Given partitions l and m in the Grassmannian Gr(k,n), and a set of fixed flags G1,...,Gd, where d=k*(k-n) - |l| - |m|. The function solves the system taking the first d-1 flags, and replacing the last one for a simpler one Gm. Then it uses homotopy continuation to track the solutions of this simpler system to solutions of the original system. This function is used to solve Simple Schubert Problems, as described in the paper: Leykin and Sottile, "Galois groups of Schubert problems via homotopy continuation", Mathematics of Computation, 78 (2009) 1749--1765.
i1 : k = 3 o1 = 3 |
i2 : n = 7 o2 = 7 |
i3 : l = {2,1,0} o3 = {2, 1, 0} o3 : List |
i4 : m = {1,1,0} o4 = {1, 1, 0} o4 : List |
i5 : d = k*(n-k)-sum(l)-sum(m); |
i6 : G = apply(d, i->matrix apply(n-k,i->apply(n,j->random CC))); |
i7 : Sols =solveSimpleSchubert((k,n),l,m,G) --This problem has 77 solutions! -- o7 = {{2.83037-1.16682*ii, .339087-1.17121*ii, -.179497+2.13795*ii, ------------------------------------------------------------------------ -.155164-1.83616*ii, .594075-1.25791*ii, .375115+1.07551*ii, ------------------------------------------------------------------------ -.767582+.750854*ii}, {.616511-1.23608*ii, 1.14351-.428504*ii, ------------------------------------------------------------------------ .61279-.237252*ii, 2.40519-.943408*ii, 1.52285+4.31747*ii, ------------------------------------------------------------------------ .158879+.160696*ii, .252295+.536086*ii}, {1.59564-3.53412*ii, ------------------------------------------------------------------------ 1.4881+.73784*ii, .103491+2.3495*ii, 1.95469-.49702*ii, ------------------------------------------------------------------------ .985635+1.07916*ii, .139753+2.59479*ii, -.572996+1.78227*ii}, ------------------------------------------------------------------------ {1.00034+1.16377*ii, .282334-.0623693*ii, -.0261111+.469647*ii, ------------------------------------------------------------------------ .84702+.456431*ii, 1.37268-.081691*ii, -.727831+.904989*ii, ------------------------------------------------------------------------ -1.29211+.851385*ii}, {1.31908-.09193*ii, 1.68667+.214682*ii, ------------------------------------------------------------------------ -.157446-.621006*ii, .512377+2.35174*ii, .727981+4.07791*ii, ------------------------------------------------------------------------ .405833-.485107*ii, -.922487-.546294*ii}, {-.177866+.2745*ii, ------------------------------------------------------------------------ -1.0785+.377005*ii, .033043-1.20722*ii, -.182675+.322299*ii, ------------------------------------------------------------------------ .972502+.039856*ii, -.844351-.697949*ii, -.586467-.339161*ii}, ------------------------------------------------------------------------ {.918986+.729155*ii, .0544907+.48707*ii, 1.76017+.099046*ii, ------------------------------------------------------------------------ -.989923+.209755*ii, 1.21071+.830863*ii, -7.50874-4.28251*ii, ------------------------------------------------------------------------ -10.5236+7.00191*ii}, {-4.8344-2.30609*ii, -1.04978-6.47208*ii, ------------------------------------------------------------------------ -4.0858+.430332*ii, -.247438-.570484*ii, .312555+.57111*ii, ------------------------------------------------------------------------ .339541+.238912*ii, .142301+1.32292*ii}, {.698366+.608809*ii, ------------------------------------------------------------------------ .95351-.341281*ii, .277886-.784558*ii, 1.09216+.199628*ii, ------------------------------------------------------------------------ -.560935-1.56259*ii, -.044748-1.86422*ii, -2.47285-1.90376*ii}, ------------------------------------------------------------------------ {.738961+1.08457*ii, -.735094+.575263*ii, .930967+.024263*ii, ------------------------------------------------------------------------ -.669722+.129668*ii, .739605+.671634*ii, .0666671-.264572*ii, ------------------------------------------------------------------------ -.69609+.75765*ii}, {.965771-.629416*ii, -.316157-.403439*ii, ------------------------------------------------------------------------ -1.51768+5.13956*ii, 1.46815-2.03927*ii, 1.07465-.506629*ii, ------------------------------------------------------------------------ -.61724-1.18922*ii, -.969504-.803241*ii}, {.091364-1.0877*ii, ------------------------------------------------------------------------ -.205565-.533216*ii, -.152703-.917898*ii, .697898+3.35222*ii, ------------------------------------------------------------------------ 4.37021-2.46295*ii, .189392-.770681*ii, .211364-.380062*ii}, ------------------------------------------------------------------------ {.970958+1.03881*ii, -.033912+.601984*ii, .421707+.091466*ii, ------------------------------------------------------------------------ -.951042+.158509*ii, 1.44278+1.4802*ii, -.200247-.439407*ii, ------------------------------------------------------------------------ -.505303+.153709*ii}, {.667701-.7178*ii, .0224798-.107527*ii, ------------------------------------------------------------------------ -1.76825+.263386*ii, 1.20835-.249924*ii, 1.49888-.221283*ii, ------------------------------------------------------------------------ -.413561+.0570748*ii, -.545814+.074177*ii}, {.611252-.923722*ii, ------------------------------------------------------------------------ .017085-.593473*ii, -3.57302-1.4063*ii, 2.50843+2.46452*ii, ------------------------------------------------------------------------ 1.81309-.66578*ii, .055338-1.25873*ii, -.035530-.931*ii}, ------------------------------------------------------------------------ {4.72049+1.33919*ii, 1.75729+.246846*ii, -.903759+3.65303*ii, ------------------------------------------------------------------------ 3.18925+.216125*ii, -1.1456+2.32551*ii, .131337+1.33329*ii, ------------------------------------------------------------------------ .72027+.505613*ii}, {.785901+.098456*ii, -.23242+.0295552*ii, ------------------------------------------------------------------------ .739642+.296011*ii, .136402-.0477763*ii, .59205+.235903*ii, ------------------------------------------------------------------------ 8.42502+6.70125*ii, 3.23608+6.26523*ii}, {-.142789-2.54626*ii, ------------------------------------------------------------------------ 1.31922-1.49604*ii, .280374-1.55877*ii, .266205-.217297*ii, ------------------------------------------------------------------------ .095254+1.73734*ii, -.372531-.340654*ii, .40685+.389897*ii}, ------------------------------------------------------------------------ {-.965005-.524762*ii, -.850863-.680911*ii, .0666037-.0517023*ii, ------------------------------------------------------------------------ -.172818-.0192463*ii, .171706+.860751*ii, -.279174+.225652*ii, ------------------------------------------------------------------------ -.135133+.986769*ii}, {-.578466-2.46713*ii, 1.24047-1.46576*ii, ------------------------------------------------------------------------ .063654-.835605*ii, .166326-.413034*ii, .443282+.686889*ii, ------------------------------------------------------------------------ .070334+1.00916*ii, -.460264+1.57546*ii}, {.958026+.682366*ii, ------------------------------------------------------------------------ .406162+.547165*ii, 1.23458-.823822*ii, -1.77795+1.15503*ii, ------------------------------------------------------------------------ 2.23779+2.10422*ii, -.817004-.111244*ii, -1.03764+.816608*ii}, ------------------------------------------------------------------------ {6.71719-.064777*ii, 5.87225-1.43526*ii, 2.57362-3.43779*ii, ------------------------------------------------------------------------ -.424259-.658763*ii, -.027455+3.22171*ii, .0617571-.256373*ii, ------------------------------------------------------------------------ .201933-.081954*ii}, {.837651-.01776*ii, .080552+1.0221*ii, ------------------------------------------------------------------------ -.048032-1.42918*ii, -.458712+.739391*ii, -.957983+1.48391*ii, ------------------------------------------------------------------------ -.0057591+.228492*ii, .0801215+.334954*ii}, {-.335264+.468981*ii, ------------------------------------------------------------------------ -.85052-.249673*ii, .078588+.709592*ii, -.0051331-.0727716*ii, ------------------------------------------------------------------------ .038605+1.00938*ii, -1.45296+2.2789*ii, .184977+1.65745*ii}, ------------------------------------------------------------------------ {1.25726+.441114*ii, -.209568+.615339*ii, .506669+1.36158*ii, ------------------------------------------------------------------------ .401848+.102313*ii, .814918+.447657*ii, -5.44325-5.09058*ii, ------------------------------------------------------------------------ .354587-2.67153*ii}, {.991809+.282546*ii, 1.31351+.625964*ii, ------------------------------------------------------------------------ -1.18737-1.67876*ii, -2.39781+2.38038*ii, -2.77821+1.32197*ii, ------------------------------------------------------------------------ -.0792051+.419036*ii, .316059+.434391*ii}, {.786403+.423102*ii, ------------------------------------------------------------------------ -.254866-.311113*ii, -.411882-.43632*ii, .569311+.437771*ii, ------------------------------------------------------------------------ 1.10602+.197325*ii, -.734128+.775301*ii, -.512867+.597257*ii}, ------------------------------------------------------------------------ {.530206+.182987*ii, .267353-.627555*ii, .136504-.0534943*ii, ------------------------------------------------------------------------ .055419-.527812*ii, -.189794-.615281*ii, .092370-.839385*ii, ------------------------------------------------------------------------ -.118704-1.5459*ii}, {.354697-2.21884*ii, .468827-.231337*ii, ------------------------------------------------------------------------ -.242795-.698118*ii, -.539902+.406311*ii, -.319066+1.10558*ii, ------------------------------------------------------------------------ .347758-.301442*ii, .234016+.762547*ii}, {.083327-1.68596*ii, ------------------------------------------------------------------------ 1.13553+.87302*ii, 4.72372-1.51816*ii, .101769-1.11528*ii, ------------------------------------------------------------------------ .721827+.439285*ii, 1.05034-2.47462*ii, 1.24974+.957912*ii}, ------------------------------------------------------------------------ {.349036+.480496*ii, -.442291+.7817*ii, .700609+.562738*ii, ------------------------------------------------------------------------ -.00782018-.0302859*ii, .561071+.642179*ii, -.012391-3.46218*ii, ------------------------------------------------------------------------ .060909-1.02606*ii}, {1.39166-.170938*ii, 1.41613-.126288*ii, ------------------------------------------------------------------------ -1.05682+.761826*ii, 7.14911+7.32129*ii, -2.13623+7.32394*ii, ------------------------------------------------------------------------ -.016024+.865674*ii, .610208+.799856*ii}, {-1.45486+3.55806*ii, ------------------------------------------------------------------------ -23.9773+10.1617*ii, 13.5641+20.0492*ii, -.318982-.96501*ii, ------------------------------------------------------------------------ -.0515422+.12378*ii, .02413-2.20945*ii, 3.00757+.650103*ii}, ------------------------------------------------------------------------ {-.518932-4.29838*ii, 2.08586-.507361*ii, .507326+1.57303*ii, ------------------------------------------------------------------------ 2.03021-.245073*ii, .622707+1.62675*ii, 2.06576+1.41166*ii, ------------------------------------------------------------------------ .425359+1.54847*ii}, {.617738-.407646*ii, -.426388-.138073*ii, ------------------------------------------------------------------------ 7.1245+.928229*ii, -2.53532-1.15553*ii, 1.12701-.548861*ii, ------------------------------------------------------------------------ -.846797-.646969*ii, -1.30067-.772363*ii}, {.412591-.247387*ii, ------------------------------------------------------------------------ 1.77301-1.87985*ii, -.593417+.034869*ii, -.35955-.11525*ii, ------------------------------------------------------------------------ -.607773-.047087*ii, .408996-.710548*ii, -.558421-.577332*ii}, ------------------------------------------------------------------------ {.448486+.466528*ii, .145996+.84424*ii, .068363-2.02747*ii, ------------------------------------------------------------------------ -1.52147+1.11863*ii, 1.39177+1.04669*ii, -1.02401+1.29007*ii, ------------------------------------------------------------------------ -2.0382+.423085*ii}, {.88539+.386903*ii, .328355-.541645*ii, ------------------------------------------------------------------------ .785763+.047963*ii, .0425523+.0898986*ii, .44867+.269815*ii, ------------------------------------------------------------------------ 1.32757+.591549*ii, 1.00823+.636428*ii}, {1.90755-.213533*ii, ------------------------------------------------------------------------ .770571+2.89298*ii, 1.05303-2.2697*ii, -.846758-.334234*ii, ------------------------------------------------------------------------ -.078368+.551162*ii, .321975-.428153*ii, -.205606+.64375*ii}, ------------------------------------------------------------------------ {1.07608-.050741*ii, 1.39746-.282675*ii, -.0928957-.132527*ii, ------------------------------------------------------------------------ -3.90339-3.90227*ii, -7.82381-8.83392*ii, .078951-.15352*ii, ------------------------------------------------------------------------ -.573059-.163445*ii}, {.816078+.275593*ii, 1.12942-.411091*ii, ------------------------------------------------------------------------ -.194283-.869602*ii, .503989-1.17673*ii, -.96889-1.17963*ii, ------------------------------------------------------------------------ -.187246-.520936*ii, -1.11249-1.19833*ii}, {.598506+.607586*ii, ------------------------------------------------------------------------ -.357378+.515676*ii, -.400765-.0240624*ii, -.91291-.169604*ii, ------------------------------------------------------------------------ .075919+.846655*ii, .058203-1.32517*ii, .129987+.485521*ii}, ------------------------------------------------------------------------ {.649368-.281033*ii, .178564+.477535*ii, 2.18503+.212868*ii, ------------------------------------------------------------------------ .0588415-.420831*ii, .937315+.335758*ii, 1.76197-3.05745*ii, ------------------------------------------------------------------------ 1.6167+.652616*ii}, {.80873+.766799*ii, .585551+.490728*ii, ------------------------------------------------------------------------ .414654-.875804*ii, -1.74775+.587978*ii, 2.92108+4.14704*ii, ------------------------------------------------------------------------ -.248523-.228073*ii, -.386832+.194522*ii}, {.689862-.080104*ii, ------------------------------------------------------------------------ .771525+.991817*ii, -.383983-1.21437*ii, -.748579+.829531*ii, ------------------------------------------------------------------------ -1.25648+.877242*ii, -.049215+.294826*ii, .189683+.531035*ii}, ------------------------------------------------------------------------ {.881167+.427014*ii, .268105-.0096692*ii, .765203+.076744*ii, ------------------------------------------------------------------------ .228207+.0175461*ii, 1.29618-.357272*ii, -.717145+.124914*ii, ------------------------------------------------------------------------ -3.51734-.227384*ii}, {1.7251+.928577*ii, 2.74863+1.97259*ii, ------------------------------------------------------------------------ 3.52728-2.47956*ii, .19062-1.59231*ii, -.843351+.684587*ii, ------------------------------------------------------------------------ -.994148-.674488*ii, -.788133-.455214*ii}, {.837498+1.39153*ii, ------------------------------------------------------------------------ 1.1407+2.58084*ii, 2.41347+.232758*ii, -.703573-.491564*ii, ------------------------------------------------------------------------ 2.38398+2.20851*ii, .225157-.254973*ii, .0682961-.110237*ii}, ------------------------------------------------------------------------ {1.44651-1.79*ii, 2.20384-1.74707*ii, -.187375-1.68581*ii, ------------------------------------------------------------------------ -.254739-.514403*ii, -.685064+1.31354*ii, -.408477-.334541*ii, ------------------------------------------------------------------------ .606803-.45191*ii}, {-2.22799+1.00517*ii, 2.20445-.436488*ii, ------------------------------------------------------------------------ 4.71575-1.86952*ii, .994985+1.72538*ii, -.107391-.92427*ii, ------------------------------------------------------------------------ 1.50233-1.6995*ii, -.309709-.0107719*ii}, {.373175-.949261*ii, ------------------------------------------------------------------------ 1.89583-.546296*ii, -.259622-2.11873*ii, -.305812-.726001*ii, ------------------------------------------------------------------------ -.219359+.900303*ii, -1.50091-2.7817*ii, 2.34084+1.06788*ii}, ------------------------------------------------------------------------ {1.33596-.229447*ii, -.117514-1.85582*ii, -1.21232+3.63221*ii, ------------------------------------------------------------------------ -.390801-1.18822*ii, .679468-.552979*ii, -.276353-.490547*ii, ------------------------------------------------------------------------ -.827168+.144626*ii}, {1.13662-.022903*ii, .40874+.247727*ii, ------------------------------------------------------------------------ .499645+1.90144*ii, .628111-1.33391*ii, .900587-.128164*ii, ------------------------------------------------------------------------ 3.69554-1.66978*ii, .797363-4.78321*ii}, {.222182-.24866*ii, ------------------------------------------------------------------------ -.30128-.930265*ii, -.193887-.187332*ii, .221136+.0942152*ii, ------------------------------------------------------------------------ -7.25787+4.50238*ii, .0564447+.057707*ii, .0722465+.0022400*ii}, ------------------------------------------------------------------------ {.716201+2.06053*ii, -1.54182+4.60067*ii, 4.35754+4.02669*ii, ------------------------------------------------------------------------ .48999-.975936*ii, .255998+.449672*ii, -.778497-1.40891*ii, ------------------------------------------------------------------------ -.042823-.573077*ii}, {1.64276+.001373*ii, .155719+1.27799*ii, ------------------------------------------------------------------------ .863647+.417604*ii, -.734295+.185368*ii, -.608821+1.99639*ii, ------------------------------------------------------------------------ -.519068-.36595*ii, -.110819+.0334624*ii}, {-3.10025+9.02146*ii, ------------------------------------------------------------------------ -9.14471+6.34299*ii, 2.88861-2.63229*ii, .640979-1.71262*ii, ------------------------------------------------------------------------ 1.20153-.499412*ii, -1.1635-1.60127*ii, .0059969-.349524*ii}, ------------------------------------------------------------------------ {2.89934-1.99216*ii, -.181976+2.17916*ii, .761259+5.06574*ii, ------------------------------------------------------------------------ 1.19648-1.13168*ii, .753024+.260107*ii, -2.38512+1.58541*ii, ------------------------------------------------------------------------ -1.88844+.696083*ii}, {2.24734-1.53041*ii, 1.27731-.976219*ii, ------------------------------------------------------------------------ -.458681-.263437*ii, -.693793-2.51314*ii, -.776581-4.66298*ii, ------------------------------------------------------------------------ -.217371+.460229*ii, -.3899+.211753*ii}, {1.04966+.19065*ii, ------------------------------------------------------------------------ 3.48792-.907152*ii, -1.95764-2.32365*ii, -.374345-1.19025*ii, ------------------------------------------------------------------------ -.665604-.076337*ii, -.176124+.0087169*ii, -.616265-.442341*ii}, ------------------------------------------------------------------------ {.66296+.455059*ii, 1.78698-.460051*ii, -.797804+.681177*ii, ------------------------------------------------------------------------ -1.37486-.754645*ii, 1.02864-.746415*ii, .101123+.387554*ii, ------------------------------------------------------------------------ .683733+.14124*ii}, {1.40291+.759787*ii, -.13922-16.7338*ii, ------------------------------------------------------------------------ -12.1419+4.8153*ii, -.199024-.746779*ii, -.0110879-.104156*ii, ------------------------------------------------------------------------ -.594961+.114508*ii, -.523647-.759479*ii}, {.384968-.947665*ii, ------------------------------------------------------------------------ -.874706-.207024*ii, .252759+.642969*ii, -.150054+.202211*ii, ------------------------------------------------------------------------ 1.49303+1.13119*ii, -.24877-.453296*ii, -.1631-.0184815*ii}, ------------------------------------------------------------------------ {-.187379-.449934*ii, -.473294+.563438*ii, -.656045+.007513*ii, ------------------------------------------------------------------------ -.441614+.514702*ii, .674192+.794424*ii, .166846-.977591*ii, ------------------------------------------------------------------------ .0267801-.0950368*ii}, {20.8103+9.22064*ii, 11.1472+18.607*ii, ------------------------------------------------------------------------ -.445469-5.25292*ii, 4.66263+1.01282*ii, 3.73826+2.39834*ii, ------------------------------------------------------------------------ -1.15462-2.31624*ii, .0768598-.0242366*ii}, {-7.58981-6.69024*ii, ------------------------------------------------------------------------ -3.67984-12.2069*ii, -4.40577+8.48214*ii, -.197522+3.42999*ii, ------------------------------------------------------------------------ -.028529+3.06459*ii, -.112578-2.26272*ii, -.203835+.205897*ii}, ------------------------------------------------------------------------ {1.13201-.09404*ii, 2.12189-.531349*ii, -1.46063+.323137*ii, ------------------------------------------------------------------------ -1.92117-.877422*ii, -1.08384-1.66549*ii, -.372825-.0162201*ii, ------------------------------------------------------------------------ -.493548+.620201*ii}, {-1.13176+1.05717*ii, -1.77255+1.67698*ii, ------------------------------------------------------------------------ 6.66399-2.53852*ii, -.722445-1.80758*ii, .551661+.148602*ii, ------------------------------------------------------------------------ -.919012-2.44407*ii, .637682-.722628*ii}, {.689389-.111679*ii, ------------------------------------------------------------------------ .426006+.157143*ii, .707821-1.73559*ii, .802128+2.63315*ii, ------------------------------------------------------------------------ 2.9584-2.4331*ii, -.730366-.863828*ii, -.444016-1.38785*ii}, ------------------------------------------------------------------------ {1.13045+.171283*ii, .0063398+.177547*ii, .002924+1.55916*ii, ------------------------------------------------------------------------ .292584-.821949*ii, .696866-.340073*ii, 1.44421+.878123*ii, ------------------------------------------------------------------------ .380053-1.36285*ii}, {-.787263+.077468*ii, .452768+.238344*ii, ------------------------------------------------------------------------ 2.08545-1.49246*ii, -2.08942+.87208*ii, 2.52232-.269961*ii, ------------------------------------------------------------------------ .759489-.770694*ii, .0325181+.321439*ii}, {-.145165+1.18519*ii, ------------------------------------------------------------------------ .55228+3.18289*ii, 2.50872+.474219*ii, -.817847-.631736*ii, ------------------------------------------------------------------------ 3.0161+1.92997*ii, .244827-.145556*ii, .0493243-.0336758*ii}, ------------------------------------------------------------------------ {-.436358-.63454*ii, -.501912+.008873*ii, .672959-1.37972*ii, ------------------------------------------------------------------------ -1.34522+2.20124*ii, 2.84431-.550887*ii, .485699-.786291*ii, ------------------------------------------------------------------------ .33601-.104919*ii}, {4.57017+.430589*ii, 1.24963+2.1152*ii, ------------------------------------------------------------------------ .225496+5.04844*ii, 1.12434-.636565*ii, 1.00494+.320403*ii, ------------------------------------------------------------------------ -1.73957-2.54459*ii, -.0571936-.351009*ii}, {.98573+.806019*ii, ------------------------------------------------------------------------ .084274+1.00173*ii, 1.30713+.059866*ii, -.285565+.197953*ii, ------------------------------------------------------------------------ 1.07001+.952795*ii, 1.0868-1.25217*ii, -.734249-.184477*ii}, ------------------------------------------------------------------------ {1.03559-.491715*ii, 5.39495+3.50224*ii, 1.80583-2.99227*ii, ------------------------------------------------------------------------ -.001327-.623464*ii, -.201381+.318454*ii, .107246+.318594*ii, ------------------------------------------------------------------------ -.51064+.018030*ii}, {.831677-1.62408*ii, 1.49992-.677834*ii, ------------------------------------------------------------------------ -.166718-1.15969*ii, -.441149-.425255*ii, -.792299+1.35599*ii, ------------------------------------------------------------------------ -1.39974-1.04683*ii, .585051-.831685*ii}} o7 : List |
i8 : E := skewSchubertVariety((k,n),l,m); 3 7 o8 : Matrix (CC [x1, x2, x3, x4, x5, x6, x7]) <--- (CC [x1, x2, x3, x4, x5, x6, x7]) 53 53 |
i9 : apply(Sols ,s-> sub(E,matrix{s}) ) o9 = {| 1 2.83037-1.16682i .339087-1.17121i -.179497+2.13795i | 0 0 1 -.155164-1.83616i | 0 0 0 0 ------------------------------------------------------------------------ 0 0 0 |, | 1 .594075-1.25791i 0 0 | | 0 1 .375115+1.07551i -.767582+.750854i | | 0 ------------------------------------------------------------------------ .616511-1.23608i 1.14351-.428504i .61279-.237252i 0 0 1 2.40519-.943408i 1.52285+4.31747i 0 0 0 1 ------------------------------------------------------------------------ 0 0 |, | 1 1.59564-3.53412i 1.4881+.73784i 0 0 | | 0 0 1 .158879+.160696i .252295+.536086i | | 0 0 0 ------------------------------------------------------------------------ .103491+2.3495i 0 0 0 |, | 1.95469-.49702i .985635+1.07916i 0 0 | | 0 1 .139753+2.59479i -.572996+1.78227i | | ------------------------------------------------------------------------ 1 1.00034+1.16377i .282334-.0623693i -.0261111+.469647i 0 0 0 1 .84702+.456431i 1.37268-.081691i 0 0 0 0 1 ------------------------------------------------------------------------ 0 0 |, | 1 1.31908-.09193i 0 0 | | 0 0 -.727831+.904989i -1.29211+.851385i | | 0 0 ------------------------------------------------------------------------ 1.68667+.214682i -.157446-.621006i 0 0 1 .512377+2.35174i .727981+4.07791i 0 0 0 1 .405833-.485107i ------------------------------------------------------------------------ 0 |, | 1 -.177866+.2745i -1.0785+.377005i 0 | | 0 0 1 -.922487-.546294i | | 0 0 0 ------------------------------------------------------------------------ .033043-1.20722i 0 0 0 -.182675+.322299i .972502+.039856i 0 0 0 1 -.844351-.697949i -.586467-.339161i ------------------------------------------------------------------------ |, | 1 .918986+.729155i .0544907+.48707i 1.76017+.099046i | | 0 0 1 -.989923+.209755i | | 0 0 0 0 ------------------------------------------------------------------------ 0 0 0 |, | 1 1.21071+.830863i 0 0 | | 0 1 -7.50874-4.28251i -10.5236+7.00191i | | 0 ------------------------------------------------------------------------ -4.8344-2.30609i -1.04978-6.47208i -4.0858+.430332i 0 0 1 -.247438-.570484i .312555+.57111i 0 0 0 1 ------------------------------------------------------------------------ 0 0 |, | 1 .698366+.608809i 0 0 | | 0 0 .339541+.238912i .142301+1.32292i | | 0 0 ------------------------------------------------------------------------ .95351-.341281i .277886-.784558i 0 0 1 1.09216+.199628i -.560935-1.56259i 0 0 0 1 -.044748-1.86422i ------------------------------------------------------------------------ 0 |, | 1 .738961+1.08457i -.735094+.575263i 0 | | 0 0 1 -2.47285-1.90376i | | 0 0 0 ------------------------------------------------------------------------ .930967+.024263i 0 0 0 |, -.669722+.129668i .739605+.671634i 0 0 | 0 1 .0666671-.264572i -.69609+.75765i | ------------------------------------------------------------------------ | 1 .965771-.629416i -.316157-.403439i -1.51768+5.13956i | 0 0 1 1.46815-2.03927i | 0 0 0 0 ------------------------------------------------------------------------ 0 0 0 |, | 1 1.07465-.506629i 0 0 | | 0 1 -.61724-1.18922i -.969504-.803241i | | 0 ------------------------------------------------------------------------ .091364-1.0877i -.205565-.533216i -.152703-.917898i 0 0 1 .697898+3.35222i 4.37021-2.46295i 0 0 0 1 ------------------------------------------------------------------------ 0 0 |, | 1 .970958+1.03881i 0 0 | | 0 0 .189392-.770681i .211364-.380062i | | 0 0 ------------------------------------------------------------------------ -.033912+.601984i .421707+.091466i 0 0 1 -.951042+.158509i 1.44278+1.4802i 0 0 0 1 -.200247-.439407i ------------------------------------------------------------------------ 0 |, | 1 .667701-.7178i .0224798-.107527i 0 | | 0 0 1 -.505303+.153709i | | 0 0 0 ------------------------------------------------------------------------ -1.76825+.263386i 0 0 0 1.20835-.249924i 1.49888-.221283i 0 0 0 1 -.413561+.0570748i -.545814+.074177i ------------------------------------------------------------------------ |, | 1 .611252-.923722i .017085-.593473i -3.57302-1.4063i | | 0 0 1 2.50843+2.46452i | | 0 0 0 0 ------------------------------------------------------------------------ 0 0 0 |, | 1 4.72049+1.33919i 1.81309-.66578i 0 0 | | 0 0 1 .055338-1.25873i -.035530-.931i | | 0 0 ------------------------------------------------------------------------ 1.75729+.246846i -.903759+3.65303i 0 0 1 3.18925+.216125i -1.1456+2.32551i 0 0 0 1 .131337+1.33329i ------------------------------------------------------------------------ 0 |, | 1 .785901+.098456i -.23242+.0295552i 0 | | 0 0 1 .72027+.505613i | | 0 0 0 ------------------------------------------------------------------------ .739642+.296011i 0 0 0 |, | .136402-.0477763i .59205+.235903i 0 0 | | 0 1 8.42502+6.70125i 3.23608+6.26523i | | ------------------------------------------------------------------------ 1 -.142789-2.54626i 1.31922-1.49604i .280374-1.55877i 0 0 0 1 .266205-.217297i .095254+1.73734i 0 0 0 0 1 ------------------------------------------------------------------------ 0 0 |, | 1 -.965005-.524762i 0 0 | | 0 0 -.372531-.340654i .40685+.389897i | | 0 0 ------------------------------------------------------------------------ -.850863-.680911i .0666037-.0517023i 0 0 1 -.172818-.0192463i .171706+.860751i 0 0 0 1 -.279174+.225652i ------------------------------------------------------------------------ 0 |, | 1 -.578466-2.46713i 1.24047-1.46576i 0 | | 0 0 1 -.135133+.986769i | | 0 0 0 ------------------------------------------------------------------------ .063654-.835605i 0 0 0 |, .166326-.413034i .443282+.686889i 0 0 | 0 1 .070334+1.00916i -.460264+1.57546i | ------------------------------------------------------------------------ | 1 .958026+.682366i .406162+.547165i 1.23458-.823822i 0 | 0 0 1 -1.77795+1.15503i 2.23779+2.10422i | 0 0 0 0 1 ------------------------------------------------------------------------ 0 0 |, | 1 6.71719-.064777i 0 0 | | 0 0 -.817004-.111244i -1.03764+.816608i | | 0 0 ------------------------------------------------------------------------ 5.87225-1.43526i 2.57362-3.43779i 0 0 1 -.424259-.658763i -.027455+3.22171i 0 0 0 1 .0617571-.256373i ------------------------------------------------------------------------ 0 |, | 1 .837651-.01776i .080552+1.0221i 0 | | 0 0 1 .201933-.081954i | | 0 0 0 ------------------------------------------------------------------------ -.048032-1.42918i 0 0 0 -.458712+.739391i -.957983+1.48391i 0 0 0 1 -.0057591+.228492i .0801215+.334954i ------------------------------------------------------------------------ |, | 1 -.335264+.468981i -.85052-.249673i .078588+.709592i | | 0 0 1 -.0051331-.0727716i | | 0 0 0 0 ------------------------------------------------------------------------ 0 0 0 |, | 1 .038605+1.00938i 0 0 | | 0 1 -1.45296+2.2789i .184977+1.65745i | | 0 ------------------------------------------------------------------------ 1.25726+.441114i -.209568+.615339i .506669+1.36158i 0 0 1 .401848+.102313i .814918+.447657i 0 0 0 1 ------------------------------------------------------------------------ 0 0 |, | 1 .991809+.282546i 0 0 | | 0 0 -5.44325-5.09058i .354587-2.67153i | | 0 0 ------------------------------------------------------------------------ 1.31351+.625964i -1.18737-1.67876i 0 0 1 -2.39781+2.38038i -2.77821+1.32197i 0 0 0 1 -.0792051+.419036i ------------------------------------------------------------------------ 0 |, | 1 .786403+.423102i -.254866-.311113i 0 | | 0 0 1 .316059+.434391i | | 0 0 0 ------------------------------------------------------------------------ -.411882-.43632i 0 0 0 |, .569311+.437771i 1.10602+.197325i 0 0 | 0 1 -.734128+.775301i -.512867+.597257i | ------------------------------------------------------------------------ | 1 .530206+.182987i .267353-.627555i .136504-.0534943i | 0 0 1 .055419-.527812i | 0 0 0 0 ------------------------------------------------------------------------ 0 0 0 |, | 1 -.189794-.615281i 0 0 | | 0 1 .092370-.839385i -.118704-1.5459i | | 0 ------------------------------------------------------------------------ .354697-2.21884i .468827-.231337i -.242795-.698118i 0 0 1 -.539902+.406311i -.319066+1.10558i 0 0 0 1 ------------------------------------------------------------------------ 0 0 |, | 1 .083327-1.68596i 0 0 | | 0 0 .347758-.301442i .234016+.762547i | | 0 0 ------------------------------------------------------------------------ 1.13553+.87302i 4.72372-1.51816i 0 0 1 .101769-1.11528i .721827+.439285i 0 0 0 1 1.05034-2.47462i ------------------------------------------------------------------------ 0 |, | 1 .349036+.480496i -.442291+.7817i 0 | | 0 0 1 1.24974+.957912i | | 0 0 0 ------------------------------------------------------------------------ .700609+.562738i 0 0 0 -.00782018-.0302859i .561071+.642179i 0 0 0 1 -.012391-3.46218i .060909-1.02606i ------------------------------------------------------------------------ |, | 1 1.39166-.170938i 1.41613-.126288i -1.05682+.761826i | | 0 0 1 7.14911+7.32129i | | 0 0 0 0 ------------------------------------------------------------------------ 0 0 0 |, | 1 -2.13623+7.32394i 0 0 | | 0 1 -.016024+.865674i .610208+.799856i | | 0 ------------------------------------------------------------------------ -1.45486+3.55806i -23.9773+10.1617i 13.5641+20.0492i 0 0 1 -.318982-.96501i -.0515422+.12378i 0 0 0 1 ------------------------------------------------------------------------ 0 0 |, | 1 -.518932-4.29838i 0 0 | | 0 0 .02413-2.20945i 3.00757+.650103i | | 0 0 ------------------------------------------------------------------------ 2.08586-.507361i .507326+1.57303i 0 0 1 2.03021-.245073i .622707+1.62675i 0 0 0 1 2.06576+1.41166i ------------------------------------------------------------------------ 0 |, | 1 .617738-.407646i -.426388-.138073i 0 | | 0 0 1 .425359+1.54847i | | 0 0 0 ------------------------------------------------------------------------ 7.1245+.928229i 0 0 0 -2.53532-1.15553i 1.12701-.548861i 0 0 0 1 -.846797-.646969i -1.30067-.772363i ------------------------------------------------------------------------ |, | 1 .412591-.247387i 1.77301-1.87985i -.593417+.034869i | | 0 0 1 -.35955-.11525i | | 0 0 0 0 ------------------------------------------------------------------------ 0 0 0 |, | 1 -.607773-.047087i 0 0 | | 0 1 .408996-.710548i -.558421-.577332i | | 0 ------------------------------------------------------------------------ .448486+.466528i .145996+.84424i .068363-2.02747i 0 0 1 -1.52147+1.11863i 1.39177+1.04669i 0 0 0 1 ------------------------------------------------------------------------ 0 0 |, | 1 .88539+.386903i 0 0 | | 0 0 -1.02401+1.29007i -2.0382+.423085i | | 0 0 ------------------------------------------------------------------------ .328355-.541645i .785763+.047963i 0 0 1 .0425523+.0898986i .44867+.269815i 0 0 0 1 1.32757+.591549i ------------------------------------------------------------------------ 0 |, | 1 1.90755-.213533i .770571+2.89298i 0 | | 0 0 1 1.00823+.636428i | | 0 0 0 ------------------------------------------------------------------------ 1.05303-2.2697i 0 0 0 |, -.846758-.334234i -.078368+.551162i 0 0 | 0 1 .321975-.428153i -.205606+.64375i | ------------------------------------------------------------------------ | 1 1.07608-.050741i 1.39746-.282675i -.0928957-.132527i | 0 0 1 -3.90339-3.90227i | 0 0 0 0 ------------------------------------------------------------------------ 0 0 0 |, | 1 -7.82381-8.83392i 0 0 | | 0 1 .078951-.15352i -.573059-.163445i | | 0 ------------------------------------------------------------------------ .816078+.275593i 1.12942-.411091i -.194283-.869602i 0 0 1 .503989-1.17673i -.96889-1.17963i 0 0 0 1 ------------------------------------------------------------------------ 0 0 |, | 1 .598506+.607586i 0 0 | | 0 0 -.187246-.520936i -1.11249-1.19833i | | 0 0 ------------------------------------------------------------------------ -.357378+.515676i -.400765-.0240624i 0 0 1 -.91291-.169604i .075919+.846655i 0 0 0 1 .058203-1.32517i ------------------------------------------------------------------------ 0 |, | 1 .649368-.281033i .178564+.477535i 0 | | 0 0 1 .129987+.485521i | | 0 0 0 ------------------------------------------------------------------------ 2.18503+.212868i 0 0 0 |, | .0588415-.420831i .937315+.335758i 0 0 | | 0 1 1.76197-3.05745i 1.6167+.652616i | | ------------------------------------------------------------------------ 1 .80873+.766799i .585551+.490728i .414654-.875804i 0 0 0 1 -1.74775+.587978i 2.92108+4.14704i 0 0 0 0 1 ------------------------------------------------------------------------ 0 0 |, | 1 .689862-.080104i 0 0 | | 0 0 -.248523-.228073i -.386832+.194522i | | 0 0 ------------------------------------------------------------------------ .771525+.991817i -.383983-1.21437i 0 0 1 -.748579+.829531i -1.25648+.877242i 0 0 0 1 -.049215+.294826i ------------------------------------------------------------------------ 0 |, | 1 .881167+.427014i .268105-.0096692i 0 | | 0 0 1 .189683+.531035i | | 0 0 0 ------------------------------------------------------------------------ .765203+.076744i 0 0 0 .228207+.0175461i 1.29618-.357272i 0 0 0 1 -.717145+.124914i -3.51734-.227384i ------------------------------------------------------------------------ |, | 1 1.7251+.928577i 2.74863+1.97259i 3.52728-2.47956i | | 0 0 1 .19062-1.59231i | | 0 0 0 0 ------------------------------------------------------------------------ 0 0 0 |, | 1 -.843351+.684587i 0 0 | | 0 1 -.994148-.674488i -.788133-.455214i | | 0 ------------------------------------------------------------------------ .837498+1.39153i 1.1407+2.58084i 2.41347+.232758i 0 0 1 -.703573-.491564i 2.38398+2.20851i 0 0 0 1 ------------------------------------------------------------------------ 0 0 |, | 1 1.44651-1.79i 2.20384-1.74707i 0 0 | | 0 0 1 .225157-.254973i .0682961-.110237i | | 0 0 0 ------------------------------------------------------------------------ -.187375-1.68581i 0 0 0 |, -.254739-.514403i -.685064+1.31354i 0 0 | 0 1 -.408477-.334541i .606803-.45191i | ------------------------------------------------------------------------ | 1 -2.22799+1.00517i 2.20445-.436488i 4.71575-1.86952i 0 | 0 0 1 .994985+1.72538i -.107391-.92427i | 0 0 0 0 1 ------------------------------------------------------------------------ 0 0 |, | 1 .373175-.949261i 0 0 | | 0 0 1.50233-1.6995i -.309709-.0107719i | | 0 0 ------------------------------------------------------------------------ 1.89583-.546296i -.259622-2.11873i 0 0 1 -.305812-.726001i -.219359+.900303i 0 0 0 1 -1.50091-2.7817i ------------------------------------------------------------------------ 0 |, | 1 1.33596-.229447i -.117514-1.85582i 0 | | 0 0 1 2.34084+1.06788i | | 0 0 0 ------------------------------------------------------------------------ -1.21232+3.63221i 0 0 0 -.390801-1.18822i .679468-.552979i 0 0 0 1 -.276353-.490547i -.827168+.144626i ------------------------------------------------------------------------ |, | 1 1.13662-.022903i .40874+.247727i .499645+1.90144i | | 0 0 1 .628111-1.33391i | | 0 0 0 0 ------------------------------------------------------------------------ 0 0 0 |, | 1 .900587-.128164i 0 0 | | 0 1 3.69554-1.66978i .797363-4.78321i | | 0 ------------------------------------------------------------------------ .222182-.24866i -.30128-.930265i -.193887-.187332i 0 0 1 .221136+.0942152i -7.25787+4.50238i 0 0 0 1 ------------------------------------------------------------------------ 0 0 |, | 1 .716201+2.06053i 0 0 | | 0 0 .0564447+.057707i .0722465+.0022400i | | 0 0 ------------------------------------------------------------------------ -1.54182+4.60067i 4.35754+4.02669i 0 0 1 .48999-.975936i .255998+.449672i 0 0 0 1 -.778497-1.40891i ------------------------------------------------------------------------ 0 |, | 1 1.64276+.001373i .155719+1.27799i 0 | | 0 0 1 -.042823-.573077i | | 0 0 0 ------------------------------------------------------------------------ .863647+.417604i 0 0 0 -.734295+.185368i -.608821+1.99639i 0 0 0 1 -.519068-.36595i -.110819+.0334624i ------------------------------------------------------------------------ |, | 1 -3.10025+9.02146i -9.14471+6.34299i 2.88861-2.63229i | | 0 0 1 .640979-1.71262i | | 0 0 0 0 ------------------------------------------------------------------------ 0 0 0 |, | 1 1.20153-.499412i 0 0 | | 0 1 -1.1635-1.60127i .0059969-.349524i | | 0 ------------------------------------------------------------------------ 2.89934-1.99216i -.181976+2.17916i .761259+5.06574i 0 0 1 1.19648-1.13168i .753024+.260107i 0 0 0 1 ------------------------------------------------------------------------ 0 0 |, | 1 2.24734-1.53041i 0 0 | | 0 0 -2.38512+1.58541i -1.88844+.696083i | | 0 0 ------------------------------------------------------------------------ 1.27731-.976219i -.458681-.263437i 0 0 1 -.693793-2.51314i -.776581-4.66298i 0 0 0 1 -.217371+.460229i ------------------------------------------------------------------------ 0 |, | 1 1.04966+.19065i 3.48792-.907152i 0 | | 0 0 1 -.3899+.211753i | | 0 0 0 ------------------------------------------------------------------------ -1.95764-2.32365i 0 0 0 -.374345-1.19025i -.665604-.076337i 0 0 0 1 -.176124+.0087169i -.616265-.442341i ------------------------------------------------------------------------ |, | 1 .66296+.455059i 1.78698-.460051i -.797804+.681177i | | 0 0 1 -1.37486-.754645i | | 0 0 0 0 ------------------------------------------------------------------------ 0 0 0 |, | 1 1.02864-.746415i 0 0 | | 0 1 .101123+.387554i .683733+.14124i | | 0 ------------------------------------------------------------------------ 1.40291+.759787i -.13922-16.7338i -12.1419+4.8153i 0 0 1 -.199024-.746779i -.0110879-.104156i 0 0 0 1 ------------------------------------------------------------------------ 0 0 |, | 1 .384968-.947665i 0 0 | | 0 0 -.594961+.114508i -.523647-.759479i | | 0 0 ------------------------------------------------------------------------ -.874706-.207024i .252759+.642969i 0 0 1 -.150054+.202211i 1.49303+1.13119i 0 0 0 1 -.24877-.453296i ------------------------------------------------------------------------ 0 |, | 1 -.187379-.449934i -.473294+.563438i 0 | | 0 0 1 -.1631-.0184815i | | 0 0 0 ------------------------------------------------------------------------ -.656045+.007513i 0 0 0 -.441614+.514702i .674192+.794424i 0 0 0 1 .166846-.977591i .0267801-.0950368i ------------------------------------------------------------------------ |, | 1 20.8103+9.22064i 11.1472+18.607i -.445469-5.25292i | | 0 0 1 4.66263+1.01282i | | 0 0 0 0 ------------------------------------------------------------------------ 0 0 0 |, | 1 3.73826+2.39834i 0 0 | | 0 1 -1.15462-2.31624i .0768598-.0242366i | | 0 ------------------------------------------------------------------------ -7.58981-6.69024i -3.67984-12.2069i -4.40577+8.48214i 0 0 1 -.197522+3.42999i -.028529+3.06459i 0 0 0 1 ------------------------------------------------------------------------ 0 0 |, | 1 1.13201-.09404i 0 0 | | 0 0 -.112578-2.26272i -.203835+.205897i | | 0 0 ------------------------------------------------------------------------ 2.12189-.531349i -1.46063+.323137i 0 0 1 -1.92117-.877422i -1.08384-1.66549i 0 0 0 1 -.372825-.0162201i ------------------------------------------------------------------------ 0 |, | 1 -1.13176+1.05717i -1.77255+1.67698i 0 | | 0 0 1 -.493548+.620201i | | 0 0 0 ------------------------------------------------------------------------ 6.66399-2.53852i 0 0 0 |, -.722445-1.80758i .551661+.148602i 0 0 | 0 1 -.919012-2.44407i .637682-.722628i | ------------------------------------------------------------------------ | 1 .689389-.111679i .426006+.157143i .707821-1.73559i 0 | 0 0 1 .802128+2.63315i 2.9584-2.4331i | 0 0 0 0 1 ------------------------------------------------------------------------ 0 0 |, | 1 1.13045+.171283i 0 0 | | 0 0 -.730366-.863828i -.444016-1.38785i | | 0 0 ------------------------------------------------------------------------ .0063398+.177547i .002924+1.55916i 0 0 1 .292584-.821949i .696866-.340073i 0 0 0 1 1.44421+.878123i ------------------------------------------------------------------------ 0 |, | 1 -.787263+.077468i .452768+.238344i 0 | | 0 0 1 .380053-1.36285i | | 0 0 0 ------------------------------------------------------------------------ 2.08545-1.49246i 0 0 0 |, -2.08942+.87208i 2.52232-.269961i 0 0 | 0 1 .759489-.770694i .0325181+.321439i | ------------------------------------------------------------------------ | 1 -.145165+1.18519i .55228+3.18289i 2.50872+.474219i 0 | 0 0 1 -.817847-.631736i 3.0161+1.92997i | 0 0 0 0 1 ------------------------------------------------------------------------ 0 0 |, | 1 -.436358-.63454i 0 0 | | 0 0 .244827-.145556i .0493243-.0336758i | | 0 0 ------------------------------------------------------------------------ -.501912+.008873i .672959-1.37972i 0 0 1 -1.34522+2.20124i 2.84431-.550887i 0 0 0 1 .485699-.786291i ------------------------------------------------------------------------ 0 |, | 1 4.57017+.430589i 1.24963+2.1152i .225496+5.04844i 0 | | 0 0 1 1.12434-.636565i .33601-.104919i | | 0 0 0 0 ------------------------------------------------------------------------ 0 0 0 |, | 1 1.00494+.320403i 0 0 | | 0 1 -1.73957-2.54459i -.0571936-.351009i | | 0 ------------------------------------------------------------------------ .98573+.806019i .084274+1.00173i 1.30713+.059866i 0 0 1 -.285565+.197953i 1.07001+.952795i 0 0 0 1 ------------------------------------------------------------------------ 0 0 |, | 1 1.03559-.491715i 0 0 | | 0 0 1.0868-1.25217i -.734249-.184477i | | 0 0 ------------------------------------------------------------------------ 5.39495+3.50224i 1.80583-2.99227i 0 0 1 -.001327-.623464i -.201381+.318454i 0 0 0 1 .107246+.318594i ------------------------------------------------------------------------ 0 |, | 1 .831677-1.62408i 1.49992-.677834i 0 | | 0 0 1 -.51064+.018030i | | 0 0 0 ------------------------------------------------------------------------ -.166718-1.15969i 0 0 0 -.441149-.425255i -.792299+1.35599i 0 0 0 1 -1.39974-1.04683i .585051-.831685i ------------------------------------------------------------------------ |} | | o9 : List |