ|BZBR ii***6J$*.&<*BA6$Ȳ$PKkxH@ؽIndex/Slide-181.iwaP)#$*(@((((((B((( B"31B/ Transitionnone?)l0X*(2(:x(@(((5,((*C2yz >  KCrC uDB%#%@A-?0(8C"58J *C`BZybzjyT q^  2:B 0B >D(  (  D*+ ++ H>*AB"* 0A2 :BPr ( T,*+C  1[Bg![@Q! B%ͳCn[ Ch*b!#=T>1% "9E :ERE(++5"(*J b%$LLWe call this deflaAl. For a given tolerance, we&ine th 1 time5rbJrdJB     > .+(en =(I(*+A .BM% ZDC!6A.=f"=C1EJ(++]<*BK%5UBJUK.7=$N!{|U8!)!Dz]!5!-a x! {b|j{v! 2Y x:xR: A+N Baj%***6V+: AB6Ⱥ>+.PB֋E@[DAj$: =f$ "9 :b++*JlWhen computing eigenvaluesHmust choose a stopp&criterio decide oG8nvergence. Typi!y,try to split|matrix L into blocks as follows:uVZ (6B:C 6DAcB.=fȡM9 :R0++5L*@,M5+@2J,Z6 , ,  > 2 $CK,KD,! =jL_=L,:L,^L+.   BMC ZDC%l#%@A-?0(8$Ah*b.=T   _6C 0>D<(( @<*BK"* (B2 :KPr :$ (B6$C 6D%~$B.$=f$!9$ B2$V$$((5$ L*B@+%&5+@J+BJ16+Z6+ -@  % *(C]p@,D)L^CBA^LC2L =jL"=L:LBZL(.nL*bJ)H8Toda, GOE data5SbJSJZH>020 $BK=uBE|WD,0 =j0"=0,:0,^0((Uzc*11x$Examples of empirical distribu:4s * H % %>R %Z***6V(.<*BAEU|AJ4BZ|% m0*( (.j >*AB6lͲl>2B ED=f "Y :^(..LThese are histogramE the deflaAZ time for a fixed tolerance (1 e-8) and matrix sizes that range from 10, 30, 50, ..., 190.* *en  >2B =f "9 (.T.0unshifted QR%(*C2"q @w@Dw C" 58J !{PCZbjT>D D 39O :CN% "R aC54CZbj 6JqPKkxHV\11Index/MasterSlide-6.iwa- J#@*@BA@@BAAB A"31B/ Transitionnone?)`0X։*A2A: RTitle - Center@B@B^Ӫ@Lorem Ipsum DolorDonec quis nunc$@ % @ d*BA   >  B $NDD%!l@%@A-?0(8$Ah*b.=T H _:D 0D >D<BB @<*BA"* (A2 :BPr :$ B@6$C 6D $B2$ =j$%=$B6$BZ$@=$ @B6$@$A6HBRHC%$%BH6$ =j$"=$:$BZ$$AA5$aB)%@NBody Level One Two2hre2 Four.!Five* I8  / ?2(Q /?:AV~=CNCC e CZ AAyHai Text*qȲ8B PKkxHpIndex/Slide-116.iwaX1+*@B B"31B/ Transitionnone?)?0X*2::@7 Q ˨*C  ?  `KBlB /D%! d@A-?0(8Ch*b>((BT  DP(  2CJ,B 0,B >D  Q*J"Summary* 2 : JPr en !G:;C9;f;JbF `-+ :;^; =;*Jb);Љ(2) Empirical universality for fluctuations of deflH times emerges from6>@of gap statisticsPhTracy-Widom point process.*1b2%!:a' I0*B  U C 6D.%A;nB]2%baf1 :^U$M*@B) 5<@J< BJ m.]  mc$  v 0*  >*AAAJAZ6A6!ZMa(ZDCA.>f"=C  V=+<*B)5)BJ)BZ) I>:B!8 NDEҽ >f8J .B(3) Each algorithm and matrix ensemble determines a class of ransorting networks. It seems reasonable to hope that 6$could holdtheBXx (this has not been tested).* ɥ$b Z [J :?A>i? I*?*f `e?*6?V?6?Au?Nh^?9 :2. B6.a.MB.>bf "5' :,b }.K(1) Several iterative eigenvalueyD$s are tiedAintegresystembڧPKkxHƖJJIndex/MasterSlide-7.iwaF J#@*BAC@@BBBB C"20B. Transitionnone?)\0X໺4*B2@: : 4RTitle & Subt d@BBbӪALorem Ipsum DolorDonec quis nunc,@A %"$ C $C@ !*BB   >  C 6D !#%l@%@A-?0(8$Bh*b.=T . B_6BB >D<BB 4<*@B"*  @2 :BPr :$C:$ C NDB$6$ =j$"9$:$BZ$$CC5$!B)%@NBody Level One Two2hre2 Four.! Five* !u8  / ?2(1 /?!] @A6)=C@2%sCC6CZ AAYH*BAE Text*QA:0CR0BYD0AF0D0D eDZ0CC=0@  >DJ L0A=C lCDN6HICNBZ b r0 @  J!  -$)8*A   B !ZD*<Ah*b >^ ` *2N  @<*AB"* A2 :BPr a^>)V)MA ))6)J 6)B:)BJ)^)R-{0* n >*AA6NAZ  q *֛CY%ACA A! ^A'C] >(Bb2lU ,2V ֛UZ*J%&XSorting networks* JJ.9I en]:DC9DcC7Aa D>H~fD `:^ 5D!E )EPNProjection onto diagonal of a Jacobi matrix evolving by Toda (Roger Brockett).*- ,ڇ6dB>d C 6D%r`AB]2JbȡU :BZ5M*@@J<BJQ.]  $  B]T DJ%!T I}%pCT2! >jnD%vQ :T^T ] *JIݨ?Angel, Holroyd, Romik, Virag-- Uniform so6J (2007)U  JUPKkxH=jjIndex/Slide-183.iwaf N' !q* @B B"31B/ Transitionnone?)p0X* 2 :Vx@   %  *B   ?  C 6D |%ې%@A-?0(8Bh*b.>T . BY6BB >D   `M*@B "* @2 :@BJ  Pr#    g$  !6VA>VMC$ZDCVA.V>fV"5VC2VVV 5V<*BK%S5NBJNKZBA-A0*  n >*Bm6pmJpBZpa  q$*C UApiC! b.$DEECY >(Bb 1C`- ,:,^ 5 *J%'3Part 2. Random matrices and empirical universality * MJ]J.] a=en 2} : A6[i Bu*C**6V ֑[=* B%o5: J:^, PKkxH2)Index/Slide-124.iwaN'#"*@''''''''B' B"31B/ Transitionnone?)l0X*'2':x'@'''0,''X**C  >  #C?ΣC 7DC%ඐ<#%@A-?0(8Ch*b=T O _:C 0C >D0** *Jb"(1) an algorithm (QR, Toda, signum) (2) a deflation tolerance in the range: 1e-2 -- 1e-12 (3) a matrix size between 10--200 (4Pensemble (1--6) *& b $ % [ \ ~  2 : 2JPr >2BM6)CB% =j"=B6BZ**5T!-$Choose:* -b) !Ken =8'i(*A* y B BA(ZD*C%AzA}=f: "9: *:*^*.<<*ABe53AJ3Bb\':3 C*."!"%"W!",\ =f" "9" ,:\,^\*.3X*J%!The numerical experiments* J=b:a *ANaMe%aa6aVr*:a BK%@UaBJ$KbaFa2aB`DeNAa#Cҽ=fa "Ya #6"V"** * Compute\wime and*AB6NͲPKkxHfttIndex/Slide-138.iwapR+.&*..ę.B.@. B"31B/ Transitionnone?)?0X*.2.::ę:@...45ę0. X *C  >  @BC *gD%,.,@A-?0(8$Ch*b=0HBT  CO CFC >D  >*J "KMain new idea: Combine analysis of Toda with coatoricspermuta!Ds.*  2 : JPr en ,.)}(*B. 9 C 6D! %!%zB=1%]fB"5oB!N$..= L*@.B)*1!@J6BJ ..A=6A-$  A M4*C2 >ERpC"A(C2bCNC" 5p8J DyCZbf05b y6Jba:ƙCY-%VB eҬ= RCf- ƙu,*JI0<Invariant manifolds: Each (n-k)-face on the polytope is i:.0such f/corresponds to block diagonal matrices with k s. * m  ,  F GJ:.A 2aM!ZD$A.=j_.6J$.ұ<*BB%ՕBJ/BZ:.8[CȺSB EC  =(fT3 _ ,:+,BR$5$v%%L,Understanding deflwP (M, E. Pujals, 2015)ՊJT!يZ !_IF***6V.]A͈UAJ/^. w * ..j >*BA%lͲlPKkxH\c!!Index/Slide-178.iwa$rJD*8Bҥ֗ B"-+B) Transitionnone?) X0*2:   *:ҥhi     r ҥ)GV*ߥC2 ?  B nC [sC\ ? >?HBAPgPC'!2??b? "26CZ?U?=*BE? M AJ.=IJRX6B~ JX X &D A%"L 6XDXہXXzX 5Y= :=R F .X~!HB!_DPX ?j " ,:V8 A2z*JfT;Particle system to ty maaTes (Flaschka, Manakov)UKJbK a** V)DDfh a9h D6DZ>*>y2y ^ }z#  qz?ǕCcACz6mzCRezaz~ua=? @.?PN?J CcoA [ACz8 Z6 !" b z !a(*חm  -CV w`$%ඐ<nB mh >(fS};2 2Jח^M*S SetV $* ߁2$?*B}2j "r .% ( PKkxHꬍIndex/Slide-194.iwa N'!q*@B B"31B/ Transitionnone?)@0X*2:@%  *²B   ?  C 6D |%ې%@A-?0(8Bh*b.>T . BY6BB >D ² `M*B@ղ"* @2 :@BJ ղPr#    g$  !6VAʲ6VMC$ZDCVA.V>fV"5VC2VVVʲ5V<*BK%S5NBJNKZBA>)V) B5)*)*)*6)V)5)=*B )*QX J+BZ+a Mm(*C qPCAR b.$DeHeAnCy >BbR 1C`MB ,:,^ 5' *J%(^\Part 3. The largest gap conjecture, gradient flows on the permutahedron and sorting networks*1JJ.i en2  _-0*ײ ײ}; >*mBu=mJbPKkxHozMMIndex/MasterSlide-10.iwaIJ#@*ABC@BBB@B B"31B/ Transitionnone?)`0X*A2@: : RTitle & BulletsBCBbت@Lorem Ipsum DolorMaecenas aliquam ma Fligula nostra, taciti. Sociis mauris in integer, a dolor netus non dui Xlet, sagittis felis sodales, 5sW ^osqu Esemper aenean suspendisse El eu libero cras`rdum at eget habitasse mayleoDcipit nec amet elementumA(@AE*AA   >  B B ZD*C%A l@%@A-?0(8$Ah*b.=T   _:*C 0*C >D<AA 4H*BA" I Text* 4A2 :BPr B60 BB60C 6D % ~0B.0=f0!90 B -P BBZ0B10<*@B%05$@$C )@63@VcMC!3Acc6cVc(@@13a6B)4@NBody Level One Two2hre2 Four.!Five* I!8  / ?2(Q /?]fACBC\% ֥=j_]6N CC<*q @ I!4B CPKkxHZM, Index/Slide-176.iwa Qjd*XB@ B"31B/ Transitionnone?) t?0X*2:: @     #    #    # )*A  ?  C `D AA%!A:%@A-?0(8A*.@  ﵪBPjA & [BB B(- (- / K.( 5D - (- P! \ s (> -E@<*AB"* ! A2 ! :BPr A )8*A  5BMC!Z ZDC Ah*b.>T ! "6C =' A'6)B:)BJ)f):)^) !))*)*)*6)YP!$%m 6)BRȲRau,* qD iA6Ay 6:6C@eF 66B 2C 6%B.>fȡ B2V[խM*@Ͱյ@JBJ Z      F֝HC@reX 6 :A& @0Cj:M mC,*C2 MB{a1 "KDɀB1C" 6p8J }DRBZbf/ }D@J ]x *C&  L l n n Cw> H~BfwC66 Ju *J $ Heuristically, the spectrum of stocha!t Airy is a random perturbation,><5Loperator, i.e. minuszeros;+ funcP* J J. en* =c A֝F 'DFFFFjFF:FM:6w  Xn>:6     f  6 ZF *   r -::2:n>~F2.2 D n a22F2.2b 2b [[B3b  *gb >Hb fb *b 6b XPThe edge-scaling limit "k (is given by St6x C .  |!2m =C^$Y >r " ^2 (Eigenvaluem i H)H.H(*A "*B!H?A=? Af*B` R &j   (j   . :"B:&N&bO0* ** >*AA" NmA. 2 AC&CȺwT3!,B92>b ""9 ,2V ]mA%&#_!: :]]]]j]!>"#6  n ՁFN.krF.2U &U D0 0 >0 r0 *U iU ":0 :P ; ; PKkxHuuIndex/Slide-175.iwaq R2m-*!mmmm@mmmmmmmBmm B"31B/ Transitionnone?)`0X*m2m:m:@mmmJ mR$mh*C2ji >  C} D $C$B%шm%@A-?0(8C"58JTZibjjiT e^ u:$B 0$B >Dm 4*mm  B BlZD*C$mh*b.=b _ *C6*CN$m @<*BA"* (A2 :BPr m R*mm9"腶Bb&AK[!" B%ඐm=a _Q mQ,*Jq^A,2,is an iteratABDscheme to compute a}adL. There are two steps:* xm JvMv(MvG I ]~.2Brh!& )AΠ =n _5:iRim.}.$(Step 1: C-|0QR factors at-e k*6!U   =aQ 5b*m*Y CE8!cc =cbcu2;%f,2: IntertwinEse5l to obtainEnex5|~yA * m: C 6s .s=f| E:^$mm5yL*@Bs 5B@J BJ m.M]m  0  mJ hgѦA`  5ͦ ,Zgbhjgz2=a :a Ra >mݶ CI:?­ =j:_ma ѶJ. ObservaatU?Z$ Z4`}jm:j ME   6 J $m12<*mK 5$NH KZ$m mW0*m m.j>i m6lNlbk PKkxH CIndex/Slide-174.iwa^71*%@B B"31B/ Transitionnone?)t?0X*2:@C      *C2*) ?  UCKLC Cp}B%!*t%@A-?0(8C"68JLZ)b*T d  t> p}B 0>D!KJ Z1Cv 1Dp)6 $Z b v0 N!),*C  B&HDl%v4Ch*b! >Bb@T ` 2J  X*bB"Consider the flow with Hamiltonian Tr(G(L)), and g= G'. Suppose we knowAf 5M.Then*1!b6M$* . >*AAE<{AJ{A.2>Cyb8B@D !Iib >j }b2bBThus, L(t) may be recovered by Stieltjes inverseyDmapping.5j  ͗,* ߍa!hB   ?(j._) ,:,R !.f*JEDGeneralized soluJ mula* GJJ.V~k 9,*A .M ZDC!A=R>f<`9R p1b V<<*BB%SUBJ42 6)AN) A| *)*)*6)V)=*B )*Q_ J+62+E]TB>+C 6.)a%rP ATB.TEfTAYT :^$M*B@%-U\@R1J1R.==     PKkxH*WIndex/Slide-173.iwaX1#,* ''''''''@''''B'' B"31B/ Transitionnone?)l0X*'2':x'@'''I,''X*C)  >  `BC @ DB%ඐ<#%@A-?0(8Ch*b=T O   _:B 0B >D0)) Θl*Jb".At the k-th step compute ,QR factoriza!:* V0b2 :JPr ')P(*C2ml )O!C~%O(CB%jO C" 58J@Zlbmjlbe u9e :eReNon  5 ,Znbojnz2 : R')(*)B y C 6DAI !B}=fB2 :R0))u L*@B+i U@JBJQ+Z+ mt  a.N,jiQ,E,)KA,$,5I,,Zibjjiv, 2Y, $:a$Ra 'B888BbV"a };DB8=b* " :R())U(*bJI'$_(1) Reduc+Rinitial matrix to tridiagonal form. (2) Use a shift while iterating as follows.* jb 4J6n8  3en4: )A6aZM! ZDC!,A.=b "9 C2V).<*BK%uBJ7KZF)B%" a=f" "U ). F. Then set5*' 0*) ).t >*AB6AJBZ:Z BE}***6V).&<*BA6$Ȳ$' _ F]aT%2KD,anp 2 =j_],:,^).Y*J%18The practical Palgorithm* Jr^ F>2>PB'f!>=j>_=>).>. !?Ɍ`parameter is often chosenbe ' 0eigenvalue of dlower right 2x2 block thatIcloserCP &D-hand corner entryL<0(Wilkinson). ڣ 'B=B )2 =j"=).3M.<Again, we find1PKkxH? Index/Slide-172.iwaN.(*B@ B"20B. Transitionnone?)d0Xy*2:@1  J  \*A   ?  BMC ZDC%%@A-?0(8Ah*b.>T   `6C 0>D  @<*KB"* B2 :KPr !.  S *C9' C|aB KD,B!v'C.'>f' "9' ,B2'V' 5%Z*J%&XSummary (part 2)* J9J.9)9 en=DA-C0* = >*Bm5mJ{B.{pA Ia2:C9BXxUAE!>]j@Q `1 6V 5!%8LNumerical experiments reveal universal fluctuations in the distribution of  efla& time for three Hamiltonian eigenvalue algorithms.* IAJA6B6C 6De^B2 >jȡ}:BZ5 M*@Ӊu@J<BJ1]bať0  :VA6V ݩ*C*CfV `9V *:*CZV=*B e RPZD;:AC.pBeA@@GDC%ඐ<~-6>j)"A pICZ)A+2A$ Such "u+4ity in haltingXs" is not restricted to1s2 Tom Trogdon performed extensive numecomputk8s on several alQH, and found similarity. Th4include a crit scalreghe conjugate gradient method kan inte=decis  . y@results are repor!Pin Deift, M. , Olver,5D(PNAS, 2014). *u,  J>C\  en PKkxHIndex/MasterSlide.iwaL%@ *ACBCBCA@C@ B"31B/ Transitionnone?)t0X*A2C:@::BRTitle, Bullets & PhotoC Media@@BCZALorem Ipsum DolorMaecenas aliquam ma   B B ZD*C%A,l@%@A-?0(8$Ah*b.=T   _:*C 0*C >D@CCe4H*BA" E$ Text* 4A2 :BPr B60 BA60@DMCC0=f0"=060V0B10  8 DHD%@%@A-?0(8C" 5*C8JTH0DZbj (C  B?@^$h*bw 8T V ":@D 0@D >DB  @)\0*A@   B|ZD*C]A=f " *C6*CN$@ pH*BA" Title Text* 4A2 !:BPr C  ] m=? BA:?!?%?N!?]b6?=f? "9? :R0BB5?0AA5$a6 BBIdD!=  8*mْ   B BnZD*4mh*b.>b ` *2N ْ @<*BA"* A2 :BPr A86)m>)C 6D!w%r 2)>f)!9) B.Z)ђ5)$M*B@%,51@R1J Z=  -$  ar   *mY} BĬA}<};DB%ඐbT "Q} :T^T 5Ra*J)P`Assume L is tridiagonal*1DJeJ.YI en]-J0* }>a m5mJ{b Ȱ>mӰ:15KZ Ӱ={2H1and that the off-18 entries are pove. * m: 2 b  "u ,: ,^  =eY*JE@Jacobi matrices5CJ |>MAɃC2J Ԓ}<*Km%WuNK.]( PKkxHXS;kkIndex/Slide-161.iwag%lCl>*2@lllBlݐl B"31B/ Transitionnone?) {?0X*l2l::::::ݐ::::::@llldeݐ )A*C  >  FBB *gD%! ,l,@A-?0(8$Ch*b=0H~BT  CO CFC >D  X*J"Conven!>al wisdom: Toda rates are determined by the smallest gap, sinceB7sP of cr rgen4o equilibrium (Moser). * 2 !5: !HJPr en ,l)(*Bl 9 C 6D!l%L!%~B=1%fB"5B!N$ll= L*@lB)*1!@J6BJ l.A=6l M/  !PAH.C9NWB!N  DeN2=Hf uxa )M.eV4occurs at slowEU Jyz(*A zlC ]C 9g@|TC%eT>~ AfaEB` R & AAM  (iq$u  D5_<*BMUBJ-BZ="Y<jmѱ6 *AA=5KAJAb: .bB,!K b=brb bq2bH'Approach to x-axisZ fast[[!* Z  iZ***6N l}A>]N~b]Jq"!D.A~ eABA5i,AHAZaz0.@.J8  * \xCdICbR] :L >'@bC0  ׏XBn$, B:2s DȺAT3s ,B 2s =b "Հ ,2HJ^64Why the larg[ ?JMY .GY *Y 8C! !G@BY !%A  Y  Y i    Y (Y 6Y $n+DMk! 3CG[@[}%%[q +[CF' 22 mJ=D;(AJMDz5ҁ~AEH:ANAPKkxHCuIndex/Slide-39.iwaD$#*C&&& &&@& C"31B/ Transitionnone?)T0X*&2&: :&: @&&& &.$&  &  D*) )) >*AB"* A2 : BPr & lx*@)   >  B4B $NDB%!l#%@A-?0(8 @h*b.=T H _:C 0C >D0))5&*OB%%POHow long does it take to compute the eigenvalues of a random symmetric matrix?5vOvA% - * C9v<"CC E;D%'%drvC=v =hnv\C_5f:vVv =vn* J)v Govind Menon (Brown University)*  JLJ.Ien $&5V*C)9U uCDU-U7Cj%E]U$))U2T~Joint work with: Christian Pfrang (Ph.D, B%pN2011) Percy Deift, Tom Trogdon (Courant Institute) Enrique Pujals (IMPA).*"2P  7 8 g h $  ! 6 E T e =: B): Aqa9PB}9=jRBuIBZ).* m9i:, Division of Applied Mathematics, -y^. *-ei   2J63-!  1 2y: )B6C 6a}R6%ajB_5:^). L*B@+ @RJJ+PV+    PKkxHpO<Index/Slide-186.iwaD$#y*''̂ '''@B B"31B/ Transitionnone?)D0X*'2': :̂ @''' )' *B*   >  C 6D }%l#%@A-?0(8$Bh*b.=T . B_6BB >D<** \L*B@,"* /@2 :BJ ,Pr" ,  d<  '  <** ** >*BAARZ!  n *CĂ 9lCXC JbFDB%ඐ<~C9=b "1 :VĂ 5&*bJ%%@?How do eigenvalue algorithms perform on large, random matrices?5fbJfJ.f) en=a7Bq=qCW>!q CEeApr+.q =(fq1 _m ,:q,^q 9q^*J)qThe guiding question* G:rA:rM!G(ZDCrAY=b "Y C-WZr*.<*KB%FUvBJ/K.v]kB$R$ B *$*$*6$V*:$ BA6$Ȳ<' PKkxH/31+ffIndex/Slide-182.iwabR+#&*@'''''''B'''' B"31B/ Transitionnone?)l0X*'2':x'@''':,''X*)C  >  BC CB%ඐ<#%@A-?0(8Ch*b=T O   _:B 0B >D0)) \*J "The random matrix ensembles:*  2 : \JPr ')@(*B) 9C,C 6D % zCB=C)fCB2S :C^C$))5E L*@+B)F*1@25 :BJ +Z6+ -0  ':2|`CIAB֏ =fL "] 26^L).*Jb)IHT(1) QR -- with and 2,BbV%@>D=f "9 :^).J-@y Empirically investigates the performance of three eigenvalue9v s on2YDces from differenti. u>b>2C g4!aC =f "9 aC6aCZ)..aPGaussian Orthogonal E (GOE)i!(Wigner5ki' Bernoulli.,4) Hermite-1.05) Jacobi uni! (JUq 6) U @doubly stochastic=V$ (UDSJ) *#.X( F ^ v   ' ><*) )) >*ABr*AJlBP^:)A 2v B$ZD*CA.=jz_]z*:z*^z).&<*A5$Ȳ$ '  :4Z4MJ4464VX ):4 BK6XBJXKZX>2$C3T!"1.z =bV "UV z ).N*.z l >323aLa>[a,9 ~j . =b3 "53 ,:,^).3a!3ET"Christ Pfrang’$sis (2011)" JybPKkxH5FIndex/Slide-146.iwaN'#"*~'~((@(o(B( B"31B/ Transitionnone?)H0X*(2(:':o:~:~@(((0o~,~~X*~m  >  CD @>DB%ඐ<#%@A-?0(8mh*b=BT O   _:B 0B >D~ Ψ*Jm":Q is an orthogonal matrix R E\CC z*B{B%AGjC"5p8J BpBZbb0lU @PN( (*+A .)BM ZDCA.)=f)"])C.V$++U+<*BKE**qABJK.V}K( m0*+ +.j >*AB6lAJlBZl( Mno:o67CbV~%7j _y 7oα727 nAssume L#< real, symmetrix!. eGram-Schmidt procedure may be viewed as thtB2 U mJ>c,en ' + ¹o:HZH qE*H*CH*6HVH+.<*BA6Ȳ:$ B*6lC 6KlB.l=f$ȁY 27Z$(**Q L*@B,%&Uf@JfBJ,b,    PKkxH_Ԉ&&Index/Slide-137.iwa"^71*%@B B"31B/ Transitionnone?)t?0X*2::@C /  +  *C2 ?  'Cv,C rCA%!;t%@A-?0(8C"68J CpBZbT  ] CB@  >D!J Uo8DC=6 $  b z$: @6 BN )0*   >*AB"* A2 : BPr A  U,*C -P<[CȺSB EC D%Ev0 Ch*b> (BT>-T3 `6,2<J5' j*J!& Normal hyperbolicity (M, Pujals)* KJIJ.I)I en2T:T.T87B`D |' T>jfT6C-daVT 5T!U)UXjEasy consequence of Schur-Horn theorem for Toda. Analogous statement requires more care2H~f 5*!)0tMain observa, : order relPs between eigenvalues!4blocks imply nNFPof invariant manifold!qSRB- c5 ',*B 9C 6aeI%rB2b B"uUBR 5$M*@B) @JLBJ1m.H  0  :VA>V (ZD*CVA.V>fV "5V *6V8<*A-S5NȺ6.6CY}}B;<h!'>ځn' ;;I}Sa\est example (enough to understand largest gap): The stable equilibrium lies at the intersec of (n-1) NM( Each suchc is5I form snratŏattracafor anyFE0at least *6  60 7  J39 6A6izMASi znSZI6zVz6z BmEPuzmJ4^ PKkxH}Index/Slide-145.iwaL%/ *@//B////0/ B"31B/ Transitionnone?)0X*/20:/:x/@///+,// $/ X*/C  >  ]CIC JbFDB%ඐ</%@A-?0(8Ch*b= T   _:B + >D4// x*J"An important feature of many numerical algorithms is that ``typical behaviour’’ is often much better48n ``worst case -=. It*AB%M* lAJBPl>o2oX"C:AoGAoCo= ACbo"ATQ_oAoCZo //Qo* ]r  T-3). 3) Smoothed analysis (Spie hTSankar, Teng, 2004).*Y4 m  ] $ mQE mAFE A l n  e 2t>t C/PCHB 1C,BW/=bt " ,:,^/QtLe MotivaE\ &=q2&=1a#(*A0 93B !3,ZD*C%a4zA=b3 "53 *:*^00<*AB%4ȲR2$/N$M!$$$6$V:/!QY<*BK6$BJHKZR=$B:H C 6D*HB.HfHAYH :{^{$005$L*/@MJUO@J+BJq~/Z6   PKkxHIndex/MasterSlide-9.iwa J#@*BCBCCBBBB B"31B/ Transitionnone?)0Xۊ*B2CRBlankBCBbتDCB۰*@B   >  C 6D % l@%@A-?0(8$Bh*b.=T . B_6BB >D<@@ @<*@B"* (@2 :BPr C6$ BA6$MC(ZDC%$A.$=f$"5$C2$V$$BB5$!B)%@NBody Level One Two2hre2 Four.!Five* )u!8  / ?2(1 /?=f: A@:E^B!NDDҥ=f "9 D2N@: BA6AC i$C  :C@V %CEy***6VC@.CH*B)C0 Title Text*ȲOPKkxHv!!Index/Slide-144.iwaP)#$*&@&&&&ϲ B& & B"20B. Transitionnone?)h0Xy*&2&:&: :ϲ @&&&2 &40ϲ &X*C)  >  BD @GDC%ඐ<#%@A-?0(8Ch*b =$CT O   _:C 0C >D4)) Є*J"Certain sorting and dynamic programmllgorithms, can also be recast as gradient flows (Brockett, Bloch, Fabyusovich). This includes, for example, aD@ that solves the assignment problem -- so “purely discrete” %sbe KDd by “continuous+@methods. Bayer% 4Lagarias found>Punderlie Karmarkar's 5-FLother interior pointq
*   2 : JPr2",en  J S c v  @&I(*A) Y BMCA(ZDC%AyzA]=f"Y6V$))U<*BKE* -DBJYKP=Y=$> )C9"v?fBʏ%"@Q~"6 =f" "9" ).2XMany numerical linearA ebrao0 were shown tX time-1 maps of completaADgrable Hamiltonianesa\Deift, Demmel, Li, Tomeie=@Nanda. Moody ChuWatkin@$vestigated#*ABE;U6AR6Z+ : .PC|aB KD,B 6=f "Y ,:a,Va  -?ZA%%DSummary (part 1)5J8:A: %C**jV _9C *6V).i<*AB%BUgȲgPKkxH+'  Index/MasterSlide-1.iwaL%@ *BC@B@@BACC C"31B/ Transitionnone?)t0X*A2B:@::>RPhoto - Horizontal@ Media@C@BZӪCLorem Ipsum DolorDonec quis nuncA,B@%ARv*CC2 >  BHB @DrfD%@%@A-?0(8C"5*C8JTDDDZbj (C   t R C@Z h*b.8T V 6C 0C >D@)L0*AB  C6Dl L M%jB=f! B6BN<AA @<*@B"* (@2 !d:BPr :$ @@6$Yq6bq@.$)Z$@=$ BA6$A$B6H BC:$!@!IH NDBH6H =jH"]H:HBZH$CC5$aBII@NBody Level One Two2hre2 Four.!Five* ILB   / ?2(Q /?a}@>%=B2)s¥:BZ@.H*BI0 Title Text*qȲC !@D , @CC PKkxHt 9ccIndex/Slide-143.iwa_ N'³ !q*ͳ ˳ ۳ ȳ Գ س @B B"31B/ Transitionnone?)@0X*Գ 2ȳ :˳ @۳ س ͳ %$ͳ    *ɳ C  ?  g%CǦC Mt-D%d@A-?0(8Ch*b>((BT  DP(  1CJ,B 0,B >Dɳ   R~*J"4Part 1. Eigenvalue algorithms as dynamical systems * 2 : JPr en ۳)g,*ܳ B 9j C 6D.%jB9j2%Hbj B"5Z:j^jܳ 5l$M* B@)n*1'@J<BJ %] .H==  -,  >VA 6VMA(ZDCVA.V>fV"5VC5fZ 5V<*BK%S5NBJNKZB..AV) BA*)*)*6)Vó }=*B )*QX J+BZ+6Ml0*  = >*mB qQmpPKkxH RIndex/Slide-177.iwa&`@:*. Ď @ݏ    B͓ Ĕ B"31B/ Transitionnone?)0X*2:͓ :   :Ďݏ @U   *Ď ݏ )p* m  ?  $CTB $KD,B%!Zd@A-?0(8mh*b.>T I `:,B 0,B >D   ^*J"Symes theorem (1980)* 2 : JPr en !T:Hm 9H85C:C e8!H %ඐjHQ `18 :H^H 5H(*Jm)K@The iterates of!V` QR algorithm are exactlysame a%0 solutions toQR flow evaluated at integer times! The Hamiltonian for2>i ^Dpectral entropy.*-$m s tRTB   acuI0* mJu0DKA(AAJA*e;Am \ﵪBPjA & [Ba  ( (! / K.( 5D - (- P! \ s (y Uv<*BAEr*qAJB.}' Fޏ ="D˱^ ޏ 6 ),*m H C 6D0 F6 >jB6:H^HU M*@B͔ U@RJ ͣbR $  FHD%UZ >:m6AMZDC2>f"* CHe2 ,6) mK6mJKZ F.B!%(J 6^ u0* n> m6 N b :  :h    ^ h  : 6m> 7 ]***6V] : Ȳ   (*C2tPvC.A HDŁDC"68J0FDZbj f5*FD2 J@:”:FDeuuuu^uu”:PKkxH Index/MasterSlide-5.iwaN'?"*B@BAAB@ACBB A"31B/ Transitionnone?)l0X*@2A:tBRQuote@ Text-1% XBBCB^ت(Bw(BB % @ *@B   >  C 6D %l?%@A-?0(8$Bh*b.=T . B_6BB >D$@ @<*@B"* (@2 :BPr A R*AB="CNDB%n" =j""9":"^"AA5 Q*B@% P–Johnny Appleseed555B:5 CCB5=5X5C2W%X5X:5X^5CC=5W!5C)5,“Type a qa$ here.” 5;CpA}BA 2a) BAr$ZD*CA.==fr "Yr *C6*CZBB=?H!?A)? Title 50A0A=0@V0MEm0060V$@@uB؉@NBody Level One Two2hre2 Four.!Five*A8  / ?2(% /?E^EaB= A@:EiD=f "Y D6DZ@aiBIU B ɻ!4D CPKkxH#Index/MasterSlide-11.iwaN-'* "-+B) Transitionnone?)<0*2RBlankV(    *   ?  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VB'C TD%ඐ<!%@A-?0(8Ch*b>(BT  P UDFC >D   u*bJ"N(2) Random matrices and empirical universality (with Pfrang, Deift, Trogdon).* b2 : JPrX$  en     2!:C 9(f1 `- ,B -,BZ!.^*1TThree stories *  25:^` !k!D  2HFH.HvB6bAJbFAB>b "1H :H^H!H2H*UFH(1) Eigenvalue algorithms as dynamical systems (flashback to the 1980s]%!  _,*A YB A,ZD*C%AQAb>b} "5} *6bVb d=*B dU JB.2+%=+ B6+ C 6D=+B.+Kf+!9+ :^  M*B@)-F@R1J Z=  $   0*   >*mB)ǕfmRZgB.lBl OD඾Q >flQ!'.'.PH&(3) A few explanas 8ujalsIbJM>(^(2.>. ARYM!jiYY6YVY ]<*BK6BJDKZPKkxHWIndex/Slide-179.iwa"X8#3*'&&& B&&&@&&&& B"20B. Transitionnone?)\0X=*&2&:0&:: x @&&&S &X  ( & %D*) )) >*AB"* A2 : BPr  n*C2 >  CTC Cp-B%!R#%@A-?0(8C"58JLZ b T c\ CB p-B  >D&-3(*C)  B0BjKD,Br Ch!h*b.=f_,2N())U *JEH:Completely integrable Hamiltonian flows on Jacobi matrices* 2UU>J.>A`,en  + 22YNY UY9C]4EYDp)Y5IY ZAYzY0.i6Y)RFY]YCBB_0  0en]Su$* m u8B#D 5H)vm} =v^v =x%xm)xaquDJ D AHҾ 6  Z z: @6ANAl),*m  B(B%LKD,B%v4mh*b.>f`9,B.N  Ґ{*J"1Spectral and inverse stheory (Stieltjes).* 2 : JPr en a )d,*m =g!gMEG ZDC'2g>fg"=gC2gJ 5i<*Km%h*1mJ4K.4=)vJn MC%'aaʎ'NanNvNdzN ǑNnCv ؁N 6N  Z~N:N @.HRNJ Ck/De/56 ,@CD)\CCZ v.0.@.JG[0* } >*Bm6NBZ  ^Fo 5aCQ!aA 6)a Z!aza0.aE1oVAB. B1+}; ඐb( "я :^яr*JEx(Given L, clearly we may find  *цJR B\.\B!\X!\ \ >\f\ 5\2]U$ [1894]: g%o %X@ , reconstructl using continued fractions&6u .BGu ^C >j!Q6GThe se data![sist ofu , eigenvaluesL%(the top rowU:m JB BvdC(a Cp)Bc6B ZBzB0.B.BJ 6l m6l C 6D / %uD2l >bD D 2N U`$M*@BH "t @J"BJCbpp$  6VmV eR *J * *C.$J$ *<*A-S5NARNZBPKkxHެo ==Index/Slide-123.iwa9P)#$*'''''B'''''@ B"31B/ Transitionnone?)l0X*'2':x'@'''5,''*C*  >  Bdg D EDB%#,@A-?0(8$Ch*b.=T    _:B 0B >D0** >*bJ"Rescaled histograms for all matrix sizes, defla!T tolerance, and ensembles (1)--(4) (GOE, Bernoulli, Gaussian Wigner, Hermite-1).* b2 !:JPr ' ) ' ,*A*  5 B B!Z$ZD*CA.=f "9 *C6*CZ$**5<*BA%*1AJ4BZ4:$B:$C 6EI^ $B.$=f$!9$ :^$**5$ L*B@,%&5+@R+J1K,Z6, mC,  :NA:NIrMCUrrr6rV+*:r BK%L5GBJGKZ<' -;0** *.j >*AB6lͲ'(*C2L@DA C㎏C"5`8J CCZbj T>u5.:R 'BݜBiBŊc=j_}:^*.y.7unshifted QR. Fit is Gamma(2,1) with exponential tail.* N JbRN@U@@?A@Cܲ@5M@A@A@A@A@~@5.w:@Z@F@*.#D.@I@*.@`.@TodaY88standard normal:''PKkxHaɮIndex/Slide-180.iwaH(##*&&&&B&&&&@ B"31B/ Transitionnone?)`0X*&2&:&:@&&&1 &4$&  & *C2 >  @]A*B rCA(D%#%@A-?0(8C"5`8J CCZbj 0T  \ 56C U >D& 0*A(  B $ZD*C!(Ah*b =f"*6J<(( @<*BA"* (A2 :BPr &  Q0*( (.j >*AB6lͲl& m*(C9- "ud ,B sN5*1 )Emp-{u1ity%{8e{bhȡ75h :hBRh (.@ L*B@+i@ @JBBJ+.=6 + 0,  > C(|Cf0/+Ί=jf _U#6N(.J6 xTheA algorithmI WilkinsonI takes few iteraV s toV\e. This behavior is inseve),matrix size.u J6n=:ARM!CfY6V(.M<*BKEUBJ$KZPKkxHIndex/MasterSlide-8.iwaP)?$*B@AA?BB??C@B B"31B/ Transitionnone?)0Xҙ*C2A:?:?R Photo - 3 Up@ Media-1?2C "T?ABbت(@x(?@%:*@B   >  C 6D !%l?%@A-?0(8$Bh*b.=T . 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