# LaTeX2HTML 0.5.3 (Wed Jan 26 1994)
# Associate image original text (scrambled) with physical files.

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$key = q/{displaymath}I=int^b_ag(x)frac<<385>>d<<385>><<386>>dx<<386>>f(x)dx=langlegvertDvertfrangle{displaymath}/;
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$key = q/{}$frac<<373>>d<<373>><<374>>dx<<374>>f(x)=g(x)${}/;
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$key = q/{tabbing}hspace<<673>>2em<<673>>Chooseaninitialguess<<1373>><<1373>>$x^<<674>>(0)<<674>>$<<1374>><<1374>>tothesolution<<1375>><<1375>>$x$<<1376>><<1376>>.hspace<<675>>2em<<675>><<676>>bffor<<676>>=<<1377>><<1377>>$k=1$<<1378>><<1378>>,2,<<1379>><<1379>>$ldots$<<1380>><<1380>>hspace<<677>>2em<<677>>><<678>>bffor<<678>>=<<1381>><<1381>>$i=1$<<1382>><<1382>>,2,<<1383>><<1383>>$ldots$<<1384>><<1384>>,<<1385>><<1385>>$n$<<1386>><<1386>>hspace<<679>>2em<<679>>>><<1387>><<1387>>$sigma=0$<<1388>><<1388>>hspace<<680>>2em<<680>>>><<681>>bffor<<681>>=<<1389>><<1389>>$j=1$<<1390>><<1390>>,2,<<1391>><<1391>>$ldots$<<1392>><<1392>>,<<1393>><<1393>>$i-1$<<1394>><<1394>>hspace<<682>>2em<<682>>>>><<1395>><<1395>>$sigma=sigma+a_<<683>>i,j<<683>>,x_j^<<684>>(k)<<684>>$<<1396>><<1396>>hspace<<685>>2em<<685>>>><<686>>bf<<686>>hspace<<687>>2em<<687>>>><<688>>bffor<<688>><<1397>><<1397>>$j=i+1$<<1398>><<1398>>,<<1399>><<1399>>$ldots$<<1400>><<1400>>,<<1401>><<1401>>$n$<<1402>><<1402>>hspace<<689>>2em<<689>>>>><<1403>><<1403>>$sigma=sigma+a_<<690>>i,j<<690>>,x_j^<<691>>(k-1)<<691>>$<<1404>><<1404>>hspace<<692>>2em<<692>>>><<693>>bf<<693>>hspace<<694>>2em<<694>>>><<1405>><<1405>>$sigma=left(b_i-sigmaright)a_<<695>>i,i<<695>>$<<1406>><<1406>>hspace<<696>>2em<<696>>>><<1407>><<1407>>$x_i^<<697>>(k)<<697>>=x_i^<<698>>(k-1)<<698>>+omegaleft(sigma-x_i^<<699>>(k-1)<<699>>right)$<<1408>><<1408>>hspace<<700>>2em<<700>>><<701>>bf<<701>>hspace<<702>>2em<<702>>>checkconvergence;continueifnecessaryhspace<<703>>2em<<703>><<704>>bf<<704>>{tabbing}/;
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$key = q/{displaymath}rho_<<623>>SOR<<623>>sim1-frac<<624>>pi<<624>><<625>>N<<625>>{displaymath}/;
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$key = q/{displaymath}leftvertlambda_<<582>>GS<<582>>rightvert_<<583>>rmmax<<583>>=left(leftvertlambda_Jrightvert_<<584>>max<<584>>right)^2sim1-2epsilon{displaymath}/;
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$key = q/{displaymath}1-epsilonmathrel<<1201>>raise.4exhbox<<1200>>rlap<<1198>><<1337>><<1337>>$<$<<1338>><<1338>><<1198>>lower1.2exhbox<<1199>><<1339>><<1339>>$sim$<<1340>><<1340>><<1199>><<1200>><<1201>>lambda_Jmathrel<<1205>>raise.4exhbox<<1204>>rlap<<1202>><<1341>><<1341>>$<$<<1342>><<1342>><<1202>>lower1.2exhbox<<1203>><<1343>><<1343>>$sim$<<1344>><<1344>><<1203>><<1204>><<1205>>-1+epsilonhspace<<577>>2em<<577>>epsilonsimfrac<<578>>pi^2<<578>><<579>>2N^2<<579>>{displaymath}/;
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$key = q/{}$x_<<535>>22<<535>>^<<536>>(k-1)<<536>>${}/;
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$key = q/{}$underline<<414>>rmboundaryconditions<<414>>${}/;
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$key = q/{}$1;html_special_mark_lt;omega;html_special_mark_lt;2${}/;
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$key = q/{displaymath}rho_<<613>>GS<<613>>=1-frac<<614>>pi^2<<614>><<615>>N^2<<615>>{displaymath}/;
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$key = q/{}$frac<<388>>d<<388>><<389>>dx<<389>>${}/;
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$key = q/{}$(l-1),${}/;
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$key = q/{}$x_<<531>>17<<531>>^<<532>>(k-1)<<532>>${}/;
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$key = q/{}$L${}/;
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$key = q/{}$odot${}/;
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$key = q/{}$x_<<533>>10<<533>>^<<534>>(k)<<534>>${}/;
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$key = q/{}$l=1ldots11${}/;
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$key = q/{}$lambda_J${}/;
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$key = q/{}$underline<<544>>x<<544>>^<<545>>(k)<<545>>${}/;
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$key = q/{displaymath}x_i^<<478>>(k)<<478>>=frac<<792>>leftlbrackb_i-sumlimits_<<479>>j;html_special_mark_lt;i<<479>>,a_<<480>>i,j<<480>>,x_j^<<481>>(k)<<481>>-sumlimits_<<482>>j;html_special_mark_gt;i<<482>>,a_<<483>>i,j<<483>>,x_j^<<484>>(k-1)<<484>>rightrbrack<<792>><<793>>a_<<485>>i,i<<485>><<793>>{displaymath}/;
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$key = q/{tabbing}hspace<<440>>2em<<440>>Chooseaninitialguess<<1251>><<1251>>$x^<<441>>(0)<<441>>$<<1252>><<1252>>tothesolution<<1253>><<1253>>$x$<<1254>><<1254>>.hspace<<442>>2em<<442>><<443>>bffor<<443>>=<<1255>><<1255>>$k=1$<<1256>><<1256>>,2,<<1257>><<1257>>$ldots$<<1258>><<1258>>hspace<<444>>2em<<444>>><<445>>bffor<<445>>=<<1259>><<1259>>$i=1$<<1260>><<1260>>,2,<<1261>><<1261>>$ldots$<<1262>><<1262>>,<<1263>><<1263>>$n$<<1264>><<1264>>hspace<<446>>2em<<446>>>><<1265>><<1265>>$sigma=0$<<1266>><<1266>>hspace<<447>>2em<<447>>>><<448>>bffor<<448>>=<<1267>><<1267>>$j=1$<<1268>><<1268>>,2,<<1269>><<1269>>$ldots$<<1270>><<1270>>,<<1271>><<1271>>$i-1$<<1272>><<1272>>hspace<<449>>2em<<449>>>>><<1273>><<1273>>$sigma=sigma+a_<<450>>i,j<<450>>,x_j^<<451>>(k)<<451>>$<<1274>><<1274>>hspace<<452>>2em<<452>>>><<453>>bf<<453>>hspace<<454>>2em<<454>>>><<455>>bffor<<455>><<1275>><<1275>>$j=i+1$<<1276>><<1276>>,<<1277>><<1277>>$ldots$<<1278>><<1278>>,<<1279>><<1279>>$n$<<1280>><<1280>>hspace<<456>>2em<<456>>>>><<1281>><<1281>>$sigma=sigma+a_<<457>>i,j<<457>>,x_j^<<458>>(k-1)<<458>>$<<1282>><<1282>>hspace<<459>>2em<<459>>>><<460>>bf<<460>>hspace<<461>>2em<<461>>>><<1283>><<1283>>$x_i^<<462>>(k)<<462>>=left(b_i-sigmaright)a_<<463>>i,i<<463>>$<<1284>><<1284>>hspace<<464>>2em<<464>>><<465>>bf<<465>>hspace<<466>>2em<<466>>>checkconvergence;continueifnecessaryhspace<<467>>2em<<467>><<468>>bf<<468>>{tabbing}/;
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$key = q/{}$D${}/;
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$key = q/{}$omega=1${}/;
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$key = q/{}$kproptofrac<<596>>1<<596>><<797>>left(1-vertlambdavert_<<597>>rmmax<<597>>right)<<797>>${}/;
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$key = q/{}$6times6${}/;
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$key = q/{displaymath}vertlambdavert^k_<<593>>max<<593>>sim.001{displaymath}/;
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$key = q/{displaymath}underline<<488>>x<<488>>^<<489>>(k)<<489>>=(D-L)^<<490>>-1<<490>>left(Uunderline<<491>>x<<491>>^<<492>>(k-1)<<492>>+underline<<493>>b<<493>>right){displaymath}/;
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$key = q/{}psfig<<524>>figure=918_09.ps,clip=true,width=4.0in<<524>>{}/;
$cached_env_img{$key} ='<IMG ALIGN=BOTTOM ALT="" SRC="_5078_tex2html_wrap1299.xbm">'; 
$key = q/{}$=D-L-U${}/;
$cached_env_img{$key} ='<IMG ALIGN=BOTTOM ALT="" SRC="_5078_tex2html_wrap1285.xbm">'; 
$key = q/{}$nabla^2=frac<<420>>d^2<<420>><<421>>dx^2<<421>>=-L,L${}/;
$cached_env_img{$key} ='<IMG ALIGN=BOTTOM ALT="" SRC="_5078_tex2html_wrap1239.xbm">'; 
$key = q/{}$M^<<570>>-1<<570>>N${}/;
$cached_env_img{$key} ='<IMG ALIGN=BOTTOM ALT="" SRC="_5078_tex2html_wrap1329.xbm">'; 
$key = q/{}$omega${}/;
$cached_env_img{$key} ='<IMG ALIGN=BOTTOM ALT="" SRC="_5078_tex2html_wrap1363.xbm">'; 
$key = q/{}$left(vertlambdavert_<<607>>max<<607>>right)${}/;
$cached_env_img{$key} ='<IMG ALIGN=BOTTOM ALT="" SRC="_5078_tex2html_wrap1355.xbm">'; 
$key = q/{}$x_<<527>>16<<527>>^<<528>>(k)<<528>>${}/;
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$key = q/{}psfig<<551>>figure=918_11.ps,clip=true,width=3.5in<<551>>{}/;
$cached_env_img{$key} ='<IMG ALIGN=BOTTOM ALT="" SRC="_5078_tex2html_wrap1327.xbm">'; 
$key = q/{}$(A,B)^dagger=B^daggerA^dagger${}/;
$cached_env_img{$key} ='<IMG ALIGN=BOTTOM ALT="" SRC="_5078_tex2html_wrap1243.xbm">'; 
$key = q/{displaymath}1(1-vertlambdavert_<<599>>rmmax<<599>>),{displaymath}/;
$cached_env_img{$key} ='<IMG ALIGN=BOTTOM ALT="" SRC="_5078_displaymath598.xbm">'; 
$key = q/{}$expleftlbrack-kleft(1-leftvertlambdarightvert_<<595>>rmmax<<595>>right)rightrbrack=.001${}/;
$cached_env_img{$key} ='<IMG ALIGN=BOTTOM ALT="" SRC="_5078_tex2html_wrap1347.xbm">'; 
$key = q/{displaymath}int^b_ag(x)frac<<407>>d<<407>><<408>>dx<<408>>f(x),dx=-int^b_af(x)frac<<409>>d<<409>><<410>>dx<<410>>g(x),dx{displaymath}/;
$cached_env_img{$key} ='<IMG ALIGN=BOTTOM ALT="" SRC="_5078_displaymath406.xbm">'; 
$key = q/{}$f(a)=f(b)${}/;
$cached_env_img{$key} ='<IMG ALIGN=BOTTOM ALT="" SRC="_5078_tex2html_wrap1233.xbm">'; 
$key = q/{}$(n)${}/;
$cached_env_img{$key} ='<IMG ALIGN=BOTTOM ALT="" SRC="_5078_tex2html_wrap1213.xbm">'; 
$key = q/{}$n${}/;
$cached_env_img{$key} ='<IMG ALIGN=BOTTOM ALT="" SRC="_5078_tex2html_wrap1249.xbm">'; 
$key = q/{displaymath}biglbrackf(x)g(x)bigrbrack^b_a=f(b),g(b)-f(a),g(a){displaymath}/;
$cached_env_img{$key} ='<IMG ALIGN=BOTTOM ALT="" SRC="_5078_displaymath394.xbm">'; 
$key = q/{}$l${}/;
$cached_env_img{$key} ='<IMG ALIGN=BOTTOM ALT="" SRC="_5078_tex2html_wrap1307.xbm">'; 
$key = q/{}$k${}/;
$cached_env_img{$key} ='<IMG ALIGN=BOTTOM ALT="" SRC="_5078_tex2html_wrap1345.xbm">'; 
$key = q/{}$L=ifrac<<412>>d<<412>><<413>>dx<<413>>${}/;
$cached_env_img{$key} ='<IMG ALIGN=BOTTOM ALT="" SRC="_5078_tex2html_wrap1229.xbm">'; 
$key = q/{}$i${}/;
$cached_env_img{$key} ='<IMG ALIGN=BOTTOM ALT="" SRC="_5078_tex2html_wrap1235.xbm">'; 
$key = q/{}psfig<<542>>figure=918_10.ps,clip=true,width=5.0in<<542>>{}/;
$cached_env_img{$key} ='<IMG ALIGN=BOTTOM ALT="" SRC="_5078_tex2html_wrap1313.xbm">'; 
$key = q/{}$h${}/;
$cached_env_img{$key} ='<IMG ALIGN=BOTTOM ALT="" SRC="_5078_tex2html_wrap1371.xbm">'; 
$key = q/{}$bullet${}/;
$cached_env_img{$key} ='<IMG ALIGN=BOTTOM ALT="" SRC="_5078_tex2html_wrap1325.xbm">'; 
$key = q/{}$ifrac<<398>>d<<398>><<399>>dx<<399>>${}/;
$cached_env_img{$key} ='<IMG ALIGN=BOTTOM ALT="" SRC="_5078_tex2html_wrap1227.xbm">'; 
$key = q/{}$x_<<529>>15<<529>>^<<530>>(k)<<530>>${}/;
$cached_env_img{$key} ='<IMG ALIGN=BOTTOM ALT="" SRC="_5078_tex2html_wrap1291.xbm">'; 
$key = q/{displaymath}rho_J=1-frac<<609>>pi^2<<609>><<610>>2N^2<<610>>{displaymath}/;
$cached_env_img{$key} ='<IMG ALIGN=BOTTOM ALT="" SRC="_5078_displaymath608.xbm">'; 
$key = q/{}$frac<<379>>d<<379>><<380>>dx<<380>>${}/;
$cached_env_img{$key} ='<IMG ALIGN=BOTTOM ALT="" SRC="_5078_tex2html_wrap1221.xbm">'; 
$key = q/{}$frac<<370>>d<<370>><<371>>dx<<371>>${}/;
$cached_env_img{$key} ='<IMG ALIGN=BOTTOM ALT="" SRC="_5078_tex2html_wrap1217.xbm">'; 
$key = q/{}$omegasim2${}/;
$cached_env_img{$key} ='<IMG ALIGN=BOTTOM ALT="" SRC="_5078_tex2html_wrap1353.xbm">'; 
$key = q/{}$lambda_<<573>>GS<<573>>${}/;
$cached_env_img{$key} ='<IMG ALIGN=BOTTOM ALT="" SRC="_5078_tex2html_wrap1331.xbm">'; 
$key = q/{}$frac<<425>>d^2<<425>><<426>>dx^2<<426>>${}/;
$cached_env_img{$key} ='<IMG ALIGN=BOTTOM ALT="" SRC="_5078_tex2html_wrap1247.xbm">'; 
$key = q/{displaymath}omega_<<618>>opt<<618>>sim2left(1-frac<<619>>pi<<619>><<620>>N<<620>>right){displaymath}/;
$cached_env_img{$key} ='<IMG ALIGN=BOTTOM ALT="" SRC="_5078_displaymath617.xbm">'; 
$key = q/{}$nabla^2${}/;
$cached_env_img{$key} ='<IMG ALIGN=BOTTOM ALT="" SRC="_5078_tex2html_wrap1245.xbm">'; 
$key = q/{}$Npi${}/;
$cached_env_img{$key} ='<IMG ALIGN=BOTTOM ALT="" SRC="_5078_tex2html_wrap1357.xbm">'; 
$key = q/{}$lambda_<<574>>GS<<574>>=lambda_<<575>>J<<575>>^2${}/;
$cached_env_img{$key} ='<IMG ALIGN=BOTTOM ALT="" SRC="_5078_tex2html_wrap1335.xbm">'; 
$key = q/{}$f(x)${}/;
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1;