# LaTeX2HTML 0.6.4 (Tues Aug 30 1994)
# Associate image original text (scrambled) with physical files.
$key = q/{}$|A^-1M|=|(I-G)^-1|leq1(1-|G|)${}/;
$cached_env_img{$key} ='
';
$key = q/{}$L=(l_*1,l_*2,ldots~)${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}$deltaH{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$|r^(i)|_M^-12,2=(r^(i)TM^-1r^(i))^12${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}e^(i)=x^(i)-x=A^-1(Ax^(i)-b)=A^-1r^(i),{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$infty${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}$K_BPS^-1S_B{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$T_k${}/;
$cached_env_img{$key} ='
';
$key = q/{}$a_i,i${}/;
$cached_env_img{$key} ='
';
$key = q/{}$A=D_A+L_A+U_A${}/;
$cached_env_img{$key} ='
';
$key = q/{}$x=A^Ty${}/;
$cached_env_img{$key} ='
';
$key = q/{}$O(h^-2)${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}$(R_E_iu_B)_j=(u_B)_j{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$|r^(i)|_infty(n|A|_infty|x^(i)|_infty+|b|_infty)${}/;
$cached_env_img{$key} ='
';
$key = q/{}$|A|_2${}/;
$cached_env_img{$key} ='
';
$key = q/{}$N${}/;
$cached_env_img{$key} ='
';
$key = q/{}$|e^(i)|_inftyleq|,|A^-1|cdot|r^(i)|,|_infty${}/;
$cached_env_img{$key} ='
';
$key = q/{}$Q_k${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}array|r|r|r|r|r|r|r|hlinettcol_ind(:,1);html_special_mark_amp;1;html_special_mark_amp;2;html_special_mark_amp;1;html_special_mark_amp;2;html_special_mark_amp;5;html_special_mark_amp;5hlinettcol_ind(:,2);html_special_mark_amp;2;html_special_mark_amp;3;html_special_mark_amp;3;html_special_mark_amp;4;html_special_mark_amp;6;html_special_mark_amp;6hlinettcol_ind(:,3);html_special_mark_amp;4;html_special_mark_amp;5;html_special_mark_amp;4;html_special_mark_amp;5;html_special_mark_amp;0;html_special_mark_amp;0hlinettcol_ind(:,4);html_special_mark_amp;0;html_special_mark_amp;0;html_special_mark_amp;0;html_special_mark_amp;6;html_special_mark_amp;0;html_special_mark_amp;0hlinearray.{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$a_i,jnot=0${}/;
$cached_env_img{$key} ='
';
$key = q/{}$nnzb${}/;
$cached_env_img{$key} ='
';
$key = q/{}$b-Ahatx${}/;
$cached_env_img{$key} ='
';
$key = q/{}$T'_i${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}hatLambda_k=left[arraycccccp^(1)^TAp^(1);html_special_mark_amp;0;html_special_mark_amp;;html_special_mark_amp;cdots;html_special_mark_amp;00;html_special_mark_amp;p^(2)^TAp^(2);html_special_mark_amp;;html_special_mark_amp;;html_special_mark_amp;vdots;html_special_mark_amp;ddots;html_special_mark_amp;ddots;html_special_mark_amp;ddots;html_special_mark_amp;vdots;html_special_mark_amp;;html_special_mark_amp;ddots;html_special_mark_amp;ddots;html_special_mark_amp;00;html_special_mark_amp;cdots;html_special_mark_amp;;html_special_mark_amp;0;html_special_mark_amp;p^(k)^TAp^(k)arrayright].{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$a_i,i+j${}/;
$cached_env_img{$key} ='
';
$key = q/{}$A^Ttildep^(k)${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}$u_I{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$Ap^(i)${}/;
$cached_env_img{$key} ='
';
$key = q/{}$(A^TA)x=tildeb${}/;
$cached_env_img{$key} ='
';
$key = q/{}$p_i${}/;
$cached_env_img{$key} ='
';
$key = q/{}$|I-M^-1A|${}/;
$cached_env_img{$key} ='
';
$key = q/{}$S_5${}/;
$cached_env_img{$key} ='
';
$key = q/{}$Mz=r${}/;
$cached_env_img{$key} ='
';
$key = q/{}$n_b${}/;
$cached_env_img{$key} ='
';
$key = q/{}$M=P_n(A)${}/;
$cached_env_img{$key} ='
';
$key = q/{}$rho${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}hatT_k=R_k^TAR_k=B_k^ThatLambda_kB_k{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$A=LQ${}/;
$cached_env_img{$key} ='
';
$key = q/{}$M=(I+L)D(I+U)${}/;
$cached_env_img{$key} ='
';
$key = q/{}$i+d${}/;
$cached_env_img{$key} ='
';
$key = q/{tabbing}xxxxxx=kill>Initially:$d_i,i=a_i,i$forall$i$>bffor=$i=1..nm$do:>>$left{arraylld_i+1,i+1=d_i+1,i+1-a_i+1,id_i,i^-1a_i,i+1;html_special_mark_amp;mboxifthereisno$k$;html_special_mark_amp;mboxsuchthat$i=kn$d_i+n,i+n=d_i+n,i+n-a_i+n,id_i,i^-1a_i,i+n;html_special_mark_amp;mboxif$i+nleqnm$arrayright.${tabbing}/;
$cached_env_img{$key} ='
';
$key = q/{}$(w^(i),v^(k))${}/;
$cached_env_img{$key} ='
';
$key = q/{}$|G|${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}$A_IB{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}-(tildea(x)u_x)_x-(tildeb(y)u_y)_y=f.{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$U_Q${}/;
$cached_env_img{$key} ='
';
$key = q/{}$y${}/;
$cached_env_img{$key} ='
';
$key = q/{}$Anot=LU${}/;
$cached_env_img{$key} ='
';
$key = q/{}$|z|${}/;
$cached_env_img{$key} ='
';
$key = q/{figure}[tp]centerleavevmodeframebox[4.5in]vboxtabbing$X_1=A_1,1$bffor=$igeq1$>let$Y_iapproxX_iinv$>$X_i+1=A_i+1,i+1-A_i+1,iY_iA_i,i+1$tabbinglabelfig:blu-fac-tricenter{figure}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}$E_i{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}$u{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$r^(i)^TM^-1r^(i)${}/;
$cached_env_img{$key} ='
';
$key = q/{}$c_1${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}A^(i+1)approxS^(i)=A^(i)_2,2-A^(i)_2,1A^(i)^-1_1,1A^(i)_1,2.{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$a_j,j${}/;
$cached_env_img{$key} ='
';
$key = q/{}$|x|_infty=max_j|x_j|${}/;
$cached_env_img{$key} ='
';
$key = q/{}$M=(D+L)D^-1(D+U)=(D+L)(I+D^-1U)${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}$S_Bv_B{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}$R_iu{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}$A{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$sqrtnvarepsilon${}/;
$cached_env_img{$key} ='
';
$key = q/{}$M_1=nobreakI${}/;
$cached_env_img{$key} ='
';
$key = q/{}$y_k${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}c_1leqx^TAxoverx^TBxleqc_2quadhboxforall$x$,{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$z^(i-1)${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}$omip{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$q${}/;
$cached_env_img{$key} ='
';
$key = q/{}$e^(i)=x^(i)-x${}/;
$cached_env_img{$key} ='
';
$key = q/{}${beta_k}${}/;
$cached_env_img{$key} ='
';
$key = q/{}$L=L_ADinv${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}left(arrayccI;html_special_mark_amp;AA^T;html_special_mark_amp;0arrayright)left(arraycrxarrayright)=left(arraycb0arrayright).{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$(i+1,i)${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}$jinI_i{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$Q_k=R_kDelta^-1${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}y_i=sum_j(A^T)_i,jx_j=sum_ja_j,ix_j,{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}$T'_i{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$ILU(0)${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}sum_j=1^na_i,jx_j=b_i,{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$u(x)=sinpix${}/;
$cached_env_img{$key} ='
';
$key = q/{}$m'timesn'${}/;
$cached_env_img{$key} ='
';
$key = q/{}$mtimesn${}/;
$cached_env_img{$key} ='
';
$key = q/{}$(A^-1R)^T=R^TA^-T${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}(I+LD^-1)z=y,qquad(D+U)x=z.{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$i${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}$2{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}$O(h^-1){displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{figure}[tp]centerleavevmodeframebox[4.5in]vboxtabbingLet$X$beabandedmatrix,factor$X=(I+L)D^-1(I+U)$,let$Y=(I-U)D(I-L)$tabbinglabelfig:apx-invcenter{figure}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}$n'_i{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}K_as^-1v=R_H^TA_H^-1R_Hv+sum_i=1^pR_i^TA'_i^-1R_iv.{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$emstop_tolleqvarepsilon${}/;
$cached_env_img{$key} ='
';
$key = q/{}$I-G${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}-Delta(u_n+1-u_n)=-(calLu_n-f).{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}$n_E_itimesn{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$lambda_max(A)${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}$M_E_i{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}x_0=M^-1b{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$hatx${}/;
$cached_env_img{$key} ='
';
$key = q/{}$A'_i,A_H${}/;
$cached_env_img{$key} ='
';
$key = q/{}$a${}/;
$cached_env_img{$key} ='
';
$key = q/{}$mboxspan{r^(0),Ar^(0),A^2r^(0),ldots}${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}$R_X_k,R^T_X_k{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$R^T_i${}/;
$cached_env_img{$key} ='
';
$key = q/{}$x^(i)=M^-1Nx^(i-1)+M^-1bequivGx^(i-1)+c${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}Ar^(i)=r^(i+1)t_i+1,i+r^(i)t_i,i+r^(i-1)t_i-1,i,{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$c_2${}/;
$cached_env_img{$key} ='
';
$key = q/{}$ttcol_ind(k)=j${}/;
$cached_env_img{$key} ='
';
$key = q/{}$fequiv(f_I,f_B)^T${}/;
$cached_env_img{$key} ='
';
$key = q/{}$(i-1,j)${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}$O(1H^2){displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$inot=j${}/;
$cached_env_img{$key} ='
';
$key = q/{}$|A|_2=(lambda_max(AA^T))^12${}/;
$cached_env_img{$key} ='
';
$key = q/{}$f(x)geq0${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}$jinE_i{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$r^(i+1)${}/;
$cached_env_img{$key} ='
';
$key = q/{}$|A|cdot|x^(i)|+|b|${}/;
$cached_env_img{$key} ='
';
$key = q/{}$deltar^(i)${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}$H{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$nnz=nnzbtimesn_b^2${}/;
$cached_env_img{$key} ='
';
$key = q/{}$(deltar^(i))_j${}/;
$cached_env_img{$key} ='
';
$key = q/{}$(i,j)${}/;
$cached_env_img{$key} ='
';
$key = q/{}$e^(i)=A^-1r^(i)${}/;
$cached_env_img{$key} ='
';
$key = q/{}$approx1-10N^2${}/;
$cached_env_img{$key} ='
';
$key = q/{}$B_k${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}R_k=[r^(0),~r^(1),~ldots,~r^(k-1)],{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$M=M_1M_2${}/;
$cached_env_img{$key} ='
';
$key = q/{}$[lambda_min(A),lambda_max(A)]${}/;
$cached_env_img{$key} ='
';
$key = q/{}$lambda_min(A)${}/;
$cached_env_img{$key} ='
';
$key = q/{}$|e^(i)|leq|,|A^-1|cdot|r^(i)|,|${}/;
$cached_env_img{$key} ='
';
$key = q/{}$I^ntimesn${}/;
$cached_env_img{$key} ='
';
$key = q/{}$Q${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}hboxifttxind(j)=ithen$ttx(j)=x_i$.{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$n_d${}/;
$cached_env_img{$key} ='
';
$key = q/{}$omegain(0,2)${}/;
$cached_env_img{$key} ='
';
$key = q/{}$ArightarrowM^-1A${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}Lu(x_i)=2u(x_i)-u(x_i-1)-u(x_i+1)=f(x_i)N^2qquadhboxfor$x_i=iN$,$i=1ldotsN-1$.{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$ktimesk${}/;
$cached_env_img{$key} ='
';
$key = q/{}$varepsilon=2^-24approx10^-7${}/;
$cached_env_img{$key} ='
';
$key = q/{}$Ly=x${}/;
$cached_env_img{$key} ='
';
$key = q/{}$n_b-1${}/;
$cached_env_img{$key} ='
';
$key = q/{}$|deltab_j|leqtolcdotf_j${}/;
$cached_env_img{$key} ='
';
$key = q/{}$|deltab|leqtolcdot|b|${}/;
$cached_env_img{$key} ='
';
$key = q/{}$I${}/;
$cached_env_img{$key} ='
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$key = q/{displaymath}$A'_i{displaymath}/;
$cached_env_img{$key} ='
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$key = q/{}$ntimesnobreakn${}/;
$cached_env_img{$key} ='
';
$key = q/{}$y=D_B^12x_B${}/;
$cached_env_img{$key} ='
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$key = q/{}$|deltab|leqemstop_tolcdot|b|${}/;
$cached_env_img{$key} ='
';
$key = q/{}$|x|${}/;
$cached_env_img{$key} ='
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$key = q/{}$kappa_2(A)=O(h^-2)${}/;
$cached_env_img{$key} ='
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$key = q/{displaymath}M=(Yinv+L)(I+YU).{displaymath}/;
$cached_env_img{$key} ='
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$key = q/{}$r^(0),~r^(1),~ldots,~r^(j-1)${}/;
$cached_env_img{$key} ='
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$key = q/{}$pm10^-6|A|${}/;
$cached_env_img{$key} ='
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$key = q/{}$y=Ax${}/;
$cached_env_img{$key} ='
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$key = q/{}$|A^-1|_inftycdot|r^(i)|_infty${}/;
$cached_env_img{$key} ='
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$key = q/{}$a_ij${}/;
$cached_env_img{$key} ='
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$key = q/{tabular}|r|r|r|r|r|hlinettjd_ptr;html_special_mark_amp;1;html_special_mark_amp;7;html_special_mark_amp;13;html_special_mark_amp;17hline{tabular}/;
$cached_env_img{$key} ='
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$key = q/{}$A${}/;
$cached_env_img{$key} ='
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$key = q/{displaymath}left(arrayrrrrrr10;html_special_mark_amp;-3;html_special_mark_amp;0;html_special_mark_amp;1;html_special_mark_amp;0;html_special_mark_amp;00;html_special_mark_amp;9;html_special_mark_amp;6;html_special_mark_amp;0;html_special_mark_amp;-2;html_special_mark_amp;03;html_special_mark_amp;0;html_special_mark_amp;8;html_special_mark_amp;7;html_special_mark_amp;0;html_special_mark_amp;00;html_special_mark_amp;6;html_special_mark_amp;0;html_special_mark_amp;7;html_special_mark_amp;5;html_special_mark_amp;40;html_special_mark_amp;0;html_special_mark_amp;0;html_special_mark_amp;0;html_special_mark_amp;9;html_special_mark_amp;130;html_special_mark_amp;0;html_special_mark_amp;0;html_special_mark_amp;0;html_special_mark_amp;5;html_special_mark_amp;-1arrayright)longrightarrowleft(arrayrrrr10;html_special_mark_amp;-3;html_special_mark_amp;19;html_special_mark_amp;6;html_special_mark_amp;-23;html_special_mark_amp;8;html_special_mark_amp;76;html_special_mark_amp;7;html_special_mark_amp;5;html_special_mark_amp;49;html_special_mark_amp;135;html_special_mark_amp;-1arrayright){displaymath}/;
$cached_env_img{$key} ='
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$key = q/{displaymath}$a(x,y){displaymath}/;
$cached_env_img{$key} ='
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$key = q/{}$y=Acdotx${}/;
$cached_env_img{$key} ='
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$key = q/{}$P_xtimesP_y${}/;
$cached_env_img{$key} ='
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$key = q/{}$|e^(i)|leqtaucdot|A^-1|${}/;
$cached_env_img{$key} ='
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$key = q/{}$K^-1v${}/;
$cached_env_img{$key} ='
';
$key = q/{}$M^-1${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}$n'_itimesn'_i{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}$delta{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}$K_VS^-1S_B{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$I_n={1,ldots,n}${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}$R^T_i,R_i,R^T_H,R_H{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$z^(i-1)^Ttilder^(i-1)approx0${}/;
$cached_env_img{$key} ='
';
$key = q/{}$A^TAx=A^Tb${}/;
$cached_env_img{$key} ='
';
$key = q/{}$|A^-1|${}/;
$cached_env_img{$key} ='
';
$key = q/{}$r^(k)=P_k(A)r^(0)${}/;
$cached_env_img{$key} ='
';
$key = q/{}$Au=f${}/;
$cached_env_img{$key} ='
';
$key = q/{}$1${}/;
$cached_env_img{$key} ='
';
$key = q/{}$rmspan{b,~Ab,~ldots,~A^k-1b}${}/;
$cached_env_img{$key} ='
';
$key = q/{}$e_j,k=|A|_infty${}/;
$cached_env_img{$key} ='
';
$key = q/{}$X=rmdiag(X_i)${}/;
$cached_env_img{$key} ='
';
$key = q/{}$Ax^(i)${}/;
$cached_env_img{$key} ='
';
$key = q/{}$bigcup_ipartialomip${}/;
$cached_env_img{$key} ='
';
$key = q/{}$v^(j)${}/;
$cached_env_img{$key} ='
';
$key = q/{}$kappa_2(B)=lambda_max(B)lambda_min(B))${}/;
$cached_env_img{$key} ='
';
$key = q/{}$X_i${}/;
$cached_env_img{$key} ='
';
$key = q/{}$M_1=M_2^t${}/;
$cached_env_img{$key} ='
';
$key = q/{}$k+1${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}$u_B{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$h^-1${}/;
$cached_env_img{$key} ='
';
$key = q/{}$r^(i-1)${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}$V_H{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}${A^kr^(0)}${}/;
$cached_env_img{$key} ='
';
$key = q/{}$|deltab|${}/;
$cached_env_img{$key} ='
';
$key = q/{}${x^TAx:x^Tx=1}${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}A^(i)=pmatrixA^(i)_1,1;html_special_mark_amp;A^(i)_1,2crA^(i)_2,1;html_special_mark_amp;A^(i)_2,2{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$max{|deltaA||A|,|deltab||b|}${}/;
$cached_env_img{$key} ='
';
$key = q/{}$a_j,i${}/;
$cached_env_img{$key} ='
';
$key = q/{}$|cdot|${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}$tildeS_B{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$d${}/;
$cached_env_img{$key} ='
';
$key = q/{}$ttval(k)=a_i,j${}/;
$cached_env_img{$key} ='
';
$key = q/{}$|,|A^-1|cdot|r^(i)|,|_infty${}/;
$cached_env_img{$key} ='
';
$key = q/{}$A^T${}/;
$cached_env_img{$key} ='
';
$key = q/{}$|b|${}/;
$cached_env_img{$key} ='
';
$key = q/{}$A+O(h^2)D_A${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}$K_BPS{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}P_ix=A_i(A^T_iA_i)^-1A^T_ix.{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}x^(i)=x^(0)+y_1v^(1)+cdots+y_iv^(i),{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$y_1${}/;
$cached_env_img{$key} ='
';
$key = q/{}$LUy=x${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}i;html_special_mark_lt;jrightarrowsigma(i);html_special_mark_lt;sigma(j){displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$(x,y)=barx^Ty${}/;
$cached_env_img{$key} ='
';
$key = q/{}$beta_i=r^(i)^Tr^(i)r^(i-1)^Tr^(i-1)${}/;
$cached_env_img{$key} ='
';
$key = q/{}$mboxspan{r^(0),ldots,A^i-1r^(0)}${}/;
$cached_env_img{$key} ='
';
$key = q/{}$Ap^(j)${}/;
$cached_env_img{$key} ='
';
$key = q/{}$(a_i,i)${}/;
$cached_env_img{$key} ='
';
$key = q/{}$n^2${}/;
$cached_env_img{$key} ='
';
$key = q/{}$R_i${}/;
$cached_env_img{$key} ='
';
$key = q/{}$(A+deltaA)x^(i)=b+deltab${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}$A'_iw_i=v_i{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$H=D_R^-12CD_B^-12${}/;
$cached_env_img{$key} ='
';
$key = q/{}$^,${}/;
$cached_env_img{$key} ='
';
$key = q/{}$T${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}$log^2(Hh){displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$|r^(i)|leqS_1equivemstop_tolcdot(|A|cdot|x^(i)|+|b|)${}/;
$cached_env_img{$key} ='
';
$key = q/{}$0;html_special_mark_lt;omega;html_special_mark_lt;1${}/;
$cached_env_img{$key} ='
';
$key = q/{}$a_i,i;html_special_mark_gt;sum_jnot=i|a_i,j|${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}$T_h{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$x^(i+1)${}/;
$cached_env_img{$key} ='
';
$key = q/{}$tildeA'_i^-1,tildeA_i^-1${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}$A_I^-1{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$(n^2-1)2${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}$tildeA_H{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}1leqtti,tti+jleqn{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$L${}/;
$cached_env_img{$key} ='
';
$key = q/{}$d-c${}/;
$cached_env_img{$key} ='
';
$key = q/{}$(x^(i)-hatx)^TA(x^(i)-hatx)${}/;
$cached_env_img{$key} ='
';
$key = q/{}$O(varepsilon)|A^-1|cdot(|A|cdot|x^(i)|+|b|)${}/;
$cached_env_img{$key} ='
';
$key = q/{}$A=LL^T${}/;
$cached_env_img{$key} ='
';
$key = q/{}$D_i+1equivmboxdiag(|r^(0)|_2,|r^(1)|_2,...,|r^(i)|_2)${}/;
$cached_env_img{$key} ='
';
$key = q/{}$itimesi${}/;
$cached_env_img{$key} ='
';
$key = q/{}$M=LU${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}M=(I+L)D(I+U),qquadhboxwhere$L=L_ADinv$,$U=D^-1U_A$.{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}AR_i=R_i+1barT_i,{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$e^(i)equivx^(i)-x${}/;
$cached_env_img{$key} ='
';
$key = q/{}$D${}/;
$cached_env_img{$key} ='
';
$key = q/{}$Tx=b${}/;
$cached_env_img{$key} ='
';
$key = q/{}$calL_1${}/;
$cached_env_img{$key} ='
';
$key = q/{}$f(x+y)leqf(x)+f(y)${}/;
$cached_env_img{$key} ='
';
$key = q/{}$T_H${}/;
$cached_env_img{$key} ='
';
$key = q/{}$O(varepsilon)|,|A^-1|cdot(|A|cdot|x^(i)|+|b|)|${}/;
$cached_env_img{$key} ='
';
$key = q/{tabbing}junkjunk=junk=junk=junk=junk=junkkill>bfuntil$igeqmaxit$or$|r^(i)|leqemstop_tolcdot|b|;;.${tabbing}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}M^-1Ax=M^-1f{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}A=left[arrayccca_1,1;html_special_mark_amp;cdots;html_special_mark_amp;a_1,nvdots;html_special_mark_amp;;html_special_mark_amp;vdotsa_m,1;html_special_mark_amp;cdots;html_special_mark_amp;a_m,narrayright]hspace2em(a_i,jincalR)hspace2emleft[arraycccA_1,1;html_special_mark_amp;cdots;html_special_mark_amp;A_1,n'vdots;html_special_mark_amp;;html_special_mark_amp;vdotsA_m',1;html_special_mark_amp;cdots;html_special_mark_amp;A_m',n'arrayright]hspace2em(A_i,jincalR^m_itimesn_i).{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}A=(D+L)D^-1(D+U)=(D+L)(I+D^-1U){displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$B_1${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}S_BequivA_B-A_IB^TA_I^-1A_IB{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$Ax=b${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}B_1=(D-omegaU)^-1(omegaL+(1-omega)D),{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}$V_k{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}$tildeA_I,tildeS_B{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$Omega_i${}/;
$cached_env_img{$key} ='
';
$key = q/{tabbing}junkjunk=junk=junk=junk=junk=junkkill>$i=0$>bfrepeat>>$i=i+1$>>Computetheapproximatesolution$x^(i)$.>>Computetheresidual$r^(i)=Ax^(i)-b$.>>Compute$|r^(i)|$and$|x^(i)|$.>bfuntil$igeqmaxit$or$|r^(i)|leqemstop_tolcdot(|A|cdot|x^(i)|+|b|)$.{tabbing}/;
$cached_env_img{$key} ='
';
$key = q/{}$r^(i)=Ax^(i)-b${}/;
$cached_env_img{$key} ='
';
$key = q/{}$w${}/;
$cached_env_img{$key} ='
';
$key = q/{}$|e^(i)|=|A^-1Mhatr^(i)|leq|A^-1M|cdot|hatr^(i)|${}/;
$cached_env_img{$key} ='
';
$key = q/{}$jrightarrowi+j${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}$A_Bv_B-A_IB^T(A_I^-1(A_IBv_B)){displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$4${}/;
$cached_env_img{$key} ='
';
$key = q/{}$(x,y)=x^Ty${}/;
$cached_env_img{$key} ='
';
$key = q/{}$Minv${}/;
$cached_env_img{$key} ='
';
$key = q/{}$D^-1U${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}$V_H=bigcup_kV_k{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}$y_i=R^T_iw_i{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}$R_i,A'_i{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}$ntimesn_H{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$R^T_H${}/;
$cached_env_img{$key} ='
';
$key = q/{}$P_n(x)=x^n${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}$A_i^-1{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$A^-1R${}/;
$cached_env_img{$key} ='
';
$key = q/{}$L^-1AA^TL^-Ty=b${}/;
$cached_env_img{$key} ='
';
$key = q/{figure}[tp]centerleavevmodeframebox[4.5in]vboxtabbing$x^(0)$isaninitialguessbffor=$j=1,2,....$>Solve$r$from$Mr=b-Ax^(0)$>$v^(1)=r|r|_2$>$s:=|r|_2e_1$>bffor=$i=1,2,...,m$>>Solve$w$from$Mw=Av^(i)$>>bffor=$k=1,...,i$>>>$h_k,i=(w,v^(k))$>>>$w=w-h_k,iv^(k)$>>bf>>$h_i+1,i=|w|_2$>>$v^(i+1)=wh_i+1,i$>>apply$J_1,...,J_i-1$on$(h_1,i,...,h_i+1,i)$>>construct$J_i$,actingon$i$thand$(i+1)$stcomponent>>of$h_.,i$,suchthat$(i+1)$stcomponentof$J_ih_.,i$is0>>$s:=J_is$>>if$s(i+1)$issmallenoughthen(UPDATE($tildex,i$)andquit)>bf>UPDATE($tildex,m$)>checkconvergence;continueifnecessarybfmbox~mboxInthisschemeUPDATE($tildex,i$)mboxreplacesthefollowingcomputations:mbox~Compute$y$asthesolutionof$Hy=tildes$,inwhichtheupper$itimesi$triangularpartof$H$has$h_i,j$asitselements(inleastsquaressenseif$H$issingular),$tildes$representsthefirst$i$componentsof$s$$tildex=x^(0)+y_1v^(1)+y_2v^(2)+...+y_iv^(i)$$s^(i+1)=|b-Atildex|_2$if$tildex$isanaccurateenoughapproximationthenquitelse$x^(0)=tildex$tabbinglabelfig:pgmrescenter{figure}/;
$cached_env_img{$key} ='
';
$key = q/{}$(D_A+L_A)D_A^-1(D_A+U_A)${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}$R_i^TA'_i^-1R_iv{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$i-pleqjleqi+q${}/;
$cached_env_img{$key} ='
';
$key = q/{}$f_j=|b_j|${}/;
$cached_env_img{$key} ='
';
$key = q/{}$i+j${}/;
$cached_env_img{$key} ='
';
$key = q/{}$L_A${}/;
$cached_env_img{$key} ='
';
$key = q/{}$I-L${}/;
$cached_env_img{$key} ='
';
$key = q/{}$tildeb=A^Tb${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}M_1^-1AM_2inv(M_2x)=M_1^-1b.{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}$u_i{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$nge250${}/;
$cached_env_img{$key} ='
';
$key = q/{}$|A^-1|cdot|r^(i)|${}/;
$cached_env_img{$key} ='
';
$key = q/{}$|A|${}/;
$cached_env_img{$key} ='
';
$key = q/{}$x^(k)${}/;
$cached_env_img{$key} ='
';
$key = q/{}$B_2${}/;
$cached_env_img{$key} ='
';
$key = q/{}$sigma${}/;
$cached_env_img{$key} ='
';
$key = q/{}$O(ttlx+ttly)${}/;
$cached_env_img{$key} ='
';
$key = q/{}$imapstoQ_i(A)P_i(A)r^(0)${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}$tildeA_H^-1A_H{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}$S_X_k{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$deltaa_j,k${}/;
$cached_env_img{$key} ='
';
$key = q/{figure}[tp]centerleavevmodeframebox[4.5in]vboxtabbingLet$S$bethenonzeroset${(i,j)colona_ijnot=0}$bffor=$i=1,2,ldots$>bfset$d_iileftarrowa_ii$bffor=$i=1,2,ldots$>bfset$d_iileftarrow1d_ii$>bffor=$j=i+1,i+2,ldots$>>if$(i,j)inS$and$(j,i)inS$then>>bfset$d_jjleftarrowd_jj-a_jid_iia_ij$tabbinglabelfig:d-ilucenter{figure}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}|A|_2cdot|A^-1|_2=fraclambda^12_max(A^TA)lambda^12_min(A^TA),{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$2n^2${}/;
$cached_env_img{$key} ='
';
$key = q/{}$d+c${}/;
$cached_env_img{$key} ='
';
$key = q/{}$x^(0)${}/;
$cached_env_img{$key} ='
';
$key = q/{}$K_as,K_BPS,K_VS${}/;
$cached_env_img{$key} ='
';
$key = q/{}$y=A^Tx${}/;
$cached_env_img{$key} ='
';
$key = q/{}$u_n(x)=sinnpix${}/;
$cached_env_img{$key} ='
';
$key = q/{}$|A^-1|_2=1lambda^12_min(AA^T)=1lambda^12_min(A^TA)${}/;
$cached_env_img{$key} ='
';
$key = q/{}$|x|_2${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}$g_B{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$(AA^T)y=b${}/;
$cached_env_img{$key} ='
';
$key = q/{}$n^22${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}p^(j)=r^(j-1)+beta_j-1p^(j-1),~j=2,~3,~ldots,~k~,{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$Ap^(i-1)${}/;
$cached_env_img{$key} ='
';
$key = q/{}$j;html_special_mark_gt;i+q${}/;
$cached_env_img{$key} ='
';
$key = q/{}$(i,j-1)${}/;
$cached_env_img{$key} ='
';
$key = q/{}$|deltaA|leqemstop_tolcdot|A|${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}$A_I,S_B{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$S_Bu_B=g_B${}/;
$cached_env_img{$key} ='
';
$key = q/{}$|deltar^(i)|leqO(varepsilon)(|A|cdot|x^(i)|+|b|)${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}y=Axqquadhboxandqquady=A^Tx.{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$|x^(i)|${}/;
$cached_env_img{$key} ='
';
$key = q/{}$2nnz+n+1${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}arraylcl|x|_infty;html_special_mark_amp;equiv;html_special_mark_amp;max_j|x_j|;,|x|_1;html_special_mark_amp;equiv;html_special_mark_amp;sum_j|x_j|;rm,and|x|_2;html_special_mark_amp;equiv;html_special_mark_amp;(sum_j|x_j|^2)^12;;.array{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}$A_B{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$L_ADinv${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}B_2=(D-omegaL)^-1(omegaU+(1-omega)D).{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}|e^(i)|leq|A^-1|cdot|r^(i)|leqemstop_tolcdot|A^-1|cdot|b|{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$omega=1${}/;
$cached_env_img{$key} ='
';
$key = q/{}$ILU(k)${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}hboxforall$j$,doforall$i$:qquady_ileftarrowy_i+a_j,ix_j.{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$a_i,j=a_j,i${}/;
$cached_env_img{$key} ='
';
$key = q/{}$A^-T${}/;
$cached_env_img{$key} ='
';
$key = q/{}$Ax${}/;
$cached_env_img{$key} ='
';
$key = q/{}$pminfty${}/;
$cached_env_img{$key} ='
';
$key = q/{}$|w^(i)|${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}$K_VS{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$n_d=nn_b${}/;
$cached_env_img{$key} ='
';
$key = q/{}$n'_i${}/;
$cached_env_img{$key} ='
';
$key = q/{}$omega${}/;
$cached_env_img{$key} ='
';
$key = q/{}$z${}/;
$cached_env_img{$key} ='
';
$key = q/{}$A=(a_i,j)${}/;
$cached_env_img{$key} ='
';
$key = q/{}$min_i{a_i,i-sum_jnot=i|a_i,j|}${}/;
$cached_env_img{$key} ='
';
$key = q/{}$Av${}/;
$cached_env_img{$key} ='
';
$key = q/{}$r^(i)^TM^-1r^(j)=0${}/;
$cached_env_img{$key} ='
';
$key = q/{}$|A^-1|_infty${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}x^(i)=M^-1Nx^(i-1)+M^-1b=Gx^(i-1)+c,{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$r^(0)${}/;
$cached_env_img{$key} ='
';
$key = q/{tabular}|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|hlinettval;html_special_mark_amp;10;html_special_mark_amp;-2;html_special_mark_amp;3;html_special_mark_amp;9;html_special_mark_amp;3;html_special_mark_amp;7;html_special_mark_amp;8;html_special_mark_amp;7;html_special_mark_amp;3$cdots$9;html_special_mark_amp;13;html_special_mark_amp;4;html_special_mark_amp;2;html_special_mark_amp;-1hlinettcol_ind;html_special_mark_amp;1;html_special_mark_amp;5;html_special_mark_amp;1;html_special_mark_amp;2;html_special_mark_amp;6;html_special_mark_amp;2;html_special_mark_amp;3;html_special_mark_amp;4;html_special_mark_amp;1$cdots$5;html_special_mark_amp;6;html_special_mark_amp;2;html_special_mark_amp;5;html_special_mark_amp;6hline{tabular}/;
$cached_env_img{$key} ='
';
$key = q/{}$h=N^-d${}/;
$cached_env_img{$key} ='
';
$key = q/{}$R_i+1D_i+1^-1${}/;
$cached_env_img{$key} ='
';
$key = q/{}$r${}/;
$cached_env_img{$key} ='
';
$key = q/{figure}[tp]centerleavevmodeframebox[4.5in]vboxtabbingbffor=$k=1,ldots,2n-1$>bfdo=bfinparallelfor$i=max(1,k+1-n),min(n,k)$>>$j=k-i+1$>>$x_i+(j-1)nleftarrow$=$D_i+(j-1)ny_i+(j-1)n$>>>$-L_i+(j-1)ni-1+(j-1)nx_i-1+(j-1)n$>>>$-L_i+(j-1)ni+(j-2)nx_i+(j-2)n$tabbinglabelfig:wave-solvecenter{figure}/;
$cached_env_img{$key} ='
';
$key = q/{}$Ap^(k)${}/;
$cached_env_img{$key} ='
';
$key = q/{}$(LU)^Ty=x${}/;
$cached_env_img{$key} ='
';
$key = q/{}$|r^(i)|leqS_5equivemstop_tolcdot|r^(0)|${}/;
$cached_env_img{$key} ='
';
$key = q/{}$p^(i)${}/;
$cached_env_img{$key} ='
';
$key = q/{}$P(0)=1${}/;
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$key = q/{}${r^(k)}${}/;
$cached_env_img{$key} ='
';
$key = q/{}$L^T${}/;
$cached_env_img{$key} ='
';
$key = q/{}$s${}/;
$cached_env_img{$key} ='
';
$key = q/{}$LQ${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}|Ax^(i)-b|_2=|D_i+1barT_iy-|r^(0)|_2e^(1)|_2{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$|deltaA|leqvarepsilon|A|${}/;
$cached_env_img{$key} ='
';
$key = q/{}$|e^(i)|leqS_5cdot|A^-1|${}/;
$cached_env_img{$key} ='
';
$key = q/{}$K_as${}/;
$cached_env_img{$key} ='
';
$key = q/{}$barx${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}array|r|r|r|r|r|r|r|hlinettval(:,1);html_special_mark_amp;10;html_special_mark_amp;9;html_special_mark_amp;3;html_special_mark_amp;6;html_special_mark_amp;9;html_special_mark_amp;5hlinettval(:,2);html_special_mark_amp;-3;html_special_mark_amp;6;html_special_mark_amp;8;html_special_mark_amp;7;html_special_mark_amp;13;html_special_mark_amp;-1hlinettval(:,3);html_special_mark_amp;1;html_special_mark_amp;-2;html_special_mark_amp;7;html_special_mark_amp;5;html_special_mark_amp;0;html_special_mark_amp;0hlinettval(:,4);html_special_mark_amp;0;html_special_mark_amp;0;html_special_mark_amp;0;html_special_mark_amp;4;html_special_mark_amp;0;html_special_mark_amp;0hlinearray,{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$delta${}/;
$cached_env_img{$key} ='
';
$key = q/{}$1:nnzb${}/;
$cached_env_img{$key} ='
';
$key = q/{}$k${}/;
$cached_env_img{$key} ='
';
$key = q/{}$A_I=blockdiagonal(A_i)${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}-(a(x,y)u_x)_x-(b(x,y)u_y)_y=f,{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$y^(i)${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}d_i,i=left{arraylla_1,1;html_special_mark_amp;mboxif$i=1$a_i,i-a_i,i-1d_i-1^-1a_i-1,i;html_special_mark_amp;mboxif$1;html_special_mark_lt;ileqn$a_i,i-a_i,i-nd_i-n^-1a_i-n,i;html_special_mark_amp;mboxif$i=kn+1$with$kgeq1$arrayrcla_i,i;html_special_mark_amp;-;html_special_mark_amp;a_i,i-1d_i-1^-1a_i-1,i;html_special_mark_amp;;html_special_mark_amp;mbox-a_i,i-nd_i-n^-1a_i-n,iarray;html_special_mark_amp;mboxotherwise.arrayright.{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$kp${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}Lz=x,qquadUy=z.{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{figure}[tp]centerleavevmodeframebox[4.5in]vboxtabbingbffor=$i=1,2,dots$>$X_ileftarrowA_ii$bffor=$i=1,2,dots$>Let$Y_iapproxX_iinv$>bffor=$j=i+1,i+2,ldots$>>if$A_ijnot=0$and$A_jinot=0$,>>then$X_jleftarrowX_j-A_jiY_iA_ij$tabbinglabelfig:blu-faccenter{figure}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}$v_B{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$a_i,j${}/;
$cached_env_img{$key} ='
';
$key = q/{}$z=A^Tcdotx${}/;
$cached_env_img{$key} ='
';
$key = q/{}$|r^(i)|leqS_3equivemstop_tolcdot|x^(i)||A^-1|${}/;
$cached_env_img{$key} ='
';
$key = q/{}$c${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}$partialomip{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$i+j=k-1${}/;
$cached_env_img{$key} ='
';
$key = q/{}$|A|_F${}/;
$cached_env_img{$key} ='
';
$key = q/{}$T_h${}/;
$cached_env_img{$key} ='
';
$key = q/{}$(A^-1)_i,jgeq0${}/;
$cached_env_img{$key} ='
';
$key = q/{}$L^-Ts${}/;
$cached_env_img{$key} ='
';
$key = q/{}$x^(k-1)${}/;
$cached_env_img{$key} ='
';
$key = q/{}$x=M^(p)y${}/;
$cached_env_img{$key} ='
';
$key = q/{}$g=D_B^-12(f_B-C^TD_R^-1f_R)${}/;
$cached_env_img{$key} ='
';
$key = q/{}$ttrow_ptr(n+1)=nnz+1${}/;
$cached_env_img{$key} ='
';
$key = q/{}$hatr^(i)${}/;
$cached_env_img{$key} ='
';
$key = q/{}$M^-1A${}/;
$cached_env_img{$key} ='
';
$key = q/{}$x^(0)=0${}/;
$cached_env_img{$key} ='
';
$key = q/{}$A'_i^-1,A_i^-1${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}$Omega{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$p^(i)^TAp^(i)${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}M=(D+L)D^-1(D+L)^T,{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}r_0leftarrowM_1^-1r_0{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{figure}[tp]centerleavevmodeframebox[4.5in]vboxtabbingLet$M=LU$and$y$begiven.bffor=$i=1,2,ldots$>$z_i=ell_ii^-1(y_i-sum_j;html_special_mark_lt;iell_ijz_j)$bffor=$i=n,n-1,n-2,ldots$>$x_i=u_ii^-1(z_i-sum_j;html_special_mark_gt;iu_ijx_j)$tabbinglabelfig:lu-solvcenter{figure}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}left[arrayccD_R;html_special_mark_amp;CO;html_special_mark_amp;D_B-C^TD_R^-1Carrayright]left[arraycx_Rx_Barrayright]=left[arraycf_Rf_B-C^TD_R^-1f_Rarrayright].{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$S={1,ldots,n}${}/;
$cached_env_img{$key} ='
';
$key = q/{}$S${}/;
$cached_env_img{$key} ='
';
$key = q/{}$Omega_i,i=1,...,p${}/;
$cached_env_img{$key} ='
';
$key = q/{}$Y=rmdiag(Y_i)${}/;
$cached_env_img{$key} ='
';
$key = q/{}$M_1${}/;
$cached_env_img{$key} ='
';
$key = q/{}$|Ax|leq|A|cdot|x|${}/;
$cached_env_img{$key} ='
';
$key = q/{}$|(I-G)^-1|leq1(1-|G|)${}/;
$cached_env_img{$key} ='
';
$key = q/{}$(A+deltaA)x^(i)=(b+deltab)${}/;
$cached_env_img{$key} ='
';
$key = q/{}$v^(1)${}/;
$cached_env_img{$key} ='
';
$key = q/{}$1:n_d+1${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}frac|e^(i)||x^(i)|leqfrac|A^-1|cdot|r^(i)||x^(i)|leqemstop_tol;;,{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}$v_i=R_iv{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{figure}[tp]centerleavevmodeframebox[4.5in]vboxtabbingLet$M=(D+L)(I+D^-1U)$and$y$begiven.bffor=$i=1,2,ldots$>$z_i=d_ii^-1(y_i-sum_j;html_special_mark_lt;iell_ijz_j)$bffor=$i=n,n-1,n-2,ldots$>$x_i=z_i-d_ii^-1sum_j;html_special_mark_gt;iu_ijx_j$tabbinglabelfig:ldu-solvcenter{figure}/;
$cached_env_img{$key} ='
';
$key = q/{}$Delta${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}arraylcl|A|_infty;html_special_mark_amp;equiv;html_special_mark_amp;max_jsum_k|a_j,k|;,|A|_1;html_special_mark_amp;equiv;html_special_mark_amp;max_ksum_j|a_j,k|;rm,and|A|_F;html_special_mark_amp;equiv;html_special_mark_amp;(sum_jk|a_j,k|^2)^12;;,array{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$|b|ll|A|cdot|x|${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}$K_as{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$x^TAx;html_special_mark_gt;0${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}$X=bigcup_kX_k{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$beta_i-1${}/;
$cached_env_img{$key} ='
';
$key = q/{}$T_i${}/;
$cached_env_img{$key} ='
';
$key = q/{}$r_0${}/;
$cached_env_img{$key} ='
';
$key = q/{}$Ax+b${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}A=D_A+L_A+U_A,qquadM=(D+L_A)D^-1(D+U_A),{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$u_n${}/;
$cached_env_img{$key} ='
';
$key = q/{}$M^(p)approxMinv${}/;
$cached_env_img{$key} ='
';
$key = q/{}$C${}/;
$cached_env_img{$key} ='
';
$key = q/{}$S_4equivmax_j(|r^(i)|_j(Ecdot|x^(i)|+f)_j)leqemstop_tol${}/;
$cached_env_img{$key} ='
';
$key = q/{}$s=L^-1r^(i)${}/;
$cached_env_img{$key} ='
';
$key = q/{tabular}|r|r|r|r|r|r|r|r|r|r|r|hlinettval(:,-1);html_special_mark_amp;0;html_special_mark_amp;0;html_special_mark_amp;3;html_special_mark_amp;6;html_special_mark_amp;0;html_special_mark_amp;5hlinettval(:,0);html_special_mark_amp;10;html_special_mark_amp;9;html_special_mark_amp;8;html_special_mark_amp;7;html_special_mark_amp;9;html_special_mark_amp;-1hlinettval(:,+1);html_special_mark_amp;0;html_special_mark_amp;-3;html_special_mark_amp;6;html_special_mark_amp;7;html_special_mark_amp;5;html_special_mark_amp;13hlinettval(:,+2);html_special_mark_amp;0;html_special_mark_amp;1;html_special_mark_amp;-2;html_special_mark_amp;0;html_special_mark_amp;4;html_special_mark_amp;0hline{tabular}/;
$cached_env_img{$key} ='
';
$key = q/{}$j;html_special_mark_lt;i${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}mboxforeach$k$,$i,j;html_special_mark_gt;k$:quada_i,jleftarrowleft{arraylla_i,j-a_i,ka_k,k^-1a_k,j;html_special_mark_amp;mboxif$(i,j)inS$a_i,j;html_special_mark_amp;mboxotherwise.arrayright.{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}|e^(i)|leq|A^-1|cdot|r^(i)|leqemstop_tolcdot|A^-1|cdot(|A|cdot|x^(i)|+|b|);.{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$Q_i${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}$tildeA_H^-1{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$ttrow_ptr(i)leqk;html_special_mark_lt;ttrow_ptr(i+1)${}/;
$cached_env_img{$key} ='
';
$key = q/{}$x^Tx=0${}/;
$cached_env_img{$key} ='
';
$key = q/{}$2times2${}/;
$cached_env_img{$key} ='
';
$key = q/{}$H,h${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}xleftarrowM_2^-1x{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$|A+B|leq|A|+|B|${}/;
$cached_env_img{$key} ='
';
$key = q/{}$|r^(i)|leqS_2equivemstop_tolcdot|b|${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}|Ax|leq|A|,|x|.{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$a_i,j=${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}$B{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$M_2${}/;
$cached_env_img{$key} ='
';
$key = q/{}$v${}/;
$cached_env_img{$key} ='
';
$key = q/{}$10^-6${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}$ntimesn'_i{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}calL=calL_1+calL_2,qquadhboxwhere$calL_1=-partial^2overpartialx^2$,$calL_2=-partial^2overpartialy^2$.{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$T'_isubsetT_h${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}y_i=sum_ja_i,jx_j{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$3${}/;
$cached_env_img{$key} ='
';
$key = q/{}$a_ii;html_special_mark_gt;sum_jnot=i|a_ij|${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}m_i,j=left{arraylla_i,i;html_special_mark_amp;mboxif$i=j$0;html_special_mark_amp;mboxotherwise.arrayright.{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$ntimesn${}/;
$cached_env_img{$key} ='
';
$key = q/{}$LDinv${}/;
$cached_env_img{$key} ='
';
$key = q/{}$|r^(0)|${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}|e^(i)|_inftyleq|,|A^-1|cdot|r^(i)|,|leqS_4cdot|,|A^-1|(E|x^(i)|+f)|_infty{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}$T_H{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$|r^(i)|_2${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}$K^12{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$p^(1)=r^(0)${}/;
$cached_env_img{$key} ='
';
$key = q/{}$n${}/;
$cached_env_img{$key} ='
';
$key = q/{}$R_k=P_kB_k${}/;
$cached_env_img{$key} ='
';
$key = q/{}$|x|_2=(sum_j|x_j|^2)^12${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}$R_E_i{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$-U${}/;
$cached_env_img{$key} ='
';
$key = q/{}$ncofm${}/;
$cached_env_img{$key} ='
';
$key = q/{}$AA^T${}/;
$cached_env_img{$key} ='
';
$key = q/{}$|x+y|leq|x|+|y|${}/;
$cached_env_img{$key} ='
';
$key = q/{}$y_sigma_k(i)${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}1llfrac|A|cdot|x||b|=frac|A|cdot|A^-1b||b|leq|A|cdot|A^-1|;;.{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}$A_H{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$Ax=f${}/;
$cached_env_img{$key} ='
';
$key = q/{}$lambda(B)=1-lambda(L)2=1-2sin^2npi(2N)${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}$X{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$f${}/;
$cached_env_img{$key} ='
';
$key = q/{}$y_i${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}$A_H,R_H{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$xleftarrowx-l_*1y_1${}/;
$cached_env_img{$key} ='
';
$key = q/{}$nnz${}/;
$cached_env_img{$key} ='
';
$key = q/{displaymath}$R^T_i{displaymath}/;
$cached_env_img{$key} ='
';
$key = q/{}$lambda_min${}/;
$cached_env_img{$key} ='
';
$key = q/{}$(A+A^T)2${}/;
$cached_env_img{$key} ='
';
$key = q/{}$ttp+ttq${}/;
$cached_env_img{$key} ='
';
$key = q/{tabular}|r|r|r|r|r|r|r|r|hlinettrow_ptr;html_special_mark_amp;1;html_special_mark_amp;3;html_special_mark_amp;6;html_special_mark_amp;9;html_special_mark_amp;13;html_special_mark_amp;17;html_special_mark_amp;20hline{tabular}/;
$cached_env_img{$key} ='
';
1;