# LaTeX2HTML 0.6.4 (Tues Aug 30 1994)
# Associate image original text (scrambled) with physical files.
$key = q/{}$p-1${}/;
$cached_env_img{$key} ='
';
$key = q/{}$n^2${}/;
$cached_env_img{$key} ='
';
$key = q/{}$gamma_i${}/;
$cached_env_img{$key} ='
';
$key = q/{}$V${}/;
$cached_env_img{$key} ='
';
$key = q/{}$T${}/;
$cached_env_img{$key} ='
';
$key = q/{}$U=[u_1,...,u_n]${}/;
$cached_env_img{$key} ='
';
$key = q/{}$P=P_r.P_c${}/;
$cached_env_img{$key} ='
';
$key = q/{}$P${}/;
$cached_env_img{$key} ='
';
$key = q/{}$N${}/;
$cached_env_img{$key} ='
';
$key = q/{}$P_rtimesP_c${}/;
$cached_env_img{$key} ='
';
$key = q/{}$beta${}/;
$cached_env_img{$key} ='
';
$key = q/{}$r=c=1${}/;
$cached_env_img{$key} ='
';
$key = q/{}$O(n^3)${}/;
$cached_env_img{$key} ='
';
$key = q/{figure}[h]centerlinehboxpsfigfile=all-sp2.ps,height=2.8in,width=2.8inhspace*.2inpsfigfile=luplots.ps,height=2.8in,width=2.8in{figure}/;
$cached_env_img{$key} ='
';
$key = q/{}$K()${}/;
$cached_env_img{$key} ='
';
$key = q/{figure}[t]centerlinehboxpsfigfile=llt-sp2.ps,height=2.8in,width=2.8inhspace*.2inpsfigfile=p150_llt.ps,height=2.8in,width=2.8in{figure}/;
$cached_env_img{$key} ='
';
$key = q/{}$2~(p-1)${}/;
$cached_env_img{$key} ='
';
$key = q/{}psfigfigure=SoftwareHierarchy.eps,width=3in{}/;
$cached_env_img{$key} ='
';
$key = q/{}$P_r${}/;
$cached_env_img{$key} ='
';
$key = q/{}$rtimesc${}/;
$cached_env_img{$key} ='
';
$key = q/{}$E(N,P)${}/;
$cached_env_img{$key} ='
';
$key = q/{}$r=2^hatr${}/;
$cached_env_img{$key} ='
';
$key = q/{}$P_rn${}/;
$cached_env_img{$key} ='
';
$key = q/{}$A=Q(ULambdaU^T)Q^T=(QU)Lambda(QU)^T=VLambdaV^T${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}leftlanglep,b,irightrangle=leftlangleleftlfloorfracmrrightrfloor,rmmod,P,,leftlfloorfraclfloorfracmrrfloorPrightrfloor,,m,rmmod,rrightrangle;.{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{figure}[h]centerlinehboxpsfigfile=lu-sp2.ps,height=2.8in,width=2.8inhspace*.2inpsfigfile=qrplots.ps,height=2.8in,width=2.8in{figure}/;
$cached_env_img{$key} ='
';
$key = q/{}$P_c${}/;
$cached_env_img{$key} ='
';
$key = q/{}$NP${}/;
$cached_env_img{$key} ='
';
$key = q/{}$P_r=usqrtP${}/;
$cached_env_img{$key} ='
';
$key = q/{}$(m,n)${}/;
$cached_env_img{$key} ='
';
$key = q/{}$u${}/;
$cached_env_img{$key} ='
';
$key = q/{}$O(n^2)${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}leftlangle(leftlfloorfracmrrightrfloor,rmmod,P_r,,leftlfloorfracncrightrfloor,rmmod,P_c),,left(leftlfloorfraclfloorfracmrrfloorP_rrightrfloor,,leftlfloorfraclfloorfracncrfloorP_crightrfloorright),,(m,rmmod,r,,n,rmmod,c)rightrangle;.{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$Ptimes1${}/;
$cached_env_img{$key} ='
';
$key = q/{}$k~times~r${}/;
$cached_env_img{$key} ='
';
$key = q/{}$K('bcast','1-tree',p,n)approxlog_2(p)${}/;
$cached_env_img{$key} ='
';
$key = q/{}$P_rP_c${}/;
$cached_env_img{$key} ='
';
$key = q/{}$(r=c)${}/;
$cached_env_img{$key} ='
';
$key = q/{}$m${}/;
$cached_env_img{$key} ='
';
$key = q/{}$i${}/;
$cached_env_img{$key} ='
';
$key = q/{}psfigfigure=code1.ps,height=4.6in,width=6.2in{}/;
$cached_env_img{$key} ='
';
$key = q/{}$i=1,2,3${}/;
$cached_env_img{$key} ='
';
$key = q/{}$P_r.P_cleqP${}/;
$cached_env_img{$key} ='
';
$key = q/{}$P_c~=~1${}/;
$cached_env_img{$key} ='
';
$key = q/{}$c${}/;
$cached_env_img{$key} ='
';
$key = q/{}$hatr${}/;
$cached_env_img{$key} ='
';
$key = q/{figure}[h]centerlinehboxpsfigfile=trd-sp2.ps,height=2.8in,width=2.8inhspace*.2inpsfigfile=brd-sp2.ps,height=2.8in,width=2.8in{figure}/;
$cached_env_img{$key} ='
';
$key = q/{}$1timesP${}/;
$cached_env_img{$key} ='
';
$key = q/{}$2~log_2(p)${}/;
$cached_env_img{$key} ='
';
$key = q/{}$P_r~=~1${}/;
$cached_env_img{$key} ='
';
$key = q/{}$T(N,P)${}/;
$cached_env_img{$key} ='
';
$key = q/{}$N=n^2${}/;
$cached_env_img{$key} ='
';
$key = q/{}$Q${}/;
$cached_env_img{$key} ='
';
$key = q/{}$gamma_3${}/;
$cached_env_img{$key} ='
';
$key = q/{}$P_cn${}/;
$cached_env_img{$key} ='
';
$key = q/{}$kinleft{0,1,ldotsfracn-1rright}${}/;
$cached_env_img{$key} ='
';
$key = q/{}$M${}/;
$cached_env_img{$key} ='
';
$key = q/{}$K${}/;
$cached_env_img{$key} ='
';
$key = q/{}$sqrtPn${}/;
$cached_env_img{$key} ='
';
$key = q/{}$T_rmseq(N)${}/;
$cached_env_img{$key} ='
';
$key = q/{}$P_r;html_special_mark_lt;P_c${}/;
$cached_env_img{$key} ='
';
$key = q/{}$T=ULambdaU^T${}/;
$cached_env_img{$key} ='
';
$key = q/{}$A${}/;
$cached_env_img{$key} ='
';
$key = q/{}$^ddagger${}/;
$cached_env_img{$key} ='
';
$key = q/{}$K('combine','1-tree',p,n)${}/;
$cached_env_img{$key} ='
';
$key = q/{}$K('bcast','S-ring',p,n)${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}leftlangle(p,q),(b,d),(i,j)rightrangle={displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{figure}[h]centerlinehboxpsfigfile=lublocksize.ps,height=2.8in,width=2.8inhspace*.2inpsfigfile=ludist.ps,height=2.8in,width=2.8in{figure}/;
$cached_env_img{$key} ='
';
$key = q/{}$v${}/;
$cached_env_img{$key} ='
';
$key = q/{}$A=QTQ^T${}/;
$cached_env_img{$key} ='
';
$key = q/{}$r${}/;
$cached_env_img{$key} ='
';
$key = q/{figure}[t]centerlinehboxpsfigfile=p150_lu.ps,height=2.9in,width=2.9inhspace*.2inpsfigfile=p150_qr.ps,height=2.9in,width=2.9in{figure}/;
$cached_env_img{$key} ='
';
$key = q/{}$p${}/;
$cached_env_img{$key} ='
';
$key = q/{}$n${}/;
$cached_env_img{$key} ='
';
$key = q/{}$c=lceilNP_crceil${}/;
$cached_env_img{$key} ='
';
$key = q/{}$gamma_1=gamma_2=0${}/;
$cached_env_img{$key} ='
';
$key = q/{}$hatP${}/;
$cached_env_img{$key} ='
';
$key = q/{}$0leqm;html_special_mark_lt;M${}/;
$cached_env_img{$key} ='
';
$key = q/{}$P=2^hatP${}/;
$cached_env_img{$key} ='
';
$key = q/{}$P_c=vsqrtP${}/;
$cached_env_img{$key} ='
';
$key = q/{}$r=lceilMP_rrceil${}/;
$cached_env_img{$key} ='
';
$key = q/{}$log_2(P_r)${}/;
$cached_env_img{$key} ='
';
$key = q/{}$^dagger${}/;
$cached_env_img{$key} ='
';
$key = q/{}$b${}/;
$cached_env_img{$key} ='
';
$key = q/{}$P_r;html_special_mark_gt;P_c${}/;
$cached_env_img{$key} ='
';
$key = q/{}$Lambda=rmdiag(lambda_1,...,lambda_n)${}/;
$cached_env_img{$key} ='
';
$key = q/{}$P_c=2P_r${}/;
$cached_env_img{$key} ='
';
$key = q/{}$gamma_2,gamma_3${}/;
$cached_env_img{$key} ='
';
$key = q/{}$T_LU(n^2,P)${}/;
$cached_env_img{$key} ='
';
$key = q/{}$P_rtimesP_cleqP${}/;
$cached_env_img{$key} ='
';
1;