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$cached_env_img{$key} = q|table1747|; $key = q/{tabular}[t]p4.0inunderlinermlargeKeyIdeas:rmABLACScontextiscreatedviaacalltoeitherttBLACS_GRIDINITorttBLACS_GRIDMAP.Noroutinerequiringacontextmaybeuseduntiloneoftheseroutineshasbeencalled.MultiplecallstottBLACS_GRIDINITorttBLACS_GRIDMAPresultinthecreationofnewcontexts.Topreserveresources,theusershouldfreeunusedcontextsbycallingttBLACS_GRIDEXIT.WhenallBLACSoperationsaredone,acalltottBLACS_EXITfreesanyremainingcontextsandshutsdownallBLACSoperations.rmPleasenotethatttBLACS_GRIDINITandttBLACS_GRIDMAPacceptsystemcontextsasinput.AdefaultsystemcontextencompassingallavailableprocessesmaybeobtainedbyacalltottBLACS_GET.rmttBLACS_SETcanonlybeusedtochangethemessageIDrangebeforethecreationofthefirstcontext.Subsequentcallswillbeignored.underlinermlargeTopologyHintsrmTopologiesallowtheusertooptimizecommunicationpatternsforaparticularoperation.Iftheuserdoesnothaveacommunicationpipetomaintain,thedefaultttTOP=''isrecommed.{tabular}/; $cached_env_img{$key} = q|tabular7062|; $key = q/{displaymath}left(arrayrx_1x_2x_3x_4x_5x_6x_7x_8x_9arrayright)=left(arrayr0frac-16frac12000frac-12frac160arrayright).{displaymath}/; $cached_env_img{$key} = q|displaymath12393|; $key = q/{_inline}$Lambda=mboxrmdiag(lambda_1,ldots,lambda_n)${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18622|; $key = q/{displaymath}A=U^HU;;rmor;;A=LL^Hmbox(intheHermitiancase),{displaymath}/; $cached_env_img{$key} = q|displaymath12935|; $key = q/{_inline}$0leqp_r;SPMlt;P_r${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline14503|; $key = q/{_inline}$P=MtimesN1000000${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline16164|; $key = q/{_inline}$omega_cleqp(n)cdotepsilon${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18334|; $key = q/{_inline}$D^-1AD^-1${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline19305|; $key = q/{tabular}[t]p4.5inlargebfMeaningofprefixestabularllS-112;SPMamp;C-113D-114;SPMamp;Z-115;SPMamp;(maynotbesupportedbyallmachines)tabularlargebfLevel2andLevel3PBLASlargebfMatrixTypesGE-GEneralSY-SYmmetricHE-HErmitianTR-TRiangularlargebfLevel2andLevel3PBLASlargebfOptionsDummyoptionsargumentsaredeclaredasCHARACTER_inline$ast$_inline1andmaybepassedascharacterstrings.TRANS_='Notranspose','Transpose','Conjugatetranspose',(_inline$X,~X^T,~X^H$_inline)UPLO='Uppertriangular','LowerTriangular'DIAG='Non-unittriangular','Unittriangular'SIDE='Left'or'Right'(Aorop(A)ontheleft,orAorop(A)ontheright)Forrealmatrices,TRANS_='T'andTRANS_='C'havethesamemeaning.ForHermitianmatrices,TRANS_='T'isnotallowed.Forcomplexsymmetricmatrices,TRANS_='C'isnotallowed.largebfArrayDescriptor,IncrementThearraydescriptor_inline$DESCA$_inlineisanintegerarrayofdimension9.Itdescribesthetwo-dimensionalblock-cyclicmappingofthematrixA.ThefirsttwoentriesarethedescriptortypeandtheBLACScontext.Thethirdandfourthentriesarethedimensionsofthematrix(row,column).Thefifthandsixthentriesaretherow-andcolumnblocksizesusedtodistributethematrix.Theseventhandeightharethecoordinatesoftheprocesscontainingthefirstentryofthematrix.Thelastentrycontainstheleadingdimensionofthelocalarraycontainingthematrixelements.Theincrementspecifiedforvectorsisalwaysglobal.Sofaronly1andDESCA(M_)aresupported.{tabular}/; $cached_env_img{$key} = q|tabular6979|; $key = q/{tabular}llValue;SPMamp;Meaninghlinett`U';SPMamp;Uppertrianglett`L';SPMamp;Lowertriangle{tabular}/; $cached_env_img{$key} = q|tabular3444|; $key = q/{_inline}$Q^T${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline13300|; $key = q/{_inline}$sigma_1geqcdotsgeqsigma_rgeq0${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline14129|; $key = q/{_inline}$A^TX=B${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline12654|; $key = q/{_inline}$|hatalpha-alpha||hatalpha|leq.1${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline17700|; $key = q/{_inline}$Sigma=mboxrmdiag(sigma_1,ldots,sigma_r)${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18852|; $key = q/{_inline}$|x|_i=|x_i|${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline17562|; $key = q/{eqnarray*}|hatx-x|_infty=2;,;;SPMamp;displaystylefrac|hatx-x|_infty|x|_infty=.02;,;;SPMamp;frac|hatx-x|_infty|hatx|_infty=.0202|hatx-x|_2=2.238;,;;SPMamp;displaystylefrac|hatx-x|_2|x|_2=.0223;,;;SPMamp;frac|hatx-x|_2|hatx|_2=.0225|hatx-x|_1=3.1;,;;SPMamp;displaystylefrac|hatx-x|_1|x|_1=.0282;,;;SPMamp;frac|hatx-x|_1|hatx|_1=.0279{eqnarray*}/; $cached_env_img{$key} = q|eqnarray4677|; $key = q/{_inline}$Az=lambdaz${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline13848|; $key = q/{_inline}$9times9${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline12402|; $key = q/{_inline}$ttBNORM=120${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline19147|; $key = q/{_inline}$kappa_infty${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18559|; $key = q/{_inline}$theta(hatcalS,calS)${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline17870|; $key = q/{_inline}$theta${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline17638|; $key = q/{displaymath}|hatlambda_i-lambda_i|mathrelraisebox-.75ex_inline$mathopsimlimits^textstyle;SPMlt;$_inlinep(n)epsiloncdot(|B|_2|A|_2+kappa_2(B)cdot|hatlambda_i|)=ttEERRBD(i);.{displaymath}/; $cached_env_img{$key} = q|displaymath19058|; $key = q/{table}[htb]centertabular||l|l|l|l||hlineDESC_();SPMamp;Symbolic;SPMamp;Scope;SPMamp;Definition;SPMamp;Name;SPMamp;;SPMamp;hline1;SPMamp;DTYPE_A;SPMamp;(global);SPMamp;Thedescriptortype(DTYPE_A=501);SPMamp;;SPMamp;;SPMamp;for_inline$1timesP_c$_inlineprocessgridfor;SPMamp;;SPMamp;;SPMamp;bandandtridiagonalmatrices;SPMamp;;SPMamp;;SPMamp;block-columndistributed.2;SPMamp;CTXT_A;SPMamp;(global);SPMamp;TheBLACScontexthandle,indicatingthe;SPMamp;;SPMamp;;SPMamp;BLACSprocessgridoverwhichtheglobal;SPMamp;;SPMamp;;SPMamp;matrixAisdistributed.Thecontextitself;SPMamp;;SPMamp;;SPMamp;isglobal,butthehandle(theintegervalue);SPMamp;;SPMamp;;SPMamp;mayvary.3;SPMamp;N_A;SPMamp;(global);SPMamp;ThenumberofcolumnsintheglobalmatrixA.4;SPMamp;NB_A;SPMamp;(global);SPMamp;Thecolumnblocksize.5;SPMamp;CSRC_A;SPMamp;(global);SPMamp;Theprocesscolumnoverwhichthefirst;SPMamp;;SPMamp;;SPMamp;columnoftheglobalmatrixAisdistributed.6;SPMamp;LLD_A;SPMamp;(local);SPMamp;Theleadingdimensionofthelocal;SPMamp;;SPMamp;;SPMamp;array.Forthetridiagonal;SPMamp;;SPMamp;;SPMamp;subroutines,thisentryisignored.7;SPMamp;;SPMamp;;SPMamp;Unused,reservedhlinetabularcenter{table}/; $cached_env_img{$key} = q|table3188|; $key = q/{_inline}$hatV+deltahatV${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18965|; $key = q/{figure}[h]centerlinehboxpsfigfigure=FIGbcol.ps,height=1.4in,width=1.4inhspace*1.0inpsfigfigure=FIGcycliccol.ps,height=1.4in,width=1.4inlabelfig:bcol{figure}/; $cached_env_img{$key} = q|figure2369|; $key = q/{_inline}$AB^-1${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline13809|; $key = q/{displaymath}frac|x-hatx|_2|x|_2mathrelraisebox-.75ex_inline$mathopsimlimits^textstyle;SPMlt;$_inlinep(n)epsilonleft{frac2kappa_2(A)cos(theta)+tan(theta)kappa_2^2(A)right},{displaymath}/; $cached_env_img{$key} = q|displaymath18467|; $key = q/{_inline}$|hatx-x|_p${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline17486|; $key = q/{_inline}$Z^-1AZ^-H=Lambda${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline19111|; $key = q/{_inline}$N_A=7${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline15334|; $key = q/{displaymath}R=bordermatrix;SPMamp;n;SPMamp;m-ncrn;SPMamp;R_11;SPMamp;R_12,quadmboxifquadn;SPMlt;m,{displaymath}/; $cached_env_img{$key} = q|displaymath13658|; $key = q/{_inline}$mgen${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline12690|; $key = q/{_inline}$NtimesNRHS${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline15189|; $key = q/{_inline}$CleftarrowalphaAB+betaC${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline16223|; $key = q/{table}[h]labeltab:bmapcentertabular||l|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c||hline_inline$I$_inline;SPMamp;1;SPMamp;2;SPMamp;3;SPMamp;4;SPMamp;5;SPMamp;6;SPMamp;7;SPMamp;8;SPMamp;9;SPMamp;10;SPMamp;11;SPMamp;12;SPMamp;13;SPMamp;14;SPMamp;15;SPMamp;16hline_inline$p$_inline;SPMamp;0;SPMamp;0;SPMamp;0;SPMamp;0;SPMamp;0;SPMamp;0;SPMamp;0;SPMamp;0;SPMamp;1;SPMamp;1;SPMamp;1;SPMamp;1;SPMamp;1;SPMamp;1;SPMamp;1;SPMamp;1_inline$l$_inline;SPMamp;0;SPMamp;0;SPMamp;0;SPMamp;0;SPMamp;0;SPMamp;0;SPMamp;0;SPMamp;0;SPMamp;0;SPMamp;0;SPMamp;0;SPMamp;0;SPMamp;0;SPMamp;0;SPMamp;0;SPMamp;0_inline$x$_inline;SPMamp;1;SPMamp;2;SPMamp;3;SPMamp;4;SPMamp;5;SPMamp;6;SPMamp;7;SPMamp;8;SPMamp;1;SPMamp;2;SPMamp;3;SPMamp;4;SPMamp;5;SPMamp;6;SPMamp;7;SPMamp;8hline_inline$l*NB+x$_inline;SPMamp;1;SPMamp;2;SPMamp;3;SPMamp;4;SPMamp;5;SPMamp;6;SPMamp;7;SPMamp;8;SPMamp;1;SPMamp;2;SPMamp;3;SPMamp;4;SPMamp;5;SPMamp;6;SPMamp;7;SPMamp;8hlinetabularcenter{table}/; $cached_env_img{$key} = q|table2416|; $key = q/{_inline}$Az_i=lambda_iBz_i${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline19103|; $key = q/{_inline}$N^2P${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline16304|; $key = q/{_inline}$(0,K+1),ldots,(0,P_c-1)${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline15267|; $key = q/{_inline}$U^HU${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline14323|; $key = q/{_inline}$(0,0),ldots,(0,K-1)${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline15253|; $key = q/{displaymath}omega_infty=frac|r|_infty|A|_inftycdot|hatx|_infty+|b|_infty;.{displaymath}/; $cached_env_img{$key} = q|displaymath18207|; $key = q/{_inline}$|x|_pneq0${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline17496|; $key = q/{_inline}$|tau-1|leq1${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline14463|; $key = q/{displaymath}R=bordermatrix;SPMamp;mcrhfillm;SPMamp;R_11crn-m;SPMamp;0,quadmboxifquadngeqm,{displaymath}/; $cached_env_img{$key} = q|displaymath13657|; $key = q/{_inline}$hatx${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline17482|; $key = q/{table}[h]indexnorm!vectorindexnorm!matrixindexnorm!onenormindexnorm!twonormindexnorm!infinitynormindexnorm!Frobeniusnormlabeltabnormscentertabular||l|l|l||hlineNorm;SPMamp;Vector;SPMamp;MatrixhlineOne-norm;SPMamp;_inline$|x|_1=sum_i|x_i|$_inline;SPMamp;_inline$|A|_1=max_jsum_i|a_ij|$_inlineTwo-norm;SPMamp;_inline$|x|_2=(sum_i|x_i|^2)^12$_inline;SPMamp;_inline$|A|_2=max_xneq0|Ax|_2|x|_2$_inlineFrobeniusnorm;SPMamp;_inline$|x|_F=|x|_2$_inline;SPMamp;_inline$|A|_F=(sum_ij|a_ij|^2)^12$_inlineInfinity-norm;SPMamp;_inline$|x|_infty=max_i|x_i|$_inline;SPMamp;_inline$|A|_infty=max_isum_j|a_ij|$_inlinehlinetabularcenter{table}/; $cached_env_img{$key} = q|table4627|; $key = q/{_inline}$|B|${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline19359|; $key = q/{_inline}$vhat_i${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline19001|; $key = q/{_inline}$C_v,N^2sqrtP${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline12074|; $key = q/{_inline}$frac14${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline16665|; $key = q/{table}[htb]centertabular||l|l|l|l||hlineDESC_();SPMamp;Symbolic;SPMamp;Scope;SPMamp;Definition;SPMamp;Name;SPMamp;;SPMamp;hline1;SPMamp;DTYPE_A;SPMamp;(global);SPMamp;DescriptortypeDTYPE_A=1for;SPMamp;;SPMamp;;SPMamp;densematrices.2;SPMamp;CTXT_A;SPMamp;(global);SPMamp;BLACScontexthandle,indicatingthe;SPMamp;;SPMamp;;SPMamp;BLACSprocessgridoverwhichtheglobal;SPMamp;;SPMamp;;SPMamp;matrixAisdistributed.Thecontextitself;SPMamp;;SPMamp;;SPMamp;isglobal,butthehandle(theintegervalue);SPMamp;;SPMamp;;SPMamp;mayvary.3;SPMamp;M_A;SPMamp;(global);SPMamp;NumberofrowsintheglobalarrayA.4;SPMamp;N_A;SPMamp;(global);SPMamp;NumberofcolumnsintheglobalarrayA.5;SPMamp;MB_A;SPMamp;(global);SPMamp;Blockingfactorusedtodistribute;SPMamp;;SPMamp;;SPMamp;therowsofthearray.6;SPMamp;NB_A;SPMamp;(global);SPMamp;Blockingfactorusedtodistribute;SPMamp;;SPMamp;;SPMamp;thecolumnsofthearray.7;SPMamp;RSRC_A;SPMamp;(global);SPMamp;Processrowoverwhichthefirstrow;SPMamp;;SPMamp;;SPMamp;ofthearrayAisdistributed.8;SPMamp;CSRC_A;SPMamp;(global);SPMamp;Processcolumnoverwhichthefirst;SPMamp;;SPMamp;;SPMamp;columnofthearrayAisdistributed.9;SPMamp;LLD_A;SPMamp;(local);SPMamp;Leadingdimensionofthelocal;SPMamp;;SPMamp;;SPMamp;array.LLD_A_inline$geq$_inlineMAX(1,LOC_inline$_r$_inline(M_A)).hlinetabularcenter{table}/; $cached_env_img{$key} = q|table2535|; $key = q/{_inline}${2x}${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline17844|; $key = q/{_inline}$R_2${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline13288|; $key = q/{_inline}$gamma${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline17636|; $key = q/{_inline}$MBneqNB${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline20032|; $key = q/{displaymath}theta(hatz_i,z_i)mathrelraisebox-.75ex_inline$mathopsimlimits^textstyle;SPMlt;$_inlinep(n)epsilonleft(frac|B|_2|A|_2(kappa_2(B))^12gap_i+kappa_2(B)right)=ttZERRBD(i);.{displaymath}/; $cached_env_img{$key} = q|displaymath19059|; $key = q/{displaymath}|b-Ax|_2=|Q^Tb-Q^TAx|_2=left|arraycc_1-Rxc_2arrayright|_2,quadmboxwherecequivleft(arraycc_1c_2arrayright)=left(arraycQ_1^TbQ_2^Tbarrayright)=Q^Tb;{displaymath}/; $cached_env_img{$key} = q|displaymath13234|; $key = q/{_inline}$|hatlambda_i-hatlambda_j|leqttORTOLcdotmax_k|hatlambda_k|${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18742|; $key = q/{_inline}$Az_i=lambda_iz_i${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18634|; $key = q/{tabular}[t]p4.0inunderlinermlargeBroadcastTopologies244245246247248249250251252253254255256257258vdots259{tabular}/; $cached_env_img{$key} = q|tabular7145|; $key = q/{_inline}$yleftarrowalphaAx+betay${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline16221|; $key = q/{_inline}$A=U_1BV_1^H${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline14157|; $key = q/{_inline}$CleftarrowC+A,B${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline16435|; $key = q/{displaymath}|fl(aoplusb)-(aoplusb)|leqepsiloncdot|aoplusb|;;,{displaymath}/; $cached_env_img{$key} = q|displaymath17196|; $key = q/{_inline}$hatU+deltahatU${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18963|; $key = q/{_inline}$Q^TQ=I${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline13210|; $key = q/{_inline}$|E|_2|A|_2leqp(m,n)epsilon${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18953|; $key = q/{_inline}$|E|${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline19353|; $key = q/{_inline}$mboxrank(A)=m${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline12696|; $key = q/{_inline}$int(fracM_BMB_B)=K${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline15293|; $key = q/{_inline}$O(N^2)${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline16659|; $key = q/{_inline}$ttRCONDB=.8326${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline19149|; $key = q/{_inline}$_N2${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline12226|; $key = q/{displaymath}Q=H_1H_2ldotsH_k.{displaymath}/; $cached_env_img{$key} = q|displaymath14415|; $key = q/{_inline}$C_f,N^3P${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline16211|; $key = q/{table}[htb]labeltabdrivellsqcentertabular||l|l|l|l|l||hlineOperation;SPMamp;multicolumn2c|SinglePrecision;SPMamp;multicolumn2c||DoublePrecisioncline2-5;SPMamp;Real;SPMamp;Complex;SPMamp;Real;SPMamp;ComplexhlinesolveLLSusing_inline$QR$_inlineor_inline$LQ$_inlinefactorization;SPMamp;PSGELSindexPSGELS;SPMamp;PCGELSindexPCGELS;SPMamp;PDGELSindexPDGELS;SPMamp;PZGELSindexPZGELShlinetabularcenter{table}/; $cached_env_img{$key} = q|table1027|; $key = q/{_inline}$10^-3|T|${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline13934|; $key = q/{table}[thb]labeltab:pblas2minipagelinewidthcentertabular||l|c|c|r|r|r|r|r||hline;SPMamp;Process;SPMamp;Block;SPMamp;multicolumn5c||Valuesof_inline$N$_inlinecline4-8;SPMamp;grid;SPMamp;size;SPMamp;2000;SPMamp;4000;SPMamp;6000;SPMamp;8000;SPMamp;10000hlineCrayT3EfootnoteAScaLAPACKprocessisrunningoneachnode.EachnodeconsistsofonecomputationalDECAlphaEV5processor,whichisusedbytheBLAS.;SPMamp;_inline$2times2$_inline;SPMamp;32;SPMamp;516;SPMamp;475;SPMamp;477;SPMamp;459;SPMamp;PSGEMV;SPMamp;_inline$4times4$_inline;SPMamp;32;SPMamp;1715;SPMamp;1422;SPMamp;2039;SPMamp;1892;SPMamp;1896;SPMamp;_inline$8times8$_inline;SPMamp;32;SPMamp;2986;SPMamp;6129;SPMamp;7157;SPMamp;5253;SPMamp;7887hlineIBMSP2footnoteAScaLAPACKprocessisrunningoneachnode.EachnodeconsistsofonecomputationalIBMPOWER2590processor,whichisusedbytheBLAS.;SPMamp;_inline$2times2$_inline;SPMamp;50;SPMamp;552;SPMamp;643;SPMamp;663;SPMamp;;SPMamp;PDGEMV;SPMamp;_inline$4times4$_inline;SPMamp;50;SPMamp;1497;SPMamp;2081;SPMamp;2244;SPMamp;2493;SPMamp;2380;SPMamp;_inline$8times8$_inline;SPMamp;50;SPMamp;1379;SPMamp;2178;SPMamp;2355;SPMamp;2708;SPMamp;2575hlineIntelXPSMPParagonfootnoteAScaLAPACKprocessisrunningoneachnode.EachnodeconsistsoftwocomputationalInteli860XPRISCprocessors,whichareusedbythemulti-threadedBLAS.;SPMamp;_inline$2times2$_inline;SPMamp;32;SPMamp;162;SPMamp;;SPMamp;;SPMamp;;SPMamp;PDGEMV;SPMamp;_inline$4times4$_inline;SPMamp;32;SPMamp;543;SPMamp;620;SPMamp;666;SPMamp;;SPMamp;;SPMamp;_inline$8times8$_inline;SPMamp;32;SPMamp;1597;SPMamp;2117;SPMamp;2356;SPMamp;2461;SPMamp;2563hlineClusterofSunUltra2sfootnoteAScaLAPACKprocessisrunningoneachnode.EachnodeconsistsoftwocomputationalUltraSPARC-1processors,butonlyoneprocessorisusedbytheBLAS.;SPMamp;_inline$2times2$_inline;SPMamp;64;SPMamp;107;SPMamp;121;SPMamp;122;SPMamp;112;SPMamp;PDGEMV;SPMamp;_inline$2times4$_inline;SPMamp;64;SPMamp;198;SPMamp;233;SPMamp;237;SPMamp;219;SPMamp;204;SPMamp;_inline$3times4$_inline;SPMamp;64;SPMamp;275;SPMamp;332;SPMamp;345;SPMamp;261;SPMamp;307hlineBerkeleyNOWfootnoteAScaLAPACKprocessisrunningoneachnode.EachnodeconsistsofonecomputationalUltraSPARC-1processor,whichisusedbytheBLAS.;SPMamp;_inline$2times2$_inline;SPMamp;32;SPMamp;118;SPMamp;123;SPMamp;121;SPMamp;;SPMamp;PDGEMV;SPMamp;_inline$2times4$_inline;SPMamp;32;SPMamp;232;SPMamp;221;SPMamp;240;SPMamp;220;SPMamp;;SPMamp;_inline$4times4$_inline;SPMamp;32;SPMamp;226;SPMamp;406;SPMamp;401;SPMamp;439;SPMamp;474;SPMamp;_inline$4times8$_inline;SPMamp;32;SPMamp;711;SPMamp;645;SPMamp;612;SPMamp;723;SPMamp;836;SPMamp;_inline$8times8$_inline;SPMamp;32;SPMamp;328;SPMamp;816;SPMamp;1033;SPMamp;1274;SPMamp;1532hlinetabularcenterminipage{table}/; $cached_env_img{$key} = q|table3850|; $key = q/{displaymath}left{arraylcl(l,m);SPMamp;=;SPMamp;displaystyle(lfloor(I-1)(P_rastMB)rfloor,lfloor(J-1)(P_castNB)rfloor),(p_r,p_c);SPMamp;=;SPMamp;displaystyle((RSRC+lfloor(I-1)MBrfloor)~rmmod~P_r,(CSRC+lfloor(J-1)NBrfloor)~rmmod~P_c),(x,y);SPMamp;=;SPMamp;displaystyle(rmmod(I-1,MB)+1,rmmod(J-1,NB)+1).arrayright.{displaymath}/; $cached_env_img{$key} = q|displaymath14670|; $key = q/{_inline}$epsilon_rmdoubleepsilon_rmsingle${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline17278|; $key = q/{figure}[thb]centertabularc|ccc|ccc|c;SPMamp;multicolumn7cProcesses;SPMamp;multicolumn3c0;SPMamp;multicolumn3c1;SPMamp;2hlineDL;SPMamp;_inline$ast$_inline;SPMamp;_inline$a_21$_inline;SPMamp;_inline$a_32$_inline;SPMamp;_inline$a_43$_inline;SPMamp;_inline$a_54$_inline;SPMamp;_inline$a_65$_inline;SPMamp;_inline$a_76$_inlineD;SPMamp;_inline$a_11$_inline;SPMamp;_inline$a_22$_inline;SPMamp;_inline$a_33$_inline;SPMamp;_inline$a_44$_inline;SPMamp;_inline$a_55$_inline;SPMamp;_inline$a_66$_inline;SPMamp;_inline$a_77$_inlineDU;SPMamp;_inline$a_12$_inline;SPMamp;_inline$a_23$_inline;SPMamp;_inline$a_34$_inline;SPMamp;_inline$a_45$_inline;SPMamp;_inline$a_56$_inline;SPMamp;_inline$a_67$_inline;SPMamp;tabularcenterlabelfig:tridiagnp7{figure}/; $cached_env_img{$key} = q|figure3064|; $key = q/{displaymath}J=p~NB+x,{displaymath}/; $cached_env_img{$key} = q|displaymath15123|; $key = q/{_inline}$(t_ft_v)${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline16376|; $key = q/{_inline}$ABz=lambdaz${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline12877|; $key = q/{_inline}$T_12${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline13706|; $key = q/{_inline}$hatx-x${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18545|; $key = q/{displaymath}Av_i=sigma_iu_iquadmboxandquadA^Tu_i=sigma_iv_iquad(rmorquadA^Hu_i=sigma_iv_i),{displaymath}/; $cached_env_img{$key} = q|displaymath12806|; $key = q/{_inline}$X^HX=I${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline17914|; $key = q/{_inline}$A^HX=B${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline12656|; $key = q/{_inline}$LL^H${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline14321|; $key = q/{_inline}$|alpha|${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline17694|; $key = q/{_inline}$O(n^2)${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18127|; $key = q/{_inline}$hatZhatLambdahatZ^T${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18680|; $key = q/{_inline}$min_x||A,x-b||${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline16768|; $key = q/{_inline}$sigma_1${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18977|; $key = q/{displaymath}theta(hatz_i,z_i)mathrelraisebox-.75ex_inline$mathopsimlimits^textstyle;SPMlt;$_inlinep(n)epsilonfrac|B^-1|_2|A|_2(kappa_2(B))^12+kappa_2(B)|hatlambda_i|gap_i=ttZERRBD(i);.{displaymath}/; $cached_env_img{$key} = q|displaymath19057|; $key = q/{_inline}$k^th${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline14705|; $key = q/{_inline}$1times3${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline14534|; $key = q/{table}[htb]centertabular||l|l|l|l||hlineDESC_();SPMamp;Symbolic;SPMamp;Scope;SPMamp;Definition;SPMamp;Name;SPMamp;;SPMamp;hline1;SPMamp;DTYPE_A;SPMamp;(global);SPMamp;DescriptortypeDTYPE_A=601;SPMamp;;SPMamp;;SPMamp;foranout-of-corematrix.2;SPMamp;CTXT_A;SPMamp;(global);SPMamp;BLACScontexthandle,indicatingthe;SPMamp;;SPMamp;;SPMamp;_inline$MPtimesNQ$_inlineBLACSprocessgridover;SPMamp;;SPMamp;;SPMamp;whichtheglobalmatrixAisdistributed.;SPMamp;;SPMamp;;SPMamp;Contextitselfisglobal,;SPMamp;;SPMamp;;SPMamp;butthehandle(theintegervalue)mayvary.3;SPMamp;M_A;SPMamp;(global);SPMamp;NumberofrowsintheglobalarrayA.4;SPMamp;N_A;SPMamp;(global);SPMamp;NumberofcolumnsintheglobalarrayA.5;SPMamp;MB_A;SPMamp;(global);SPMamp;Blockingfactorusedtodistribute;SPMamp;;SPMamp;;SPMamp;therowsofthe_inline$MMBtimesNNB$_inlinesubmatrix.6;SPMamp;NB_A;SPMamp;(global);SPMamp;Blockingfactorusedtodistribute;SPMamp;;SPMamp;;SPMamp;thecolumnsofthe_inline$MMBtimesNNB$_inlinesubmatrix.7;SPMamp;RSRC_A;SPMamp;(global);SPMamp;Processrowoverwhichthefirstrow;SPMamp;;SPMamp;;SPMamp;ofthearrayAisdistributed.8;SPMamp;CSRC_A;SPMamp;(global);SPMamp;Processcolumnoverwhichthefirst;SPMamp;;SPMamp;;SPMamp;columnofthearrayAisdistributed.9;SPMamp;LLD_A;SPMamp;(global);SPMamp;Theconceptualleadingdimensionoftheglobal;SPMamp;;SPMamp;;SPMamp;array.10;SPMamp;IODEV_A;SPMamp;global;SPMamp;IOunitdevicenumberassociatedwith;SPMamp;;SPMamp;;SPMamp;theout-of-corematrixA.11;SPMamp;SIZE_A;SPMamp;local;SPMamp;Amountoflocalin-corememoryavailablefor;SPMamp;;SPMamp;;SPMamp;thefactorizationofA.hlinetabularcenter{table}/; $cached_env_img{$key} = q|table3366|; $key = q/{displaymath}|hatx-betax|_2=fracsqrt2sinthetasqrt1+costhetaapproxtheta;.{displaymath}/; $cached_env_img{$key} = q|displaymath17415|; $key = q/{displaymath}|x||_2=|Qx|_2,quadmboxif_inline$Q$_inlineisorthogonalorunitary.{displaymath}/; $cached_env_img{$key} = q|displaymath13200|; $key = q/{_inline}$AastX-B${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18507|; $key = q/{_inline}$t_N2${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline12220|; $key = q/{_inline}$c_1${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline13312|; $key = q/{_inline}$BAz=lambdaz${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline12879|; $key = q/{_inline}$T_21${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline13708|; $key = q/{_inline}$|fracdeltaz|${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18053|; $key = q/{_inline}$t_f${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline12202|; $key = q/{_inline}$|hatA-A|_p|A|_p${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline17528|; $key = q/{tabular}llmulticolumn2cbfExpertDriverRoutinesforStandardandGeneralizedSymmetricEigenvalueProblems[0.1in]MatrixProblemType;SPMamp;RoutinehlineSymmetric;SPMamp;34Eigenvaluesvectors;SPMamp;35;SPMamp;36Hermitian;SPMamp;37Eigenvaluesvectors;SPMamp;38;SPMamp;39Symmetric;SPMamp;40Eigenvaluesvectors;SPMamp;41;SPMamp;42Hermitian;SPMamp;43Eigenvaluesvectors;SPMamp;44;SPMamp;45;SPMamp;46;SPMamp;{tabular}/; $cached_env_img{$key} = q|tabular6894|; $key = q/{_inline}${betax;,;betarm~a~scalar}${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline17604|; $key = q/{tabular}[t]lunderlinelargeOptions240241242243{tabular}/; $cached_env_img{$key} = q|tabular7141|; $key = q/{_inline}$f_pq(m,n)${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline17746|; $key = q/{_inline}$|hatx-betax|_2mathrelraisebox-.75ex$mathopsimlimits^textstyle;SPMlt;$bartheta${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline17662|; $key = q/{_inline}$kappa_infty(B)${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18099|; $key = q/{_inline}$t_v${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline12228|; $key = q/{tabular}ll|lName;SPMamp;Operation;SPMamp;PrefixeshlineP_SWAP;SPMamp;_inline$xleftrightarrowy$_inline;SPMamp;S,D,C,ZP_SCAL;SPMamp;_inline$xleftarrowalphax$_inline;SPMamp;S,D,C,Z,CS,ZDP_COPY;SPMamp;_inline$yleftarrowx$_inline;SPMamp;S,D,C,ZP_AXPY;SPMamp;_inline$yleftarrowalphax+y$_inline;SPMamp;S,D,C,ZP_DOT;SPMamp;_inline$dotleftarrowx^Ty$_inline;SPMamp;S,DP_DOTU;SPMamp;_inline$dotuleftarrowx^Ty$_inline;SPMamp;C,ZP_DOTC;SPMamp;_inline$dotcleftarrowx^Hy$_inline;SPMamp;C,ZP_NRM2;SPMamp;_inline$norm2leftarrow|x|_2$_inline;SPMamp;S,D,SC,DZP_ASUM;SPMamp;_inline$asumleftarrow|re(x)|_1+|im(x)|_1$_inline;SPMamp;S,D,SC,DZP_AMAX;SPMamp;_inline$indxleftarrow1^st~kni|Re(x_k)|+|Im(x_k)|$_inline;SPMamp;S,D,C,Z;SPMamp;111_inline$=max(|Re(x_i)|+|Im(x_i)|)=amax$_inline;SPMamp;P_GEMV;SPMamp;_inline$yleftarrowalphaop(A)x+betay,op(A)=A,A^T,A^H,A-mtimesn$_inline;SPMamp;S,D,C,ZP_HEMV;SPMamp;_inline$yleftarrowalphaAx+betay$_inline;SPMamp;C,ZP_SYMV;SPMamp;_inline$yleftarrowalphaAx+betay$_inline;SPMamp;S,DP_TRMV;SPMamp;_inline$xleftarrowAx,xleftarrowA^Tx,xleftarrowA^Hx,$_inline;SPMamp;S,D,C,ZP_TRSV;SPMamp;_inline$xleftarrowalphaA^-1x,xleftarrowalphaA^-Tx,xleftarrowalphaA^-Hx,$_inline;SPMamp;S,D,C,ZP_GER;SPMamp;_inline$Aleftarrowalphaxy^T+A,A-mtimesn$_inline;SPMamp;S,DP_GERU;SPMamp;_inline$Aleftarrowalphaxy^T+A,A-mtimesn$_inline;SPMamp;C,ZP_GERC;SPMamp;_inline$Aleftarrowalphaxy^H+A,A-mtimesn$_inline;SPMamp;C,ZP_HER;SPMamp;_inline$Aleftarrowalphaxx^H+A$_inline;SPMamp;C,ZP_HER2;SPMamp;_inline$Aleftarrowalphaxy^H+y(alphax)^H+A$_inline;SPMamp;C,ZP_SYR;SPMamp;_inline$Aleftarrowalphaxx^T+A$_inline;SPMamp;S,DP_SYR2;SPMamp;_inline$Aleftarrowalphaxy^T+alphayx^T+A$_inline;SPMamp;S,DP_GEMM;SPMamp;_inline$Cleftarrowalphaop(A)op(B)+betaC,op(X)=X,X^T,X^H,C-mtimesn$_inline;SPMamp;S,D,C,ZP_SYMM;SPMamp;_inline$CleftarrowalphaAB+betaC,CleftarrowalphaBA+betaC,C-mtimesn,A=A^T$_inline;SPMamp;S,D,C,ZP_HEMM;SPMamp;_inline$CleftarrowalphaAB+betaC,CleftarrowalphaBA+betaC,C-mtimesn,A=A^H$_inline;SPMamp;C,ZP_SYRK;SPMamp;_inline$CleftarrowalphaAA^T+betaC,CleftarrowalphaA^TA+betaC,C-ntimesn$_inline;SPMamp;S,D,C,ZP_HERK;SPMamp;_inline$CleftarrowalphaAA^H+betaC,CleftarrowalphaA^HA+betaC,C-ntimesn$_inline;SPMamp;C,ZP_SYR2K;SPMamp;_inline$CleftarrowalphaAB^T+alphaBA^T+betaC,CleftarrowalphaA^TB+alphaB^TA+betaC,C-ntimesn$_inline;SPMamp;S,D,C,ZP_HER2K;SPMamp;_inline$CleftarrowalphaAB^H+baralphaBA^H+betaC,CleftarrowalphaA^HB+baralphaB^HA+betaC,C-ntimesn$_inline;SPMamp;C,ZP_TRAN;SPMamp;_inline$CleftarrowbetaC+alphaA^T,A-ntimesm,C-mtimesn$_inline;SPMamp;S,DP_TRANU;SPMamp;_inline$CleftarrowbetaC+alphaA^T,A-ntimesm,C-mtimesn$_inline;SPMamp;C,ZP_TRANC;SPMamp;_inline$CleftarrowbetaC+alphaA^H,A-ntimesm,C-mtimesn$_inline;SPMamp;C,ZP_TRMM;SPMamp;_inline$Bleftarrowalphaop(A)B,BleftarrowalphaBop(A),op(A)=A,A^T,A^H,B-mtimesn$_inline;SPMamp;S,D,C,ZP_TRSM;SPMamp;_inline$Bleftarrowalphaop(A^-1)B,BleftarrowalphaBop(A^-1),op(A)=A,A^T,A^H,B-mtimesn$_inline;SPMamp;S,D,C,Z{tabular}/; $cached_env_img{$key} = q|tabular6927|; $key = q/{_inline}$p^th${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline15153|; $key = q/{figure}[htb]centerlinehboxpsfigfigure=mat9.ps,height=1.6in,width=1.8inhspace*1.0inpsfigfigure=grid9.ps,height=1.6in,width=1.8in{figure}/; $cached_env_img{$key} = q|figure2562|; $key = q/{_inline}$kappa_p(A)equiv|A|_pcdot|A^-1|_p${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline17542|; $key = q/{_inline}$N~=~16${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline14797|; $key = q/{_inline}$hatSigma${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline14247|; $key = q/{_inline}$hatr${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18362|; $key = q/{table}[htbp]centeringtabular||l|l||hlineScope;SPMamp;MeaninghlineRow;SPMamp;Allprocessesinaprocessrowparticipate.hlineColumn;SPMamp;Allprocessesinaprocesscolumnparticipate.hlineAll;SPMamp;Allprocessesintheprocessgridparticipate.hlinetabular{table}/; $cached_env_img{$key} = q|table2322|; $key = q/{displaymath}maxleft(frac|E|_infty|A|_infty,frac|f|_infty|b|_inftyright){displaymath}/; $cached_env_img{$key} = q|displaymath18206|; $key = q/{table}[htb]labeltableofldascentertabular||c|c|c|c||hlineProcessCoordinates;SPMamp;LLD_;SPMamp;LOC_inline$_r$_inline(M_);SPMamp;LOC_inline$_c$_inline(N_)hline(0,0);SPMamp;5;SPMamp;5;SPMamp;4(0,1);SPMamp;5;SPMamp;5;SPMamp;3(0,2);SPMamp;5;SPMamp;5;SPMamp;2(1,0);SPMamp;4;SPMamp;4;SPMamp;4(1,1);SPMamp;4;SPMamp;4;SPMamp;3(1,2);SPMamp;4;SPMamp;4;SPMamp;2hlinetabularcenter{table}/; $cached_env_img{$key} = q|table2576|; $key = q/{displaymath}A=PLU,{displaymath}/; $cached_env_img{$key} = q|displaymath12933|; $key = q/{displaymath}B^-1A=Z^T(T^-1R){displaymath}/; $cached_env_img{$key} = q|displaymath13661|; $key = q/{displaymath}AleftarrowA-UX^T-XU^T,{displaymath}/; $cached_env_img{$key} = q|displaymath16854|; $key = q/{displaymath}frac|x-hatx|_infty|x|_inftymathrelraisebox-.75ex_inline$mathopsimlimits^textstyle;SPMlt;$_inline2cdotomega_inftycdotkappa_infty(A)=ttERRBD;.{displaymath}/; $cached_env_img{$key} = q|displaymath18208|; $key = q/{_inline}$eta${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18027|; $key = q/{_inline}${-x}${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline17842|; $key = q/{}fboxparboxtextwidthTheroyaltiesfromthesalesofthisbookarebeingplacedinafundtohelpstudentsattSIAMmeetingsandotherSIAM-relatedactivities.ThisfundisadministeredbySIAMandqualifiedindividualsareencouragedtowritedirectlytoSIAMforguidelines.{}/; $cached_env_img{$key} = q|tex2html_wrap12059|; $key = q/{tabular}[t]lmulticolumn1lLargebfCInterface[0.28in]underlinermlargeInitialization197int*underlinemypnum,int*underlinenprocs)198int*underlinemypnum,int*underlinenprocs)199intictxt,intwhat,int*underlineval)200intictxt,intwhat,int*val)201int*underlineictxt,202intnprow,intnpcol)203int*underlineictxt,int*pmap,intldpmap,intnprow,intnpcol)underlinermlargeDestruction204intictxt,intwait)205intictxt)206intictxt,interrornum)207intdoneflag)underlinermlargeSingvoidC_inline$Box$_inlinegesd2d(intictxt,208intm,intn,TYPE*A,intlda,209intrdest,intcdest)voidC_inline$Box$_inlinegebs2d(intictxt,char*scope,char*top,210intm,intn,TYPE*A,intlda)voidC_inline$Box$_inlinetrsd2d(intictxt,char*uplo,char*diag,211intm,intn,TYPE*A,intlda,212intrdest,intcdest)voidC_inline$Box$_inlinetrbs2d(intictxt,char*scope,char*top,char*uplo,char*diag,intm,intn,TYPE*A,intlda)underlinermlargeReceivingvoidC_inline$Box$_inlinegerv2d(intictxt,213intm,intn,TYPE*underlineA,intlda,214intrsrc,intcsrc)voidC_inline$Box$_inlinegebr2d(intictxt,char*scope,char*top,215intm,intn,TYPE*underlineA,intlda,216intrsrc,intcsrc)voidC_inline$Box$_inlinetrrv2d(intictxt,char*uplo,char*diag,217intm,intn,TYPE*underlineA,intlda,218intrsrc,intcsrc)voidC_inline$Box$_inlinetrbr2d(intictxt,char*scope,char*top,char*uplo,char*diag,intm,intn,TYPE*underlineA,intlda,219intrsrc,intcsrc)underlinelargeCombineOperationsvoidC_inline$Box$_inlinegamx2d(intictxt,char*scope,char*top,intm,intn,TYPE*underlineA,intlda,int*underlineRA,int*underlineCA,220intRCflag,intrdest,intcdest)voidC_inline$Box$_inlinegamn2d(intictxt,char*scope,char*top,intm,intn,TYPE*underlineA,intlda,int*underlineRA,int*underlineCA,221intRCflag,intrdest,intcdest)voidC_inline$Box$_inlinegsum2d(intictxt,char*scope,char*top,intm,intn,TYPE*underlineA,intlda,222intrdest,intcdest){tabular}/; $cached_env_img{$key} = q|tabular7078|; $key = q/{_inline}$ttRCOND=4.712cdot10^-2${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18499|; $key = q/{displaymath}left|fracrmalg(z)-f(z)f(z)right|approxleft|fracf'(z)cdotzf(z)right|cdotleft|fracdeltazright|equivkappa(f,z)cdotleft|fracdeltazright|.{displaymath}/; $cached_env_img{$key} = q|displaymath17998|; $key = q/{_inline}$y=L^Hz${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline14341|; $key = q/{_inline}$hatalpha${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline17436|; $key = q/{_inline}$A=QTQ^T${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline13866|; $key = q/{_inline}$U_2${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline14183|; $key = q/{_inline}$4.504cdot10^-8${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18302|; $key = q/{displaymath}.9frac|hatalpha-alpha||hatalpha|leqfrac|hatalpha-alpha||alpha|leq1.1frac|hatalpha-alpha||hatalpha|;;,{displaymath}/; $cached_env_img{$key} = q|displaymath17417|; $key = q/{displaymath}Aleftarrow(I-VT^TV^T)(A-XV^T).{displaymath}/; $cached_env_img{$key} = q|displaymath16856|; $key = q/{_inline}$r=min(m,n)${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline14127|; $key = q/{_inline}$|hatx|_2=1${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline17646|; $key = q/{figure}[htb]centerlinepsfigfigure=mata9.ps,height=1.5in,width=2.0inlabelfig:mata9{figure}/; $cached_env_img{$key} = q|figure705|; $key = q/{_inline}$_N${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline12218|; $key = q/{_inline}$Q^HQ=I${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline13214|; $key = q/{_inline}$kappa_infty(A)=|A|_inftycdot|A^-1|_infty${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18081|; $key = q/{_inline}$hatc_1${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline13502|; $key = q/{_inline}$rho=|Ahatx-b|_2${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18533|; $key = q/{_inline}$betax${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline17580|; $key = q/{displaymath}omega_c=max_ifrac|r_i|(|A|cdot|hatx|+|b|)_i;.{displaymath}/; $cached_env_img{$key} = q|displaymath18210|; $key = q/{displaymath}|fl(apmb)-(apmb)|leqepsiloncdot(|a|+|b|).{displaymath}/; $cached_env_img{$key} = q|displaymath17198|; $key = q/{_inline}$|deltahatV|leqp(m,n)epsilon${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18969|; $key = q/{_inline}$_MM${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline12090|; $key = q/{_inline}$0,1,ldots,P-1${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline14493|; $key = q/{displaymath}AleftarrowA-UX^T-YV^T,{displaymath}/; $cached_env_img{$key} = q|displaymath16855|; $key = q/{displaymath}max_i,j,kleft(frac|e_ij||a_ij|,frac|f_k||b_k|right){displaymath}/; $cached_env_img{$key} = q|displaymath18209|; $key = q/{displaymath}maxleft(frac|E|_2|A|_2,frac|f|_2|b|_2right)leqp(n)epsilon{displaymath}/; $cached_env_img{$key} = q|displaymath18466|; $key = q/{displaymath}|fl(sqrta)-(sqrta)|leqepsiloncdot|sqrta|;.{displaymath}/; $cached_env_img{$key} = q|displaymath17197|; $key = q/{_inline}$~times~${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline14845|; $key = q/{displaymath}|hatsigma_i-sigma_i|leqttSERRBD;;.{displaymath}/; $cached_env_img{$key} = q|displaymath18830|; $key = q/{_inline}$p(n)leq10n${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18240|; $key = q/{_inline}$rmalg(z)=f(z+delta)${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18007|; $key = q/{table}[h]labeltableVectorNormFpqcenterValuesof_inline$f_pq(n)$_inlinesuchthat_inline$|x|_pleqf_pq(n)|x|_q$_inline,where_inline$x$_inlineisan_inline$n$_inline-vectortabular||cc|c|c|c||hline;SPMamp;;SPMamp;multicolumn3|c||_inline$q$_inline;SPMamp;;SPMamp;multicolumn1|c1;SPMamp;multicolumn1c2;SPMamp;multicolumn1c||_inline$infty$_inlinehline;SPMamp;1;SPMamp;1;SPMamp;_inline$sqrtn$_inline;SPMamp;_inline$n$_inlinecline3-5_inline$p$_inline;SPMamp;2;SPMamp;1;SPMamp;1;SPMamp;_inline$sqrtn$_inlinecline3-5;SPMamp;_inline$infty$_inline;SPMamp;1;SPMamp;1;SPMamp;1hlinetabularcenter{table}/; $cached_env_img{$key} = q|table4862|; $key = q/{_inline}$1timesP${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline15088|; $key = q/{_inline}$b-Ahatx${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18364|; $key = q/{_inline}$lambdahat_i${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18698|; $key = q/{_inline}$epsilon=2^-24=5.960cdot10^-8${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18304|; $key = q/{_inline}$A^H${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline12724|; $key = q/{tabular}llmulticolumn2cbfExpertDriverRoutinesforLinearEquations[0.1in]MatrixType;SPMamp;RoutinehlineGeneral;SPMamp;22;SPMamp;23;SPMamp;24;SPMamp;25;SPMamp;26;SPMamp;27SymmetricHermitian;SPMamp;28PositiveDefinite;SPMamp;29;SPMamp;30;SPMamp;31;SPMamp;32;SPMamp;33;SPMamp;{tabular}/; $cached_env_img{$key} = q|tabular6887|; $key = q/{displaymath}frac|x-hatx|_infty|hatx|_inftyleqttFERR=frac|,|A^-1|(|hatr|+nepsilon(|A|cdot|hatx|+|b|))|_infty|hatx|_infty;;.{displaymath}/; $cached_env_img{$key} = q|displaymath18212|; $key = q/{_inline}$sigma_max(A)${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline17806|; $key = q/{displaymath}hatx=left(arrayc1.19911arrayright)cdot(1.1^2+99^2+11^2)^-12;rmso;|hatx|_2=1;;rmand;;x=left(arrayc11009arrayright){displaymath}/; $cached_env_img{$key} = q|displaymath17416|; $key = q/{_inline}$A_k+b:N,k+b:N${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline14595|; $key = q/{_inline}$MB_B${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline15281|; $key = q/{_inline}$hatsigma_1geqcdotsgeqhatsigma_r${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18870|; $key = q/{_inline}$ttORTOL=10^-3${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18744|; $key = q/{_inline}${hatz_i,,,iincalI}${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18722|; $key = q/{_inline}$max_i|lambda_i|=|A|_2${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18702|; $key = q/{_inline}$aoplusb${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline17240|; $key = q/{_inline}$Q^H${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline13302|; $key = q/{_inline}$v_1,v_2,ldots,v_K${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline16800|; $key = q/{_inline}$yleftarrowy+A,x${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline16483|; $key = q/{table}[htb]labeltab:pblas3minipagelinewidthcentertabular||l|c|c|r|r|r|r|r||hline;SPMamp;Process;SPMamp;Block;SPMamp;multicolumn5c||Valuesof_inline$N$_inlinecline4-8;SPMamp;grid;SPMamp;size;SPMamp;2000;SPMamp;4000;SPMamp;6000;SPMamp;8000;SPMamp;10000hlineCrayT3EfootnoteAScaLAPACKprocessisrunningoneachnode.EachnodeconsistsofonecomputationalDECAlphaEV5processor,whichisusedbytheBLAS.;SPMamp;_inline$2times2$_inline;SPMamp;32;SPMamp;1055;SPMamp;1070;SPMamp;1075;SPMamp;;SPMamp;PSGEMM;SPMamp;_inline$4times4$_inline;SPMamp;32;SPMamp;3630;SPMamp;4005;SPMamp;4258;SPMamp;4171;SPMamp;4292;SPMamp;_inline$8times8$_inline;SPMamp;32;SPMamp;13456;SPMamp;14287;SPMamp;15419;SPMamp;15858;SPMamp;16755hlineIBMSP2footnoteAScaLAPACKprocessisrunningoneachnode.EachnodeconsistsofonecomputationalIBMPOWER2590processor,whichisusedbytheBLAS.;SPMamp;_inline$2times2$_inline;SPMamp;50;SPMamp;755;SPMamp;;SPMamp;;SPMamp;;SPMamp;PDGEMM;SPMamp;_inline$4times4$_inline;SPMamp;50;SPMamp;2514;SPMamp;2850;SPMamp;3040;SPMamp;;SPMamp;;SPMamp;_inline$8times8$_inline;SPMamp;50;SPMamp;6205;SPMamp;8709;SPMamp;9862;SPMamp;10468;SPMamp;10774hlineIntelXPSMPParagonfootnoteAScaLAPACKprocessisrunningoneachnode.EachnodeconsistsoftwocomputationalInteli860XPRISCprocessors,whichareusedbythemulti-threadedBLAS.;SPMamp;_inline$2times2$_inline;SPMamp;32;SPMamp;330;SPMamp;;SPMamp;;SPMamp;;SPMamp;PDGEMM;SPMamp;_inline$4times4$_inline;SPMamp;32;SPMamp;1233;SPMamp;1281;SPMamp;1334;SPMamp;;SPMamp;;SPMamp;_inline$8times8$_inline;SPMamp;32;SPMamp;4496;SPMamp;4864;SPMamp;5030;SPMamp;5103;SPMamp;5257hlineClusterofSunUltra2sfootnoteAScaLAPACKprocessisrunningoneachnode.EachnodeconsistsoftwocomputationalUltraSPARC-1processors,butonlyoneprocessorisusedbytheBLAS.;SPMamp;_inline$2times2$_inline;SPMamp;64;SPMamp;406;SPMamp;442;SPMamp;;SPMamp;;SPMamp;PDGEMM;SPMamp;_inline$2times4$_inline;SPMamp;64;SPMamp;529;SPMamp;770;SPMamp;851;SPMamp;;SPMamp;;SPMamp;_inline$3times4$_inline;SPMamp;64;SPMamp;500;SPMamp;940;SPMamp;1119;SPMamp;1220;SPMamp;hlineBerkeleyNOWfootnoteAScaLAPACKprocessisrunningoneachnode.EachnodeconsistsofonecomputationalUltraSPARC-1processor,whichisusedbytheBLAS.;SPMamp;_inline$2times2$_inline;SPMamp;32;SPMamp;463;SPMamp;470;SPMamp;;SPMamp;;SPMamp;PDGEMM;SPMamp;_inline$2times4$_inline;SPMamp;32;SPMamp;496;SPMamp;524;SPMamp;;SPMamp;;SPMamp;;SPMamp;_inline$4times4$_inline;SPMamp;32;SPMamp;926;SPMamp;1031;SPMamp;632;SPMamp;1754;SPMamp;;SPMamp;_inline$4times8$_inline;SPMamp;32;SPMamp;2490;SPMamp;2822;SPMamp;3316;SPMamp;3306;SPMamp;3450;SPMamp;_inline$8times8$_inline;SPMamp;32;SPMamp;4130;SPMamp;5457;SPMamp;6041;SPMamp;6360;SPMamp;6647hlinetabularcenterminipage{table}/; $cached_env_img{$key} = q|table3877|; $key = q/{displaymath}A=Qleft(arraycR0arrayright),quadmboxif_inline$mgen$_inline,{displaymath}/; $cached_env_img{$key} = q|displaymath13230|; $key = q/{displaymath}A^-1approxleft(arrayccc-.667;SPMamp;-1.333;SPMamp;1-.667;SPMamp;3.667;SPMamp;-21;SPMamp;-2;SPMamp;1arrayright);;rmand;;kappa_infty(A)=158.33;;.{displaymath}/; $cached_env_img{$key} = q|displaymath17412|; $key = q/{_inline}$NB_A~=~3${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline15352|; $key = q/{_inline}$int(fracN_ANB_A)=K${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline15251|; $key = q/{_inline}$max_i,j,kleft(frac|e_ij||a_ij|,frac|f_k||b_k|right)${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18330|; $key = q/{_inline}$R_12=0${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline13551|; $key = q/{_inline}$frac43n^3${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18123|; $key = q/{table}[htb]labeltab:luperfminipagelinewidthcentertabular||l|c|c|r|r|r|r|r||hline;SPMamp;Process;SPMamp;Block;SPMamp;multicolumn5c||Valuesof_inline$N$_inlinecline4-8;SPMamp;Grid;SPMamp;Size;SPMamp;2000;SPMamp;5000;SPMamp;7500;SPMamp;10000;SPMamp;15000hlineCrayT3EfootnoteAScaLAPACKprocessisrunningoneachnode.EachnodeconsistsofonecomputationalDECAlphaEV5processor,whichisusedbytheBLAS.;SPMamp;_inline$1times4$_inline;SPMamp;32;SPMamp;702;SPMamp;884;SPMamp;932;SPMamp;;SPMamp;PSGESV;SPMamp;_inline$2times8$_inline;SPMamp;32;SPMamp;1608;SPMamp;2680;SPMamp;3218;SPMamp;3356;SPMamp;3602;SPMamp;_inline$4times16$_inline;SPMamp;32;SPMamp;2419;SPMamp;6912;SPMamp;9028;SPMamp;10299;SPMamp;12547hlineIBMSP2footnoteAScaLAPACKprocessisrunningoneachnode.EachnodeconsistsofonecomputationalIBMPOWER2590processor,whichisusedbytheBLAS.;SPMamp;_inline$1times4$_inline;SPMamp;50;SPMamp;421;SPMamp;603;SPMamp;;SPMamp;;SPMamp;PDGESV;SPMamp;_inline$2times8$_inline;SPMamp;50;SPMamp;722;SPMamp;1543;SPMamp;1903;SPMamp;2149;SPMamp;;SPMamp;_inline$4times16$_inline;SPMamp;50;SPMamp;924;SPMamp;3017;SPMamp;4295;SPMamp;5596;SPMamp;7057hlineIntelXPSMPParagonfootnoteAScaLAPACKprocessisrunningoneachnode.EachnodeconsistsoftwocomputationalInteli860XPRISCprocessors,whichareusedbythemulti-threadedBLAS.;SPMamp;_inline$1times4$_inline;SPMamp;32;SPMamp;212;SPMamp;282;SPMamp;;SPMamp;;SPMamp;PDGESV;SPMamp;_inline$2times8$_inline;SPMamp;32;SPMamp;460;SPMamp;865;SPMamp;1010;SPMamp;;SPMamp;;SPMamp;_inline$4times16$_inline;SPMamp;32;SPMamp;721;SPMamp;2084;SPMamp;2837;SPMamp;3344;SPMamp;3963hlineIntelASCIRedSupercomputerfootnoteAScaLAPACKprocessisrunningoneachnode.EachnodeconsistsoftwocomputationalIntelPentiumProprocessors,whichareusedbythemulti-threadedBLAS.;SPMamp;_inline$1times4$_inline;SPMamp;32;SPMamp;566;SPMamp;;SPMamp;;SPMamp;;SPMamp;PDGESV;SPMamp;_inline$2times8$_inline;SPMamp;32;SPMamp;1379;SPMamp;2497;SPMamp;;SPMamp;;SPMamp;;SPMamp;_inline$4times8$_inline;SPMamp;32;SPMamp;1912;SPMamp;4030;SPMamp;;SPMamp;;SPMamp;;SPMamp;_inline$4times16$_inline;SPMamp;32;SPMamp;2389;SPMamp;6329;SPMamp;;SPMamp;9731;SPMamp;hlineClusterofSunUltra2sfootnoteAScaLAPACKprocessisrunningoneachnode.EachnodeconsistsoftwocomputationalUltraSPARC-1processors,butonlyoneprocessorisusedbytheBLAS.;SPMamp;_inline$2times2$_inline;SPMamp;64;SPMamp;203;SPMamp;340;SPMamp;380;SPMamp;;SPMamp;PDGESV;SPMamp;_inline$2times4$_inline;SPMamp;64;SPMamp;196;SPMamp;499;SPMamp;618;SPMamp;689;SPMamp;;SPMamp;_inline$2times6$_inline;SPMamp;64;SPMamp;212;SPMamp;608;SPMamp;816;SPMamp;934;SPMamp;hlineBerkeleyNOWfootnoteAScaLAPACKprocessisrunningoneachnode.EachnodeconsistsofonecomputationalUltraSPARC-1processor,whichisusedbytheBLAS.;SPMamp;_inline$1times4$_inline;SPMamp;32;SPMamp;350;SPMamp;;SPMamp;;SPMamp;;SPMamp;PDGESV;SPMamp;_inline$1times8$_inline;SPMamp;32;SPMamp;560;SPMamp;744;SPMamp;;SPMamp;;SPMamp;;SPMamp;_inline$2times8$_inline;SPMamp;32;SPMamp;811;SPMamp;1310;SPMamp;1472;SPMamp;1547;SPMamp;;SPMamp;_inline$4times8$_inline;SPMamp;32;SPMamp;948;SPMamp;1913;SPMamp;2149;SPMamp;2324;SPMamp;2436;SPMamp;_inline$4times16$_inline;SPMamp;32;SPMamp;1171;SPMamp;3091;SPMamp;3937;SPMamp;4263;SPMamp;4560hlinetabularcenterminipage{table}/; $cached_env_img{$key} = q|table3947|; $key = q/{_inline}$V_1${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline14163|; $key = q/{_inline}$Pgeq9${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline17053|; $key = q/{displaymath}theta(hatcalS,calS)=arccossigma_min(hatX^HX);;.{displaymath}/; $cached_env_img{$key} = q|displaymath17418|; $key = q/{_inline}$omega_inftyleqp(n)cdotepsilon${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18288|; $key = q/{_inline}$B=LL^T${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline14317|; $key = q/{_inline}$T=SLambdaS^T${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline13884|; $key = q/{table}[htb]labeltab:suggcentertabular||l|c|l||hlineDriver;SPMamp;ProcessGrid;SPMamp;CommentshlinePxGESV;SPMamp;_inline$P_rllP_c$_inline;SPMamp;1Ddistributionforsmall_inline$P$_inline(_inline$leq8$_inline)PxPOSV;SPMamp;_inline$P_r=P_c$_inline;SPMamp;PxGELS;SPMamp;_inline$P_rllP_c$_inlineif_inline$MgeN$_inline;SPMamp;1Ddistributionforsmall_inline$P$_inline(_inline$leq8$_inline);SPMamp;_inline$P_rggP_c$_inlineotherwise;SPMamp;PxSYEVX;SPMamp;_inline$P_rleqP_c$_inline;SPMamp;_inline$lcm(P_r,P_c);SPMlt;10timesmax(P_r,P_c)$_inlinePxSYEV;SPMamp;_inline$P_rleqP_c$_inline;SPMamp;_inline$lcm(P_r,P_c);SPMlt;10timesmax(P_r,P_c)$_inlinePxGESVD;SPMamp;_inline$P_r=P_c$_inline;SPMamp;_inline$lcm(P_r,P_c);SPMlt;10timesmax(P_r,P_c)$_inlinePxLAHQR;SPMamp;_inline$P_r=P_c$_inline;SPMamp;hlinetabularcenter{table}/; $cached_env_img{$key} = q|table4271|; $key = q/{_inline}$N_A${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline14833|; $key = q/{_inline}$Ax=lambdax${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline17574|; $key = q/{_inline}$NBgelceilNPrceil${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline15090|; $key = q/{_inline}$|Ax|_pleq|A|_pcdot|x|_p${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline17702|; $key = q/{_inline}$a_ij${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18188|; $key = q/{_inline}$-(100asti+j)${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline16134|; $key = q/{displaymath}AB^-1=(RT^-1)Z^T{displaymath}/; $cached_env_img{$key} = q|displaymath13752|; $key = q/{_inline}$|x|_infty${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline17462|; $key = q/{_inline}$ZLambdaZ^H${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline13906|; $key = q/{_inline}$(K+1,0),ldots,(P_r-1,0)${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline15309|; $key = q/{table}[htb]labeltab:node2centertabular||l|c|c||hline;SPMamp;IntelXPS;SPMamp;IntelASCIRed;SPMamp;MPParagon;SPMamp;SupercomputerhlineProcessor;SPMamp;i860XP;SPMamp;IntelPentiumProhlineClockspeed(MHz);SPMamp;50;SPMamp;200hlineProcessorspernode;SPMamp;2;SPMamp;2hlineMemorypernode(MB);SPMamp;64;SPMamp;128hlineOperatingsystem;SPMamp;R1.4-2;SPMamp;AlphaReleasehlineBLAS;SPMamp;BasicMathLibrary;SPMamp;AlphaRelease;SPMamp;Software(Release5.0);SPMamp;hlineBLACS;SPMamp;NXBLACS1.0;SPMamp;NXBLACS1.0hlineCommunicationSoftware;SPMamp;NX;SPMamp;NXonPortalshlineCcompiler;SPMamp;icc(R5.0.2);SPMamp;pgccRel1.3-4ahlineCflags;SPMamp;-O4;SPMamp;-OhlineFortrancompiler;SPMamp;if77(R5.0.2);SPMamp;pgf77Rel1.3-4ahlineFortranflags;SPMamp;-O4;SPMamp;-OhlinePrecision;SPMamp;double(64-bit);SPMamp;double(64-bit)hlinetabularcenter{table}/; $cached_env_img{$key} = q|table3773|; $key = q/{_inline}$lfloorktcrfloor${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline14569|; $key = q/{_inline}$v_1${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline14451|; $key = q/{_inline}$ast${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline15263|; $key = q/{displaymath}theta(hatcalS,calS)mathrelraisebox-.75ex_inline$mathopsimlimits^textstyle;SPMlt;$_inlinefracp(m,n)epsilon|A|_2gap_calI.{displaymath}/; $cached_env_img{$key} = q|displaymath18943|; $key = q/{_inline}$max_1leqileqn|x_i|${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline17460|; $key = q/{_inline}$cos(theta)${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18549|; $key = q/{_inline}${hatx}${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline17616|; $key = q/{displaymath}A=PU^TDUP^T;;rmor;;A=PLDL^TP^Tmbox(inthesymmetriccase){displaymath}/; $cached_env_img{$key} = q|displaymath12940|; $key = q/{_inline}$ttERRBD=1.5cdot10^-2${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18250|; $key = q/{_inline}$Sigma${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline12820|; $key = q/{_inline}$|x-hatx|_infty|hatx|_infty${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18358|; $key = q/{_inline}$R_12${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline13555|; $key = q/{_inline}$theta(gammahatx,betax)=.0209${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline17632|; $key = q/{table}[h]labeltab:bcblockmap0centertabular||l|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c||hline_inline$J$_inline;SPMamp;1;SPMamp;2;SPMamp;3;SPMamp;4;SPMamp;5;SPMamp;6;SPMamp;7;SPMamp;8;SPMamp;9;SPMamp;10;SPMamp;11;SPMamp;12;SPMamp;13;SPMamp;14;SPMamp;15;SPMamp;16hline_inline$p$_inline;SPMamp;1;SPMamp;1;SPMamp;1;SPMamp;1;SPMamp;1;SPMamp;1;SPMamp;1;SPMamp;1;SPMamp;0;SPMamp;0;SPMamp;0;SPMamp;0;SPMamp;0;SPMamp;0;SPMamp;0;SPMamp;0_inline$x$_inline;SPMamp;1;SPMamp;2;SPMamp;3;SPMamp;4;SPMamp;5;SPMamp;6;SPMamp;7;SPMamp;8;SPMamp;1;SPMamp;2;SPMamp;3;SPMamp;4;SPMamp;5;SPMamp;6;SPMamp;7;SPMamp;8hlinetabularcenter{table}/; $cached_env_img{$key} = q|table2738|; $key = q/{_inline}$NB=tc=lceilNPrceil${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline14603|; $key = q/{_inline}$sub(~A~)${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline12098|; $key = q/{displaymath}R=bordermatrix;SPMamp;n-m;SPMamp;mcrm;SPMamp;0;SPMamp;R_12,quadmboxifquadmleqn,{displaymath}/; $cached_env_img{$key} = q|displaymath13748|; $key = q/{_inline}$P_r;SPMlt;P_c${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline17130|; $key = q/{table}[thb]labeltabgstcentertabular||l|c|l|l|l||hline;SPMamp;Typeof;SPMamp;multicolumn1c|Factorization;SPMamp;multicolumn1c|Reduction;SPMamp;multicolumn1c||Recoveryof;SPMamp;Problem;SPMamp;multicolumn1c|of_inline$B$_inline;SPMamp;;SPMamp;multicolumn1c||Eigenvectorshline1.;SPMamp;_inline$Az=lambdaBz$_inline;SPMamp;_inline$B=LL^T$_inline;SPMamp;_inline$C=L^-1AL^-T$_inline;SPMamp;_inline$z=L^-Ty$_inline;SPMamp;;SPMamp;_inline$B=U^TU$_inline;SPMamp;_inline$C=U^-TAU^-1$_inline;SPMamp;_inline$z=U^-1y$_inlinehline2.;SPMamp;_inline$ABz=lambdaz$_inline;SPMamp;_inline$B=LL^T$_inline;SPMamp;_inline$C=L^TAL$_inline;SPMamp;_inline$z=L^-Ty$_inline;SPMamp;;SPMamp;_inline$B=U^TU$_inline;SPMamp;_inline$C=UAU^T$_inline;SPMamp;_inline$z=U^-1y$_inlinehline3.;SPMamp;_inline$BAz=lambdaz$_inline;SPMamp;_inline$B=LL^T$_inline;SPMamp;_inline$C=L^TAL$_inline;SPMamp;_inline$z=Ly$_inline;SPMamp;;SPMamp;_inline$B=U^TU$_inline;SPMamp;_inline$C=UAU^T$_inline;SPMamp;_inline$z=U^Ty$_inlinehlinetabularcenter{table}/; $cached_env_img{$key} = q|table2129|; $key = q/{_inline}$A=QHQ^T${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline14044|; $key = q/{table}[thb]labeltabdrivelineqcentertabular||l|l|l|l|l|l||hlineTypeofmatrix;SPMamp;Operation;SPMamp;multicolumn2c|Singleprecision;SPMamp;multicolumn2c||Doubleprecisioncline3-6andstoragescheme;SPMamp;;SPMamp;Real;SPMamp;Complex;SPMamp;Real;SPMamp;Complexhlinegeneral;SPMamp;simpledriver;SPMamp;PSGESVindexPSGESV;SPMamp;PCGESVindexPCGESV;SPMamp;PDGESVindexPDGESV;SPMamp;PZGESVindexPZGESV(partialpivoting);SPMamp;expertdriver;SPMamp;PSGESVXindexPSGESVX;SPMamp;PCGESVXindexPCGESVX;SPMamp;PDGESVXindexPDGESVX;SPMamp;PZGESVXindexPZGESVXhlinegeneralband;SPMamp;simpledriver;SPMamp;PSGBSVindexPSGBSV;SPMamp;PCGBSVindexPCGBSV;SPMamp;PDGBSVindexPDGBSV;SPMamp;PZGBSVindexPZGBSV(partialpivoting);SPMamp;;SPMamp;;SPMamp;;SPMamp;;SPMamp;hlinegeneralband;SPMamp;simpledriver;SPMamp;PSDBSVindexPSDBSV;SPMamp;PCDBSVindexPCDBSV;SPMamp;PDDBSVindexPDDBSV;SPMamp;PZDBSVindexPZDBSV(nopivoting);SPMamp;;SPMamp;;SPMamp;;SPMamp;;SPMamp;hlinegeneraltridiagonal;SPMamp;simpledriver;SPMamp;PSDTSVindexPSDTSV;SPMamp;PCDTSVindexPCDTSV;SPMamp;PDDTSVindexPDDTSV;SPMamp;PZDTSVindexPZDTSV(nopivoting);SPMamp;;SPMamp;;SPMamp;;SPMamp;;SPMamp;hlinesymmetricHermitian;SPMamp;simpledriver;SPMamp;PSPOSVindexPSPOSV;SPMamp;PCPOSVindexPCPOSV;SPMamp;PDPOSVindexPDPOSV;SPMamp;PZPOSVindexPZPOSVpositivedefinite;SPMamp;expertdriver;SPMamp;PSPOSVXindexPSPOSVX;SPMamp;PCPOSVXindexPCPOSVX;SPMamp;PDPOSVXindexPDPOSVX;SPMamp;PZPOSVXindexPZPOSVXhlinesymmetricHermitian;SPMamp;simpledriver;SPMamp;PSPBSVindexPSPBSV;SPMamp;PCPBSVindexPCPBSV;SPMamp;PDPBSVindexPDPBSV;SPMamp;PZPBSVindexPZPBSVpositivedefiniteband;SPMamp;;SPMamp;;SPMamp;;SPMamp;;SPMamp;hlinesymmetricHermitian;SPMamp;simpledriver;SPMamp;PSPTSVindexPSPTSV;SPMamp;PCPTSVindexPCPTSV;SPMamp;PDPTSVindexPDPTSV;SPMamp;PZPTSVindexPZPTSVpositivedefinite;SPMamp;;SPMamp;;SPMamp;;SPMamp;;SPMamp;tridiagonal;SPMamp;;SPMamp;;SPMamp;;SPMamp;;SPMamp;hlinetabularcenter{table}/; $cached_env_img{$key} = q|table931|; $key = q/{displaymath}gap_calI=min_iincalIatopjnotincalI|lambda_i-lambda_j|{displaymath}/; $cached_env_img{$key} = q|displaymath18602|; $key = q/{_inline}$A^-1${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline13063|; $key = q/{_inline}$C_v,N^2${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline16256|; $key = q/{table}[htb]labeltab:node3centertabular||l|c|c||hline;SPMamp;Clusterof;SPMamp;Berkeley;SPMamp;SunUltra2's;SPMamp;NOWhlineProcessor;SPMamp;UltraSPARC-1;SPMamp;UltraSPARC-1hlineClockspeed(MHz);SPMamp;167;SPMamp;167hlineProcessorspernode;SPMamp;2;SPMamp;1hlineMemorypernode(MB);SPMamp;256;SPMamp;128hlineOperatingsystem;SPMamp;SunOS2.5.1;SPMamp;SunOS2.5.1hlineBLAS;SPMamp;;SPMamp;SunPerformanceLibraryv1.2hlineBLACS;SPMamp;_inline$1.1alpha$_inline;SPMamp;_inline$1.1alpha$_inlinehlineCommunicationSoftware;SPMamp;MPICH1.0.13;SPMamp;MPIonAM-IIhlineCcompiler;SPMamp;gcc(v2.7.2.1);SPMamp;cc(SunWorkshop4.2)hlineCflags;SPMamp;-O3;SPMamp;-dalign-xarch=v8plusa;SPMamp;;SPMamp;-xchip=ultra-fast;SPMamp;;SPMamp;-xtarget=native-xO5hlineFortrancompiler;SPMamp;f77(SC4.0);SPMamp;f77(SC4.0)hlineFortranflags;SPMamp;-O4-f;SPMamp;-dalign-xarch=v8plusa;SPMamp;;SPMamp;-xchip=ultra-fast;SPMamp;;SPMamp;-xtarget=native-xO5hlinePrecision;SPMamp;double(64-bit);SPMamp;double(64-bit)hlinetabularcenter{table}/; $cached_env_img{$key} = q|table3782|; $key = q/{_inline}$i^th${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline19434|; $key = q/{_inline}$2times2${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline14859|; $key = q/{_inline}$alpha${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline16235|; $key = q/{_inline}$|A|_2${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline17802|; $key = q/{_inline}$|delta|_infty${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18154|; $key = q/{displaymath}A=left(arrayccL;SPMamp;0arrayright)QquadLongleftrightarrowquadA^T=Q^Tleft(arraycL^T0arrayright).{displaymath}/; $cached_env_img{$key} = q|displaymath13357|; $key = q/{table}[htb]labeltab:lsperfminipagelinewidthcentertabular||l|c|c|r|r|r|r|r||hline;SPMamp;Process;SPMamp;Block;SPMamp;multicolumn5c||Valuesof_inline$N$_inlinecline4-8;SPMamp;Grid;SPMamp;Size;SPMamp;2000;SPMamp;5000;SPMamp;7500;SPMamp;10000;SPMamp;15000hlineCrayT3EfootnoteAScaLAPACKprocessisrunningoneachnode.EachnodeconsistsofonecomputationalDECAlphaEV5processor,whichisusedbytheBLAS.;SPMamp;_inline$1times4$_inline;SPMamp;32;SPMamp;568;SPMamp;650;SPMamp;659;SPMamp;692;SPMamp;PSGELS;SPMamp;_inline$2times8$_inline;SPMamp;32;SPMamp;1652;SPMamp;2290;SPMamp;2434;SPMamp;2594;SPMamp;2674;SPMamp;_inline$4times16$_inline;SPMamp;32;SPMamp;3433;SPMamp;7020;SPMamp;8072;SPMamp;9012;SPMamp;9746hlineIBMSP2footnoteAScaLAPACKprocessisrunningoneachnode.EachnodeconsistsofonecomputationalIBMPOWER2590processor,whichisusedbytheBLAS.;SPMamp;_inline$1times4$_inline;SPMamp;50;SPMamp;387;SPMamp;564;SPMamp;;SPMamp;;SPMamp;PDGELS;SPMamp;_inline$2times8$_inline;SPMamp;50;SPMamp;725;SPMamp;1506;SPMamp;1829;SPMamp;2041;SPMamp;;SPMamp;_inline$4times16$_inline;SPMamp;50;SPMamp;1157;SPMamp;3367;SPMamp;4715;SPMamp;5836;SPMamp;7182hlineIntelXPSMPParagonfootnoteAScaLAPACKprocessisrunningoneachnode.EachnodeconsistsoftwocomputationalInteli860XPRISCprocessors,whichareusedbythemulti-threadedBLAS.;SPMamp;_inline$1times4$_inline;SPMamp;32;SPMamp;201;SPMamp;255;SPMamp;;SPMamp;;SPMamp;PDGELS;SPMamp;_inline$2times8$_inline;SPMamp;32;SPMamp;528;SPMamp;825;SPMamp;898;SPMamp;930;SPMamp;;SPMamp;_inline$4times16$_inline;SPMamp;32;SPMamp;1004;SPMamp;2354;SPMamp;2937;SPMamp;3263;SPMamp;3598hlineClusterofSunUltra2sfootnoteAScaLAPACKprocessisrunningoneachnode.EachnodeconsistsoftwocomputationalUltraSPARC-1processors,butonlyoneprocessorisusedbytheBLAS.;SPMamp;_inline$2times2$_inline;SPMamp;64;SPMamp;206;SPMamp;281;SPMamp;318;SPMamp;;SPMamp;PDGELS;SPMamp;_inline$2times4$_inline;SPMamp;64;SPMamp;220;SPMamp;519;SPMamp;585;SPMamp;612;SPMamp;;SPMamp;_inline$2times6$_inline;SPMamp;64;SPMamp;241;SPMamp;616;SPMamp;794;SPMamp;905;SPMamp;hlineBerkeleyNOWfootnoteAScaLAPACKprocessisrunningoneachnode.EachnodeconsistsofonecomputationalUltraSPARC-1processor,whichisusedbytheBLAS.;SPMamp;_inline$1times4$_inline;SPMamp;32;SPMamp;238;SPMamp;363;SPMamp;;SPMamp;;SPMamp;PDGELS;SPMamp;_inline$1times8$_inline;SPMamp;32;SPMamp;482;SPMamp;635;SPMamp;682;SPMamp;;SPMamp;;SPMamp;_inline$2times8$_inline;SPMamp;32;SPMamp;836;SPMamp;1246;SPMamp;1351;SPMamp;1415;SPMamp;;SPMamp;_inline$4times8$_inline;SPMamp;32;SPMamp;1213;SPMamp;2026;SPMamp;2575;SPMamp;2433;SPMamp;;SPMamp;_inline$8times8$_inline;SPMamp;32;SPMamp;1513;SPMamp;3446;SPMamp;4296;SPMamp;4754;SPMamp;5242hlinetabularcenterminipage{table}/; $cached_env_img{$key} = q|table4001|; $key = q/{_inline}$y=L^Tz${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline14335|; $key = q/{_inline}$NB_A${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline14839|; $key = q/{_inline}$H=STS^T${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline14066|; $key = q/{table}[h]labeltabIEEEvaluescentertabular||l|l|l||hlineMachineParameter;SPMamp;SinglePrecision(32bits);SPMamp;DoublePrecision(64bits)hlineRelativemachineprecision_inline$epsilon$_inline;SPMamp;_inline$2^-24approx5.96cdot10^-8$_inline;SPMamp;_inline$2^-53approx1.11cdot10^-16$_inlinehspace*.25in=PxLAMCH(ICTXT,'E');SPMamp;;SPMamp;hlineUnderflowthreshold;SPMamp;_inline$2^-126approx1.18cdot10^-38$_inline;SPMamp;_inline$2^-1022approx2.23cdot10^-308$_inlinehspace*.25in=PxLAMCH(ICTXT,'U');SPMamp;;SPMamp;hlineOverflowthreshold;SPMamp;_inline$2^128(1-epsilon)approx3.40cdot10^38$_inline;SPMamp;_inline$2^1024(1-epsilon)approx1.79cdot10^308$_inlinehspace*.25in=PxLAMCH(ICTXT,'O');SPMamp;;SPMamp;hlinetabularcenter{table}/; $cached_env_img{$key} = q|table4445|; $key = q/{_inline}$|A|_pleqf_pq(m,n)|A|_q${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline17748|; $key = q/{displaymath}AP=Qleft(arrayccT_11;SPMamp;00;SPMamp;0arrayright)Z^T{displaymath}/; $cached_env_img{$key} = q|displaymath13544|; $key = q/{_inline}$Cy=lambday${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline14329|; $key = q/{_inline}$mlln${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline14233|; $key = q/{_inline}$R_21${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline13795|; $key = q/{_inline}$P_rtimes1${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline15275|; $key = q/{tabular}||c|c||c|c||c|c||hline_inline$i$_inline;SPMamp;_inline$lambda_i$_inline;SPMamp;ttEERRBD_inline$(i)$_inline;SPMamp;true_inline$|hatlambda_i-lambda_i|$_inline;SPMamp;ttZERRBD_inline$(i)$_inline;SPMamp;true_inline$theta(hatv_i,v_i)$_inlinehline1;SPMamp;_inline$-4.817cdot10^6$_inline;SPMamp;_inline$1.3$_inline;SPMamp;_inline$6.0cdot10^-3$_inline;SPMamp;_inline$1.7cdot10^-7$_inline;SPMamp;_inline$7.0cdot10^-9$_inline2;SPMamp;_inline$8.094cdot10^6$_inline;SPMamp;_inline$1.6$_inline;SPMamp;_inline$1.5$_inline;SPMamp;_inline$3.4cdot10^-7$_inline;SPMamp;_inline$3.3cdot10^-8$_inline3;SPMamp;_inline$1.219cdot10^7$_inline;SPMamp;_inline$1.9$_inline;SPMamp;_inline$4.5$_inline;SPMamp;_inline$3.4cdot10^-7$_inline;SPMamp;_inline$4.7cdot10^-8$_inlinehline{tabular}/; $cached_env_img{$key} = q|tabular6068|; $key = q/{_inline}$R=USigmaV^T${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline14227|; $key = q/{displaymath}left(arrayrx_3x_4x_7x_8arrayright)qquadleft(arrayrb_1b_2b_3b_4arrayright)=left(arrayrfrac120frac-12frac16arrayright).{displaymath}/; $cached_env_img{$key} = q|displaymath12395|; $key = q/{_inline}$t_m${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline12208|; $key = q/{displaymath}Rx=c_1,{displaymath}/; $cached_env_img{$key} = q|displaymath13235|; $key = q/{_inline}$|delta|${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18065|; $key = q/{_inline}$z_i${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18630|; $key = q/{displaymath}|lambdahat_i-lambda_i|leqp(n)cdotepsiloncdotkappa_2(H)cdotlambda_i,{displaymath}/; $cached_env_img{$key} = q|displaymath18603|; $key = q/{displaymath}A=QRquadmboxandquadB=QTZ,{displaymath}/; $cached_env_img{$key} = q|displaymath13656|; $key = q/{tabular}[t]p4.0inunderlinermlargeGlobalTopologies260261262263vdots264265266267268269270271272{tabular}/; $cached_env_img{$key} = q|tabular7149|; $key = q/{table}[htb]labeltabtypescentertabularllDB;SPMamp;generalband(diagonallydominant-like)DT;SPMamp;generaltridiagonal(diagonallydominant-like)GB;SPMamp;generalbandGE;SPMamp;general(i.e.,unsymmetric,insomecasesrectangular)GG;SPMamp;generalmatrices,generalizedproblem(i.e.,apairofgeneralmatrices)HE;SPMamp;(complex)HermitianOR;SPMamp;(real)orthogonalPB;SPMamp;symmetricorHermitianpositivedefinitebandPO;SPMamp;symmetricorHermitianpositivedefinitePT;SPMamp;symmetricorHermitianpositivedefinitetridiagonalST;SPMamp;(real)symmetrictridiagonalSY;SPMamp;symmetricTR;SPMamp;triangular(orinsomecasesquasi-triangular)TZ;SPMamp;trapezoidalUN;SPMamp;(complex)unitarytabularcenter{table}/; $cached_env_img{$key} = q|table891|; $key = q/{tabular}llValue;SPMamp;Meaninghlinett`L';SPMamp;Multiplygeneraldistributedmatrixbysymmetricortriangular;SPMamp;distributedmatrixontheleft.tt`R';SPMamp;Multiplygeneraldistributedmatrixbysymmetricortriangular;SPMamp;distributedmatrixontheright.{tabular}/; $cached_env_img{$key} = q|tabular3422|; $key = q/{displaymath}LOC_r()approxlceillceilM_AMB_ArceilP_rrceilastMB_A.{displaymath}/; $cached_env_img{$key} = q|displaymath14672|; $key = q/{_inline}$mod(M_B,MB_B)=0${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline15277|; $key = q/{displaymath}LOC_c()approxdisplaystylefracdisplaystylefracN_A+NB_A-1NB_A+P_c-1P_castNB_A{displaymath}/; $cached_env_img{$key} = q|displaymath14673|; $key = q/{_inline}$|hatalpha-alpha||hatalpha|${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline17450|; $key = q/{_inline}$Z^TB^-1Z=I${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline12905|; $key = q/{_inline}$Z^-1AZ^-H=I${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline12915|; $key = q/{_inline}$omega_c${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18356|; $key = q/{tabular}[t]lmulticolumn1lLargebfFortranInterface[0.28in]underlinelargeInitialization116underlineMYPNUM,underlineNPROCS)117underlineMYPNUM,underlineNPROCS)118ICTXT,WHAT,underlineVAL)119ICTXT,WHAT,VAL)120underlineICTXT,121NPROW,NPCOL)122underlineICTXT,USERMAP,LDUMAP,NPROW,NPCOL)underlinelargeDestruction123124125126underlinelargeSingtt_inline$Box$_inlineGESD2D(ICTXT,127M,N,A,LDA,RDEST,CDEST)tt_inline$Box$_inlineGEBS2D(ICTXT,128M,N,A,LDA129)tt_inline$Box$_inlineTRSD2D(ICTXT,130UPLO,DIAG,M,N,A,LDA,RDEST,CDEST)tt_inline$Box$_inlineTRBS2D(ICTXT,SCOPE,TOP,UPLO,DIAG,M,N,A,LDA131)underlinelargeReceivingtt_inline$Box$_inlineGERV2D(ICTXT,132M,N,underlineA,LDA,RSRC,CSRC)tt_inline$Box$_inlineGEBR2D(ICTXT,133M,N,underlineA,LDA,RSRC,CSRC)tt_inline$Box$_inlineTRRV2D(ICTXT,134M,N,underlineA,LDA,RSRC,CSRC)tt_inline$Box$_inlineTRBR2D(ICTXT,135M,N,underlineA,LDA,RSRC,CSRC)underlinelargeCombineOperationstt_inline$Box$_inlineGAMX2D(ICTXT,SCOPE,TOP,M,N,underlineA,LDA,underlineRA,underlineCA,RCFLAG,RDEST,CDEST)tt_inline$Box$_inlineGAMN2D(ICTXT,SCOPE,TOP,M,N,underlineA,LDA,underlineRA,underlineCA,RCFLAG,RDEST,CDEST)tt_inline$Box$_inlineGSUM2D(ICTXT,SCOPE,TOP,M,N,underlineA,LDA,136RDEST,CDEST)itAllroutinesprecededbya_inline$Box$_inlinehavethefollowingitprefixes:ttS,D,C,Z,I.{tabular}/; $cached_env_img{$key} = q|tabular7001|; $key = q/{displaymath}T=bordermatrix;SPMamp;pcrn-p;SPMamp;T_11crhfillp;SPMamp;T_21,quadmboxifquadn;SPMgt;p,{displaymath}/; $cached_env_img{$key} = q|displaymath13660|; $key = q/{displaymath}A=ZLambdaZ^T,{displaymath}/; $cached_env_img{$key} = q|displaymath12774|; $key = q/{_inline}$mod(M_B,MB_B)neq0${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline15287|; $key = q/{_inline}$z+delta${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18021|; $key = q/{_inline}$|f'(z)|${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18069|; $key = q/{_inline}${hatv_i,,,iincalI}${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline19021|; $key = q/{_inline}$A-lambdaB${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18087|; $key = q/{_inline}$NPROWtimesNPCOL${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline12614|; $key = q/{displaymath}x=PZleft(arraycT_11^-1hatc_10arrayright).{displaymath}/; $cached_env_img{$key} = q|displaymath13545|; $key = q/{_inline}$Az=lambdaBz${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline12875|; $key = q/{_inline}$Ahatx${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18539|; $key = q/{table}[htb]labeltab:evxperfminipagelinewidthcentertabular||l|c|c|r|r|r|r|r||hline;SPMamp;Process;SPMamp;Block;SPMamp;multicolumn5c||Valuesof_inline$N$_inlinecline4-8;SPMamp;Grid;SPMamp;Size;SPMamp;1000;SPMamp;2000;SPMamp;3000;SPMamp;4000;SPMamp;5000hlineCrayT3EfootnoteAScaLAPACKprocessisrunningoneachnode.EachnodeconsistsofonecomputationalDECAlphaEV5processor,whichisusedbytheBLAS.;SPMamp;_inline$1times4$_inline;SPMamp;32;SPMamp;15;SPMamp;76;SPMamp;;SPMamp;;SPMamp;PSSYEVX;SPMamp;_inline$2times8$_inline;SPMamp;32;SPMamp;7;SPMamp;29;SPMamp;76;SPMamp;164;SPMamp;;SPMamp;_inline$4times16$_inline;SPMamp;32;SPMamp;5;SPMamp;17;SPMamp;34;SPMamp;64;SPMamp;123hlineIBMSP2footnoteAScaLAPACKprocessisrunningoneachnode.EachnodeconsistsofonecomputationalIBMPOWER2590processor,whichisusedbytheBLAS.;SPMamp;_inline$1times4$_inline;SPMamp;50;SPMamp;36;SPMamp;;SPMamp;;SPMamp;;SPMamp;PDSYEVX;SPMamp;_inline$2times8$_inline;SPMamp;50;SPMamp;18;SPMamp;76;SPMamp;;SPMamp;;SPMamp;;SPMamp;_inline$4times16$_inline;SPMamp;50;SPMamp;16;SPMamp;52;SPMamp;112;SPMamp;213;SPMamp;397hlineIntelXPSMPParagonfootnoteAScaLAPACKprocessisrunningoneachnode.EachnodeconsistsoftwocomputationalInteli860XPRISCprocessors,whichareusedbythemulti-threadedBLAS.;SPMamp;_inline$1times4$_inline;SPMamp;32;SPMamp;96;SPMamp;;SPMamp;;SPMamp;;SPMamp;PDSYEVX;SPMamp;_inline$2times8$_inline;SPMamp;32;SPMamp;35;SPMamp;142;SPMamp;;SPMamp;;SPMamp;;SPMamp;_inline$4times16$_inline;SPMamp;32;SPMamp;19;SPMamp;60;SPMamp;132;SPMamp;260;SPMamp;hlineClusterofSunUltra2sfootnoteAScaLAPACKprocessisrunningoneachnode.EachnodeconsistsoftwocomputationalUltraSPARC-1processors,butonlyoneprocessorisusedbytheBLAS.;SPMamp;_inline$2times2$_inline;SPMamp;64;SPMamp;45;SPMamp;263;SPMamp;;SPMamp;;SPMamp;PDSYEVX;SPMamp;_inline$2times4$_inline;SPMamp;64;SPMamp;44;SPMamp;389;SPMamp;;SPMamp;;SPMamp;;SPMamp;_inline$2times6$_inline;SPMamp;64;SPMamp;79;SPMamp;849;SPMamp;1486;SPMamp;;SPMamp;hlineBerkeleyNOWfootnoteAScaLAPACKprocessisrunningoneachnode.EachnodeconsistsofonecomputationalUltraSPARC-1processor,whichisusedbytheBLAS.;SPMamp;_inline$1times4$_inline;SPMamp;32;SPMamp;28;SPMamp;;SPMamp;;SPMamp;;SPMamp;PDSYEVX;SPMamp;_inline$1times8$_inline;SPMamp;32;SPMamp;23;SPMamp;;SPMamp;;SPMamp;;SPMamp;;SPMamp;_inline$2times8$_inline;SPMamp;32;SPMamp;15;SPMamp;64;SPMamp;172;SPMamp;;SPMamp;;SPMamp;_inline$4times8$_inline;SPMamp;32;SPMamp;14;SPMamp;47;SPMamp;128;SPMamp;608;SPMamp;663hlinetabularcenterminipage{table}/; $cached_env_img{$key} = q|table4060|; $key = q/{_inline}$|deltahatU|leqp(m,n)epsilon${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18967|; $key = q/{_inline}$Z^HAZ=Lambda${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline12913|; $key = q/{_inline}$|A|_1=max_1leqjleqnsum_i=1^m|a_ij|${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline17522|; $key = q/{_inline}$T_rmseq(N)${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline16300|; $key = q/{table}[htbp]labeltabcompsvdcentertabular||l|l|l|l|l|l||hlineTypeof;SPMamp;Operation;SPMamp;multicolumn2c|SinglePrecision;SPMamp;multicolumn2c||DoublePrecisioncline3-6Matrix;SPMamp;;SPMamp;Real;SPMamp;Complex;SPMamp;Real;SPMamp;Complexhlinegeneral;SPMamp;bidiagonalreduction;SPMamp;PSGEBRDindexPSGEBRD;SPMamp;PCGEBRDindexPCGEBRD;SPMamp;PDGEBRDindexPDGEBRD;SPMamp;PZGEBRDindexPZGEBRDhlineorthogonal;SPMamp;multiplymatrixafter;SPMamp;PSORMBRindexPSORMBR;SPMamp;PCUNMBRindexPCUNMBR;SPMamp;PDORMBRindexPDORMBR;SPMamp;PZUNMBRindexPZUNMBRunitary;SPMamp;bidiagonalreduction;SPMamp;;SPMamp;;SPMamp;;SPMamp;hlinetabularcenter{table}/; $cached_env_img{$key} = q|table2093|; $key = q/{_inline}$}${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline17834|; $key = q/{_inline}$|hatx-betax|_2${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline17590|; $key = q/{_inline}$frac23n^3${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18115|; $key = q/{_inline}$P_r${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline12172|; $key = q/{_inline}$Q_1${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline13270|; $key = q/{_inline}$MBtimesNB${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline14817|; $key = q/{table}[h]labeltab:blockmapcentertabular||l|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c||hline_inline$J$_inline;SPMamp;1;SPMamp;2;SPMamp;3;SPMamp;4;SPMamp;5;SPMamp;6;SPMamp;7;SPMamp;8;SPMamp;9;SPMamp;10;SPMamp;11;SPMamp;12;SPMamp;13;SPMamp;14;SPMamp;15;SPMamp;16hline_inline$p$_inline;SPMamp;0;SPMamp;0;SPMamp;0;SPMamp;0;SPMamp;0;SPMamp;0;SPMamp;0;SPMamp;0;SPMamp;1;SPMamp;1;SPMamp;1;SPMamp;1;SPMamp;1;SPMamp;1;SPMamp;1;SPMamp;1_inline$x$_inline;SPMamp;1;SPMamp;2;SPMamp;3;SPMamp;4;SPMamp;5;SPMamp;6;SPMamp;7;SPMamp;8;SPMamp;1;SPMamp;2;SPMamp;3;SPMamp;4;SPMamp;5;SPMamp;6;SPMamp;7;SPMamp;8hlinetabularcenter{table}/; $cached_env_img{$key} = q|table2713|; $key = q/{displaymath}A=left(arrayccc1;SPMamp;2;SPMamp;34;SPMamp;5;SPMamp;67;SPMamp;8;SPMamp;10arrayright);;,;;hatA=left(arrayccc.44;SPMamp;2.36;SPMamp;3.043.09;SPMamp;5.87;SPMamp;6.667.36;SPMamp;7.77;SPMamp;9.07arrayright){displaymath}/; $cached_env_img{$key} = q|displaymath17411|; $key = q/{_inline}$|A|=max_i,j|a_ij|${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline17512|; $key = q/{_inline}$h_ii=1${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18776|; $key = q/{_inline}$ttRCOND=3.907cdot10^-6${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18246|; $key = q/{_inline}$Z^T${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline13563|; $key = q/{displaymath}left(arrayrx_1x_2x_5x_6x_9arrayright)qquadleft(arrayrb_1b_2b_3b_4b_5arrayright)=left(arrayr0frac-16000arrayright),{displaymath}/; $cached_env_img{$key} = q|displaymath12394|; $key = q/{tabular}|l|l|l|hline_;SPMamp;DataandPrecision;SPMamp;Typehlines;SPMamp;single-precisionreal;SPMamp;floatd;SPMamp;double-precisionreal;SPMamp;doublec;SPMamp;single-precisioncomplex;SPMamp;floatz;SPMamp;double-precisioncomplex;SPMamp;doublei;SPMamp;integer;SPMamp;inthline{tabular}/; $cached_env_img{$key} = q|tabular6773|; $key = q/{_inline}$ttFERR=3.0cdot10^-5${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18260|; $key = q/{_inline}$7times7${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline15332|; $key = q/{tabular}|lp0.75in|p4.5in|hlinemulticolumn2|c|Routine;SPMamp;multicolumn1c|Descriptioncline1-2Real;SPMamp;Complex;SPMamp;hlinehlinePSLAHQR;SPMamp;;SPMamp;ComputestheeigenvaluesandSchurfactorizationofanupperHessenbergmatrix,usingthedouble-shiftsingle-shift_inline$QR$_inlinealgorithm.PSLAHRD;SPMamp;PCLAHRD;SPMamp;Reducesthefirst_inline$nb$_inlinecolumnsofageneralrectangularmatrix_inline$A$_inlinesothatelementsbelowthe_inline$k^th$_inlinesubdiagonalarezero,byanorthogonalunitarytransformation,andreturnsauxiliarymatriceswhichareneededtoapplythetransformationtotheunreducedpartof_inline$A$_inline.PSLAIECT;SPMamp;;SPMamp;ExploitsIEEEarithmetictoacceleratethecomputationsofeigenvalues.PSLAMCHindexPSLAMCHindexPDLAMCH;SPMamp;;SPMamp;Determinesmachineparametersforfloating-pointarithmetic.PSLANGE;SPMamp;PCLANGE;SPMamp;Returnsthevalueofthe1-norm,Frobeniusnorm,infinity-norm,orthelargestabsolutevalueofanyelement,ofageneralrectangularmatrix.PSLANHS;SPMamp;PCLANHS;SPMamp;Returnsthevalueofthe1-norm,Frobeniusnorm,infinity-norm,orthelargestabsolutevalueofanyelement,ofanupperHessenbergmatrix.PSLANSY;SPMamp;PCLANSYPCLANHE;SPMamp;Returnsthevalueofthe1-norm,Frobeniusnorm,infinity-norm,orthelargestabsolutevalueofanyelement,ofarealsymmetriccomplexsymmetriccomplexHermitianmatrix.PSLANTR;SPMamp;PCLANTR;SPMamp;Returnsthevalueofthe1-norm,Frobeniusnorm,infinity-norm,orthelargestabsolutevalueofanyelement,ofatriangularmatrix.PSLAPIV;SPMamp;PCLAPIV;SPMamp;Appliesapermutationmatrixtoageneraldistributedmatrix,resultinginroworcolumnpivoting.PSLAQGE;SPMamp;PCLAQGE;SPMamp;Scalesageneralrectangularmatrix,usingrowandcolumnscalingfactorscomputedbyPSGEEQUindexPSGEEQUPCGEEQUindexPCGEEQU.PSLAQSY;SPMamp;PCLAQSY;SPMamp;ScalesasymmetricHermitianmatrix,usingscalingfactorscomputedbyPSPOEQUindexPSPOEQUPCPOEQUindexPCPOEQU.PSLARED1D;SPMamp;;SPMamp;Redistributesanarrayassumingthattheinputarray,BYCOL,isdistributedacrossrowsandthatallprocesscolumnscontainthesamecopyofBYCOL.PSLARED2D;SPMamp;;SPMamp;Redistributesanarrayassumingthattheinputarray,BYROW,isdistributedacrosscolumnsandthatallprocessrowscontainthesamecopyofBYROW.Theoutputarray,BYALL,willbeidenticalonallprocesses.PSLARF;SPMamp;PCLARF;SPMamp;Appliesanelementaryreflectortoageneralrectangularmatrix.PSLARFB;SPMamp;PCLARFB;SPMamp;Appliesablockreflectororitstransposeconjugate-transposetoageneralrectangularmatrix.;SPMamp;PCLARFC;SPMamp;(complex)Applies(multipliesby)theconjugatetransposeofanelementaryreflectortoageneralmatrix.PSLARFG;SPMamp;PCLARFG;SPMamp;Generatesanelementaryreflector(Householdermatrix).PSLARFT;SPMamp;PCLARFT;SPMamp;Formsthetriangularfactor_inline$T$_inlineofablockreflector_inline$H=I-VTV^H$_inline.hline{tabular}/; $cached_env_img{$key} = q|tabular6653|; $key = q/{displaymath}LOC_r()approxdisplaystylefracdisplaystylefracM_A+MB_A-1MB_A+P_r-1P_rastMB_A{displaymath}/; $cached_env_img{$key} = q|displaymath14671|; $key = q/{_inline}$hatV${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline14255|; $key = q/{_inline}$u^HA=lambdau^H${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline13986|; $key = q/{_inline}$D=mboxrmdiag(b_11^-12,ldots,b_nn^-12)${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline19283|; $key = q/{_inline}$rmmod(k,P)${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline14707|; $key = q/{_inline}$ttRNORM=8.843${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18497|; $key = q/{displaymath}H^Hx=beta(1,0,ldots,0)^T{displaymath}/; $cached_env_img{$key} = q|displaymath14416|; $key = q/{_inline}$ttPSLAMCH(ICTXT,'E')=2^-24=5.961cdot10^-8${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18242|; $key = q/{_inline}$LLD_A${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline12116|; $key = q/{_inline}$lfloorsqrtPrfloor${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline17061|; $key = q/{_inline}$0;SPMlt;=p_c;SPMlt;P_c${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline14633|; $key = q/{_inline}$hatz_i${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18638|; $key = q/{_inline}$mboxrank(A)=n${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline12692|; $key = q/{figure}[thb]centertabularccc|ccc|cmulticolumn7cProcessesmulticolumn3c0;SPMamp;multicolumn3c1;SPMamp;2hline_inline$a_11$_inline;SPMamp;_inline$a_22$_inline;SPMamp;_inline$a_33$_inline;SPMamp;_inline$a_44$_inline;SPMamp;_inline$a_55$_inline;SPMamp;_inline$a_66$_inline;SPMamp;_inline$a_77$_inline_inline$a_21$_inline;SPMamp;_inline$a_32$_inline;SPMamp;_inline$a_43$_inline;SPMamp;_inline$a_54$_inline;SPMamp;_inline$a_65$_inline;SPMamp;_inline$a_76$_inline;SPMamp;_inline$ast$_inline_inline$a_31$_inline;SPMamp;_inline$a_42$_inline;SPMamp;_inline$a_53$_inline;SPMamp;_inline$a_64$_inline;SPMamp;_inline$a_75$_inline;SPMamp;_inline$ast$_inline;SPMamp;_inline$ast$_inlinetabularcenterlabelfig:Lsymmband7{figure}/; $cached_env_img{$key} = q|figure2938|; $key = q/{displaymath}A=left(arrayccc4;SPMamp;3;SPMamp;52;SPMamp;5;SPMamp;83;SPMamp;6;SPMamp;104;SPMamp;5;SPMamp;11arrayright);rmand;b=left(arrayc100.1.1.01.01arrayright);,{displaymath}/; $cached_env_img{$key} = q|displaymath18464|; $key = q/{_inline}$p_r${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline14643|; $key = q/{tabular}llValue;SPMamp;Meaninghlinett`U';SPMamp;Unittriangulartt`R';SPMamp;Nonunittriangular{tabular}/; $cached_env_img{$key} = q|tabular3453|; $key = q/{figure}[thb]centertabularc|ccc|ccc|c;SPMamp;multicolumn7cProcesses;SPMamp;multicolumn3c0;SPMamp;multicolumn3c1;SPMamp;2hlineD;SPMamp;_inline$a_11$_inline;SPMamp;_inline$a_22$_inline;SPMamp;_inline$a_33$_inline;SPMamp;_inline$a_44$_inline;SPMamp;_inline$a_55$_inline;SPMamp;_inline$a_66$_inline;SPMamp;_inline$a_77$_inlineE;SPMamp;_inline$a_21$_inline;SPMamp;_inline$a_32$_inline;SPMamp;_inline$a_43$_inline;SPMamp;_inline$a_54$_inline;SPMamp;_inline$a_65$_inline;SPMamp;_inline$a_76$_inline;SPMamp;tabularcenterlabelfig:Lbidiag7{figure}/; $cached_env_img{$key} = q|figure3109|; $key = q/{_inline}$lfloor(k-1)NBrfloor,mod~P${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline14601|; $key = q/{displaymath}x=Q^Tleft(arraycL^-1b0arrayright){displaymath}/; $cached_env_img{$key} = q|displaymath13358|; $key = q/{_inline}$fracN_ANB_A=P_c${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline15237|; $key = q/{displaymath}T=Q^THQ{displaymath}/; $cached_env_img{$key} = q|displaymath16857|; $key = q/{_inline}$delta${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18009|; $key = q/{displaymath}LOC_c()approxlceillceilN_ANB_ArceilP_crceilastNB_A.{displaymath}/; $cached_env_img{$key} = q|displaymath14674|; $key = q/{_inline}$Z^-1AZ^-T=Lambda${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline19107|; $key = q/{_inline}$gap_i=min_jneqi|lambda_i-lambda_j|${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18712|; $key = q/{tabular}[t]p4.0inunderlinelargeGlobalTopologies178179180181vdots182183184185186187188189190underlinelargeNotationrmUnderlinedparametersareoutputarguments.Ifaroutineisunderlined,itisafunctionthatreturnsavalue.TheprefixbfPusuallystandsforprocess.Otherstandardnotationsare191192193194195196{tabular}/; $cached_env_img{$key} = q|tabular7056|; $key = q/{_inline}$ttBNORM=100.1${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18495|; $key = q/{_inline}$(A+E)-lambda(B+F)${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline19345|; $key = q/{_inline}$|deltahatZ|_2leqp(n)epsilon${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18690|; $key = q/{figure}[thb]centertabularccc|ccc|cmulticolumn7cProcessesmulticolumn3c0;SPMamp;multicolumn3c1;SPMamp;2hline_inline$ast$_inline;SPMamp;_inline$ast$_inline;SPMamp;_inline$a_13$_inline;SPMamp;_inline$a_24$_inline;SPMamp;_inline$a_35$_inline;SPMamp;_inline$a_46$_inline;SPMamp;_inline$a_57$_inline_inline$ast$_inline;SPMamp;_inline$a_12$_inline;SPMamp;_inline$a_23$_inline;SPMamp;_inline$a_34$_inline;SPMamp;_inline$a_45$_inline;SPMamp;_inline$a_56$_inline;SPMamp;_inline$a_67$_inline_inline$a_11$_inline;SPMamp;_inline$a_22$_inline;SPMamp;_inline$a_33$_inline;SPMamp;_inline$a_44$_inline;SPMamp;_inline$a_55$_inline;SPMamp;_inline$a_66$_inline;SPMamp;_inline$a_77$_inline_inline$a_21$_inline;SPMamp;_inline$a_32$_inline;SPMamp;_inline$a_43$_inline;SPMamp;_inline$a_54$_inline;SPMamp;_inline$a_65$_inline;SPMamp;_inline$a_76$_inline;SPMamp;_inline$ast$_inline_inline$a_31$_inline;SPMamp;_inline$a_42$_inline;SPMamp;_inline$a_53$_inline;SPMamp;_inline$a_64$_inline;SPMamp;_inline$a_75$_inline;SPMamp;_inline$ast$_inline;SPMamp;_inline$ast$_inlinetabularcenterlabelfig:nonsymmbandnp7{figure}/; $cached_env_img{$key} = q|figure2837|; $key = q/{_inline}$sigmahat_i${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18971|; $key = q/{_inline}$_rmseq()${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline16302|; $key = q/{_inline}$O(epsilon)${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18764|; $key = q/{_inline}$=1.5cdot10^-3${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18252|; $key = q/{_inline}$A=QTQ^H${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline13874|; $key = q/{_inline}$theta(hatcalS,calS)=.01414${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline17920|; $key = q/{_inline}$1P^th${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline14591|; $key = q/{_inline}$|F|${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline19357|; $key = q/{_inline}$frac12${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline16667|; $key = q/{_inline}$P_c=3${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline12424|; $key = q/{_inline}$P=P_rtimesP_c${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline17184|; $key = q/{_inline}$gap_n${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18997|; $key = q/{_inline}$4.6cdot10^-7${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18503|; $key = q/{_inline}$sigma_1geqsigma_2geqcdotsgeqsigma_rgeq0${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18854|; $key = q/{_inline}$|E|_infty|A|_infty${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18077|; $key = q/{displaymath}frac|Aastx-b|(|x|ast|A|astepsastN){displaymath}/; $cached_env_img{$key} = q|displaymath12396|; $key = q/{_inline}$18N^3${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline16994|; $key = q/{displaymath}R=bordermatrix;SPMamp;n,crm-n;SPMamp;R_11crhfilln;SPMamp;R_21,quadmboxifquadm;SPMgt;n,{displaymath}/; $cached_env_img{$key} = q|displaymath13749|; $key = q/{tabular}|lp0.75in|p4.5in|hlinemulticolumn2|c|Routine;SPMamp;multicolumn1c|Descriptioncline1-2Real;SPMamp;Complex;SPMamp;hlinehlinePSORGR2;SPMamp;PCUNGR2;SPMamp;Generatesallorpartoftheorthogonalunitarymatrix_inline$Q$_inlinefroman_inline$RQ$_inlinefactorizationdeterminedbyPSGERQFindexPSGERQFPCGERQFindexPCGERQF(unblockedalgorithm).PSORM2L;SPMamp;PCUNM2L;SPMamp;Multipliesageneralmatrixbytheorthogonalunitarymatrixfroma_inline$QL$_inlinefactorizationdeterminedbyPSGEQLFindexPSGEQLFPCGEQLFindexPCGEQLF(unblockedalgorithm).PSORM2R;SPMamp;PCUNM2R;SPMamp;Multipliesageneralmatrixbytheorthogonalunitarymatrixfroma_inline$QR$_inlinefactorizationdeterminedbyPSGEQRFindexPSGEQRFPCGEQRFindexPCGEQRF(unblockedalgorithm).PSORML2;SPMamp;PCUNML2;SPMamp;Multipliesageneralmatrixbytheorthogonalunitarymatrixfroman_inline$LQ$_inlinefactorizationdeterminedbyPSGELQFindexPSGELQFPCGELQFindexPCGELQF(unblockedalgorithm).PSORMR2;SPMamp;PCUNMR2;SPMamp;Multipliesageneralmatrixbytheorthogonalunitarymatrixfroman_inline$RQ$_inlinefactorizationdeterminedbyPSGERQFindexPSGERQFPCGERQFindexPCGERQF(unblockedalgorithm).PSPBTRSV;SPMamp;PCPBTRSV;SPMamp;SolvesasingletriangularlinearsystemviafrontsolveorbacksolvewherethetriangularmatrixisafactorofabandedmatrixcomputedbyPSPBTRFPCPBTRF.PSPTTRSV;SPMamp;PCPTTRSV;SPMamp;SolvesasingletriangularlinearsystemviafrontsolveorbacksolvewherethetriangularmatrixisafactorofatridiagonalmatrixcomputedbyPSPTTRFPCPTTRF.PSPOTF2;SPMamp;PCPOTF2;SPMamp;ComputestheCholeskyfactorizationofasymmetricHermitianpositivedefinitematrix(localunblockedalgorithm).PSRSCL;SPMamp;PCSRSCL;SPMamp;Multipliesavectorbythereciprocalofarealscalar.PSSYGS2;SPMamp;PCHEGS2;SPMamp;ReducesasymmetricHermitiandefinitegeneralizedeigenproblem_inline$Ax=lambdaBx$_inline,_inline$ABx=lambdax$_inline,or_inline$BAx=lambdax$_inline,tostandardform,where_inline$B$_inlinehasbeenfactorizedbyPSPOTRFindexPSPOTRFPCPOTRFindexPCPOTRF(localunblockedalgorithm).PSSYTD2;SPMamp;PCHETD2;SPMamp;ReducesasymmetricHermitianmatrixtorealsymmetrictridiagonalformbyanorthogonalunitarysimilaritytransformation(localunblockedalgorithm).PSTRTI2;SPMamp;PCTRTI2;SPMamp;Computestheinverseofatriangularmatrix(localunblockedalgorithm).PXERBLAindexPXERBLA;SPMamp;;SPMamp;Error-handlingroutinecalledbyScaLAPACKroutinesifaninputparameterhasaninvalidvalue.SDBTF2;SPMamp;CDBTF2;SPMamp;Computesan_inline$LU$_inlinefactorizationofageneralbandmatrixwithnopivoting(localunblockedalgorithm).SDBTRF;SPMamp;CDBTRF;SPMamp;Computesan_inline$LU$_inlinefactorizationofageneralbandmatrixwithnopivoting(localblockedalgorithm).hline{tabular}/; $cached_env_img{$key} = q|tabular6694|; $key = q/{figure}[thb]centertabularc|ccc|ccc|c;SPMamp;multicolumn7cProcesses;SPMamp;multicolumn3c0;SPMamp;multicolumn3c1;SPMamp;2hlineD;SPMamp;_inline$a_11$_inline;SPMamp;_inline$a_22$_inline;SPMamp;_inline$a_33$_inline;SPMamp;_inline$a_44$_inline;SPMamp;_inline$a_55$_inline;SPMamp;_inline$a_66$_inline;SPMamp;_inline$a_77$_inlineE;SPMamp;_inline$a_12$_inline;SPMamp;_inline$a_23$_inline;SPMamp;_inline$a_34$_inline;SPMamp;_inline$a_45$_inline;SPMamp;_inline$a_56$_inline;SPMamp;_inline$a_67$_inline;SPMamp;tabularcenterlabelfig:Ubidiag7{figure}/; $cached_env_img{$key} = q|figure3141|; $key = q/{_inline}$f_k${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18190|; $key = q/{_inline}$P_r=P_c${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline16172|; $key = q/{_inline}$|B-AX|_2${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline12748|; $key = q/{_inline}$K;SPMlt;P_r${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline15307|; $key = q/{_inline}$|A|_infty=max_1leqileqmsum_j=1^n|a_ij|${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline17514|; $key = q/{_inline}$min_xneq0|Ax|_2|x|_2${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline17814|; $key = q/{displaymath}|hatlambda_i-lambda_i|mathrelraisebox-.75ex_inline$mathopsimlimits^textstyle;SPMlt;$_inlinep(n)epsiloncdot(|B^-1|_2|A|_2+kappa_2(B)cdot|hatlambda_i|)=ttEERRBD(i);.{displaymath}/; $cached_env_img{$key} = q|displaymath19056|; $key = q/{_inline}$1sqrt(x_i^HAx_i)^2+(x_i^HBx_i)^2${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline19363|; $key = q/{displaymath}chi(hatlambda_i,lambda_i)=frac|hatlambda_i-lambda_i|sqrt1+hatlambda_i^2cdotsqrt1+lambda_i^2.{displaymath}/; $cached_env_img{$key} = q|displaymath19060|; $key = q/{_inline}$mboxrank(A)=k${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline13482|; $key = q/{figure}[htb]centerlinepsfigfigure=FIGview.ps,height=1.0in,width=2.0inlabelfig:view{figure}/; $cached_env_img{$key} = q|figure2606|; $key = q/{_inline}$u_i${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline12840|; $key = q/{displaymath}A=PU^HUP^T;;rmor;;A=PLL^HP^Tmbox(intheHermitiancase),{displaymath}/; $cached_env_img{$key} = q|displaymath12939|; $key = q/{_inline}$O(n^3)${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline13936|; $key = q/{table}[htb]labeltabdriveseigcentertabular||c|l|l|l|l|l||hlineTypeof;SPMamp;Function;SPMamp;multicolumn2c|SinglePrecision;SPMamp;multicolumn2c||DoublePrecisioncline3-6Problem;SPMamp;;SPMamp;Real;SPMamp;Complex;SPMamp;Real;SPMamp;ComplexhlineSEP;SPMamp;simpledriver;SPMamp;PSSYEVindexPSSYEV;SPMamp;;SPMamp;PDSYEVindexPDSYEV;SPMamp;;SPMamp;expertdriver;SPMamp;PSSYEVXindexPSSYEVX;SPMamp;PCHEEVXindexPCHEEVX;SPMamp;PDSYEVXindexPDSYEVX;SPMamp;PZHEEVXindexPZHEEVXhlineSVD;SPMamp;singularvaluesvectors;SPMamp;PSGESVDindexPSGESVD;SPMamp;;SPMamp;PDGESVDindexPDGESVD;SPMamp;hlinetabularcenter{table}/; $cached_env_img{$key} = q|table1093|; $key = q/{_inline}$theta(hatx,x)${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline17626|; $key = q/{_inline}$sin(theta)=rho|b|_2${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18535|; $key = q/{displaymath}A=U^TU;;rmor;;A=LL^Tmbox(inthesymmetriccase){displaymath}/; $cached_env_img{$key} = q|displaymath12934|; $key = q/{tabular}[t]lunderlinelargeInformationalandMiscellaneousBLACS_GRIDINFO(ICTXT,underlineNPROW,underlineNPCOL,underlineMYROW,underlineMYCOL)underlineBLACS_PNUM137BLACS_PCOORD138(ICTXT,PNUM,139underlinePROW,140underlinePCOL)141underlinermlargeNon-standard142underlineDCPUTIME00()underlineDWALLTIME00()underlineKSENDID143underlineKRECVID144underlineKBSID145underlineKBRID146underlinelargeDeclarations147148149150151152153154155156157underlinelargeOptions158159160161{tabular}/; $cached_env_img{$key} = q|tabular7032|; $key = q/{_inline}$kappa_c(A,b,hatx)equiv|,|A^-1|(|A|cdot|hatx|+|b|),|_infty|hatx|_infty${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18352|; $key = q/{displaymath}frac|hatx-x|_infty|x|_inftyleqttERRBD{displaymath}/; $cached_env_img{$key} = q|displaymath18202|; $key = q/{_inline}$rmalg(z)=f(z+delta)+eta${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18023|; $key = q/{table}[htb]labeltab:estperfminipagelinewidthcentertabular||l|c|r|r|r|r|r|r|r|r|r|r||hlineIBMSP2footnoteOneprocessspawnedpernodeandonecomputationalIBMPOWER2590processorpernode.;SPMamp;_inline$P$_inline;SPMamp;multicolumn10c||Valuesof_inline$N$_inlinecline3-12;SPMamp;;SPMamp;multicolumn2c|2000;SPMamp;multicolumn2c|5000;SPMamp;multicolumn2c|7500;SPMamp;multicolumn2c|10000;SPMamp;multicolumn2c||15000cline3-12;SPMamp;;SPMamp;Est;SPMamp;Obt;SPMamp;Est;SPMamp;Obt;SPMamp;Est;SPMamp;Obt;SPMamp;Est;SPMamp;Obt;SPMamp;Est;SPMamp;ObthlinePDGESV;SPMamp;4;SPMamp;357;SPMamp;421;SPMamp;632;SPMamp;603;SPMamp;;SPMamp;;SPMamp;;SPMamp;;SPMamp;;SPMamp;;SPMamp;16;SPMamp;497;SPMamp;722;SPMamp;1581;SPMamp;1543;SPMamp;2116;SPMamp;1903;SPMamp;2424;SPMamp;2149;SPMamp;;SPMamp;;SPMamp;64;SPMamp;502;SPMamp;924;SPMamp;2432;SPMamp;3017;SPMamp;4235;SPMamp;4295;SPMamp;5793;SPMamp;5596;SPMamp;7992;SPMamp;7057hlinePDPOSV;SPMamp;4;SPMamp;530;SPMamp;462;SPMamp;669;SPMamp;615;SPMamp;;SPMamp;;SPMamp;;SPMamp;;SPMamp;;SPMamp;;SPMamp;16;SPMamp;1315;SPMamp;1081;SPMamp;2083;SPMamp;1811;SPMamp;2366;SPMamp;2118;SPMamp;2535;SPMamp;2312;SPMamp;;SPMamp;;SPMamp;64;SPMamp;2577;SPMamp;1807;SPMamp;5327;SPMamp;4431;SPMamp;6709;SPMamp;5727;SPMamp;7661;SPMamp;6826;SPMamp;8887;SPMamp;8084hlinetabularcenterminipage{table}/; $cached_env_img{$key} = q|table4184|; $key = q/{_inline}$pminfty${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline17191|; $key = q/{tabular}[t]lunderlinermlargeInformationalandMiscellaneousvoidCblacs_gridinfo(intictxt,int*underlinenprow,int*underlinenpcol,int*underlinemyprow,int*underlinemypcol)223underlineCblacs_pnum224(intictxt,225int226prow,int227pcol)voidCblacs_pcoord228(intictxt,intpnum,int*underlineprow,int*underlinepcol)voidCblacs_barrier(intictxt,char*scope)underlinermlargeNon-standard229intntasks,int*tids)230231232underlineCksid233234underlineCkrecvid235236underlineCkbsid237238underlineCkbrid239{tabular}/; $cached_env_img{$key} = q|tabular7125|; $key = q/{displaymath}hatX=left(arraycc-.79996;SPMamp;.60005-.59997;SPMamp;-.79990-.01;SPMamp;-.01arrayright);;rmand;;X=left(arraycc1;SPMamp;00;SPMamp;10;SPMamp;0arrayright){displaymath}/; $cached_env_img{$key} = q|displaymath17419|; $key = q/{_inline}$O(nm^2)${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18748|; $key = q/{displaymath}H=I-tauvv^T,{displaymath}/; $cached_env_img{$key} = q|displaymath16782|; $key = q/{_inline}$|Ax-b|_2${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18475|; $key = q/{_inline}$_r${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline12112|; $key = q/{_inline}$A+E=(hatU+deltahatU)hatSigma(hatV+deltahatV)${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18961|; $key = q/{_inline}$hatX^HhatX=I${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline17906|; $key = q/{_inline}$theta(hatx,x)leqbarthetall1${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline17656|; $key = q/{figure}[htb]centerlinepsfigfigure=gridb9.ps,height=1.5in,width=2.0inlabelfig:gridb9{figure}/; $cached_env_img{$key} = q|figure717|; $key = q/{displaymath}A=Q_1R,{displaymath}/; $cached_env_img{$key} = q|displaymath13232|; $key = q/{_inline}$hatlambda_1leqcdotsleqhatlambda_n${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18636|; $key = q/{_inline}$0;SPMlt;=p_r;SPMlt;P_r${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline14631|; $key = q/{_inline}$(1t_f)${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline16378|; $key = q/{_inline}$NB~=~3${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline14795|; $key = q/{_inline}$ttRCOND=3.454cdot10^-3${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18316|; $key = q/{_inline}$Z^HBZ=I${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline12921|; $key = q/{_inline}$Z^TAZ=Lambda${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline12895|; $key = q/{_inline}$c=Q^Tb${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline13506|; $key = q/{table}[tbh]labeltabcompgeigcentertabular||l|l|l|l|l|l||hlineTypeofMatrix;SPMamp;Operation;SPMamp;multicolumn2c|SinglePrecision;SPMamp;multicolumn2c||DoublePrecisioncline3-6;SPMamp;;SPMamp;Real;SPMamp;Complex;SPMamp;Real;SPMamp;ComplexhlinesymmetricHermitian;SPMamp;reduction;SPMamp;PSSYGSTindexPSSYGST;SPMamp;PCHEGSTindexPCHEGST;SPMamp;PDSYGSTindexPDSYGST;SPMamp;PZHEGSTindexPZHEGSThlinetabularcenter{table}/; $cached_env_img{$key} = q|table2167|; $key = q/{_inline}$P_rtimesP_cleqP${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline17180|; $key = q/{_inline}$mod(N_A,NB_A)=0${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline15235|; $key = q/{displaymath}A=left(arrayccc1;SPMamp;2;SPMamp;32;SPMamp;4;SPMamp;53;SPMamp;5;SPMamp;6arrayright);,{displaymath}/; $cached_env_img{$key} = q|displaymath18598|; $key = q/{_inline}$Av=lambdav${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline13982|; $key = q/{_inline}$mod(N_A,NB_A)neq0${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline15245|; $key = q/{_inline}$|b-Ax|_2${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline12736|; $key = q/{displaymath}A=Qleft(arraycR0arrayright)P^T,quadmgen,{displaymath}/; $cached_env_img{$key} = q|displaymath13452|; $key = q/{displaymath}T=bordermatrix;SPMamp;p;SPMamp;n-pcrp;SPMamp;T_11;SPMamp;T_12,quadmboxifquadp;SPMlt;nn{displaymath}/; $cached_env_img{$key} = q|displaymath13751|; $key = q/{_inline}$A=ZTZ^T${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline14010|; $key = q/{tabular}[t]p4.0inunderlinermlargeKeyIdeas:rmABLACScontextiscreatedviaacalltoeitherttCblacs_gridinitorttCblacs_gridmap.Noroutinerequiringacontextmaybeuseduntiloneoftheseroutineshasbeencalled.MultiplecallstottCblacs_gridinitorttCblacs_gridmapresultinthecreationofnewcontexts.Topreserveresources,theusershouldfreeunusedcontextsbycallingttCblacs_gridexit.WhenallBLACSoperationsaredone,acalltottCblacs_exitfreesanyremainingcontexts,andshutsdownallBLACSoperations.PleasenotethatttCblacs_gridinitandttCblacs_gridmapacceptsystemcontextsasinput.AdefaultsystemcontextencompassingallavailableprocessesmaybeobtainedbyacalltottCblacs_get.rmttCblacs_setcanonlybeusedtosettheBLACS'messageIDrangebeforethecreationofthefirstcontext.Subsequentcallswillbeignored.underlinermlargeTopologyHintsrmTopologiesallowtheusertooptimizecommunicationpatternsforaparticularoperation.Iftheuserdoesnothaveacommunicationpipetomaintain,thedefaultTOP=;SPMquot;;SPMquot;isrecommed.{tabular}/; $cached_env_img{$key} = q|tabular7158|; $key = q/{_inline}$calI${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18724|; $key = q/{_inline}$X=[x_1,...,x_k]${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline17912|; $key = q/{table}[htb]labeltabdrivegeigcentertabular||c|l|l|l|l|l||hlineTypeof;SPMamp;Function;SPMamp;multicolumn2c|Singleprecision;SPMamp;multicolumn2c||Doubleprecisioncline3-6problem;SPMamp;;SPMamp;Real;SPMamp;Complex;SPMamp;Real;SPMamp;ComplexhlineGSEP;SPMamp;expertdriver;SPMamp;PSSYGVXindexPSSYGVX;SPMamp;PCHEGVXindexPCHEGVX;SPMamp;PDSYGVXindexPDSYGVX;SPMamp;PZHEGVXindexPZHEGVXhlinetabularcenter{table}/; $cached_env_img{$key} = q|table1135|; $key = q/{_inline}$hatu_i${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18874|; $key = q/{_inline}$epsiloncdotkappa_infty(A)=epsiloncdotmax_i|a_ii|min_i|a_ii|${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18166|; $key = q/{_inline}$kappa_1${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18561|; $key = q/{_inline}$zne0${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline12780|; $key = q/{_inline}$lfloorkNBrfloor${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline15094|; $key = q/{_inline}$F_MM${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline12088|; $key = q/{_inline}$P_r,P_c${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline12110|; $key = q/{_inline}$N^3${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline17067|; $key = q/{table}[htbp]labeltabcompeig2centertabular||l|l|l|l|l|l||hlineTypeof;SPMamp;Operation;SPMamp;multicolumn2c|SinglePrecision;SPMamp;multicolumn2c||DoublePrecisioncline3-6Matrix;SPMamp;;SPMamp;Real;SPMamp;Complex;SPMamp;Real;SPMamp;Complexhlinegeneral;SPMamp;Hessenbergreduction;SPMamp;PSGEHRDindexPSGEHRD;SPMamp;PCGEHRDindexPCGEHRD;SPMamp;PDGEHRDindexPDGEHRD;SPMamp;PZGEHRDindexPZGEHRDorthogonal;SPMamp;multiplymatrixafter;SPMamp;PSORMHRindexPSORMHR;SPMamp;PCUNMHRindexPCUNMHR;SPMamp;PDORMHRindexPDORMHR;SPMamp;PZUNMHRindexPZUNMHRunitary;SPMamp;Hessenbergreduction;SPMamp;;SPMamp;;SPMamp;;SPMamp;hlineHessenberg;SPMamp;eigenvaluesand;SPMamp;PSLAHQRindexPSLAHQR;SPMamp;;SPMamp;PDLAHQRindexPDLAHQR;SPMamp;;SPMamp;Schurdecomposition;SPMamp;;SPMamp;;SPMamp;;SPMamp;hlinetabularcenter{table}/; $cached_env_img{$key} = q|table2014|; $key = q/{_inline}$tau${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline14435|; $key = q/{_inline}$infty${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline17548|; $key = q/{_inline}$ABz_i=lambda_iz_i${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline19119|; $key = q/{_inline}$A=QHQ^H${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline14050|; $key = q/{tabular}[t]p4.0inunderlinelargeBroadcastTopologies162163164165166167168169170171172173174175176vdots177{tabular}/; $cached_env_img{$key} = q|tabular7052|; $key = q/{_inline}$H_i${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline14421|; $key = q/{_inline}$lambda_1leqlambda_2leqcdotsleqlambda_n${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18624|; $key = q/{_inline}$r=b-Ahatx${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18270|; $key = q/{table}[h]labeltab:bcmapcentertabular||l|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c||hline_inline$I$_inline;SPMamp;1;SPMamp;2;SPMamp;3;SPMamp;4;SPMamp;5;SPMamp;6;SPMamp;7;SPMamp;8;SPMamp;9;SPMamp;10;SPMamp;11;SPMamp;12;SPMamp;13;SPMamp;14;SPMamp;15;SPMamp;16hline_inline$p$_inline;SPMamp;0;SPMamp;0;SPMamp;0;SPMamp;1;SPMamp;1;SPMamp;1;SPMamp;0;SPMamp;0;SPMamp;0;SPMamp;1;SPMamp;1;SPMamp;1;SPMamp;0;SPMamp;0;SPMamp;0;SPMamp;1_inline$l$_inline;SPMamp;0;SPMamp;0;SPMamp;0;SPMamp;0;SPMamp;0;SPMamp;0;SPMamp;1;SPMamp;1;SPMamp;1;SPMamp;1;SPMamp;1;SPMamp;1;SPMamp;2;SPMamp;2;SPMamp;2;SPMamp;2_inline$x$_inline;SPMamp;1;SPMamp;2;SPMamp;3;SPMamp;1;SPMamp;2;SPMamp;3;SPMamp;1;SPMamp;2;SPMamp;3;SPMamp;1;SPMamp;2;SPMamp;3;SPMamp;1;SPMamp;2;SPMamp;3;SPMamp;1hline_inline$l*NB+x$_inline;SPMamp;1;SPMamp;2;SPMamp;3;SPMamp;1;SPMamp;2;SPMamp;3;SPMamp;4;SPMamp;5;SPMamp;6;SPMamp;4;SPMamp;5;SPMamp;6;SPMamp;7;SPMamp;8;SPMamp;9;SPMamp;7hlinetabularcenter{table}/; $cached_env_img{$key} = q|table2440|; $key = q/{_inline}$maxleft(frac|E|_infty|A|_infty,frac|f|_infty|b|_inftyright)${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18284|; $key = q/{displaymath}x=Pleft(arraycR_11^-1hatc_10arrayright),{displaymath}/; $cached_env_img{$key} = q|displaymath13456|; $key = q/{displaymath}R=left(arrayccR_11;SPMamp;R_120;SPMamp;R_22arrayright)simeqleft(arrayccR_11;SPMamp;R_120;SPMamp;0arrayright).{displaymath}/; $cached_env_img{$key} = q|displaymath13455|; $key = q/{_inline}$epsilon=${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18308|; $key = q/{table}[htb]labeltabcomplineq1centertabular||l|l|l|l|l|l||hlineMatrixType;SPMamp;Operation;SPMamp;multicolumn2c|SinglePrecision;SPMamp;multicolumn2c||DoublePrecisioncline3-6andStorage;SPMamp;;SPMamp;Real;SPMamp;Complex;SPMamp;Real;SPMamp;Complexhlinegeneral;SPMamp;factorize;SPMamp;PSGETRFindexPSGETRF;SPMamp;PCGETRFindexPCGETRF;SPMamp;PDGETRFindexPDGETRF;SPMamp;PZGETRFindexPZGETRFpartialpivoting;SPMamp;solveusingfact.;SPMamp;PSGETRSindexPSGETRS;SPMamp;PCGETRSindexPCGETRS;SPMamp;PDGETRSindexPDGETRS;SPMamp;PZGETRSindexPZGETRS;SPMamp;est.cond.number;SPMamp;PSGECONindexPSGECON;SPMamp;PCGECONindexPCGECON;SPMamp;PDGECONindexPDGECON;SPMamp;PZGECONindexPZGECON;SPMamp;errorbds.forsoln.;SPMamp;PSGERFSindexPSGERFS;SPMamp;PCGERFSindexPCGERFS;SPMamp;PDGERFSindexPDGERFS;SPMamp;PZGERFSindexPZGERFS;SPMamp;invertusingfact.;SPMamp;PSGETRIindexPSGETRI;SPMamp;PCGETRIindexPCGETRI;SPMamp;PDGETRIindexPDGETRI;SPMamp;PZGETRIindexPZGETRI;SPMamp;equilibrate;SPMamp;PSGEEQUindexPSGEEQU;SPMamp;PCGEEQUindexPCGEEQU;SPMamp;PDGEEQUindexPDGEEQU;SPMamp;PZGEEQUindexPZGEEQUhlinegeneralband;SPMamp;factorize;SPMamp;PSGBTRFindexPSGBTRF;SPMamp;PCGBTRFindexPCGBTRF;SPMamp;PDGBTRFindexPDGBTRF;SPMamp;PZGBTRFindexPZGBTRFpartialpivoting;SPMamp;solveusingfact.;SPMamp;PSGBTRSindexPSGBTRS;SPMamp;PCGBTRSindexPCGBTRS;SPMamp;PDGBTRSindexPDGBTRS;SPMamp;PZGBTRSindexPZGBTRShlinegeneralband;SPMamp;factorize;SPMamp;PSDBTRFindexPSDBTRF;SPMamp;PCDBTRFindexPCDBTRF;SPMamp;PDDBTRFindexPDDBTRF;SPMamp;PZDBTRFindexPZDBTRFnopivoting;SPMamp;solveusingfact.;SPMamp;PSDBTRSindexPSDBTRS;SPMamp;PCDBTRSindexPCDBTRS;SPMamp;PDDBTRSindexPDDBTRS;SPMamp;PZDBTRSindexPZDBTRShlinegeneral;SPMamp;factorize;SPMamp;PSDTTRFindexPSDTTRF;SPMamp;PCDTTRFindexPCDTTRF;SPMamp;PDDTTRFindexPDDTTRF;SPMamp;PZDTTRFindexPZDTTRFtridiagonal;SPMamp;;SPMamp;;SPMamp;;SPMamp;;SPMamp;nopivoting;SPMamp;solveusingfact.;SPMamp;PSDTTRSindexPSDTTRS;SPMamp;PCDTTRSindexPCDTTRS;SPMamp;PDDTTRSindexPDDTTRS;SPMamp;PZDTTRSindexPZDTTRShlinesymmetric;SPMamp;factorize;SPMamp;PSPOTRFindexPSPOTRF;SPMamp;PCPOTRFindexPCPOTRF;SPMamp;PDPOTRFindexPDPOTRF;SPMamp;PZPOTRFindexPZPOTRFHermitian;SPMamp;solveusingfact.;SPMamp;PSPOTRSindexPSPOTRS;SPMamp;PCPOTRSindexPCPOTRS;SPMamp;PDPOTRSindexPDPOTRS;SPMamp;PZPOTRSindexPZPOTRSpositivedefinite;SPMamp;est.cond.number;SPMamp;PSPOCONindexPSPOCON;SPMamp;PCPOCONindexPCPOCON;SPMamp;PDPOCONindexPDPOCON;SPMamp;PZPOCONindexPZPOCON;SPMamp;errorbds.forsoln.;SPMamp;PSPORFSindexPSPORFS;SPMamp;PCPORFSindexPCPORFS;SPMamp;PDPORFSindexPDPORFS;SPMamp;PZPORFSindexPZPORFS;SPMamp;invertusingfact.;SPMamp;PSPOTRIindexPSPOTRI;SPMamp;PCPOTRIindexPCPOTRI;SPMamp;PDPOTRIindexPDPOTRI;SPMamp;PZPOTRIindexPZPOTRI;SPMamp;equilibrate;SPMamp;PSPOEQUindexPSPOEQU;SPMamp;PCPOEQUindexPCPOEQU;SPMamp;PDPOEQUindexPDPOEQU;SPMamp;PZPOEQUindexPZPOEQUhlinesymmetric;SPMamp;factorize;SPMamp;PSPBTRFindexPSPBTRF;SPMamp;PCPBTRFindexPCPBTRF;SPMamp;PDPBTRFindexPDPBTRF;SPMamp;PZPBTRFindexPZPBTRFHermitian;SPMamp;solveusingfact.;SPMamp;PSPBTRSindexPSPBTRS;SPMamp;PCPBTRSindexPCPBTRS;SPMamp;PDPBTRSindexPDPBTRS;SPMamp;PZPBTRSindexPZPBTRSpositivedefinite;SPMamp;;SPMamp;;SPMamp;;SPMamp;;SPMamp;band;SPMamp;;SPMamp;;SPMamp;;SPMamp;;SPMamp;hlinesymmetric;SPMamp;factorize;SPMamp;PSPTTRFindexPSPTTRF;SPMamp;PCPTTRFindexPCPTTRF;SPMamp;PDPTTRFindexPDPTTRF;SPMamp;PZPTTRFindexPZPTTRFHermitian;SPMamp;solveusingfact.;SPMamp;PSPTTRSindexPSPTTRS;SPMamp;PCPTTRSindexPCPTTRS;SPMamp;PDPTTRSindexPDPTTRS;SPMamp;PZPTTRSindexPZPTTRSpositivedefinite;SPMamp;;SPMamp;;SPMamp;;SPMamp;;SPMamp;tridiagonal;SPMamp;;SPMamp;;SPMamp;;SPMamp;;SPMamp;hlinetriangular;SPMamp;solve;SPMamp;PSTRTRSindexPSTRTRS;SPMamp;PCTRTRSindexPCTRTRS;SPMamp;PDTRTRSindexPDTRTRS;SPMamp;PZTRTRSindexPZTRTRS;SPMamp;est.cond.number;SPMamp;PSTRCONindexPSTRCON;SPMamp;PCTRCONindexPCTRCON;SPMamp;PDTRCONindexPDTRCON;SPMamp;PZTRCONindexPZTRCON;SPMamp;errorbds.forsoln.;SPMamp;PSTRRFSindexPSTRRFS;SPMamp;PCTRRFSindexPCTRRFS;SPMamp;PDTRRFSindexPDTRRFS;SPMamp;PZTRRFSindexPZTRRFS;SPMamp;invert;SPMamp;PSTRTRIindexPSTRTRI;SPMamp;PCTRTRIindexPCTRTRI;SPMamp;PDTRTRIindexPDTRTRI;SPMamp;PZTRTRIindexPZTRTRIhlinetabularcenter{table}/; $cached_env_img{$key} = q|table1358|; $key = q/{_inline}$B=U^TU${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline14319|; $key = q/{tabular}||c|c||c|c||c|c||hline_inline$i$_inline;SPMamp;_inline$lambda_i$_inline;SPMamp;ttEERRBD_inline$(i)$_inline;SPMamp;true_inline$|hatlambda_i-lambda_i|$_inline;SPMamp;ttZERRBD_inline$(i)$_inline;SPMamp;true_inline$theta(hatv_i,v_i)$_inlinehline1;SPMamp;_inline$-500.0$_inline;SPMamp;_inline$1.3cdot10^-4$_inline;SPMamp;_inline$9.0cdot10^-6$_inline;SPMamp;_inline$9.8cdot10^-8$_inline;SPMamp;_inline$1.0cdot10^-9$_inline2;SPMamp;_inline$1000.$_inline;SPMamp;_inline$1.7cdot10^-4$_inline;SPMamp;_inline$4.6cdot10^-5$_inline;SPMamp;_inline$1.9cdot10^-5$_inline;SPMamp;_inline$1.2cdot10^-7$_inline3;SPMamp;_inline$1010.$_inline;SPMamp;_inline$1.7cdot10^-4$_inline;SPMamp;_inline$1.0cdot10^-4$_inline;SPMamp;_inline$1.9cdot10^-5$_inline;SPMamp;_inline$1.1cdot10^-7$_inlinehline{tabular}/; $cached_env_img{$key} = q|tabular6044|; $key = q/{displaymath}A=Qleft(arrayccR_1;SPMamp;R_2arrayright),quadmboxif_inline$m;SPMlt;n$_inline,{displaymath}/; $cached_env_img{$key} = q|displaymath13233|; $key = q/{displaymath}r=b-Ax=Qleft(arrayc0c_2arrayright){displaymath}/; $cached_env_img{$key} = q|displaymath13236|; $key = q/{table}[htb]labeltab:evperfminipagelinewidthcentertabular||l|c|c|r|r|r|r|r||hline;SPMamp;Process;SPMamp;Block;SPMamp;multicolumn5c||Valuesof_inline$N$_inlinecline4-8;SPMamp;Grid;SPMamp;Size;SPMamp;1000;SPMamp;2000;SPMamp;3000;SPMamp;4000;SPMamp;5000hlineCrayT3EfootnoteAScaLAPACKprocessisrunningoneachnode.EachnodeconsistsofonecomputationalDECAlphaEV5processor,whichisusedbytheBLAS.;SPMamp;_inline$1times4$_inline;SPMamp;32;SPMamp;38;SPMamp;263;SPMamp;;SPMamp;;SPMamp;PSSYEV;SPMamp;_inline$2times8$_inline;SPMamp;32;SPMamp;15;SPMamp;81;SPMamp;239;SPMamp;552;SPMamp;;SPMamp;_inline$4times16$_inline;SPMamp;32;SPMamp;9;SPMamp;34;SPMamp;102;SPMamp;183;SPMamp;366hlineIBMSP2footnoteAScaLAPACKprocessisrunningoneachnode.EachnodeconsistsofonecomputationalIBMPOWER2590processor,whichisusedbytheBLAS.;SPMamp;_inline$1times4$_inline;SPMamp;50;SPMamp;49;SPMamp;;SPMamp;;SPMamp;;SPMamp;PDSYEV;SPMamp;_inline$2times8$_inline;SPMamp;50;SPMamp;27;SPMamp;121;SPMamp;;SPMamp;;SPMamp;;SPMamp;_inline$4times16$_inline;SPMamp;50;SPMamp;23;SPMamp;75;SPMamp;162;SPMamp;344;SPMamp;hlineIntelXPSMPParagonfootnoteAScaLAPACKprocessisrunningoneachnode.EachnodeconsistsoftwocomputationalInteli860XPRISCprocessors,whichareusedbythemulti-threadedBLAS.;SPMamp;_inline$1times4$_inline;SPMamp;32;SPMamp;189;SPMamp;;SPMamp;;SPMamp;;SPMamp;PDSYEV;SPMamp;_inline$2times8$_inline;SPMamp;32;SPMamp;84;SPMamp;645;SPMamp;;SPMamp;;SPMamp;;SPMamp;_inline$4times16$_inline;SPMamp;32;SPMamp;60;SPMamp;344;SPMamp;980;SPMamp;;SPMamp;hlineClusterofSunUltra2sfootnoteAScaLAPACKprocessisrunningoneachnode.EachnodeconsistsoftwocomputationalUltraSPARC-1processors,butonlyoneprocessorisusedbytheBLAS.;SPMamp;_inline$2times2$_inline;SPMamp;64;SPMamp;81;SPMamp;;SPMamp;;SPMamp;;SPMamp;PDSYEV;SPMamp;_inline$2times4$_inline;SPMamp;64;SPMamp;66;SPMamp;;SPMamp;;SPMamp;;SPMamp;;SPMamp;_inline$2times6$_inline;SPMamp;64;SPMamp;110;SPMamp;933;SPMamp;2308;SPMamp;;SPMamp;hlineBerkeleyNOWfootnoteAScaLAPACKprocessisrunningoneachnode.EachnodeconsistsofonecomputationalUltraSPARC-1processor,whichisusedbytheBLAS.;SPMamp;_inline$1times4$_inline;SPMamp;32;SPMamp;42;SPMamp;;SPMamp;;SPMamp;;SPMamp;PDSYEV;SPMamp;_inline$1times8$_inline;SPMamp;32;SPMamp;30;SPMamp;;SPMamp;;SPMamp;;SPMamp;;SPMamp;_inline$2times8$_inline;SPMamp;32;SPMamp;19;SPMamp;99;SPMamp;285;SPMamp;;SPMamp;;SPMamp;_inline$4times8$_inline;SPMamp;32;SPMamp;15;SPMamp;67;SPMamp;271;SPMamp;829;SPMamp;650hlinetabularcenterminipage{table}/; $cached_env_img{$key} = q|table4085|; $key = q/{_inline}$H=STS^H${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline14070|; $key = q/{_inline}$A=ZLambdaZ^T${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline13904|; $key = q/{displaymath}A=PU^TUP^T;;rmor;;A=PLL^TP^Tmbox(inthesymmetriccase){displaymath}/; $cached_env_img{$key} = q|displaymath12938|; $key = q/{_inline}$hatlambda_i${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline19245|; $key = q/{_inline}$V_2${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline14185|; $key = q/{figure}[h]centerlinepsfigfile=iso_lu.ps,height=2.5in,width=4.0inlabelfig:iso_pgon{figure}/; $cached_env_img{$key} = q|figure3673|; $key = q/{_inline}$tc=lceilNPrceil${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline14571|; $key = q/{displaymath}x=hatVhatSigma^-1hatc_1,{displaymath}/; $cached_env_img{$key} = q|displaymath14109|; $key = q/{_inline}$O(1epsilon)${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18131|; $key = q/{_inline}$BAz_i=lambda_iz_i${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline19135|; $key = q/{_inline}$(p_r,p_c)${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline14629|; $key = q/{_inline}$|tau|^2||v||_2^2=2rmRe(tau${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline14441|; $key = q/{_inline}$SRC~=~1${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline14793|; $key = q/{_inline}${sum_i=1^kbeta_ix_i${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline17830|; $key = q/{_inline}$C=L^-1AL^-T${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline14333|; $key = q/{displaymath}A=left(arrayccc4;SPMamp;16000;SPMamp;170002;SPMamp;5;SPMamp;83;SPMamp;6;SPMamp;10arrayright);;rmand;;b=left(arrayc100.1.1.01arrayright);.;;{displaymath}/; $cached_env_img{$key} = q|displaymath18203|; $key = q/{_inline}$|x_i|_2=1${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline19349|; $key = q/{_inline}$mgeqn${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline13306|; $key = q/{_inline}$AB-lambdaI${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18089|; $key = q/{_inline}$uhat_i${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18979|; $key = q/{_inline}$U=U_1U_2${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline14191|; $key = q/{_inline}$p_r,p_c${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline14501|; $key = q/{displaymath}T=bordermatrix;SPMamp;ncrhfilln;SPMamp;T_11crp-n;SPMamp;0,quadmboxifquadpgeqn,{displaymath}/; $cached_env_img{$key} = q|displaymath13750|; $key = q/{_inline}$F_mm${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline12204|; $key = q/{_inline}$x_i${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline17852|; $key = q/{_inline}$Lambda${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline12784|; $key = q/{_inline}$|hatx-x|_2${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline17588|; $key = q/{_inline}$div${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline17236|; $key = q/{_inline}$Q^TB${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline13730|; $key = q/{_inline}$chi(hatlambda_i,lambda_i)=|sin(hattheta-theta)|${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline19317|; $key = q/{_inline}$|AB|_pleq|A|_pcdot|B|_p${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline17704|; $key = q/{displaymath}E(N,P)=1overPT_rmseq(N)overT(N,P),{displaymath}/; $cached_env_img{$key} = q|displaymath16290|; $key = q/{_inline}$kappa(f,z)${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18051|; $key = q/{_inline}$ttANORM=120231${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline19145|; $key = q/{_inline}$LOC_r${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline12132|; $key = q/{_inline}${x}${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline17620|; $key = q/{_inline}$yleftarrowalphax+y${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline16219|; $key = q/{_inline}$|hatlambda_i-lambda_i|leqttEERRBD${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18736|; $key = q/{_inline}$C_f,N^3${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline12066|; $key = q/{displaymath}theta(hatcalS,calS)=theta(hatx,x)equivarccosfrac|hatx^Tx||hatx|_2cdot|x|_2;;.{displaymath}/; $cached_env_img{$key} = q|displaymath17413|; $key = q/{_inline}$1leqrmRe(tau)leq2${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline14461|; $key = q/{displaymath}Az=lambdaz,quadA=A^T,mboxwhereAmboxisreal.{displaymath}/; $cached_env_img{$key} = q|displaymath12772|; $key = q/{_inline}$BA-lambdaI${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18091|; $key = q/{_inline}$T_rmseq${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline12198|; $key = q/{_inline}$MtimesN${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline15127|; $key = q/{_inline}$hatcalS${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline17614|; $key = q/{_inline}$Z^H${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline13565|; $key = q/{_inline}$t_N${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline12212|; $key = q/{_inline}$lambda${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline12778|; $key = q/{_inline}$hatx=left(arrayc-.3974-.3349.3211arrayright)${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18258|; $key = q/{_inline}$min_xneq0|A^Tx|_2|x|_2${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline17818|; $key = q/{displaymath}theta(hatz_i,z_i)leqttZERRBD(i);.{displaymath}/; $cached_env_img{$key} = q|displaymath18597|; $key = q/{_inline}$2times3${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline14922|; $key = q/{displaymath}A=left(arrayccQ_1;SPMamp;Q_2arrayright)left(arraycR0arrayright),{displaymath}/; $cached_env_img{$key} = q|displaymath13231|; $key = q/{displaymath}I=k~NB+x=(l~P+p)*NB+x,{displaymath}/; $cached_env_img{$key} = q|displaymath14669|; $key = q/{_inline}$K_${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline12126|; $key = q/{displaymath}x=left(arrayc-.3974-.3349.3211arrayright);,;;hatx=left(arrayc-.3968-.3344.3207arrayright);,{displaymath}/; $cached_env_img{$key} = q|displaymath18204|; $key = q/{_inline}$|hatalpha-alpha||alpha|${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline17448|; $key = q/{_inline}${hatu_i,,,iincalI}${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline19007|; $key = q/{tabular}|ll|p4.5in|hlinemulticolumn2|c|Routine;SPMamp;multicolumn1c|Descriptioncline1-2Real;SPMamp;Complex;SPMamp;hlinehline;SPMamp;PCLACGV;SPMamp;Conjugatesacomplexvector.;SPMamp;itPCMAX1;SPMamp;Findstheindexoftheelementwhoserealparthasmaximumabsolutevalue(similartotheLevel1PBLASPCAMAX,butusingtheabsolutevalueoftherealpart).;SPMamp;itPSCSUM1;SPMamp;Formsthe1-normofacomplexvector(similartotheLevel1PBLASPSCASUM,butusingthetrueabsolutevalue).PSDBTRSV;SPMamp;PCDBTRSV;SPMamp;IscalledbyPSDBTRSPCDBTRS.PSDTTRSV;SPMamp;PCDTTRSV;SPMamp;IscalledbyPSDTTRSPCDTTRS.PSGEBD2;SPMamp;PCGEBD2;SPMamp;Reducesageneralrectangularmatrixtorealbidiagonalformbyanorthogonalunitarytransformation(unblockedalgorithm).PSGEHD2;SPMamp;PCGEHD2;SPMamp;ReducesageneralmatrixtoupperHessenbergformbyanorthogonalunitarysimilaritytransformation(unblockedalgorithm).PSGELQ2;SPMamp;PCGELQ2;SPMamp;Computesan_inline$LQ$_inlinefactorizationofageneralrectangularmatrix(unblockedalgorithm).PSGEQL2;SPMamp;PCGEQL2;SPMamp;Computesa_inline$QL$_inlinefactorizationofageneralrectangularmatrix(unblockedalgorithm).PSGEQR2;SPMamp;PCGEQR2;SPMamp;Computesa_inline$QR$_inlinefactorizationofageneralrectangularmatrix(unblockedalgorithm).PSGERQ2;SPMamp;PCGERQ2;SPMamp;Computesan_inline$RQ$_inlinefactorizationofageneralrectangularmatrix(unblockedalgorithm).PSGETF2;SPMamp;PCGETF2;SPMamp;Computesan_inline$LU$_inlinefactorizationofageneralmatrix,usingpartialpivotingwithrowinterchanges(localunblockedalgorithm).PSLABAD;SPMamp;;SPMamp;Returnsthesquarerootoftheunderflowandoverflowthresholdsiftheexponent-rangeisverylarge.PSLABRD;SPMamp;PCLABRD;SPMamp;Reducesthefirst_inline$nb$_inlinerowsandcolumnsofageneralrectangularmatrix_inline$A$_inlinetorealbidiagonalformbyanorthogonalunitarytransformation,andreturnsauxiliarymatricesthatareneededtoapplythetransformationtotheunreducedpartof_inline$A$_inline.PSLACHKIEE;SPMamp;;SPMamp;(real)performsasimplecheckforthefeaturesoftheIEEEstandardPSLACON;SPMamp;PCLACON;SPMamp;Estimatesthe1-normofasquarematrix,usingreversecommunicationforevaluatingmatrix-vectorproducts.PSLACONSB;SPMamp;;SPMamp;(real)looksfortwoconsecutivesmallsubdiagonalelementsPSLACP2;SPMamp;PCLACP2;SPMamp;copiesallorpartofadistributedmatrixtoanotherdistributedmatrixPSLACP3;SPMamp;;SPMamp;(real)copiesfromaglobalparallelarrayintoalocal;SPMamp;;SPMamp;replicatedarrayorviceversa.PSLACPY;SPMamp;PCLACPY;SPMamp;Copiesallorpartofonetwo-dimensionalarraytoanother.PSLAEVSWP;SPMamp;PCLAEVSWP;SPMamp;Movestheeigenvectorsfromwheretheyarecomputed,toaScaLAPACKstandardblockcyclicarray.hline{tabular}/; $cached_env_img{$key} = q|tabular6639|; $key = q/{_inline}$|A|_ij=|a_ij|${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline17566|; $key = q/{_inline}$P_rtimesP_c${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline12182|; $key = q/{_inline}$R_22${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline13484|; $key = q/{table}[htb]labeltab:nodecentertabular||l|c|c||hline;SPMamp;CrayT3E;SPMamp;IBMSP2hlineProcessor;SPMamp;DecAlphaEV5;SPMamp;POWER2590hlineClockspeed(MHz);SPMamp;300;SPMamp;66hlineProcessorspernode;SPMamp;1;SPMamp;1hlineMemorypernode(MB);SPMamp;256;SPMamp;128hlineOperatingsystem;SPMamp;UNICOSmk1.4.1;SPMamp;AIX4.1.4hlineBLAS;SPMamp;LIBSCI3.0;SPMamp;ESSL2.2.2.2hlineBLACS;SPMamp;MPIBLACS_inline$1.1alpha$_inline;SPMamp;MPIBLACS_inline$1.1alpha$_inlinehlineCommunicationSoftware;SPMamp;CrayMPI;SPMamp;POE(2.1.0.12)hlineCcompiler;SPMamp;cc;SPMamp;mpcc(3.1.4.3)hlineCflags;SPMamp;-O3;SPMamp;-qarch=pwr2-qtune=pwr2s-O2hlineFortrancompiler;SPMamp;f90;SPMamp;mpxlf(3.2.4.2)hlineFortranflags;SPMamp;-O3;SPMamp;-qarch=pwr2-qtune=pwr2s-O2hlinePrecision;SPMamp;single(64-bit);SPMamp;double(64-bit)hlinetabularcenter{table}/; $cached_env_img{$key} = q|table3764|; $key = q/{displaymath}|sigmahat_i-sigma_i|leqp(m,n)cdotepsiloncdotsigma_1=ttSERRBD;,{displaymath}/; $cached_env_img{$key} = q|displaymath18941|; $key = q/{_inline}$100asti+j${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline19471|; $key = q/{_inline}$fl(aoplusb)${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline17238|; $key = q/{_inline}$x_i=b_ia_ii${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18164|; $key = q/{displaymath}|lambdahat_i-lambda_i|leqp(n)cdotepsiloncdot|A|_2=ttEERRBD;.{displaymath}/; $cached_env_img{$key} = q|displaymath18599|; $key = q/{_inline}$|hatx-x|_p|hatx|_p${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline17498|; $key = q/{table}[htb]labeltab:svdperfminipagelinewidthcentertabular||l|c|c|r|r|r|r|r||hline;SPMamp;Process;SPMamp;Block;SPMamp;multicolumn5c||Valuesof_inline$N$_inlinecline4-8;SPMamp;Grid;SPMamp;Size;SPMamp;1000;SPMamp;2000;SPMamp;3000;SPMamp;4000;SPMamp;5000hlineCrayT3EfootnoteAScaLAPACKprocessisrunningoneachnode.EachnodeconsistsofonecomputationalDECAlphaEV5processor,whichisusedbytheBLAS.;SPMamp;_inline$1times4$_inline;SPMamp;32;SPMamp;71;SPMamp;508;SPMamp;1810;SPMamp;;SPMamp;PSGESVD;SPMamp;_inline$2times8$_inline;SPMamp;32;SPMamp;24;SPMamp;145;SPMamp;454;SPMamp;1020;SPMamp;1970;SPMamp;_inline$4times16$_inline;SPMamp;32;SPMamp;13;SPMamp;55;SPMamp;141;SPMamp;314;SPMamp;542hlineIBMSP2footnoteAScaLAPACKprocessisrunningoneachnode.EachnodeconsistsofonecomputationalIBMPOWER2590processor,whichisusedbytheBLAS.;SPMamp;_inline$1times4$_inline;SPMamp;50;SPMamp;86;SPMamp;1031;SPMamp;;SPMamp;;SPMamp;PDGESVD;SPMamp;_inline$2times8$_inline;SPMamp;50;SPMamp;37;SPMamp;190;SPMamp;547;SPMamp;;SPMamp;;SPMamp;_inline$4times16$_inline;SPMamp;50;SPMamp;28;SPMamp;104;SPMamp;242;SPMamp;;SPMamp;hlineClusterofSunUltra2sfootnoteAScaLAPACKprocessisrunningoneachnode.EachnodeconsistsoftwocomputationalUltraSPARC-1processors,butonlyoneprocessorisusedbytheBLAS.;SPMamp;_inline$2times2$_inline;SPMamp;64;SPMamp;473;SPMamp;4417;SPMamp;;SPMamp;;SPMamp;PDGESVD;SPMamp;_inline$2times4$_inline;SPMamp;64;SPMamp;283;SPMamp;2375;SPMamp;;SPMamp;;SPMamp;;SPMamp;_inline$2times6$_inline;SPMamp;64;SPMamp;225;SPMamp;1819;SPMamp;;SPMamp;;SPMamp;hlineBerkeleyNOWfootnoteAScaLAPACKprocessisrunningoneachnode.EachnodeconsistsofonecomputationalUltraSPARC-1processor,whichisusedbytheBLAS.;SPMamp;_inline$1times4$_inline;SPMamp;32;SPMamp;153;SPMamp;1250;SPMamp;;SPMamp;;SPMamp;PDGESVD;SPMamp;_inline$1times8$_inline;SPMamp;32;SPMamp;88;SPMamp;671;SPMamp;1999;SPMamp;;SPMamp;;SPMamp;_inline$2times8$_inline;SPMamp;32;SPMamp;46;SPMamp;324;SPMamp;1004;SPMamp;;SPMamp;;SPMamp;_inline$4times8$_inline;SPMamp;32;SPMamp;25;SPMamp;189;SPMamp;584;SPMamp;;SPMamp;hlinetabularcenterminipage{table}/; $cached_env_img{$key} = q|table4106|; $key = q/{_inline}$hatx_i${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline17850|; $key = q/{displaymath}left(arrayrrrrrrrrr19;SPMamp;3;SPMamp;1;SPMamp;12;SPMamp;1;SPMamp;16;SPMamp;1;SPMamp;3;SPMamp;11-19;SPMamp;3;SPMamp;1;SPMamp;12;SPMamp;1;SPMamp;16;SPMamp;1;SPMamp;3;SPMamp;11-19;SPMamp;-3;SPMamp;1;SPMamp;12;SPMamp;1;SPMamp;16;SPMamp;1;SPMamp;3;SPMamp;11-19;SPMamp;-3;SPMamp;-1;SPMamp;12;SPMamp;1;SPMamp;16;SPMamp;1;SPMamp;3;SPMamp;11-19;SPMamp;-3;SPMamp;-1;SPMamp;-12;SPMamp;1;SPMamp;16;SPMamp;1;SPMamp;3;SPMamp;11-19;SPMamp;-3;SPMamp;-1;SPMamp;-12;SPMamp;-1;SPMamp;16;SPMamp;1;SPMamp;3;SPMamp;11-19;SPMamp;-3;SPMamp;-1;SPMamp;-12;SPMamp;-1;SPMamp;-16;SPMamp;1;SPMamp;3;SPMamp;11-19;SPMamp;-3;SPMamp;-1;SPMamp;-12;SPMamp;-1;SPMamp;-16;SPMamp;-1;SPMamp;3;SPMamp;11-19;SPMamp;-3;SPMamp;-1;SPMamp;-12;SPMamp;-1;SPMamp;-16;SPMamp;-1;SPMamp;-3;SPMamp;11arrayright)left(arrayrx_1x_2x_3x_4x_5x_6x_7x_8x_9arrayright)=left(arrayr001000000arrayright){displaymath}/; $cached_env_img{$key} = q|displaymath12392|; $key = q/{_inline}$AastX=B${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline12404|; $key = q/{_inline}$leq${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline14970|; $key = q/{_inline}$4.3cdot10^-7${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18262|; $key = q/{_inline}$calS${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline17602|; $key = q/{_inline}$sigma_min(A)${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline17810|; $key = q/{displaymath}A=Qleft(arrayc0Larrayright),quadmboxif_inline$mgeqn$_inline,{displaymath}/; $cached_env_img{$key} = q|displaymath13582|; $key = q/{displaymath}A=USigmaV^T,quad(A=USigmaV^Hquadmboxinthecomplexcase),{displaymath}/; $cached_env_img{$key} = q|displaymath12804|; $key = q/{_inline}$(t_mt_f)${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline16374|; $key = q/{_inline}$|r|_2^2${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline13320|; $key = q/{displaymath}theta(hatcalS,calS)mathrelraisebox-.75ex_inline$mathopsimlimits^textstyle;SPMlt;$_inlinefracp(n)epsilon|A|_2gap_calI,{displaymath}/; $cached_env_img{$key} = q|displaymath18601|; $key = q/{_inline}$P_c${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline12162|; $key = q/{_inline}$v_1=1${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline14449|; $key = q/{figure}[h]centerlinehboxpsfigfigure=FIGbcycliccol.ps,height=1.4in,width=1.4inhspace*1.0inpsfigfigure=FIGbcyclicroc.ps,height=1.4in,width=1.4inlabelfig:bcycliccol{figure}/; $cached_env_img{$key} = q|figure2387|; $key = q/{displaymath}x=hatx=left(arrayc38.4921.59-23.88arrayright);;,{displaymath}/; $cached_env_img{$key} = q|displaymath18465|; $key = q/{_inline}$mboxrank(A);SPMlt;min(m,n)${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline12728|; $key = q/{eqnarray*}|hatA-A|_infty=2.44;,;;SPMamp;displaystylefrac|hatA-A|_infty|A|_infty=.0976;,;;SPMamp;frac|hatA-A|_infty|hatA|_infty=.1008|hatA-A|_2=1.75;,;;SPMamp;displaystylefrac|hatA-A|_2|A|_2=.1007;,;;SPMamp;frac|hatA-A|_2|hatA|_2=.1020|hatA-A|_1=1.83;,;;SPMamp;displaystylefrac|hatA-A|_1|A|_1=.0963;,;;SPMamp;frac|hatA-A|_1|hatA|_1=.0975|hatA-A|_F=1.87;,;;SPMamp;displaystylefrac|hatA-A|_F|A|_F=.1075;,;;SPMamp;frac|hatA-A|_F|hatA|_F=.1082{eqnarray*}/; $cached_env_img{$key} = q|eqnarray4739|; $key = q/{_inline}$sigma_i${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline12826|; $key = q/{displaymath}frac|x-hatx|_infty|hatx|_inftyleqomega_ccdotkappa_c(A,b,hatx).{displaymath}/; $cached_env_img{$key} = q|displaymath18211|; $key = q/{figure}[thb]centertabularccc|ccc|cmulticolumn7cProcessesmulticolumn3c0;SPMamp;multicolumn3c1;SPMamp;2hlineF;SPMamp;F;SPMamp;F;SPMamp;F;SPMamp;F;SPMamp;F;SPMamp;FF;SPMamp;F;SPMamp;F;SPMamp;F;SPMamp;F;SPMamp;F;SPMamp;FF;SPMamp;F;SPMamp;F;SPMamp;F;SPMamp;F;SPMamp;F;SPMamp;FF;SPMamp;F;SPMamp;F;SPMamp;F;SPMamp;F;SPMamp;F;SPMamp;F_inline$ast$_inline;SPMamp;_inline$ast$_inline;SPMamp;_inline$a_13$_inline;SPMamp;_inline$a_24$_inline;SPMamp;_inline$a_35$_inline;SPMamp;_inline$a_46$_inline;SPMamp;_inline$a_57$_inline_inline$ast$_inline;SPMamp;_inline$a_12$_inline;SPMamp;_inline$a_23$_inline;SPMamp;_inline$a_34$_inline;SPMamp;_inline$a_45$_inline;SPMamp;_inline$a_56$_inline;SPMamp;_inline$a_67$_inline_inline$a_11$_inline;SPMamp;_inline$a_22$_inline;SPMamp;_inline$a_33$_inline;SPMamp;_inline$a_44$_inline;SPMamp;_inline$a_55$_inline;SPMamp;_inline$a_66$_inline;SPMamp;_inline$a_77$_inline_inline$a_21$_inline;SPMamp;_inline$a_32$_inline;SPMamp;_inline$a_43$_inline;SPMamp;_inline$a_54$_inline;SPMamp;_inline$a_65$_inline;SPMamp;_inline$a_76$_inline;SPMamp;_inline$ast$_inline_inline$a_31$_inline;SPMamp;_inline$a_42$_inline;SPMamp;_inline$a_53$_inline;SPMamp;_inline$a_64$_inline;SPMamp;_inline$a_75$_inline;SPMamp;_inline$ast$_inline;SPMamp;_inline$ast$_inlinetabularcenterlabelfig:nonsymmbandpp7{figure}/; $cached_env_img{$key} = q|figure2886|; $key = q/{_inline}$lambda_1leqcdotsleqlambda_n${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline19079|; $key = q/{_inline}$Q_2${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline13276|; $key = q/{displaymath}T=bordermatrix;SPMamp;p-n;SPMamp;ncrn;SPMamp;0;SPMamp;T_12,quadmboxifquadnleqp,{displaymath}/; $cached_env_img{$key} = q|displaymath13659|; $key = q/{_inline}$BQ^T${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline13821|; $key = q/{eqnarray*}theta(hatcalS,calS);SPMamp;equiv;SPMamp;max_hatxinhatcalSatophatxneq0min_xincalSatopxneq0theta(hatx,x);;.{eqnarray*}/; $cached_env_img{$key} = q|eqnarray4963|; $key = q/{displaymath}A=PU^HDUP^T;;rmor;;A=PLDL^HP^Tmbox(intheHermitiancase),{displaymath}/; $cached_env_img{$key} = q|displaymath12941|; $key = q/{displaymath}A=left(arrayccccccca_11;SPMamp;a_12;SPMamp;0;SPMamp;0;SPMamp;0;SPMamp;0;SPMamp;0a_21;SPMamp;a_22;SPMamp;a_23;SPMamp;0;SPMamp;0;SPMamp;0;SPMamp;00;SPMamp;a_32;SPMamp;a_33;SPMamp;a_34;SPMamp;0;SPMamp;0;SPMamp;00;SPMamp;0;SPMamp;a_43;SPMamp;a_44;SPMamp;a_45;SPMamp;0;SPMamp;00;SPMamp;0;SPMamp;0;SPMamp;a_54;SPMamp;a_55;SPMamp;a_56;SPMamp;00;SPMamp;0;SPMamp;0;SPMamp;0;SPMamp;a_65;SPMamp;a_66;SPMamp;a_670;SPMamp;0;SPMamp;0;SPMamp;0;SPMamp;0;SPMamp;a_76;SPMamp;a_77arrayright){displaymath}/; $cached_env_img{$key} = q|displaymath15653|; $key = q/{_inline}$a_25${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline14869|; $key = q/{figure}[thb]centertabularccc|ccc|cmulticolumn7cProcessesmulticolumn3c0;SPMamp;multicolumn3c1;SPMamp;2hline_inline$ast$_inline;SPMamp;_inline$ast$_inline;SPMamp;_inline$a_31$_inline;SPMamp;_inline$a_42$_inline;SPMamp;_inline$a_53$_inline;SPMamp;_inline$a_64$_inline;SPMamp;_inline$a_75$_inline_inline$ast$_inline;SPMamp;_inline$a_21$_inline;SPMamp;_inline$a_32$_inline;SPMamp;_inline$a_43$_inline;SPMamp;_inline$a_54$_inline;SPMamp;_inline$a_65$_inline;SPMamp;_inline$a_76$_inline_inline$a_11$_inline;SPMamp;_inline$a_22$_inline;SPMamp;_inline$a_33$_inline;SPMamp;_inline$a_44$_inline;SPMamp;_inline$a_55$_inline;SPMamp;_inline$a_66$_inline;SPMamp;_inline$a_77$_inlinetabularcenterlabelfig:Usymmband7{figure}/; $cached_env_img{$key} = q|figure2975|; $key = q/{_inline}$A=USigmaV^T${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline14119|; $key = q/{_inline}$A=(QU)SigmaV^T${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline14231|; $key = q/{_inline}$p_c${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline14637|; $key = q/{displaymath}A=left(arrayccc4;SPMamp;3;SPMamp;52;SPMamp;5;SPMamp;83;SPMamp;6;SPMamp;104;SPMamp;5;SPMamp;11arrayright);,{displaymath}/; $cached_env_img{$key} = q|displaymath18831|; $key = q/{_inline}$oplus${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline17228|; $key = q/{table}[htb]labelstandardflopcountvspace0.1incentertabular||l|l|r|c|c||hlineDriver;SPMamp;Options;SPMamp;multicolumn1c|_inline$C_f$_inline;SPMamp;multicolumn1c|_inline$C_v$_inline;SPMamp;multicolumn1c||_inline$C_m$_inlinehlinePxGESV;SPMamp;1righthandside;SPMamp;_inline$23$_inline;SPMamp;_inline$3+14,log_2P$_inline;SPMamp;_inline$NB,(6+log_2P)$_inlinePxPOSV;SPMamp;1righthandside;SPMamp;_inline$13$_inline;SPMamp;_inline$2+12,log_2P$_inline;SPMamp;_inline$4+log_2P$_inlinePxGELS;SPMamp;1righthandside;SPMamp;_inline$43$_inline;SPMamp;_inline$3+log_2P$_inline;SPMamp;_inline$2,(NB,log_2P+1)$_inlinePxSYEVX;SPMamp;eigenvaluesonly;SPMamp;_inline$43$_inline;SPMamp;_inline$52log_2P$_inline;SPMamp;_inline$172,NB+2$_inlinePxSYEVX;SPMamp;eigenvaluesandeigenvectors;SPMamp;_inline$103$_inline;SPMamp;_inline$5log_2P$_inline;SPMamp;_inline$172,NB+2$_inlinePxSYEV;SPMamp;eigenvaluesonly;SPMamp;_inline$43$_inline;SPMamp;_inline$52log_2P$_inline;SPMamp;_inline$172,NB+2$_inlinePxSYEV;SPMamp;eigenvaluesandeigenvectors;SPMamp;_inline$223$_inline;SPMamp;_inline$5log_2P$_inline;SPMamp;_inline$172,NB+2$_inlinePxGESVD;SPMamp;singularvaluesonly;SPMamp;263;SPMamp;_inline$10log_2P$_inline;SPMamp;_inline$17NB$_inlinePxGESVD;SPMamp;singularvaluesandleftand;SPMamp;;SPMamp;;SPMamp;;SPMamp;rightsingularvectors;SPMamp;383;SPMamp;_inline$14log_2P$_inline;SPMamp;_inline$17NB$_inlinePxLAHQR;SPMamp;eigenvaluesonly;SPMamp;_inline$5$_inline;SPMamp;_inline$92,(sqrtP)astlog_2P$_inline;SPMamp;_inline$9,(2+log_2P),N$_inline;SPMamp;;SPMamp;;SPMamp;_inline$+8,NNB$_inline;SPMamp;PxLAHQR;SPMamp;fullSchurform;SPMamp;_inline$18$_inline;SPMamp;_inline$92,(sqrtP)astlog_2P$_inline;SPMamp;_inline$9,(2+log_2P),N$_inline;SPMamp;;SPMamp;;SPMamp;_inline$+8,NNB$_inline;SPMamp;hlinetabularcenter{table}/; $cached_env_img{$key} = q|table3908|; $key = q/{figure}[htb]centerlinepsfigfigure=SoftwareHierarchy.eps,width=3inlabelfig:hierarchy{figure}/; $cached_env_img{$key} = q|figure398|; $key = q/{_inline}$|A|${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline19355|; $key = q/{_inline}$kappa_2(A)=sigma_max(A)sigma_min(A)${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18531|; $key = q/{_inline}$LLD_B${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline12120|; $key = q/{_inline}$B^-1${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline13720|; $key = q/{_inline}$|x|_2${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline12704|; $key = q/{displaymath}theta(uhat_i,u_i)mathrelraisebox-.75ex_inline$mathopsimlimits^textstyle;SPMlt;$_inlinefracp(m,n)epsilon|A|_2gap_i=ttUERRBD(i){displaymath}/; $cached_env_img{$key} = q|displaymath18942|; $key = q/{displaymath}A=left(arrayccc100000;SPMamp;10099;SPMamp;210910099;SPMamp;100020;SPMamp;100122109;SPMamp;10112;SPMamp;-48461arrayright);;rmand;;B=left(arrayccc99;SPMamp;10;SPMamp;210;SPMamp;100;SPMamp;102;SPMamp;10;SPMamp;100arrayright){displaymath}/; $cached_env_img{$key} = q|displaymath19055|; $key = q/{_inline}$j^th${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline19432|; $key = q/{_inline}$f_pq(n)${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline17716|; $key = q/{_inline}$_f${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline12068|; $key = q/{_inline}$_rmseq${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline12200|; $key = q/{_inline}$C_f${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline16191|; $key = q/{_inline}$mggn${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline14213|; $key = q/{_inline}$lcm(P_r,P_c)${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline12108|; $key = q/{_inline}$B=U_2SigmaV_2^T${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline14181|; $key = q/{_inline}$hatX=[hatx_1,...,hatx_k]${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline17904|; $key = q/{_inline}$betaneq1${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline17592|; $key = q/{figure}[thb]centerlinepsfigfigure=FIGgauss.ps,height=2.2in,width=2.2inlabelfig:gauss{figure}/; $cached_env_img{$key} = q|figure2362|; $key = q/{_inline}$kappa_2(B)${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline19277|; $key = q/{_inline}$^T${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline13041|; $key = q/{displaymath}|hatlambda_i-lambda_i|leqttEERRBD;;.{displaymath}/; $cached_env_img{$key} = q|displaymath18596|; $key = q/{_inline}$_v${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline12076|; $key = q/{displaymath}x=left(arrayc11009arrayright);;rmand;;hatx=left(arrayc1.19911arrayright){displaymath}/; $cached_env_img{$key} = q|displaymath17414|; $key = q/{_inline}$C_v${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline16270|; $key = q/{_inline}$max(epsilon,10^-5)${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline17220|; $key = q/{tabular}[t]p4.0inunderlinermlargeNotationrmUnderlinedparametersareoutputarguments.Ifaroutineisunderlineditisafunctionthatreturnsavalue.Theprefixbfpusuallystandsforprocess.Otherstandardnotationsare:273274275276277278{tabular}/; $cached_env_img{$key} = q|tabular7153|; $key = q/{_inline}$chi(hatlambda_i,lambda_i)${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline19329|; $key = q/{_inline}$R_1${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline13286|; $key = q/{_inline}$K;SPMlt;P_c${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline15265|; $key = q/{_inline}$P_c=4${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline14509|; $key = q/{_inline}$betaapprox.01${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline17666|; $key = q/{_inline}$4.6cdot10^-8${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18366|; $key = q/{_inline}$p(n)epsilon${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18543|; $key = q/{_inline}$5times5${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline14857|; $key = q/{_inline}$A=ZTZ^H${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline14002|; $key = q/{_inline}$min(min_jneqn(|sigma_n-sigma_j|,sigma_n))${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18999|; $key = q/{_inline}$LDL^T${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline13039|; $key = q/{_inline}$Z^TBZ=I${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline12903|; $key = q/{_inline}$P_r=2${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline12422|; $key = q/{_inline}$t_Napprox8,t_N2${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline17101|; $key = q/{_inline}$lceilNPrceil${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline15096|; $key = q/{_inline}$x-hatx${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18296|; $key = q/{_inline}$geq${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline14966|; $key = q/{_inline}$P_r=P_c=2${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline14651|; $key = q/{_inline}$lambda_i=tantheta${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline19313|; $key = q/{_inline}$_c${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline12114|; $key = q/{_inline}$hatZ+deltahatZ${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18686|; $key = q/{_inline}$downarrow${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline19869|; $key = q/{displaymath}frac|hatx-x|_infty|x|_inftyleqttFERR{displaymath}/; $cached_env_img{$key} = q|displaymath18205|; $key = q/{table}[htb]labeltab:lltperfminipagelinewidthcentertabular||l|c|c|r|r|r|r|r||hline;SPMamp;Process;SPMamp;Block;SPMamp;multicolumn5c||Valuesof_inline$N$_inlinecline4-8;SPMamp;Grid;SPMamp;Size;SPMamp;2000;SPMamp;5000;SPMamp;7500;SPMamp;10000;SPMamp;15000hlineCrayT3EfootnoteAScaLAPACKprocessisrunningoneachnode.EachnodeconsistsofonecomputationalDECAlphaEV5processor,whichisusedbytheBLAS.;SPMamp;_inline$2times2$_inline;SPMamp;32;SPMamp;822;SPMamp;988;SPMamp;1018;SPMamp;1039;SPMamp;PSPOSV;SPMamp;_inline$4times4$_inline;SPMamp;32;SPMamp;2203;SPMamp;3322;SPMamp;3586;SPMamp;3828;SPMamp;4050;SPMamp;_inline$8times8$_inline;SPMamp;32;SPMamp;4664;SPMamp;9520;SPMamp;11534;SPMamp;12809;SPMamp;14048hlineIBMSP2footnoteAScaLAPACKprocessisrunningoneachnode.EachnodeconsistsofonecomputationalIBMPOWER2590processor,whichisusedbytheBLAS.;SPMamp;_inline$2times2$_inline;SPMamp;50;SPMamp;462;SPMamp;615;SPMamp;;SPMamp;;SPMamp;PDPOSV;SPMamp;_inline$4times4$_inline;SPMamp;50;SPMamp;1081;SPMamp;1811;SPMamp;2118;SPMamp;2312;SPMamp;;SPMamp;_inline$8times8$_inline;SPMamp;50;SPMamp;1807;SPMamp;4431;SPMamp;5727;SPMamp;6826;SPMamp;8084hlineIntelXPSMPParagonfootnoteAScaLAPACKprocessisrunningoneachnode.EachnodeconsistsoftwocomputationalInteli860XPRISCprocessors,whichareusedbythemulti-threadedBLAS.;SPMamp;_inline$2times2$_inline;SPMamp;32;SPMamp;193;SPMamp;255;SPMamp;;SPMamp;;SPMamp;PDPOSV;SPMamp;_inline$4times4$_inline;SPMamp;32;SPMamp;499;SPMamp;822;SPMamp;942;SPMamp;964;SPMamp;;SPMamp;_inline$8times8$_inline;SPMamp;32;SPMamp;1092;SPMamp;2258;SPMamp;2829;SPMamp;3220;SPMamp;3725hlineClusterofSunUltra2sfootnoteAScaLAPACKprocessisrunningoneachnode.EachnodeconsistsoftwocomputationalUltraSPARC-1processors,butonlyoneprocessorisusedbytheBLAS.;SPMamp;_inline$2times2$_inline;SPMamp;64;SPMamp;177;SPMamp;347;SPMamp;397;SPMamp;;SPMamp;PDPOSV;SPMamp;_inline$2times4$_inline;SPMamp;64;SPMamp;141;SPMamp;415;SPMamp;569;SPMamp;669;SPMamp;;SPMamp;_inline$3times4$_inline;SPMamp;64;SPMamp;100;SPMamp;410;SPMamp;610;SPMamp;755;SPMamp;hlineBerkeleyNOWfootnoteAScaLAPACKprocessisrunningoneachnode.EachnodeconsistsofonecomputationalUltraSPARC-1processor,whichisusedbytheBLAS.;SPMamp;_inline$2times2$_inline;SPMamp;32;SPMamp;341;SPMamp;;SPMamp;;SPMamp;;SPMamp;PDPOSV;SPMamp;_inline$2times4$_inline;SPMamp;32;SPMamp;491;SPMamp;703;SPMamp;769;SPMamp;;SPMamp;;SPMamp;_inline$4times4$_inline;SPMamp;32;SPMamp;828;SPMamp;1301;SPMamp;1446;SPMamp;1542;SPMamp;;SPMamp;_inline$4times8$_inline;SPMamp;32;SPMamp;890;SPMamp;1914;SPMamp;2332;SPMamp;2602;SPMamp;2889;SPMamp;_inline$8times8$_inline;SPMamp;32;SPMamp;1487;SPMamp;3263;SPMamp;4047;SPMamp;4667;SPMamp;5369hlinetabularcenterminipage{table}/; $cached_env_img{$key} = q|table3969|; $key = q/{_inline}$(A+E)hatx=b+f${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18075|; $key = q/{_inline}$p(n)=O(n^3)${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline17934|; $key = q/{_inline}$|x|_2=1${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline17596|; $key = q/{_inline}$|(A+E)hatx-(b+f)|_2${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18519|; $key = q/{eqnarray*}theta(hatv_i,v_i);SPMamp;leq;SPMamp;ttVERRBD(i)theta(hatu_i,u_i);SPMamp;leq;SPMamp;ttUERRBD(i);;.{eqnarray*}/; $cached_env_img{$key} = q|eqnarray5899|; $key = q/{displaymath}|hatlambda_i-lambda_i|leqttEERRBD(i);.{displaymath}/; $cached_env_img{$key} = q|displaymath19053|; $key = q/{_inline}$|hatA-A|_p${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline17524|; $key = q/{_inline}$rmalg${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18011|; $key = q/{_inline}$chi${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline19333|; $key = q/{_inline}$T_11${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline13799|; $key = q/{displaymath}A=left(arrayccccccca_11;SPMamp;a_12;SPMamp;a_13;SPMamp;0;SPMamp;0;SPMamp;0;SPMamp;0a_21;SPMamp;a_22;SPMamp;a_23;SPMamp;a_24;SPMamp;0;SPMamp;0;SPMamp;0a_31;SPMamp;a_32;SPMamp;a_33;SPMamp;a_34;SPMamp;a_35;SPMamp;0;SPMamp;00;SPMamp;a_42;SPMamp;a_43;SPMamp;a_44;SPMamp;a_45;SPMamp;a_46;SPMamp;00;SPMamp;0;SPMamp;a_53;SPMamp;a_54;SPMamp;a_55;SPMamp;a_56;SPMamp;a_570;SPMamp;0;SPMamp;0;SPMamp;a_64;SPMamp;a_65;SPMamp;a_66;SPMamp;a_670;SPMamp;0;SPMamp;0;SPMamp;0;SPMamp;a_75;SPMamp;a_76;SPMamp;a_77arrayright){displaymath}/; $cached_env_img{$key} = q|displaymath15326|; $key = q/{_inline}$kappa_2(H)${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18786|; $key = q/{_inline}$U_1^Tb${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline14267|; $key = q/{_inline}$A=ZLambdaZ^H${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18618|; $key = q/{_inline}$epsilon${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline17202|; $key = q/{_inline}$sqrtP${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline17063|; $key = q/{_inline}$|delta_i|=O(epsilon)|z_i|${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18176|; $key = q/{_inline}$hatA${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline17526|; $key = q/{_inline}$Sigma=mboxrmdiag(sigma_1,ldotssigma_r)${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline14125|; $key = q/{figure}[htb]centerlinepsfigfigure=mat5.ps,height=1.8in,width=4.0in{figure}/; $cached_env_img{$key} = q|figure2490|; $key = q/{_inline}$1timesP_c${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline15233|; $key = q/{_inline}$C=L^-1AL^-H${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline14339|; $key = q/{_inline}$kappa_infty(A)${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18121|; $key = q/{_inline}$P_r=P_c=lfloorsqrtPrfloor${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline17051|; $key = q/{_inline}$M_A=N_A=5${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline14851|; $key = q/{table}[htb]centertabular||l|l|l|l||hlineDESC_();SPMamp;Symbolic;SPMamp;Scope;SPMamp;Definition;SPMamp;Name;SPMamp;;SPMamp;hline1;SPMamp;DTYPE_B;SPMamp;(global);SPMamp;Thedescriptortype(DTYPE_B=502);SPMamp;;SPMamp;;SPMamp;for_inline$P_rtimes1$_inlineprocessgridfor;SPMamp;;SPMamp;;SPMamp;block-rowdistributedmatrices.2;SPMamp;CTXT_B;SPMamp;(global);SPMamp;TheBLACScontexthandle,indicatingthe;SPMamp;;SPMamp;;SPMamp;BLACSprocessgridoverwhichtheglobal;SPMamp;;SPMamp;;SPMamp;matrixBisdistributed.Thecontextitself;SPMamp;;SPMamp;;SPMamp;isglobal,butthehandle(theintegervalue);SPMamp;;SPMamp;;SPMamp;mayvary.3;SPMamp;M_B;SPMamp;(global);SPMamp;ThenumberofrowsintheglobalmatrixB.4;SPMamp;MB_B;SPMamp;(global);SPMamp;Therowblocksize.5;SPMamp;RSRC_B;SPMamp;(global);SPMamp;Theprocessrowoverwhichthefirst;SPMamp;;SPMamp;;SPMamp;rowoftheglobalmatrixBisdistributed.6;SPMamp;LLD_B;SPMamp;(local);SPMamp;Theleadingdimensionofthelocal;SPMamp;;SPMamp;;SPMamp;array.LLD_B_inline$geq$_inlineMAX(1,LOCr(M_B)).;SPMamp;;SPMamp;;SPMamp;Forthetridiagonalsubroutines,this;SPMamp;;SPMamp;;SPMamp;entryisignored.7;SPMamp;;SPMamp;;SPMamp;Reservedhlinetabularcenter{table}/; $cached_env_img{$key} = q|table3220|; $key = q/{tabular}llmulticolumn2lLargebfSimpleDriversmulticolumn2cbfSimpleDriverRoutinesforLinearEquations[0.05in]MatrixType;SPMamp;RoutinehlineGeneral;SPMamp;3;SPMamp;4;SPMamp;GeneralBand;SPMamp;5(nopivoting);SPMamp;6;SPMamp;GeneralBand;SPMamp;7(partialpivoting);SPMamp;8;SPMamp;GeneralTridiagonal;SPMamp;9(nopivoting);SPMamp;10;SPMamp;SymmetricHermitian;SPMamp;11PositiveDefinite;SPMamp;12;SPMamp;SymmetricHermitian;SPMamp;13PositiveDefiniteBand;SPMamp;14;SPMamp;SymmetricHermitian;SPMamp;15PositiveDefiniteTridiagonal;SPMamp;16;SPMamp;multicolumn2cbfSimpleDriverRoutinesforStandardandGeneralizedLinearLeastSquaresProblems[0.05in]ProblemType;SPMamp;RoutinehlineSolveUsingOrthogonalFactor,;SPMamp;17AssumingFullRank;SPMamp;18;SPMamp;multicolumn2cbfSimpleDriverRoutinesforStandardEigenvalueandSingularValueProblems[0.025in]MatrixProblemType;SPMamp;RoutinehlineSymmetricHermitian;SPMamp;19Eigenvaluesvectors;SPMamp;General;SPMamp;20SingularValuesVectors;SPMamp;21;SPMamp;{tabular}/; $cached_env_img{$key} = q|tabular6871|; $key = q/{displaymath}|r|_2=|b-Ax|_2=|c_2|_2.{displaymath}/; $cached_env_img{$key} = q|displaymath13237|; $key = q/{displaymath}rmalg(z)-f(z)=f(z+delta)-f(z)approxf'(z)cdotdelta.{displaymath}/; $cached_env_img{$key} = q|displaymath17997|; $key = q/{_inline}$|hatalpha-alpha|${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline17444|; $key = q/{_inline}$2cdotomega_infty${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18300|; $key = q/{_inline}$1leqtauleq2${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline14453|; $key = q/{_inline}$kappa(A)=|A|.|A^-1|${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline13057|; $key = q/{displaymath}A=left(arraycc0;SPMamp;Rarrayright)Q,quadmboxif_inline$mleqn$_inline.{displaymath}/; $cached_env_img{$key} = q|displaymath13583|; $key = q/{_inline}$A=U_1BV_1^T${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline14153|; $key = q/{_inline}$v_i${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline12842|; $key = q/{_inline}$kappa_2${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18563|; $key = q/{_inline}$B^-1A${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline13718|; $key = q/{table}[htb]centertabular||l|l||hlineDESC_(DTYPE_);SPMamp;Designationhline1;SPMamp;densematrices501;SPMamp;narrowbandandtridiagonalcoefficientmatrices502;SPMamp;narrowbandandtridiagonalright-hand-sidematrices601;SPMamp;out-of-coredensematriceshlinetabularcenter{table}/; $cached_env_img{$key} = q|table2340|; $key = q/{displaymath}A=left(arrayccL;SPMamp;0arrayright)Q=left(arrayccL;SPMamp;0arrayright)left(arraycQ_1Q_2arrayright)=LQ_1,quadmboxif_inline$mlen$_inline,{displaymath}/; $cached_env_img{$key} = q|displaymath13356|; $key = q/{_inline}$omega_ccdotkappa_c(A,b,hatx)${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18360|; $key = q/{table}[h]labeltab:bcmap0centertabular||l|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c||hline_inline$I$_inline;SPMamp;1;SPMamp;2;SPMamp;3;SPMamp;4;SPMamp;5;SPMamp;6;SPMamp;7;SPMamp;8;SPMamp;9;SPMamp;10;SPMamp;11;SPMamp;12;SPMamp;13;SPMamp;14;SPMamp;15;SPMamp;16hline_inline$p$_inline;SPMamp;1;SPMamp;1;SPMamp;1;SPMamp;0;SPMamp;0;SPMamp;0;SPMamp;1;SPMamp;1;SPMamp;1;SPMamp;0;SPMamp;0;SPMamp;0;SPMamp;1;SPMamp;1;SPMamp;1;SPMamp;0_inline$l$_inline;SPMamp;0;SPMamp;0;SPMamp;0;SPMamp;0;SPMamp;0;SPMamp;0;SPMamp;1;SPMamp;1;SPMamp;1;SPMamp;1;SPMamp;1;SPMamp;1;SPMamp;2;SPMamp;2;SPMamp;2;SPMamp;2_inline$x$_inline;SPMamp;1;SPMamp;2;SPMamp;3;SPMamp;1;SPMamp;2;SPMamp;3;SPMamp;1;SPMamp;2;SPMamp;3;SPMamp;1;SPMamp;2;SPMamp;3;SPMamp;1;SPMamp;2;SPMamp;3;SPMamp;1hline_inline$l*NB+x$_inline;SPMamp;1;SPMamp;2;SPMamp;3;SPMamp;1;SPMamp;2;SPMamp;3;SPMamp;4;SPMamp;5;SPMamp;6;SPMamp;4;SPMamp;5;SPMamp;6;SPMamp;7;SPMamp;8;SPMamp;9;SPMamp;7hlinetabularcenter{table}/; $cached_env_img{$key} = q|table2461|; $key = q/{_inline}$2^-532^-24approx10^-9${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline17280|; $key = q/{_inline}$lastNB+x${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline14721|; $key = q/{_inline}$|f|_infty|b|_infty${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18079|; $key = q/{_inline}$tan(theta)${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18551|; $key = q/{_inline}$MB_A=NB_A=2${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline14855|; $key = q/{_inline}$H_1,H_2,ldots,H_K${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline16802|; $key = q/{table}[h]labeltab:cmapcentertabular||l|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c||hline_inline$I$_inline;SPMamp;1;SPMamp;2;SPMamp;3;SPMamp;4;SPMamp;5;SPMamp;6;SPMamp;7;SPMamp;8;SPMamp;9;SPMamp;10;SPMamp;11;SPMamp;12;SPMamp;13;SPMamp;14;SPMamp;15;SPMamp;16hline_inline$p$_inline;SPMamp;0;SPMamp;1;SPMamp;0;SPMamp;1;SPMamp;0;SPMamp;1;SPMamp;0;SPMamp;1;SPMamp;0;SPMamp;1;SPMamp;0;SPMamp;1;SPMamp;0;SPMamp;1;SPMamp;0;SPMamp;1_inline$l$_inline;SPMamp;0;SPMamp;0;SPMamp;1;SPMamp;1;SPMamp;2;SPMamp;2;SPMamp;3;SPMamp;3;SPMamp;4;SPMamp;4;SPMamp;5;SPMamp;5;SPMamp;6;SPMamp;6;SPMamp;7;SPMamp;7_inline$x$_inline;SPMamp;1;SPMamp;1;SPMamp;1;SPMamp;1;SPMamp;1;SPMamp;1;SPMamp;1;SPMamp;1;SPMamp;1;SPMamp;1;SPMamp;1;SPMamp;1;SPMamp;1;SPMamp;1;SPMamp;1;SPMamp;1hline_inline$l*NB+x$_inline;SPMamp;1;SPMamp;1;SPMamp;2;SPMamp;2;SPMamp;3;SPMamp;3;SPMamp;4;SPMamp;4;SPMamp;5;SPMamp;5;SPMamp;6;SPMamp;6;SPMamp;7;SPMamp;7;SPMamp;8;SPMamp;8hlinetabularcenter{table}/; $cached_env_img{$key} = q|table2428|; $key = q/{_inline}$mboxrank(A)=min(m,n)${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline12706|; $key = q/{_inline}$(0,0),ldots,(K-1,0)${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline15295|; $key = q/{displaymath}left(arrayccR_11;SPMamp;R_12arrayright)Z=left(arrayccT_11;SPMamp;0arrayright).{displaymath}/; $cached_env_img{$key} = q|displaymath13543|; $key = q/{displaymath}Az=lambdaBzquadRightarrowquad(L^-1AL^-T)(L^Tz)=lambda(L^Tz).{displaymath}/; $cached_env_img{$key} = q|displaymath14301|; $key = q/{_inline}$ttANORM=3.300cdot10^4${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18244|; $key = q/{displaymath}H_1H_2ldotsH_K=I-VTV^T,{displaymath}/; $cached_env_img{$key} = q|displaymath16792|; $key = q/{_inline}$Z^-1AZ^-T=I${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline12897|; $key = q/{_inline}$tau=0${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline14455|; $key = q/{displaymath}Az=lambdaz,quadA=A^H.{displaymath}/; $cached_env_img{$key} = q|displaymath12773|; $key = q/{_inline}$C_m,NNB${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline12070|; $key = q/{tabular}llValue;SPMamp;Meaninghlinett`N';SPMamp;Operatewiththedistributedmatrix.tt`T';SPMamp;Operatewiththetransposeofthedistributedmatrix.tt`C';SPMamp;Operatewiththeconjugatetransposeofthedistributedmatrix.{tabular}/; $cached_env_img{$key} = q|tabular3431|; $key = q/{_inline}$P~=~2${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline14791|; $key = q/{_inline}$2^n-1${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline17950|; $key = q/{table}[htb]labeltab:pdlahqrminipagelinewidthcentertabular||l|c|c|r|r|r|r|r||hline;SPMamp;Process;SPMamp;Block;SPMamp;multicolumn5c||Valuesof_inline$N$_inlinecline4-8;SPMamp;Grid;SPMamp;Size;SPMamp;100;SPMamp;300;SPMamp;500;SPMamp;1000;SPMamp;1500hlineIntelXPSMPParagonfootnoteAScaLAPACKprocessisrunningoneachnode.OnlyonecomputationalInteli860XPRISCprocessorcpuwasusedpernodeforthesetimings.;SPMamp;_inline$1times1$_inline;SPMamp;50;SPMamp;3.2;SPMamp;53.7;SPMamp;;SPMamp;;SPMamp;;SPMamp;_inline$2times2$_inline;SPMamp;50;SPMamp;2.9;SPMamp;25.9;SPMamp;89.8;SPMamp;;SPMamp;;SPMamp;_inline$3times3$_inline;SPMamp;50;SPMamp;3.4;SPMamp;17.9;SPMamp;60.2;SPMamp;300.5;SPMamp;;SPMamp;_inline$4times4$_inline;SPMamp;50;SPMamp;4.5;SPMamp;17.7;SPMamp;47.8;SPMamp;200.2;SPMamp;514.5hlineIntelASCIOptionRedSupercomputerfootnoteAScaLAPACKprocessisrunningoneachnode.OnlyonecomputationalIntelPentiumProprocessorcpuwasusedpernodeforthesetimings.;SPMamp;_inline$1times1$_inline;SPMamp;50;SPMamp;0.5;SPMamp;11.2;SPMamp;54.0;SPMamp;;SPMamp;;SPMamp;_inline$2times2$_inline;SPMamp;50;SPMamp;0.4;SPMamp;4.7;SPMamp;19.5;SPMamp;131.2;SPMamp;442.5;SPMamp;_inline$3times3$_inline;SPMamp;50;SPMamp;0.5;SPMamp;3.2;SPMamp;12.3;SPMamp;74.2;SPMamp;220.4;SPMamp;_inline$4times4$_inline;SPMamp;50;SPMamp;0.7;SPMamp;3.0;SPMamp;8.7;SPMamp;43.8;SPMamp;137.1;SPMamp;_inline$5times5$_inline;SPMamp;50;SPMamp;0.8;SPMamp;3.0;SPMamp;6.5;SPMamp;31.0;SPMamp;95.6hlinetabularcenterminipage{table}/; $cached_env_img{$key} = q|table4130|; $key = q/{tabular}||c|c||c|c||c|c||hlinei;SPMamp;_inline$hatlambda_i$_inline;SPMamp;ttEERRBD;SPMamp;true_inline$|hatlambda_i-lambda_i|$_inline;SPMamp;ttZERRBD(i);SPMamp;true_inline$theta(hatz_i,z_i)$_inlinehline1;SPMamp;_inline$-.5157$_inline;SPMamp;_inline$6.7cdot10^-7$_inline;SPMamp;_inline$1.6cdot10^-7$_inline;SPMamp;_inline$9.8cdot10^-7$_inline;SPMamp;_inline$1.2cdot10^-7$_inline2;SPMamp;_inline$.1709$_inline;SPMamp;_inline$6.7cdot10^-7$_inline;SPMamp;_inline$3.2cdot10^-7$_inline;SPMamp;_inline$9.8cdot10^-7$_inline;SPMamp;_inline$7.0cdot10^-8$_inline3;SPMamp;_inline$11.34$_inline;SPMamp;_inline$6.7cdot10^-7$_inline;SPMamp;_inline$2.8cdot10^-6$_inline;SPMamp;_inline$6.1cdot10^-8$_inline;SPMamp;_inline$9.7cdot10^-8$_inlinehline{tabular}/; $cached_env_img{$key} = q|tabular5436|; $key = q/{_inline}${hatx_1,...,hatx_k}${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline17846|; $key = q/{displaymath}|r_11|ge|r_22|geldotsge|r_nn|{displaymath}/; $cached_env_img{$key} = q|displaymath13453|; $key = q/{_inline}$U_1${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline14161|; $key = q/{_inline}$A+E=(hatZ+deltahatZ)hatLambda(hatZ+deltahatZ)^T${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18684|; $key = q/{_inline}$theta(hatz_i,z_i)leqttZERRBD${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18738|; $key = q/{_inline}$left|fracrmalg(z)-f(z)f(z)right|${_inline}/; $cached_env_img{$key} = q|tex2html_wrap_inline18045|; 1; .