# LaTeX2HTML 0.4 (Wed Dec 8 1993) # Associate image original text (scrambled) with physical files. $key = q/{}$infty${}/; $cached_env_img{$key} ='

'; $key = q/{}$lambda_<<3400>>max<<3400>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$tilde<<3245>>K<<3245>>_<<3246>>VS<<3246>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$S_<<3135>>E_i<<3135>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$lambda_<<1419>>min<<1419>>${}/; $cached_env_img{$key} ='

'; $key = q/{}${x^<<2028>>(i)<<2028>>}${}/; $cached_env_img{$key} ='

'; $key = q/{}$T_k${}/; $cached_env_img{$key} ='

'; $key = q/{}$A=D_A+L_A+U_A${}/; $cached_env_img{$key} ='

'; $key = q/{}$S_<<3130>>E<<3130>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$r^<<510>>(i)<<510>>=b-Ax^<<511>>(i)<<511>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$ntimesn_H${}/; $cached_env_img{$key} ='

'; $key = q/{}$|A|_E${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}sum_<<212>>j=1<<212>>^na_<<213>>i,j<<213>>x_j=b_i,{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$L^<<1932>>-1<<1932>>AA^TL^<<1933>>-T<<1933>>y=b${}/; $cached_env_img{$key} ='

'; $key = q/{}[A^<<3283>>(i+1)<<3283>>approxS^<<3284>>(i)<<3284>>=A^<<3285>>(i)<<3285>>_<<3286>>2,2<<3286>>-A^<<3287>>(i)<<3287>>_<<3288>>2,1<<3288>>A^<<3289>>(i)^<<3686>>-1<<3686>><<3289>>_<<3290>>1,1<<3290>>A^<<3291>>(i)<<3291>>_<<3292>>1,2<<3292>>.]{}/; $cached_env_img{$key} ='

'; $key = q/{}$|r^<<2248>>(i)<<2248>>|leqS_5equiv<<2249>>emstop_tol<<2249>>cdot|r^<<2250>>(0)<<2250>>|${}/; $cached_env_img{$key} ='

'; $key = q/{}$a_<<1736>>i,j<<1736>>=0${}/; $cached_env_img{$key} ='

'; $key = q/{}$N${}/; $cached_env_img{$key} ='

'; $key = q/{}$e^<<2320>>(i)<<2320>>=A^<<2321>>-1<<2321>>r^<<2322>>(i)<<2322>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$T'_i${}/; $cached_env_img{$key} ='

'; $key = q/{}$K^<<3172>>12<<3172>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$A_<<3107>>I<<3107>>^<<3108>>-1<<3108>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$<<2514>>ttp<<2514>>+<<2515>>ttq<<2515>>${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}p^<<2880>>(j)<<2880>>=r^<<2881>>(j-1)<<2881>>+beta_jp^<<2882>>(j-1)<<2882>>,~j=2,~3,~ldots,~k~,{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$|r^<<2255>>(0)<<2255>>|${}/; $cached_env_img{$key} ='

'; $key = q/{}$r^<<2912>>(0)<<2912>>,~r^<<2913>>(1)<<2913>>,~ldots,~r^<<2914>>(j-1)<<2914>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$r^<<2404>>(i)<<2404>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$Mz=r${}/; $cached_env_img{$key} ='

'; $key = q/{}$n_b${}/; $cached_env_img{$key} ='

'; $key = q/{}$rho${}/; $cached_env_img{$key} ='

'; $key = q/{}$|deltaA|leq<<2168>>emstop_tol<<2168>>cdot|A|${}/; $cached_env_img{$key} ='

'; $key = q/{}$A=LQ${}/; $cached_env_img{$key} ='

'; $key = q/{}$(deltar^<<2403>>(i)<<2403>>)_j${}/; $cached_env_img{$key} ='

'; $key = q/{}$ArightarrowM^<<1648>>-1<<1648>>A${}/; $cached_env_img{$key} ='

'; $key = q/{}$<<2677>>ttval(diag_ptr(i))<<2677>>=a_<<2678>>i,i<<2678>>${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}<<2576>>array<<2576>><<2577>>|r|r|r|r|r|r|r|<<2577>>hline<<2578>>ttval<<2578>>(:,1)&10&9&3&6&9&5hline<<2579>>ttval<<2579>>(:,2)&-3&6&8&7&13&-1hline<<2580>>ttval<<2580>>(:,3)&1&-2&7&5&0&0hline<<2581>>ttval<<2581>>(:,4)&0&0&0&4&0&0hline<<2582>>array<<2582>>,{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$0<omega<1${}/; $cached_env_img{$key} ='

'; $key = q/{}$|r^<<2306>>(i)<<2306>>|_<<3549>>M^<<2307>>-12<<2307>>,2<<3549>>|r^<<2308>>(0)<<2308>>|_<<3550>>M^<<2309>>-12<<2309>>,2<<3550>>leqtol${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}C=(Yinv+L)(I+YU).{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$|A^<<2338>>-1<<2338>>|_2=1lambda^<<2339>>12<<2339>>_<<2340>>min<<2340>>(AA^T)=1lambda^<<2341>>12<<2341>>_<<2342>>min<<2342>>(A^TA)${}/; $cached_env_img{$key} ='

'; $key = q/{tabular}<<2607>>|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|<<2607>>hline<<2608>>ttjdiag<<2608>>&1&3&7&8&10&2;&9&9&8<<6597>><<6597>>$cdots$<<6598>><<6598>>-1;&9&6&7&5;&13hline<<2609>>ttcol_ind<<2609>>&1&1&2&3&1&5;&4&2&3<<6599>><<6599>>$cdots$<<6600>><<6600>>6;&5&3&4&5;&6hline{tabular}/; $cached_env_img{$key} ='

'; $key = q/{}$|z|${}/; $cached_env_img{$key} ='

'; $key = q/{tabbing}junkjunk=junk=junk=junk=junk=junkkill><<2069>>bfuntil<<2069>><<5959>><<5959>>$igeqmaxit$<<5960>><<5960>>or<<5961>><<5961>>$|r^<<2070>>(i)<<2070>>|leq<<2071>>emstop_tol<<2071>>cdot|b|;;.$<<5962>><<5962>>{tabbing}/; $cached_env_img{$key} ='

'; $key = q/{}$c_1${}/; $cached_env_img{$key} ='

'; $key = q/{}$omega>1${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}C=(X+L)(I+X^<<3678>>-1<<3678>>U){displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$log^2(Hh)${}/; $cached_env_img{$key} ='

'; $key = q/{}$r^<<2143>>(i)<<2143>>=Ax^<<2144>>(i)<<2144>>-b${}/; $cached_env_img{$key} ='

'; $key = q/{}$y_k${}/; $cached_env_img{$key} ='

'; $key = q/{}$v_i=R_iv${}/; $cached_env_img{$key} ='

'; $key = q/{}$q${}/; $cached_env_img{$key} ='

'; $key = q/{}$v^<<2852>>(j)<<2852>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$u(x)=sinpix${}/; $cached_env_img{$key} ='

'; $key = q/{}$a_<<3376>>i,j<<3376>>=a_<<3377>>j,i<<3377>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$mtimesn${}/; $cached_env_img{$key} ='

'; $key = q/{}$|A^<<2333>>-1<<2333>>|_2=lambda^<<2334>>-1<<2334>>_<<2335>>min<<2335>>(A)${}/; $cached_env_img{$key} ='

'; $key = q/{}$A_I,S_B${}/; $cached_env_img{$key} ='

'; $key = q/{}$A_H${}/; $cached_env_img{$key} ='

'; $key = q/{}$|x|_<<2110>>B,alpha<<2110>>equiv|Bx|_<<2111>>alpha<<2111>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$(I+LD^<<3677>>-1<<3677>>)(D+U)${}/; $cached_env_img{$key} ='

'; $key = q/{}$I-G${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}R_k=[r^<<2869>>(0)<<2869>>,~r^<<2870>>(1)<<2870>>,~ldots,~r^<<2871>>(k-1)<<2871>>],{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}<<1988>>calL<<1988>>=<<1989>>calL<<1989>>_1+<<1990>>calL<<1990>>_2,qquadhbox<<3534>>where<<5813>><<5813>>$<<1991>>calL<<1991>>_1=-<<1992>>partial^2overpartialx^2<<1992>>$<<5814>><<5814>>,<<5815>><<5815>>$<<1993>>calL<<1993>>_2=-<<1994>>partial^2overpartialy^2<<1994>>$<<5816>><<5816>><<3534>>.{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$a_<<1749>>i,j<<1749>>not=0${}/; $cached_env_img{$key} ='

'; $key = q/{}$hatx${}/; $cached_env_img{$key} ='

'; $key = q/{}$(R_<<3120>>E_i<<3120>>u_B)_j=(u_B)_j${}/; $cached_env_img{$key} ='

'; $key = q/{}$a${}/; $cached_env_img{$key} ='

'; $key = q/{}$R^T_i${}/; $cached_env_img{$key} ='

'; $key = q/{}$n_<<3118>>E_i<<3118>>timesn${}/; $cached_env_img{$key} ='

'; $key = q/{}${X_i}_<<1896>>i=1..n<<1896>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$c_2${}/; $cached_env_img{$key} ='

'; $key = q/{}$lambda_<<1428>>min<<1428>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$<<2396>>emstop_tol<<2396>>leqvarepsilon${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}d_<<1774>>i,i<<1774>>=left{<<1775>>array<<1775>><<1776>>ll<<1776>>a_<<1777>>1,1<<1777>>&mbox<<1778>>if<<5679>><<5679>>$i=1$<<5680>><<5680>><<1778>>a_<<1779>>i,i<<1779>>-a_<<1780>>i,i-1<<1780>>d_<<1781>>i-1<<1781>>^<<3667>>-1<<3667>>a_<<1782>>i-1,i<<1782>>&mbox<<1783>>if<<5681>><<5681>>$1<ileqn$<<5682>><<5682>><<1783>>a_<<1784>>i,i<<1784>>-a_<<1785>>i,i-n<<1785>>d_<<1786>>i-n<<1786>>^<<3668>>-1<<3668>>a_<<1787>>i-n,i<<1787>>&mbox<<1788>>if<<5683>><<5683>>$i=kn+1$<<5684>><<5684>>with<<5685>><<5685>>$k>1$<<5686>><<5686>><<1788>><<1789>>array<<1789>><<1790>>rcl<<1790>>a_<<1791>>i,i<<1791>>&-&a_<<1792>>i,i-1<<1792>>d_<<1793>>i-1<<1793>>^<<3669>>-1<<3669>>a_<<1794>>i-1,i<<1794>>&&mbox<<1795>><<1795>>-a_<<1796>>i,i-n<<1796>>d_<<1797>>i-n<<1797>>^<<3670>>-1<<3670>>a_<<1798>>i-n,i<<1798>><<1799>>array<<1799>>&mbox<<1800>>otherwise<<1800>>.<<1801>>array<<1801>>right.{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$inot=j${}/; $cached_env_img{$key} ='

'; $key = q/{}$A_<<3070>>I<<3070>>=blockdiagonal(A_<<3071>>i<<3071>>)${}/; $cached_env_img{$key} ='

'; $key = q/{}$r^<<613>>(i)^T<<613>>M^<<3647>>-1<<3647>>r^<<614>>(j)<<614>>=0${}/; $cached_env_img{$key} ='

'; $key = q/{}$<<1996>>calL<<1996>>_1${}/; $cached_env_img{$key} ='

'; $key = q/{}$A=(a_<<2506>>i,j<<2506>>)${}/; $cached_env_img{$key} ='

'; $key = q/{}$p^<<2908>>(1)<<2908>>,~p^<<2909>>(2)<<2909>>,~ldots,~p^<<2910>>(j)<<2910>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$B_k${}/; $cached_env_img{$key} ='

'; $key = q/{}$|A|_2=(lambda_<<2129>>max<<2129>>(AA^T))^<<2130>>12<<2130>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$h^<<658>>-1<<658>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$r^<<2040>>(i)<<2040>>=Ax^<<2041>>(i)<<2041>>-b${}/; $cached_env_img{$key} ='

'; $key = q/{}$Q${}/; $cached_env_img{$key} ='

'; $key = q/{}$r^<<550>>(j)<<550>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$lambda_<<634>>min<<634>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$|r^<<2164>>(i)<<2164>>|leqS_1equiv<<2165>>emstop_tol<<2165>>cdot(|A|cdot|x^<<2166>>(i)<<2166>>|+|b|)${}/; $cached_env_img{$key} ='

'; $key = q/{}$K_<<3159>>BPS<<3159>>^<<3160>>-1<<3160>>S_B${}/; $cached_env_img{$key} ='

'; $key = q/{}$n_d${}/; $cached_env_img{$key} ='

'; $key = q/{}$omegain(0,2)${}/; $cached_env_img{$key} ='

'; $key = q/{}$M^<<1421>>-1<<1421>>A${}/; $cached_env_img{$key} ='

'; $key = q/{}$(A^<<2370>>-1<<2370>>R)^T=R^TA^<<2371>>-T<<2371>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$ktimesk${}/; $cached_env_img{$key} ='

'; $key = q/{}$(deltar^<<2402>>(i)<<2402>>)_j${}/; $cached_env_img{$key} ='

'; $key = q/{}$A^<<2245>>-1<<2245>>${}/; $cached_env_img{$key} ='

'; $key = q/{}[frac<<3540>>|e^<<2203>>(i)<<2203>>|<<3540>><<3541>>|x^<<2204>>(i)<<2204>>|<<3541>>leqfrac<<3542>>|A^<<2205>>-1<<2205>>|cdot|r^<<2206>>(i)<<2206>>|<<3542>><<3543>>|x^<<2207>>(i)<<2207>>|<<3543>>leq<<2208>>emstop_tol<<2208>>;;,]{}/; $cached_env_img{$key} ='

'; $key = q/{}$<<3202>>A'<<3202>>_i^<<3203>>-1<<3203>>,A_i^<<3204>>-1<<3204>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$(A+deltaA)x^<<2141>>(i)<<2141>>=(b+deltab)${}/; $cached_env_img{$key} ='

'; $key = q/{}$x_i^<<3431>>(k)<<3431>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$varepsilon=2^<<2048>>-24<<2048>>approx10^<<2049>>-7<<2049>>${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}|Ax^<<757>>(i)<<757>>-b|_2=|D_<<758>>i+1<<758>>bar<<759>>T<<759>>_iy-|r^<<760>>(0)<<760>>|_2e^<<761>>(1)<<761>>|_2{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$n_b-1${}/; $cached_env_img{$key} ='

'; $key = q/{}$|deltab|leqtolcdot|b|${}/; $cached_env_img{$key} ='

'; $key = q/{}$h=N^<<285>>-d<<285>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$(A+deltaA)x^<<2027>>(i)<<2027>>=b+deltab${}/; $cached_env_img{$key} ='

'; $key = q/{}$|A^<<2353>>-1<<2353>>|_<<2354>>infty<<2354>>cdot|r^<<2355>>(i)<<2355>>|_<<2356>>infty<<2356>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$y=Ax${}/; $cached_env_img{$key} ='

'; $key = q/{}$A_H^<<3224>>-1<<3224>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$A${}/; $cached_env_img{$key} ='

'; $key = q/{}$L^<<2734>>-1<<2734>>r^<<2735>>(i)<<2735>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$y=Acdotx${}/; $cached_env_img{$key} ='

'; $key = q/{}$10^<<2045>>-6<<2045>>${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}d_<<1773>>i,i<<1773>>=left{<<1774>>array<<1774>><<1775>>ll<<1775>>a_<<1776>>1,1<<1776>>&mbox<<1777>>if<<5675>><<5675>>$i=1$<<5676>><<5676>><<1777>>a_<<1778>>i,i<<1778>>-a_<<1779>>i,i-1<<1779>>d_<<1780>>i-1<<1780>>^<<3665>>-1<<3665>>a_<<1781>>i-1,i<<1781>>&mbox<<1782>>if<<5677>><<5677>>$1<ileqn$<<5678>><<5678>><<1782>>a_<<1783>>i,i<<1783>>-a_<<1784>>i,i-n<<1784>>d_<<1785>>i-n<<1785>>^<<3666>>-1<<3666>>a_<<1786>>i-n,i<<1786>>&mbox<<1787>>if<<5679>><<5679>>$i=kn+1$<<5680>><<5680>>with<<5681>><<5681>>$k>1$<<5682>><<5682>><<1787>><<1788>>array<<1788>><<1789>>rcl<<1789>>a_<<1790>>i,i<<1790>>&-&a_<<1791>>i,i-1<<1791>>d_<<1792>>i-1<<1792>>^<<3667>>-1<<3667>>a_<<1793>>i-1,i<<1793>>&&mbox<<1794>><<1794>>-a_<<1795>>i,i-n<<1795>>d_<<1796>>i-n<<1796>>^<<3668>>-1<<3668>>a_<<1797>>i-n,i<<1797>><<1798>>array<<1798>>&mbox<<1799>>otherwise<<1799>>.<<1800>>array<<1800>>right.{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$a_<<2680>>i,j<<2680>>${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}mbox<<1662>>solve<<5497>><<5497>>$u$<<5498>><<5498>>from<<5499>><<5499>>$Mu=v$<<5500>><<5500>><<1662>>,{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$a_<<2510>>i,j<<2510>>not=0${}/; $cached_env_img{$key} ='

'; $key = q/{}$P_i^2(A)r^<<1226>>(0)<<1226>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$bar<<239>>x<<239>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$tilde<<3247>>A<<3247>>_i${}/; $cached_env_img{$key} ='

'; $key = q/{}$a_<<395>>i,j<<395>>=a_<<396>>j,i<<396>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$lambda_<<3401>>max<<3401>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$a_<<3402>>i,j<<3402>>=0${}/; $cached_env_img{$key} ='

'; $key = q/{}$O(h^<<3102>>-1<<3102>>)${}/; $cached_env_img{$key} ='

'; $key = q/{}$I_n={1,ldots,n}${}/; $cached_env_img{$key} ='

'; $key = q/{}$R_<<3117>>E_i<<3117>>${}/; $cached_env_img{$key} ='

'; $key = q/{}${r^<<2877>>(k)<<2877>>}${}/; $cached_env_img{$key} ='

'; $key = q/{}$S_<<3130>>E_i<<3130>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$deltar^<<2390>>(i)<<2390>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$g_B${}/; $cached_env_img{$key} ='

'; $key = q/{}$Av^<<2854>>(1)<<2854>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$1${}/; $cached_env_img{$key} ='

'; $key = q/{}$r^<<2396>>(i)<<2396>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$r^<<526>>(i)^T<<526>>A^<<527>>-1<<527>>r^<<528>>(i)<<528>>${}/; $cached_env_img{$key} ='

'; $key = q/{}${Y_i}_<<1904>>i=1..n<<1904>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$xleftarrowx-l_<<2704>>*1<<2704>>y_1${}/; $cached_env_img{$key} ='

'; $key = q/{}$alpha_i${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}hat<<2898>>Lambda<<2898>>_k=left[<<2899>>array<<2899>><<2900>>ccccc<<2900>>p^<<2901>>(1)^T<<2901>>Ap^<<2902>>(1)<<2902>>&0&&cdots&00&p^<<2903>>(2)^T<<2903>>Ap^<<2904>>(2)<<2904>>&&&vdots&ddots&ddots&ddots&vdots&&ddots&ddots&00&cdots&&0&p^<<2905>>(k)^T<<2905>>Ap^<<2906>>(k)<<2906>><<2907>>array<<2907>>right].{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$M^<<2848>>-1<<2848>>Av^<<2849>>(j)<<2849>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$e_<<2232>>j,k<<2232>>=|a_<<2233>>j,k<<2233>>|${}/; $cached_env_img{$key} ='

'; $key = q/{}$j<i${}/; $cached_env_img{$key} ='

'; $key = q/{}$|deltar^<<2392>>(i)<<2392>>|leqO(varepsilon)(|A|cdot|x^<<2393>>(i)<<2393>>|+|b|)${}/; $cached_env_img{$key} ='

'; $key = q/{}$|deltab|${}/; $cached_env_img{$key} ='

'; $key = q/{}$I+alphaL_1${}/; $cached_env_img{$key} ='

'; $key = q/{}$mbox<<615>>span<<615>>{r^<<616>>(0)<<616>>,ldots,A^<<617>>i-1<<617>>r^<<618>>(0)<<618>>}${}/; $cached_env_img{$key} ='

'; $key = q/{}${x^TAx:x^Tx=1}${}/; $cached_env_img{$key} ='

'; $key = q/{}$x^<<600>>(i)<<600>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$max{|deltaA||A|,|deltab||b|}${}/; $cached_env_img{$key} ='

'; $key = q/{}$sqrt<<2135>>n<<2135>>${}/; $cached_env_img{$key} ='

'; $key = q/{tabular}<<2532>>|r|r|r|r|r|r|r|r|r|r|r|<<2532>>hline<<2533>>ttval(:,-1)<<2533>>&0&3&7&8&9&2hline<<2534>>ttval(:,0)<<2534>>&10&9&8&7&9&-1hline<<2535>>ttval(:,+1)<<2535>>&-3&6&7&5&13&0hline{tabular}/; $cached_env_img{$key} ='

'; $key = q/{}$x^<<2726>>(i+1)<<2726>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$|e^<<2286>>(i)<<2286>>|=|A^<<2287>>-1<<2287>>M<<3546>>hat<<2288>>r<<2288>><<3546>>^<<2289>>(i)<<2289>>|leq|A^<<2290>>-1<<2290>>M|cdot|<<3547>>hat<<2291>>r<<2291>><<3547>>^<<2292>>(i)<<2292>>|${}/; $cached_env_img{$key} ='

'; $key = q/{}$d${}/; $cached_env_img{$key} ='

'; $key = q/{}$z^<<552>>(i-1)<<552>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$bar<<767>>T<<767>>_i${}/; $cached_env_img{$key} ='

'; $key = q/{}$A^T${}/; $cached_env_img{$key} ='

'; $key = q/{}$A+O(h^2)D_A${}/; $cached_env_img{$key} ='

'; $key = q/{}$(A^TA)x=tilde<<184>>b<<184>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$|x|_<<2380>>infty<<2380>>=max_j|x_j|${}/; $cached_env_img{$key} ='

'; $key = q/{}$z^<<1029>>(i-1)^T<<1029>>tilde<<1030>>r<<1030>>^<<1031>>(i-1)<<1031>>approx0${}/; $cached_env_img{$key} ='

'; $key = q/{}$(x,y)=barx^Ty${}/; $cached_env_img{$key} ='

'; $key = q/{}$e^<<2397>>(i)<<2397>>=A^<<2398>>-1<<2398>>r^<<2399>>(i)<<2399>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$LUy=x${}/; $cached_env_img{$key} ='

'; $key = q/{}[1leq<<2662>>tti<<2662>>,<<2663>>tti+j<<2663>>leqn.]{}/; $cached_env_img{$key} ='

'; $key = q/{}$e^<<2096>>(i)<<2096>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$R_H,R^T_H${}/; $cached_env_img{$key} ='

'; $key = q/{}$|e^<<2310>>(i)<<2310>>|leq|A^<<2311>>-1<<2311>>M^<<2312>>12<<2312>>|cdot|r^<<2313>>(i)<<2313>>|_<<3551>>M^<<2314>>-12<<2314>>,2<<3551>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$M^<<2283>>-1<<2283>>Ax=M^<<2284>>-1<<2284>>b${}/; $cached_env_img{$key} ='

'; $key = q/{}$alpha${}/; $cached_env_img{$key} ='

'; $key = q/{}$M_<<3173>>X_k<<3173>>${}/; $cached_env_img{$key} ='

'; $key = q/{}[frac<<3538>>|e^<<2202>>(i)<<2202>>|<<3538>><<3539>>|x^<<2203>>(i)<<2203>>|<<3539>>leqfrac<<3540>>|A^<<2204>>-1<<2204>>|cdot|r^<<2205>>(i)<<2205>>|<<3540>><<3541>>|x^<<2206>>(i)<<2206>>|<<3541>>leq<<2207>>emstop_tol<<2207>>;;,]{}/; $cached_env_img{$key} ='

'; $key = q/{}${A^ix}_<<622>>igeq0<<622>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$n_<<3117>>E_i<<3117>>timesn${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}hat<<2893>>T<<2893>>_k=R_k^TAR_k=B_k^That<<2894>>Lambda<<2894>>_kB_k{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}P_k=[p^<<2887>>(1)<<2887>>,~p^<<2888>>(2)<<2888>>,~ldots,~p^<<2889>>(k)<<2889>>].{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$T${}/; $cached_env_img{$key} ='

'; $key = q/{}$y_<<2542>>sigma_k(i)<<2542>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$|A^<<2367>>-1<<2367>>R|_<<2368>>infty<<2368>>${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}bar<<712>>T<<712>>_i=<<713>>array<<713>><<714>>cc<<714>><<715>>array<<715>><<716>>ccc<<716>>leftarrow&i&rightarrow<<717>>array<<717>>&<<718>>array<<718>><<719>>c<<719>>mbox<<720>>~~<<720>><<721>>array<<721>>left(<<722>>array<<722>><<723>>cccc<<723>>ddots&ddots&&ddots&ddots&ddots&&ddots&ddots&ddots&&ddots&ddots&&&ddots<<724>>array<<724>>right).&<<725>>array<<725>><<726>>c<<726>>uparrowmbox<<727>>~<<727>>i+1mbox<<728>>~<<728>>downarrow<<729>>array<<729>><<730>>array<<730>>{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$deltaa_<<2212>>j,k<<2212>>${}/; $cached_env_img{$key} ='

'; $key = q/{}[LUy=xqquadhbox<<2672>>and<<2672>>qquad(LU)^Ty=x.]{}/; $cached_env_img{$key} ='

'; $key = q/{}$r^<<2259>>(i)<<2259>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$x^<<2161>>(i)<<2161>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$|A||A^<<3648>>-1<<3648>>|=lambda^<<650>>12<<650>>_<<651>>max<<651>>(A^TA)lambda^<<652>>12<<652>>_<<653>>min<<653>>(A^TA)${}/; $cached_env_img{$key} ='

'; $key = q/{}$d-c${}/; $cached_env_img{$key} ='

'; $key = q/{}$|(I-G)^<<2331>>-1<<2331>>|leq1(1-|G|)${}/; $cached_env_img{$key} ='

'; $key = q/{}$O(1+log^2(Hh))${}/; $cached_env_img{$key} ='

'; $key = q/{}$M^<<632>>-1<<632>>A${}/; $cached_env_img{$key} ='

'; $key = q/{}$itimesi${}/; $cached_env_img{$key} ='

'; $key = q/{}$(a_<<3429>>i,i<<3429>>)${}/; $cached_env_img{$key} ='

'; $key = q/{}$K^<<3031>>-1<<3031>>v${}/; $cached_env_img{$key} ='

'; $key = q/{}$M=LU${}/; $cached_env_img{$key} ='

'; $key = q/{}$p^<<547>>(i)<<547>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$<<3558>>tilde<<3210>>A'<<3210>><<3558>>_i^<<3211>>-1<<3211>>,tilde<<3212>>A<<3212>>_i^<<3213>>-1<<3213>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$S_<<3138>>E_i<<3138>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$Lambda_k${}/; $cached_env_img{$key} ='

'; $key = q/{}$r^<<2254>>(0)<<2254>>=b${}/; $cached_env_img{$key} ='

'; $key = q/{}$x^<<2252>>(0)<<2252>>=0${}/; $cached_env_img{$key} ='

'; $key = q/{}$D${}/; $cached_env_img{$key} ='

'; $key = q/{}$Tx=b${}/; $cached_env_img{$key} ='

'; $key = q/{}$deltaH${}/; $cached_env_img{$key} ='

'; $key = q/{}$|e^<<2157>>(i)<<2157>>|leq|,|A^<<2158>>-1<<2158>>|cdot|r^<<2159>>(i)<<2159>>|,|${}/; $cached_env_img{$key} ='

'; $key = q/{}$T_H${}/; $cached_env_img{$key} ='

'; $key = q/{}$v^<<2850>>(1)<<2850>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$|deltar^<<2406>>(i)<<2406>>|leqO(varepsilon)(|A|cdot|x^<<2407>>(i)<<2407>>|+|b|)${}/; $cached_env_img{$key} ='

'; $key = q/{}$<<1997>>calL<<1997>>_1${}/; $cached_env_img{$key} ='

'; $key = q/{}$lambda_<<1429>>min<<1429>>${}/; $cached_env_img{$key} ='

'; $key = q/{}${p^<<2892>>(j)<<2892>>}${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}C=(X+L)(I+X^<<3680>>-1<<3680>>U){displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$imapstoP_i^2(A)r^<<1313>>(0)<<1313>>${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}tilder^<<940>>(i)^T<<940>>r^<<941>>(j)<<941>>=tildep^<<942>>(i)^T<<942>>Ap^<<943>>(j)<<943>>=0qquadhbox<<944>>if<<4955>><<4955>>$inot=j$<<4956>><<4956>><<944>>.{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}r^<<911>>(i)<<911>>=r^<<912>>(i-1)<<912>>-alpha_iAp^<<913>>(i)<<913>>,qquadtilder^<<914>>(i)<<914>>=tilder^<<915>>(i-1)<<915>>-alpha_iA^Ttildep^<<916>>(i)<<916>>,{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$w${}/; $cached_env_img{$key} ='

'; $key = q/{}$x^<<2796>>(i)<<2796>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$x=M_2^<<3661>>-1<<3661>>y${}/; $cached_env_img{$key} ='

'; $key = q/{}$|x|_<<2379>>infty<<2379>>=max_j|x_j|${}/; $cached_env_img{$key} ='

'; $key = q/{}$I_i${}/; $cached_env_img{$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/{}[LUy=xqquadhbox<<2673>>and<<2673>>qquad(LU)^Ty=x.]{}/; $cached_env_img{$key} ='

'; $key = q/{}$mbox<<799>>span<<799>>{r^<<800>>(0)<<800>>,Ar^<<801>>(0)<<801>>,A^2r^<<802>>(0)<<802>>,ldots}${}/; $cached_env_img{$key} ='

'; $key = q/{}$lambda(B)=1-12lambda(L)=1-sin^2npi(2N)${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}R_k=[r^<<2868>>(0)<<2868>>,~r^<<2869>>(1)<<2869>>,~ldots,~r^<<2870>>(k-1)<<2870>>],{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$|e^<<2137>>(i)<<2137>>|${}/; $cached_env_img{$key} ='

'; $key = q/{}$f_j=|b_j|${}/; $cached_env_img{$key} ='

'; $key = q/{}$alpha=(sqrt<<644>>kappa_2<<644>>-1)(sqrt<<645>>kappa_2<<645>>+1)${}/; $cached_env_img{$key} ='

'; $key = q/{}$I-L${}/; $cached_env_img{$key} ='

'; $key = q/{}$tilde<<185>>b<<185>>=A^Tb${}/; $cached_env_img{$key} ='

'; $key = q/{}$|A|${}/; $cached_env_img{$key} ='

'; $key = q/{}$A_<<3071>>I<<3071>>=blockdiagonal(A_<<3072>>i<<3072>>)${}/; $cached_env_img{$key} ='

'; $key = q/{}$<<3203>>A'<<3203>>_i^<<3204>>-1<<3204>>,A_i^<<3205>>-1<<3205>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$x^<<2252>>(0)<<2252>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$|A^<<2196>>-1<<2196>>|cdot|r^<<2197>>(i)<<2197>>|${}/; $cached_env_img{$key} ='

'; $key = q/{}$r^<<2093>>(i)<<2093>>=Ax^<<2094>>(i)<<2094>>-b${}/; $cached_env_img{$key} ='

'; $key = q/{}$x=M_2^<<3659>>-1<<3659>>y${}/; $cached_env_img{$key} ='

'; $key = q/{}$r^<<1201>>(0)<<1201>>${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}left[<<2967>>array<<2967>><<2968>>cc<<2968>>I&O-C^TD_R^<<2969>>-1<<2969>>&I<<2970>>array<<2970>>right]left[<<2971>>array<<2971>><<2972>>cc<<2972>>D_R&CC^T&D_B<<2973>>array<<2973>>right]left[<<2974>>array<<2974>><<2975>>c<<2975>>x_Rx_B<<2976>>array<<2976>>right]=left[<<2977>>array<<2977>><<2978>>cc<<2978>>I&O-C^TD_R^<<2979>>-1<<2979>>&I<<2980>>array<<2980>>right]left[<<2981>>array<<2981>><<2982>>c<<2982>>f_Rf_B<<2983>>array<<2983>>right]{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$(A_<<1895>>i,i<<1895>>)${}/; $cached_env_img{$key} ='

'; $key = q/{}$p^<<2799>>(i)<<2799>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$M^<<1432>>-1<<1432>>A${}/; $cached_env_img{$key} ='

'; $key = q/{}$Q_k=R_kDelta^<<2915>>-1<<2915>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$2n^2${}/; $cached_env_img{$key} ='

'; $key = q/{}$tilde<<1036>>p<<1036>>^<<1037>>(i)^T<<1037>>q^<<1038>>(i)<<1038>>approx0${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}Ar^<<699>>(i)<<699>>=r^<<700>>(i+1)<<700>>t_<<701>>i+1,i<<701>>+r^<<702>>(i)<<702>>t_<<703>>i,i<<703>>+r^<<704>>(i-1)<<704>>t_<<705>>i-1,i<<705>>,{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$|A^<<2194>>-1<<2194>>|${}/; $cached_env_img{$key} ='

'; $key = q/{}$|e^<<2318>>(i)<<2318>>|${}/; $cached_env_img{$key} ='

'; $key = q/{}$y=A^Tx${}/; $cached_env_img{$key} ='

'; $key = q/{}$A+alphaI${}/; $cached_env_img{$key} ='

'; $key = q/{}$u_n(x)=sinnpix${}/; $cached_env_img{$key} ='

'; $key = q/{}$deltar^<<2391>>(i)<<2391>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$approx1-5N^2${}/; $cached_env_img{$key} ='

'; $key = q/{}$sqrt<<648>>kappa_2<<648>>${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}sum_<<213>>j=1<<213>>^na_<<214>>i,j<<214>>x_j=b_i,{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$(AA^T)y=b${}/; $cached_env_img{$key} ='

'; $key = q/{}$|x|_<<2134>>infty<<2134>>leq1${}/; $cached_env_img{$key} ='

'; $key = q/{}$L^<<2740>>-T<<2740>>s${}/; $cached_env_img{$key} ='

'; $key = q/{}$a_<<2682>>j,j<<2682>>${}/; $cached_env_img{$key} ='

'; $key = q/{}[|e^<<2187>>(i)<<2187>>|leq|A^<<2188>>-1<<2188>>|cdot|r^<<2189>>(i)<<2189>>|leq<<2190>>emstop_tol<<2190>>cdot|A^<<2191>>-1<<2191>>|cdot|b|]{}/; $cached_env_img{$key} ='

'; $key = q/{}$min_i{a_<<3380>>i,i<<3380>>-sum_<<3381>>jnot=i<<3381>>|a_<<3382>>i,j<<3382>>|}${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}Lu(x_i)=2u(x_i)-u(x_<<275>>i-1<<275>>)-u(x_<<276>>i+1<<276>>)=f(x_i)qquadhbox<<277>>for<<4373>><<4373>>$x_i=iN$<<4374>><<4374>>,<<4375>><<4375>>$i=1ldotsN-1$<<4376>><<4376>><<277>>.{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{tabbing}junkjunk=junk=junk=junk=junk=junkkill><<2068>>bfuntil<<2068>><<5955>><<5955>>$igeqmaxit$<<5956>><<5956>>or<<5957>><<5957>>$|r^<<2069>>(i)<<2069>>|leq<<2070>>emstop_tol<<2070>>cdot|b|;;.$<<5958>><<5958>>{tabbing}/; $cached_env_img{$key} ='

'; $key = q/{}$r^<<2391>>(i)<<2391>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$O(h^<<1836>>-2<<1836>>)${}/; $cached_env_img{$key} ='

'; $key = q/{}$$M_<<3139>>E_i<<3139>>=alpha_<<3140>>E_i<<3140>>K^<<3141>>12<<3141>>$${}/; $cached_env_img{$key} ='

'; $key = q/{}[|e^<<2236>>(i)<<2236>>|_<<2237>>infty<<2237>>leq|,|A^<<2238>>-1<<2238>>|cdot|r^<<2239>>(i)<<2239>>|,|leqS_4cdot|,|A^<<2240>>-1<<2240>>|(E|x^<<2241>>(i)<<2241>>|+f)|_<<2242>>infty<<2242>>]{}/; $cached_env_img{$key} ='

'; $key = q/{}$M_1=M_2^T${}/; $cached_env_img{$key} ='

'; $key = q/{}$<<2436>>ttval(k)<<2436>>=a_<<2437>>i,j<<2437>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$S_<<3171>>X_k<<3171>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$imapstoP_i^2(A)r^<<1314>>(0)<<1314>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$|A|_2=(lambda_<<2130>>max<<2130>>(AA^T))^<<2131>>12<<2131>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$0<alpha<nobreak1${}/; $cached_env_img{$key} ='

'; $key = q/{}$omega=1${}/; $cached_env_img{$key} ='

'; $key = q/{}$tilde<<3226>>A<<3226>>_H^<<3227>>-1<<3227>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$O(h^<<3103>>-2<<3103>>)${}/; $cached_env_img{$key} ='

'; $key = q/{}$<<3543>>hat<<2271>>r<<2271>><<3543>>^<<2272>>(i)<<2272>>=x^<<2273>>(i)<<2273>>-Gx^<<2274>>(i)<<2274>>-c=M^<<2275>>-1<<2275>>(Ax^<<2276>>(i)<<2276>>-b)=M^<<2277>>-1<<2277>>r^<<2278>>(i)<<2278>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$(A^<<2371>>-1<<2371>>R)^T=R^TA^<<2372>>-T<<2372>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$D_<<749>>i+1<<749>>equivmbox<<750>>diag<<750>>(|r^<<751>>(0)<<751>>|_2,|r^<<752>>(1)<<752>>|_2,...,|r^<<753>>(i)<<753>>|_2)${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}left(<<781>>array<<781>><<782>>cc<<782>>I&AA^T&0<<783>>array<<783>>right)left(<<784>>array<<784>><<785>>c<<785>>rx<<786>>array<<786>>right)=left(<<787>>array<<787>><<788>>c<<788>>b0<<789>>array<<789>>right).{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$v^<<2851>>(2)<<2851>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$n'_i${}/; $cached_env_img{$key} ='

'; $key = q/{}$r^<<525>>(i)^T<<525>>A^<<526>>-1<<526>>r^<<527>>(i)<<527>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$omega${}/; $cached_env_img{$key} ='

'; $key = q/{}$R_<<3169>>X_k<<3169>>,R^T_<<3170>>X_k<<3170>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$|deltab|leq<<2168>>emstop_tol<<2168>>cdot|b|${}/; $cached_env_img{$key} ='

'; $key = q/{}$z${}/; $cached_env_img{$key} ='

'; $key = q/{}$e^<<2091>>(i)<<2091>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$x^<<600>>(0)<<600>>+mbox<<601>>span<<601>>{r^<<602>>(0)<<602>>,ldots,A^<<603>>i-1<<603>>r^<<604>>(0)<<604>>}${}/; $cached_env_img{$key} ='

'; $key = q/{}$bar<<349>>x<<349>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$(A^<<3686>>-1<<3686>>)_<<3384>>i,j<<3384>>geq0${}/; $cached_env_img{$key} ='

'; $key = q/{}$a_<<1831>>i,i<<1831>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$jinE_i${}/; $cached_env_img{$key} ='

'; $key = q/{}$Av${}/; $cached_env_img{$key} ='

'; $key = q/{}$deltaa_<<2213>>j,k<<2213>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$S_<<3167>>X_k<<3167>>${}/; $cached_env_img{$key} ='

'; $key = q/{tabular}<<2551>>|r|r|r|r|r|r|r|r|r|r|r|<<2551>>hline<<2552>>ttval(:,-1)<<2552>>&0&0&3&6&0&5hline<<2553>>ttval(:,0)<<2553>>&10&9&8&7&9&-1hline<<2554>>ttval(:,+1)<<2554>>&0&-3&6&7&5&13hline<<2555>>ttval(:,+2)<<2555>>&0&1&-2&0&4&0hline{tabular}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}tilder^<<941>>(i)^T<<941>>r^<<942>>(j)<<942>>=tildep^<<943>>(i)^T<<943>>Ap^<<944>>(j)<<944>>=0qquadhbox<<945>>if<<4959>><<4959>>$inot=j$<<4960>><<4960>><<945>>.{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$x^<<2140>>(i)<<2140>>${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}M(omega)=<<1708>>1over2-omega<<1708>>(<<1709>>1overomega<<1709>>D+L)(<<1710>>1overomega<<1710>>D)^<<3663>>-1<<3663>>(<<1711>>1overomega<<1711>>D+L)^T.{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$(LU)^Ty=x${}/; $cached_env_img{$key} ='

'; $key = q/{}$P(0)=1${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}<<271>>calL<<271>>u=-u_<<272>>xx<<272>>=fqquadhbox<<273>>on<<4375>><<4375>>$(0,1)$<<4376>><<4376>><<273>>,qquadu(0)=u_0,quadu(1)=u_1,{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$A^<<2373>>-T<<2373>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$tilde<<3249>>A<<3249>>_H^<<3250>>-1<<3250>>A_H${}/; $cached_env_img{$key} ='

'; $key = q/{}$r^<<553>>(i-1)<<553>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$hat<<611>>x<<611>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$a_<<3351>>i,j<<3351>>=${}/; $cached_env_img{$key} ='

'; $key = q/{}$partialOmega_i${}/; $cached_env_img{$key} ='

'; $key = q/{}$j${}/; $cached_env_img{$key} ='

'; $key = q/{}$g=D_B^<<3008>>-12<<3008>>(f_B-C^TD_R^<<3009>>-1<<3009>>f_R)${}/; $cached_env_img{$key} ='

'; $key = q/{}$a_<<2681>>j,i<<2681>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$xnot=0${}/; $cached_env_img{$key} ='

'; $key = q/{}$imapstoQ_i(A)P_i(A)r^<<1315>>(0)<<1315>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$|r^<<2261>>(i)<<2261>>|${}/; $cached_env_img{$key} ='

'; $key = q/{}$(A+deltaA)x^<<2028>>(i)<<2028>>=b+deltab${}/; $cached_env_img{$key} ='

'; $key = q/{}$sqrt<<2134>>n<<2134>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$d_<<1883>>i,i<<1883>>=1a_<<1884>>i,i<<1884>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$|A^<<2195>>-1<<2195>>|${}/; $cached_env_img{$key} ='

'; $key = q/{}$sqrt<<1192>>i+1<<1192>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$tilde<<3240>>A<<3240>>_I,tilde<<3241>>S<<3241>>_B${}/; $cached_env_img{$key} ='

'; $key = q/{}[P_ix=A_i(A^T_iA_i)^<<3687>>-1<<3687>>A^T_ix.]{}/; $cached_env_img{$key} ='

'; $key = q/{}$deltaA${}/; $cached_env_img{$key} ='

'; $key = q/{}$omega=2-O(h)${}/; $cached_env_img{$key} ='

'; $key = q/{}$bigcup_<<3054>>i<<3054>>partialomip${}/; $cached_env_img{$key} ='

'; $key = q/{}$M^<<2282>>-1<<2282>>Ax=M^<<2283>>-1<<2283>>b${}/; $cached_env_img{$key} ='

'; $key = q/{}$r^<<1190>>(i)<<1190>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$M=(D+L)D^<<3666>>-1<<3666>>(D+U)${}/; $cached_env_img{$key} ='

'; $key = q/{}$|e^<<2287>>(i)<<2287>>|=|A^<<2288>>-1<<2288>>M<<3548>>hat<<2289>>r<<2289>><<3548>>^<<2290>>(i)<<2290>>|leq|A^<<2291>>-1<<2291>>M|cdot|<<3549>>hat<<2292>>r<<2292>><<3549>>^<<2293>>(i)<<2293>>|${}/; $cached_env_img{$key} ='

'; $key = q/{}$$tilde<<3216>>K<<3216>>_<<3217>>as<<3217>>^<<3218>>-1<<3218>>v=R_H^Ttilde<<3219>>A<<3219>>_H^<<3220>>-1<<3220>>R_Hv+sum_<<3221>>i=1<<3221>>^pR_i^T<<3557>>tilde<<3222>>A'<<3222>><<3557>>_i^<<3223>>-1<<3223>>R_iv.$${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}x_i^<<343>>(k)<<343>>=omegabar<<344>>x<<344>>_i^<<345>>(k)<<345>>+(1-omega)x_i^<<346>>(k-1)<<346>>{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$r^<<612>>(i)<<612>>=b-Ax^<<613>>(i)<<613>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$varepsilon${}/; $cached_env_img{$key} ='

'; $key = q/{}$x^<<240>>(k-1)<<240>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$|e^<<2084>>(i)<<2084>>||x^<<2085>>(i)<<2085>>|${}/; $cached_env_img{$key} ='

'; $key = q/{}$a_<<2679>>i,j<<2679>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$Sw${}/; $cached_env_img{$key} ='

'; $key = q/{}$A=I-B${}/; $cached_env_img{$key} ='

'; $key = q/{}$S={(i,sigma(i));~iinIsubseteqI_n}${}/; $cached_env_img{$key} ='

'; $key = q/{}$bar<<238>>x<<238>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$|,|A^<<2357>>-1<<2357>>|cdot|r^<<2358>>(i)<<2358>>|,|_<<2359>>infty<<2359>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$S_<<3129>>E_i<<3129>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$(a_<<3428>>i,i<<3428>>)${}/; $cached_env_img{$key} ='

'; $key = q/{}$beta${}/; $cached_env_img{$key} ='

'; $key = q/{}$(A+deltaA)x^<<2217>>(i)<<2217>>=b+deltab${}/; $cached_env_img{$key} ='

'; $key = q/{}$tilde<<3235>>A<<3235>>_i${}/; $cached_env_img{$key} ='

'; $key = q/{}$K_<<3193>>VS<<3193>>^<<3194>>-1<<3194>>S_B${}/; $cached_env_img{$key} ='

'; $key = q/{}$A^<<3679>>-1<<3679>>=sum_<<1938>>k=0<<1938>>^inftyB^k${}/; $cached_env_img{$key} ='

'; $key = q/{}$n_<<3141>>E_i<<3141>>timesn_<<3142>>E_i<<3142>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$a_<<1827>>i,k<<1827>>a_<<1828>>k,k<<1828>>^<<3671>>-1<<3671>>a_<<1829>>k,j<<1829>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$sqrt<<649>>kappa_2<<649>>${}/; $cached_env_img{$key} ='

'; $key = q/{tabular}<<2613>>|r|r|r|r|r|r|r|<<2613>>hline<<2614>>ttperm<<2614>>&5&2&3&4&1&6hline{tabular}/; $cached_env_img{$key} ='

'; $key = q/{}${p^<<2891>>(j)<<2891>>}${}/; $cached_env_img{$key} ='

'; $key = q/{}$|x|_2=(sum_j|x_j|^2)^<<2381>>12<<2381>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$M=LL^T${}/; $cached_env_img{$key} ='

'; $key = q/{}$|A||A^<<3650>>-1<<3650>>|=lambda^<<651>>12<<651>>_<<652>>max<<652>>(A^TA)lambda^<<653>>12<<653>>_<<654>>min<<654>>(A^TA)${}/; $cached_env_img{$key} ='

'; $key = q/{}[y=Axqquadhbox<<2643>>and<<2643>>qquady=A^Tx.]{}/; $cached_env_img{$key} ='

'; $key = q/{}$Ax^<<2389>>(i)<<2389>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$|deltaA|leq<<2167>>emstop_tol<<2167>>cdot|A|${}/; $cached_env_img{$key} ='

'; $key = q/{}$r^<<2388>>(i)<<2388>>=Ax^<<2389>>(i)<<2389>>-b${}/; $cached_env_img{$key} ='

'; $key = q/{}$e^<<2089>>(i)<<2089>>=x^<<2090>>(i)<<2090>>-x${}/; $cached_env_img{$key} ='

'; $key = q/{}$-L${}/; $cached_env_img{$key} ='

'; $key = q/{}$a_<<3362>>i,j<<3362>>=0${}/; $cached_env_img{$key} ='

'; $key = q/{}$10^<<2044>>-6<<2044>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$(j,sigma(j))${}/; $cached_env_img{$key} ='

'; $key = q/{}$L_1${}/; $cached_env_img{$key} ='

'; $key = q/{}$A^Ttilde<<1048>>p<<1048>>^<<1049>>(k)<<1049>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$nvarepsilon${}/; $cached_env_img{$key} ='

'; $key = q/{}$O(n^2)${}/; $cached_env_img{$key} ='

'; $key = q/{}$|e^<<2136>>(i)<<2136>>|${}/; $cached_env_img{$key} ='

'; $key = q/{}$omip${}/; $cached_env_img{$key} ='

'; $key = q/{}$E_<<3132>>i<<3132>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$v^<<2853>>(1)<<2853>>${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}|Ax^<<744>>(i)<<744>>-b|_2=|AR_ibar<<745>>y<<745>>-b|_2=|R_<<746>>i+1<<746>>bar<<747>>T<<747>>_iy-b|_2.{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$m${}/; $cached_env_img{$key} ='

'; $key = q/{}[P_ix=A_i(A^T_iA_i)^<<3685>>-1<<3685>>A^T_ix.]{}/; $cached_env_img{$key} ='

'; $key = q/{}$kappa(M_<<3533>>omega_<<1713>>rmopt<<1713>><<3533>>^<<3664>>-1<<3664>>A)=O(sqrt<<1714>>kappa(A)<<1714>>)${}/; $cached_env_img{$key} ='

'; $key = q/{}[hbox<<2652>>forall<<6621>><<6621>>$j$<<6622>><<6622>>,doforall<<6623>><<6623>>$i$<<6624>><<6624>>:<<2652>>qquady_ileftarrowy_i+a_<<2653>>j,i<<2653>>x_j.]{}/; $cached_env_img{$key} ='

'; $key = q/{}$|deltab|leqvarepsilon|b|${}/; $cached_env_img{$key} ='

'; $key = q/{tabbing}xxxxxx=kill>Initially:<<5687>><<5687>>$d_<<1804>>i,i<<1804>>=a_<<1805>>i,i<<1805>>$<<5688>><<5688>>forall<<5689>><<5689>>$i$<<5690>><<5690>>><<1806>>bffor<<1806>>=<<5691>><<5691>>$i=1..nm$<<5692>><<5692>>do:>><<5693>><<5693>>$left{<<1807>>array<<1807>><<1808>>ll<<1808>>d_<<1809>>i+1,i+1<<1809>>=d_<<1810>>i+1,i+1<<1810>>-a_<<1811>>i+1,i<<1811>>d_<<1812>>i,i<<1812>>^<<3671>>-1<<3671>>a_<<1813>>i,i+1<<1813>>&mbox<<1814>>ifthereisno$k$<<1814>>&mbox<<1815>>suchthat$i=kn$<<1815>>d_<<1816>>i+n,i+n<<1816>>=d_<<1817>>i+n,i+n<<1817>>-a_<<1818>>i+n,i<<1818>>d_<<1819>>i,i<<1819>>^<<3672>>-1<<3672>>a_<<1820>>i,i+n<<1820>>&mbox<<1821>>if$i+nleqnm$<<1821>><<1822>>array<<1822>>right.$<<5694>><<5694>>{tabbing}/; $cached_env_img{$key} ='

'; $key = q/{}$gamma${}/; $cached_env_img{$key} ='

'; $key = q/{}$A^Tp^<<2721>>(i)<<2721>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$A^Ttilde<<1047>>p<<1047>>^<<1048>>(k)<<1048>>${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}x^<<506>>(i)<<506>>=x^<<507>>(i-1)<<507>>+alpha_ip^<<508>>(i)<<508>>.{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$(A_<<1894>>i,i<<1894>>)${}/; $cached_env_img{$key} ='

'; $key = q/{}$<<2917>>rmspan<<2917>>{b,~Ab,~ldots,~A^<<2918>>k-1<<2918>>b}${}/; $cached_env_img{$key} ='

'; $key = q/{}$x^<<2236>>(i)<<2236>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$Ap^<<541>>(i-1)<<541>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$|e^<<2257>>(i)<<2257>>|leqS_5cdot|A^<<2258>>-1<<2258>>|${}/; $cached_env_img{$key} ='

'; $key = q/{}$K_<<3228>>VS<<3228>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$L_2${}/; $cached_env_img{$key} ='

'; $key = q/{}$|e^<<2317>>(i)<<2317>>|${}/; $cached_env_img{$key} ='

'; $key = q/{}$r^<<1201>>(i)<<1201>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$sqrt<<2395>>n<<2395>>varepsilon${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}A=left[<<3408>>array<<3408>><<3409>>ccc<<3409>>a_<<3410>>1,1<<3410>>&cdots&a_<<3411>>1,n<<3411>>vdots&&vdotsa_<<3412>>m,1<<3412>>&cdots&a_<<3413>>m,n<<3413>><<3414>>array<<3414>>right]hspace<<3415>>2em<<3415>>a_<<3416>>i,j<<3416>>in<<3417>>calR<<3417>>.{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}left[<<2966>>array<<2966>><<2967>>cc<<2967>>I&O-C^TD_R^<<2968>>-1<<2968>>&I<<2969>>array<<2969>>right]left[<<2970>>array<<2970>><<2971>>cc<<2971>>D_R&CC^T&D_B<<2972>>array<<2972>>right]left[<<2973>>array<<2973>><<2974>>c<<2974>>x_Rx_B<<2975>>array<<2975>>right]=left[<<2976>>array<<2976>><<2977>>cc<<2977>>I&O-C^TD_R^<<2978>>-1<<2978>>&I<<2979>>array<<2979>>right]left[<<2980>>array<<2980>><<2981>>c<<2981>>f_Rf_B<<2982>>array<<2982>>right]{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$maxit${}/; $cached_env_img{$key} ='

'; $key = q/{}$A^<<2374>>-T<<2374>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$x^<<2027>>(i)<<2027>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$v^<<2849>>(1)<<2849>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$n=1ldotsN-1${}/; $cached_env_img{$key} ='

'; $key = q/{}$|r^<<2198>>(i)<<2198>>|leqS_3equiv<<2199>>emstop_tol<<2199>>cdot|x^<<2200>>(i)<<2200>>||A^<<2201>>-1<<2201>>|${}/; $cached_env_img{$key} ='

'; $key = q/{}$I^<<3427>>ntimesn<<3427>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$|y|^2_Aequiv(y,Ay)${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}B_2=(D-omegaL)^<<3631>>-1<<3631>>(omegaU+(1-omega)D).{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}m_<<1667>>i,j<<1667>>=left{<<1668>>array<<1668>><<1669>>ll<<1669>>a_<<1670>>i,i<<1670>>&mbox<<1671>>if<<5497>><<5497>>$i=j$<<5498>><<5498>><<1671>>0&mbox<<1672>>otherwise<<1672>>.<<1673>>array<<1673>>right.{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$tilde<<3248>>A<<3248>>_i${}/; $cached_env_img{$key} ='

'; $key = q/{}$x^<<2255>>(0)<<2255>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$kappa_2(B)=lambda_<<635>>max<<635>>(B)lambda_<<636>>min<<636>>(B))${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}bar<<713>>T<<713>>_i=<<714>>array<<714>><<715>>cc<<715>><<716>>array<<716>><<717>>ccc<<717>>leftarrow&i&rightarrow<<718>>array<<718>>&<<719>>array<<719>><<720>>c<<720>>mbox<<721>>~~<<721>><<722>>array<<722>>left(<<723>>array<<723>><<724>>cccc<<724>>ddots&ddots&&ddots&ddots&ddots&&ddots&ddots&ddots&&ddots&ddots&&&ddots<<725>>array<<725>>right).&<<726>>array<<726>><<727>>c<<727>>uparrowmbox<<728>>~<<728>>i+1mbox<<729>>~<<729>>downarrow<<730>>array<<730>><<731>>array<<731>>{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$r^<<2097>>(i)<<2097>>${}/; $cached_env_img{$key} ='

'; $key = q/{}${r^<<2915>>(j)<<2915>>}${}/; $cached_env_img{$key} ='

'; $key = q/{}$pm10^<<2046>>-6<<2046>>|A|${}/; $cached_env_img{$key} ='

'; $key = q/{}$lambda_<<1420>>min<<1420>>${}/; $cached_env_img{$key} ='

'; $key = q/{tabular}<<3325>>ll<<3325>>sharfilename&contentshlinesctemplates.shar&SingleprecisionCroutinesdctemplates.shar&DoubleprecisionCroutinessftemplates.shar&SinglePrecisionFortran77routinesdftemplates.shar&DoublePrecisionFortran77routinesmltemplates.shar&MATLABroutines{tabular}/; $cached_env_img{$key} ='

'; $key = q/{}$P_i(A)${}/; $cached_env_img{$key} ='

'; $key = q/{}$q^<<2799>>(i)<<2799>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$r^<<2262>>(i)<<2262>>=Ax^<<2263>>(i)<<2263>>-b${}/; $cached_env_img{$key} ='

'; $key = q/{}$M${}/; $cached_env_img{$key} ='

'; $key = q/{}$x^<<3360>>(i)<<3360>>${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}M(omega)=<<1707>>1over2-omega<<1707>>(<<1708>>1overomega<<1708>>D+L)(<<1709>>1overomega<<1709>>D)^<<3661>>-1<<3661>>(<<1710>>1overomega<<1710>>D+L)^T.{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$<<2678>>ttval(diag_ptr(i))<<2678>>=a_<<2679>>i,i<<2679>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$x_<<2541>>sigma_k(i)<<2541>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$lambda_<<2331>>max<<2331>>(A)${}/; $cached_env_img{$key} ='

'; $key = q/{}$lambda_<<632>>max<<632>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$M^<<3649>>-1<<3649>>A${}/; $cached_env_img{$key} ='

'; $key = q/{}$e^<<2411>>(i)<<2411>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$a_<<400>>i,j<<400>>leq0${}/; $cached_env_img{$key} ='

'; $key = q/{tabbing}xxxxxx=kill>Initially:<<5683>><<5683>>$d_<<1803>>i,i<<1803>>=a_<<1804>>i,i<<1804>>$<<5684>><<5684>>forall<<5685>><<5685>>$i$<<5686>><<5686>>><<1805>>bffor<<1805>>=<<5687>><<5687>>$i=1..nm$<<5688>><<5688>>do:>><<5689>><<5689>>$left{<<1806>>array<<1806>><<1807>>ll<<1807>>d_<<1808>>i+1,i+1<<1808>>=d_<<1809>>i+1,i+1<<1809>>-a_<<1810>>i+1,i<<1810>>d_<<1811>>i,i<<1811>>^<<3669>>-1<<3669>>a_<<1812>>i,i+1<<1812>>&mbox<<1813>>ifthereisno$k$<<1813>>&mbox<<1814>>suchthat$i=kn$<<1814>>d_<<1815>>i+n,i+n<<1815>>=d_<<1816>>i+n,i+n<<1816>>-a_<<1817>>i+n,i<<1817>>d_<<1818>>i,i<<1818>>^<<3670>>-1<<3670>>a_<<1819>>i,i+n<<1819>>&mbox<<1820>>if$i+nleqnm$<<1820>><<1821>>array<<1821>>right.$<<5690>><<5690>>{tabbing}/; $cached_env_img{$key} ='

'; $key = q/{}$|r^<<2165>>(i)<<2165>>|leqS_1equiv<<2166>>emstop_tol<<2166>>cdot(|A|cdot|x^<<2167>>(i)<<2167>>|+|b|)${}/; $cached_env_img{$key} ='

'; $key = q/{}$a_<<3350>>i,j<<3350>>=${}/; $cached_env_img{$key} ='

'; $key = q/{}$V_k${}/; $cached_env_img{$key} ='

'; $key = q/{}$|e^<<2155>>(i)<<2155>>|leqtaucdot|A^<<2156>>-1<<2156>>|${}/; $cached_env_img{$key} ='

'; $key = q/{tabular}<<3326>>ll<<3326>>sharfilename&contentshlinesctemplates.shar&SingleprecisionCroutinesdctemplates.shar&DoubleprecisionCroutinessftemplates.shar&SinglePrecisionFortran77routinesdftemplates.shar&DoublePrecisionFortran77routinesmltemplates.shar&MATLABroutines{tabular}/; $cached_env_img{$key} ='

'; $key = q/{}$lambda_<<2131>>max<<2131>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$varepsilon=2^<<2050>>-53<<2050>>approx10^<<2051>>-16<<2051>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$|A^<<2334>>-1<<2334>>|_2=lambda^<<2335>>-1<<2335>>_<<2336>>min<<2336>>(A)${}/; $cached_env_img{$key} ='

'; $key = q/{}$|deltab_<<2221>>j<<2221>>|leqtolcdotf_<<2222>>j<<2222>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$u_i${}/; $cached_env_img{$key} ='

'; $key = q/{}$a_<<396>>i,j<<396>>=a_<<397>>j,i<<397>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$p^<<2729>>(i)<<2729>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$A_Bv_B-A_<<3104>>IB<<3104>>^T(A_I^<<3105>>-1<<3105>>(A_<<3106>>IB<<3106>>v_B))${}/; $cached_env_img{$key} ='

'; $key = q/{}$A^HAx=A^Hb${}/; $cached_env_img{$key} ='

'; $key = q/{}$y_i=R^T_iw_i${}/; $cached_env_img{$key} ='

'; $key = q/{}$A^<<2372>>-1<<2372>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$|r^<<2249>>(i)<<2249>>|leqS_5equiv<<2250>>emstop_tol<<2250>>cdot|r^<<2251>>(0)<<2251>>|${}/; $cached_env_img{$key} ='

'; $key = q/{}$r^<<548>>(i)<<548>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$ArightarrowM^<<1647>>-1<<1647>>A${}/; $cached_env_img{$key} ='

'; $key = q/{tabbing}junkjunk=junk=junk=junk=junk=junkkill><<5937>><<5937>>$i=0$<<5938>><<5938>>><<2055>>bfrepeat<<2055>>>><<5939>><<5939>>$i=i+1$<<5940>><<5940>>>>Computetheapproximatesolution<<5941>><<5941>>$x^<<2056>>(i)<<2056>>$<<5942>><<5942>>.>>Computetheresidual<<5943>><<5943>>$r^<<2057>>(i)<<2057>>=Ax^<<2058>>(i)<<2058>>-b$<<5944>><<5944>>.>>Compute<<5945>><<5945>>$|r^<<2059>>(i)<<2059>>|$<<5946>><<5946>>and<<5947>><<5947>>$|x^<<2060>>(i)<<2060>>|$<<5948>><<5948>>.><<2061>>bfuntil<<2061>><<5949>><<5949>>$igeqmaxit$<<5950>><<5950>>or<<5951>><<5951>>$|r^<<2062>>(i)<<2062>>|leq<<2063>>emstop_tol<<2063>>cdot(|A|cdot|x^<<2064>>(i)<<2064>>|+|b|)$<<5952>><<5952>>.{tabbing}/; $cached_env_img{$key} ='

'; $key = q/{}$S_<<3174>>X_k<<3174>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$calL${}/; $cached_env_img{$key} ='

'; $key = q/{}$pm10^<<2045>>-6<<2045>>|A|${}/; $cached_env_img{$key} ='

'; $key = q/{}$(Ap,Ap)${}/; $cached_env_img{$key} ='

'; $key = q/{}$z^<<553>>(i-1)<<553>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$Y=<<1912>>rmdiag<<1912>>(Y_i)${}/; $cached_env_img{$key} ='

'; $key = q/{}$e^<<2407>>(i)<<2407>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$u^<<2005>>(m)<<2005>>${}/; $cached_env_img{$key} ='

'; $key = q/{}[y=Axqquadhbox<<2644>>and<<2644>>qquady=A^Tx.]{}/; $cached_env_img{$key} ='

'; $key = q/{}$|,|A^<<2358>>-1<<2358>>|cdot|r^<<2359>>(i)<<2359>>|,|_<<2360>>infty<<2360>>${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}|Ax^<<758>>(i)<<758>>-b|_2=|D_<<759>>i+1<<759>>bar<<760>>T<<760>>_iy-|r^<<761>>(0)<<761>>|_2e^<<762>>(1)<<762>>|_2{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$p${}/; $cached_env_img{$key} ='

'; $key = q/{}$deltab${}/; $cached_env_img{$key} ='

'; $key = q/{}$tilde<<3214>>A<<3214>>_H^<<3215>>-<<3215>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$S=bigcup_iS_i${}/; $cached_env_img{$key} ='

'; $key = q/{}$|r^<<2260>>(i)<<2260>>|${}/; $cached_env_img{$key} ='

'; $key = q/{}[A^<<3274>>(i)<<3274>>=pmatrix<<3560>>A^<<3275>>(i)<<3275>>_<<3276>>1,1<<3276>>&A^<<3277>>(i)<<3277>>_<<3278>>1,2<<3278>>crA^<<3279>>(i)<<3279>>_<<3280>>2,1<<3280>>&A^<<3281>>(i)<<3281>>_<<3282>>2,2<<3282>><<3560>>]{}/; $cached_env_img{$key} ='

'; $key = q/{}$a_<<1750>>i,j<<1750>>not=0${}/; $cached_env_img{$key} ='

'; $key = q/{}$D-L${}/; $cached_env_img{$key} ='

'; $key = q/{}$A=LU${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}c_1leq<<1970>>x^TAxoverx^TBx<<1970>>leqc_2quadhbox<<1971>>forall<<5809>><<5809>>$x$<<5810>><<5810>><<1971>>,{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$tilde<<186>>b<<186>>=A^Tb${}/; $cached_env_img{$key} ='

'; $key = q/{}$x^TAx>0${}/; $cached_env_img{$key} ='

'; $key = q/{}$r^<<1225>>(i)<<1225>>${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}|e^<<2170>>(i)<<2170>>|leq|A^<<2171>>-1<<2171>>|cdot|r^<<2172>>(i)<<2172>>|leq<<2173>>emstop_tol<<2173>>cdot|A^<<2174>>-1<<2174>>|cdot(|A|cdot|x^<<2175>>(i)<<2175>>|+|b|);.{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$y_<<2541>>sigma_k(i)<<2541>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$M=I${}/; $cached_env_img{$key} ='

'; $key = q/{}$e^<<2094>>(i)<<2094>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$|r^<<2199>>(i)<<2199>>|leqS_3equiv<<2200>>emstop_tol<<2200>>cdot|x^<<2201>>(i)<<2201>>||A^<<2202>>-1<<2202>>|${}/; $cached_env_img{$key} ='

'; $key = q/{}$|X|${}/; $cached_env_img{$key} ='

'; $key = q/{}$x^<<303>>(k)<<303>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$M^<<1430>>-1<<1430>>A${}/; $cached_env_img{$key} ='

'; $key = q/{}$n'_itimesn'_i${}/; $cached_env_img{$key} ='

'; $key = q/{}$p^<<2925>>(i)^T<<2925>>p^<<2926>>(i)<<2926>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$i+j=k${}/; $cached_env_img{$key} ='

'; $key = q/{}$(p,A^TAp)${}/; $cached_env_img{$key} ='

'; $key = q/{}$M_1=I${}/; $cached_env_img{$key} ='

'; $key = q/{}$|r^<<1189>>(i)<<1189>>|${}/; $cached_env_img{$key} ='

'; $key = q/{}$K_<<3227>>BPS<<3227>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$|e^<<2073>>(i)<<2073>>|leq|A^<<2074>>-1<<2074>>|cdot|r^<<2075>>(i)<<2075>>|${}/; $cached_env_img{$key} ='

'; $key = q/{}$A^mv^<<2855>>(1)<<2855>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$mbox<<616>>span<<616>>{r^<<617>>(0)<<617>>,ldots,A^<<618>>i-1<<618>>r^<<619>>(0)<<619>>}${}/; $cached_env_img{$key} ='

'; $key = q/{}$1,1${}/; $cached_env_img{$key} ='

'; $key = q/{}$(m)${}/; $cached_env_img{$key} ='

'; $key = q/{}$omega_i${}/; $cached_env_img{$key} ='

'; $key = q/{}$Ainv${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}left(<<2567>>array<<2567>><<2568>>rrrrrr<<2568>>10&-3&0&1&0&00&9&6&0&-2&03&0&8&7&0&00&6&0&7&5&40&0&0&0&9&130&0&0&0&5&-1<<2569>>array<<2569>>right)longrightarrowleft(<<2570>>array<<2570>><<2571>>rrrr<<2571>>10&-3&19&6&-23&8&76&7&5&49&135&-1<<2572>>array<<2572>>right){displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$<<2442>>ttrow_ptr(i)<<2442>>leqk<<<2443>>ttrow_ptr(i+1)<<2443>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$g=D_B^<<3007>>-12<<3007>>(f_B-C^TD_R^<<3008>>-1<<3008>>f_R)${}/; $cached_env_img{$key} ='

'; $key = q/{}$U^T${}/; $cached_env_img{$key} ='

'; $key = q/{tabbing}xxxxxx=kill><<5739>><<5739>>$X_1=A_<<1898>>1,1<<1898>>$<<5740>><<5740>>><<1899>>bffor<<1899>>=<<5741>><<5741>>$igeq1$<<5742>><<5742>>>>let<<5743>><<5743>>$Y_iapproxX_iinv$<<5744>><<5744>>>><<5745>><<5745>>$X_<<1900>>i+1<<1900>>=A_<<1901>>i+1,i+1<<1901>>-A_<<1902>>i+1,i<<1902>>Y_iA_<<1903>>i,i+1<<1903>>$<<5746>><<5746>>{tabbing}/; $cached_env_img{$key} ='

'; $key = q/{}$j<i-p${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}M_1^<<3657>>-1<<3657>>AM_2inv(M_2x)=M_1^<<3658>>-1<<3658>>b.{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$S_<<3136>>E_i<<3136>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$u_B${}/; $cached_env_img{$key} ='

'; $key = q/{}$xleftarrowx-l_<<2703>>*1<<2703>>y_1${}/; $cached_env_img{$key} ='

'; $key = q/{}$M_1^<<3660>>-1<<3660>>AM_2inv${}/; $cached_env_img{$key} ='

'; $key = q/{}psfig<<20>>file=cover.ps<<20>>{}/; $cached_env_img{$key} ='

'; $key = q/{}$<<2440>>ttval(k)<<2440>>=a_<<2441>>i,j<<2441>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$|A^<<2243>>-1<<2243>>|${}/; $cached_env_img{$key} ='

'; $key = q/{}$(i+1)timesi${}/; $cached_env_img{$key} ='

'; $key = q/{}$bar<<348>>x<<348>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$max{|lambda(A)|}${}/; $cached_env_img{$key} ='

'; $key = q/{}$(A+deltaA)x^<<3361>>(i)<<3361>>=b+deltab${}/; $cached_env_img{$key} ='

'; $key = q/{}$n_<<3118>>E_i<<3118>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$K_<<3229>>VS<<3229>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$$K_<<3037>>as<<3037>>^<<3038>>-1<<3038>>v=R_H^TA_H^<<3039>>-1<<3039>>R_Hv+sum_<<3040>>i=1<<3040>>^pR_i^T<<3041>>A'<<3041>>_i^<<3042>>-1<<3042>>R_iv.$${}/; $cached_env_img{$key} ='

'; $key = q/{}$n_H${}/; $cached_env_img{$key} ='

'; $key = q/{tabbing}xxxxxx=kill><<4811>><<4811>>$w^<<803>>(i)<<803>>=Av^<<804>>(i)<<804>>$<<4812>><<4812>>><<805>>bffor<<805>>=<<4813>><<4813>>$k=1,ldots,i$<<4814>><<4814>>>><<4815>><<4815>>$w^<<806>>(i)<<806>>=w^<<807>>(i)<<807>>-(w^<<808>>(i)<<808>>,v^<<809>>(k)<<809>>)v^<<810>>(k)<<810>>$<<4816>><<4816>>><<811>>bf<<811>>><<4817>><<4817>>$v^<<812>>(i+1)<<812>>=w^<<813>>(i)<<813>>|w^<<814>>(i)<<814>>|$<<4818>><<4818>>{tabbing}/; $cached_env_img{$key} ='

'; $key = q/{}$(A^<<3627>>-1<<3627>>)_<<401>>i,j<<401>>geq0${}/; $cached_env_img{$key} ='

'; $key = q/{}$a_<<3373>>i,j<<3373>>=0${}/; $cached_env_img{$key} ='

'; $key = q/{}$x^<<2139>>(i)<<2139>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$D_B${}/; $cached_env_img{$key} ='

'; $key = q/{}$A^<<2318>>-1<<2318>>${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}x_i^<<344>>(k)<<344>>=omegabar<<345>>x<<345>>_i^<<346>>(k)<<346>>+(1-omega)x_i^<<347>>(k-1)<<347>>{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$p^<<546>>(i)<<546>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$hat<<610>>x<<610>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$x^<<2727>>(i+1)<<2727>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$(w^<<819>>(i)<<819>>,v^<<820>>(k)<<820>>)${}/; $cached_env_img{$key} ='

'; $key = q/{}$a_<<2818>>i,j<<2818>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$kappa_2${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}x^<<505>>(i)<<505>>=x^<<506>>(i-1)<<506>>+alpha_ip^<<507>>(i)<<507>>.{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$O(varepsilon)${}/; $cached_env_img{$key} ='

'; $key = q/{}$Ap^<<1046>>(k)<<1046>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$M=(D+L)D^<<3664>>-1<<3664>>(D+U)${}/; $cached_env_img{$key} ='

'; $key = q/{}$r^<<2911>>(0)<<2911>>,~r^<<2912>>(1)<<2912>>,~ldots,~r^<<2913>>(j-1)<<2913>>${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}<<2102>>array<<2102>><<2103>>lcl<<2103>>|x|_<<2104>>infty<<2104>>&equiv&max_j|x_j|;,|x|_<<2105>>1<<2105>>&equiv&sum_j|x_j|;<<2106>>rm,and<<2106>>|x|_<<2107>>2<<2107>>&equiv&(sum_j|x_j|^2)^<<2108>>12<<2108>>;;.<<2109>>array<<2109>>{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$tilde<<3241>>S<<3241>>_B${}/; $cached_env_img{$key} ='

'; $key = q/{}$s${}/; $cached_env_img{$key} ='

'; $key = q/{}$(A+deltaA)x^<<2218>>(i)<<2218>>=b+deltab${}/; $cached_env_img{$key} ='

'; $key = q/{}$delta${}/; $cached_env_img{$key} ='

'; $key = q/{}$<<2395>>emstop_tol<<2395>>leqvarepsilon${}/; $cached_env_img{$key} ='

'; $key = q/{}$|w^<<820>>(i)<<820>>|${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}x^<<2325>>(i)<<2325>>=M^<<2326>>-1<<2326>>Nx^<<2327>>(i-1)<<2327>>+M^<<2328>>-1<<2328>>b=Gx^<<2329>>(i-1)<<2329>>+c,{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$|b-Ax^<<828>>(i)<<828>>|${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}Ar^<<698>>(i)<<698>>=r^<<699>>(i+1)<<699>>t_<<700>>i+1,i<<700>>+r^<<701>>(i)<<701>>t_<<702>>i,i<<702>>+r^<<703>>(i-1)<<703>>t_<<704>>i-1,i<<704>>,{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}-(a(x,y)u_x)_x-(b(x,y)u_y)_y=f,{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$f_j=|b|_<<2225>>infty<<2225>>${}/; $cached_env_img{$key} ='

'; $key = q/{tabular}<<2531>>|r|r|r|r|r|r|r|r|r|r|r|<<2531>>hline<<2532>>ttval(:,-1)<<2532>>&0&3&7&8&9&2hline<<2533>>ttval(:,0)<<2533>>&10&9&8&7&9&-1hline<<2534>>ttval(:,+1)<<2534>>&-3&6&7&5&13&0hline{tabular}/; $cached_env_img{$key} ='

'; $key = q/{}$lambda_<<2132>>max<<2132>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$p^<<2797>>(i)<<2797>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$A^<<2373>>-1<<2373>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$Ap^<<2721>>(i)<<2721>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$M_<<3145>>E_i<<3145>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$c${}/; $cached_env_img{$key} ='

'; $key = q/{}$tilde<<3231>>K<<3231>>_<<3232>>VS<<3232>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$lambda_<<2333>>min<<2333>>(A)${}/; $cached_env_img{$key} ='

'; $key = q/{}$sqrt<<2394>>n<<2394>>varepsilon${}/; $cached_env_img{$key} ='

'; $key = q/{}$h^<<657>>-1<<657>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$K_<<3047>>as<<3047>>^<<3048>>-1<<3048>>A${}/; $cached_env_img{$key} ='

'; $key = q/{}$p^<<2800>>(i)<<2800>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$x_i^<<3430>>(k)<<3430>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$R_i,A'_i${}/; $cached_env_img{$key} ='

'; $key = q/{}$|e^<<2151>>(i)<<2151>>|leq|A^<<2152>>-1<<2152>>|cdot|r^<<2153>>(i)<<2153>>|${}/; $cached_env_img{$key} ='

'; $key = q/{}$|A^<<2244>>-1<<2244>>|${}/; $cached_env_img{$key} ='

'; $key = q/{}${r^<<2914>>(j)<<2914>>}${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}|e^<<2171>>(i)<<2171>>|leq|A^<<2172>>-1<<2172>>|cdot|r^<<2173>>(i)<<2173>>|leq<<2174>>emstop_tol<<2174>>cdot|A^<<2175>>-1<<2175>>|cdot(|A|cdot|x^<<2176>>(i)<<2176>>|+|b|);.{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}(1+alphaL_1)(1+alphaL_2)u^<<2000>>(m+1)<<2000>>=left[(1+betaL_1)(1+betaL_2)right]u^<<2001>>(m)<<2001>>{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$|e^<<2311>>(i)<<2311>>|leq|A^<<2312>>-1<<2312>>M^<<2313>>12<<2313>>|cdot|r^<<2314>>(i)<<2314>>|_<<3553>>M^<<2315>>-12<<2315>>,2<<3553>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$r^<<556>>(0)<<556>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$<<2999>>D_B<<2999>>^<<3000>>-12<<3000>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$e^<<2319>>(i)<<2319>>=A^<<2320>>-1<<2320>>r^<<2321>>(i)<<2321>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$alpha=alpha_i=r^<<521>>(i)^T<<521>>r^<<522>>(i)<<522>>p^<<523>>(i)^T<<523>>Ap^<<524>>(i)<<524>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$S${}/; $cached_env_img{$key} ='

'; $key = q/{}$K_<<3207>>as<<3207>>,K_<<3208>>BPS<<3208>>,K_<<3209>>VS<<3209>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$Omega_i,i=1,...,p${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}mbox<<1754>>foreach<<5629>><<5629>>$k$<<5630>><<5630>>,<<5631>><<5631>>$i,j>k$<<5632>><<5632>>:<<1754>>quada_<<1755>>i,j<<1755>>leftarrowleft{<<1756>>array<<1756>><<1757>>ll<<1757>>a_<<1758>>i,j<<1758>>-a_<<1759>>i,k<<1759>>a_<<1760>>k,k<<1760>>^<<3665>>-1<<3665>>a_<<1761>>k,j<<1761>>&mbox<<1762>>if<<5633>><<5633>>$(i,j)inS$<<5634>><<5634>><<1762>>a_<<1763>>i,j<<1763>>&mbox<<1764>>otherwise<<1764>>.<<1765>>array<<1765>>right.{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$|,|A^<<2361>>-1<<2361>>|cdot|r^<<2362>>(i)<<2362>>|,|_<<2363>>infty<<2363>>=|A^<<2364>>-1<<2364>>R|_<<2365>>infty<<2365>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$M_1${}/; $cached_env_img{$key} ='

'; $key = q/{}$|Ax|leq|A|cdot|x|${}/; $cached_env_img{$key} ='

'; $key = q/{}$[lambda_<<3371>>min<<3371>>(A),lambda_<<3372>>max<<3372>>(A)]${}/; $cached_env_img{$key} ='

'; $key = q/{}$1:n_d+1${}/; $cached_env_img{$key} ='

'; $key = q/{}$lambda=2sin^2npi(2N)${}/; $cached_env_img{$key} ='

'; $key = q/{}$O(varepsilon)|A^<<2400>>-1<<2400>>|cdot(|A|cdot|x^<<2401>>(i)<<2401>>|+|b|)${}/; $cached_env_img{$key} ='

'; $key = q/{}$|b|ll|A|cdot|x|${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}(I-L)^<<3674>>-1<<3674>>=cases<<1856>>I+L+L^2+L^3+cdotscr(I+L)(I+L^2)(I+L^4)cdots<<1856>>{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$|e^<<2083>>(i)<<2083>>||x^<<2084>>(i)<<2084>>|${}/; $cached_env_img{$key} ='

'; $key = q/{}$r^<<2727>>(i+1)<<2727>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$lambda_<<1429>>max<<1429>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$s=L^<<2738>>-1<<2738>>r^<<2739>>(i)<<2739>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$r^<<1296>>(0)<<1296>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$Ax+b${}/; $cached_env_img{$key} ='

'; $key = q/{}$tilde<<3246>>A<<3246>>_H${}/; $cached_env_img{$key} ='

'; $key = q/{}$u_n${}/; $cached_env_img{$key} ='

'; $key = q/{}$K_<<3046>>as<<3046>>^<<3047>>-1<<3047>>A${}/; $cached_env_img{$key} ='

'; $key = q/{}$C${}/; $cached_env_img{$key} ='

'; $key = q/{}[e^<<2146>>(i)<<2146>>=x^<<2147>>(i)<<2147>>-x=A^<<2148>>-1<<2148>>(Ax^<<2149>>(i)<<2149>>-b)=A^<<2150>>-1<<2150>>r^<<2151>>(i)<<2151>>,]{}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}hat<<2897>>Lambda<<2897>>_k=left[<<2898>>array<<2898>><<2899>>ccccc<<2899>>p^<<2900>>(1)^T<<2900>>Ap^<<2901>>(1)<<2901>>&0&&cdots&00&p^<<2902>>(2)^T<<2902>>Ap^<<2903>>(2)<<2903>>&&&vdots&ddots&ddots&ddots&vdots&&ddots&ddots&00&cdots&&0&p^<<2904>>(k)^T<<2904>>Ap^<<2905>>(k)<<2905>><<2906>>array<<2906>>right].{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$|b-Ax^<<827>>(i)<<827>>|${}/; $cached_env_img{$key} ='

'; $key = q/{}$|A^<<2337>>-1<<2337>>|_2=1lambda^<<2338>>12<<2338>>_<<2339>>min<<2339>>(AA^T)=1lambda^<<2340>>12<<2340>>_<<2341>>min<<2341>>(A^TA)${}/; $cached_env_img{$key} ='

'; $key = q/{}$Q_i${}/; $cached_env_img{$key} ='

'; $key = q/{}$r^<<554>>(i-1)<<554>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$K_<<3162>>BPS<<3162>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$r^<<1223>>(0)<<1223>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$x^<<241>>(k)<<241>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$A'_iw_i=v_i${}/; $cached_env_img{$key} ='

'; $key = q/{}$H,h${}/; $cached_env_img{$key} ='

'; $key = q/{}$M_2${}/; $cached_env_img{$key} ='

'; $key = q/{}$a_<<2681>>i,j<<2681>>${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}P_k=[p^<<2888>>(1)<<2888>>,~p^<<2889>>(2)<<2889>>,~ldots,~p^<<2890>>(k)<<2890>>].{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$v${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}alpha_i=<<3516>>tilder^<<930>>(i-1)^T<<930>>r^<<931>>(i-1)<<931>>overtildep^<<932>>(i)^T<<932>>Ap^<<933>>(i)<<933>><<3516>>,qquadbeta_i=<<3517>>tilder^<<934>>(i)^T<<934>>r^<<935>>(i)<<935>>overtilder^<<936>>(i-1)^T<<936>>r^<<937>>(i-1)<<937>><<3517>>{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$T'_isubsetT_h${}/; $cached_env_img{$key} ='

'; $key = q/{}$Ap^<<549>>(j)<<549>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$A^<<2319>>-1<<2319>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$V_H${}/; $cached_env_img{$key} ='

'; $key = q/{}$E=bigcup_iE_i${}/; $cached_env_img{$key} ='

'; $key = q/{}$3${}/; $cached_env_img{$key} ='

'; $key = q/{}$Ap^<<550>>(j)<<550>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$p^<<2884>>(1)<<2884>>=r^<<2885>>(0)<<2885>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$S_<<3131>>E_i<<3131>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$tilde<<3244>>K<<3244>>_<<3245>>VS<<3245>>${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}alpha_i=<<3514>>tilder^<<929>>(i-1)^T<<929>>r^<<930>>(i-1)<<930>>overtildep^<<931>>(i)^T<<931>>Ap^<<932>>(i)<<932>><<3514>>,qquadbeta_i=<<3515>>tilder^<<933>>(i)^T<<933>>r^<<934>>(i)<<934>>overtilder^<<935>>(i-1)^T<<935>>r^<<936>>(i-1)<<936>><<3515>>{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$R_k=P_kB_k${}/; $cached_env_img{$key} ='

'; $key = q/{}$partialomip${}/; $cached_env_img{$key} ='

'; $key = q/{}$-U${}/; $cached_env_img{$key} ='

'; $key = q/{}$<<3545>>hat<<2284>>r<<2284>><<3545>>^<<2285>>(i)<<2285>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$r^<<2397>>(i)<<2397>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$M^<<1422>>-1<<1422>>A${}/; $cached_env_img{$key} ='

'; $key = q/{}$R_<<3168>>X_k<<3168>>,R^T_<<3169>>X_k<<3169>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$AA^T${}/; $cached_env_img{$key} ='

'; $key = q/{}$alpha_<<3143>>E_i<<3143>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$O(varepsilon)|,|A^<<2409>>-1<<2409>>|cdot(|A|cdot|x^<<2410>>(i)<<2410>>|+|b|)|${}/; $cached_env_img{$key} ='

'; $key = q/{}$|deltab_<<2220>>j<<2220>>|leqtolcdotf_<<2221>>j<<2221>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$a_<<398>>ii<<398>>>sum_<<399>>jnot=i<<399>>|a_<<400>>ij<<400>>|${}/; $cached_env_img{$key} ='

'; $key = q/{}$f${}/; $cached_env_img{$key} ='

'; $key = q/{}$|x|_<<2386>>infty<<2386>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$R^<<2866>>(k)<<2866>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$e^<<2030>>(i)<<2030>>equivx^<<2031>>(i)<<2031>>-x${}/; $cached_env_img{$key} ='

'; $key = q/{}$tilde<<3236>>A<<3236>>_i${}/; $cached_env_img{$key} ='

'; $key = q/{}$tilde<<3230>>K<<3230>>_<<3231>>BPS<<3231>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$|A^<<2343>>-1<<2343>>|_<<2344>>infty<<2344>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$x^<<2210>>(i)<<2210>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$varepsilon=2^<<2049>>-24<<2049>>approx10^<<2050>>-7<<2050>>${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}<<1989>>calL<<1989>>=<<1990>>calL<<1990>>_1+<<1991>>calL<<1991>>_2,qquadhbox<<3536>>where<<5817>><<5817>>$<<1992>>calL<<1992>>_1=-<<1993>>partial^2overpartialx^2<<1993>>$<<5818>><<5818>>,<<5819>><<5819>>$<<1994>>calL<<1994>>_2=-<<1995>>partial^2overpartialy^2<<1995>>$<<5820>><<5820>><<3536>>.{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$(A+A^T)2${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}left(<<782>>array<<782>><<783>>cc<<783>>I&AA^T&0<<784>>array<<784>>right)left(<<785>>array<<785>><<786>>c<<786>>rx<<787>>array<<787>>right)=left(<<788>>array<<788>><<789>>c<<789>>b0<<790>>array<<790>>right).{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$(A^TA)x=tilde<<185>>b<<185>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$pm10^<<2047>>-6<<2047>>|b|${}/; $cached_env_img{$key} ='

'; $key = q/{}$z^<<555>>(0)<<555>>${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}mbox<<1753>>foreach<<5625>><<5625>>$k$<<5626>><<5626>>,<<5627>><<5627>>$i,j>k$<<5628>><<5628>>:<<1753>>quada_<<1754>>i,j<<1754>>leftarrowleft{<<1755>>array<<1755>><<1756>>ll<<1756>>a_<<1757>>i,j<<1757>>-a_<<1758>>i,k<<1758>>a_<<1759>>k,k<<1759>>^<<3663>>-1<<3663>>a_<<1760>>k,j<<1760>>&mbox<<1761>>if<<5629>><<5629>>$(i,j)inS$<<5630>><<5630>><<1761>>a_<<1762>>i,j<<1762>>&mbox<<1763>>otherwise<<1763>>.<<1764>>array<<1764>>right.{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$tilde<<3229>>K<<3229>>_<<3230>>BPS<<3230>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$(x^<<606>>(i)<<606>>-hat<<607>>x<<607>>)^TA(x^<<608>>(i)<<608>>-hat<<609>>x<<609>>)${}/; $cached_env_img{$key} ='

'; $key = q/{}$e^<<2097>>(i)<<2097>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$x=A^Ty${}/; $cached_env_img{$key} ='

'; $key = q/{}$bigcup_<<3055>>i<<3055>>partialomip${}/; $cached_env_img{$key} ='

'; $key = q/{}$|A|_<<2127>>B,alpha<<2127>>equiv|BAB^<<2128>>-1<<2128>>|_<<2129>>alpha<<2129>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$rho_<<2797>>i+1<<2797>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$E_i${}/; $cached_env_img{$key} ='

'; $key = q/{}$M_<<3174>>X_k<<3174>>${}/; $cached_env_img{$key} ='

'; $key = q/{tabbing}xxxxxx=kill><<4815>><<4815>>$w^<<804>>(i)<<804>>=Av^<<805>>(i)<<805>>$<<4816>><<4816>>><<806>>bffor<<806>>=<<4817>><<4817>>$k=1,ldots,i$<<4818>><<4818>>>><<4819>><<4819>>$w^<<807>>(i)<<807>>=w^<<808>>(i)<<808>>-(w^<<809>>(i)<<809>>,v^<<810>>(k)<<810>>)v^<<811>>(k)<<811>>$<<4820>><<4820>>><<812>>bf<<812>>><<4821>><<4821>>$v^<<813>>(i+1)<<813>>=w^<<814>>(i)<<814>>|w^<<815>>(i)<<815>>|$<<4822>><<4822>>{tabbing}/; $cached_env_img{$key} ='

'; $key = q/{tabular}<<2462>>|r|r|r|r|r|r|r|r|<<2462>>hline<<2463>>ttrow_ptr<<2463>>&1&3&6&9&13&17&20hline{tabular}/; $cached_env_img{$key} ='

'; $key = q/{}$|A|_2${}/; $cached_env_img{$key} ='

'; $key = q/{}$S_<<3132>>E_i<<3132>>^<<3133>>-1<<3133>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$A^<<2369>>-1<<2369>>R${}/; $cached_env_img{$key} ='

'; $key = q/{}$Q_k${}/; $cached_env_img{$key} ='

'; $key = q/{}$alpha_<<3144>>E_i<<3144>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$A_Bv_B-A_<<3105>>IB<<3105>>^T(A_I^<<3106>>-1<<3106>>(A_<<3107>>IB<<3107>>v_B))${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}M^<<2949>>-1<<2949>>Ax=M^<<2950>>-1<<2950>>f{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$|I-M^<<3685>>-1<<3685>>A|${}/; $cached_env_img{$key} ='

'; $key = q/{}$nnzb${}/; $cached_env_img{$key} ='

'; $key = q/{}$x^<<2162>>(i)<<2162>>${}/; $cached_env_img{$key} ='

'; $key = q/{tabular}<<55>>ll<<55>><<4137>><<4137>>$A,ldots,Z$<<4138>><<4138>>&matrices<<4139>><<4139>>$a,ldots,z$<<4140>><<4140>>&vectors<<4141>><<4141>>$alpha,beta,ldots,omega$<<4142>><<4142>>&scalars<<4143>><<4143>>$A^<<56>>T<<56>>$<<4144>><<4144>>&matrixtranspose<<4145>><<4145>>$A^<<57>>H<<57>>$<<4146>><<4146>>&conjugatetranspose(Hermitian)of<<4147>><<4147>>$A$<<4148>><<4148>><<4149>><<4149>>$A^<<58>>-1<<58>>$<<4150>><<4150>>&matrixinverse<<4151>><<4151>>$A^<<59>>-T<<59>>$<<4152>><<4152>>&theinverseof<<4153>><<4153>>$A^T$<<4154>><<4154>><<4155>><<4155>>$a_<<60>>i,j<<60>>$<<4156>><<4156>>&matrixelement<<4157>><<4157>>$a_<<61>>.,j<<61>>$<<4158>><<4158>>&<<4159>><<4159>>$j$<<4160>><<4160>>thmatrixcolumn<<4161>><<4161>>$A_<<62>>i,j<<62>>$<<4162>><<4162>>&matrixsubblock<<4163>><<4163>>$a_<<63>>i<<63>>$<<4164>><<4164>>&vectorelement<<4165>><<4165>>$u_x,u_<<64>>xx<<64>>$<<4166>><<4166>>&first,secondderivativewithrespectto~<<4167>><<4167>>$x$<<4168>><<4168>><<4169>><<4169>>$(x,y)$<<4170>><<4170>>,<<4171>><<4171>>$x^Ty$<<4172>><<4172>>&vectordotproduct(innerproduct)<<4173>><<4173>>$x^<<65>>(i)<<65>>_j$<<4174>><<4174>>&<<4175>><<4175>>$j$<<4176>><<4176>>thcomponentofvector<<4177>><<4177>>$x$<<4178>><<4178>>inthe<<4179>><<4179>>$i$<<4180>><<4180>>thiterationdiag(<<4181>><<4181>>$A$<<4182>><<4182>>)&diagonalofmatrix<<4183>><<4183>>$A$<<4184>><<4184>>diag(<<4185>><<4185>>$alpha,betaldots$<<4186>><<4186>>)&diagonalmatrixconstructedfromscalars<<4187>><<4187>>$alpha,betaldots$<<4188>><<4188>>span(<<4189>><<4189>>$a,bldots$<<4190>><<4190>>)&spanningspaceofvectors<<4191>><<4191>>$a,bldots$<<4192>><<4192>><<4193>><<4193>>$calR$<<4194>><<4194>>&setofrealnumbers<<4195>><<4195>>$<<66>>calR<<66>>^<<67>>n<<67>>$<<4196>><<4196>>&real<<4197>><<4197>>$n$<<4198>><<4198>>-space<<4199>><<4199>>$||x||_<<68>>2<<68>>$<<4200>><<4200>>&2-norm<<4201>><<4201>>$||x||_<<69>>p<<69>>$<<4202>><<4202>>&<<4203>><<4203>>$p$<<4204>><<4204>>-norm<<4205>><<4205>>$||x||_<<70>>A<<70>>$<<4206>><<4206>>&the``<<4207>><<4207>>$A$<<4208>><<4208>>-norm'',definedas<<4209>><<4209>>$(Ax,x)^<<71>>12<<71>>$<<4210>><<4210>><<4211>><<4211>>$lambda_<<72>>max<<72>>(A),lambda_<<73>>min<<73>>(A)$<<4212>><<4212>>&eigenvaluesof<<4213>><<4213>>$A$<<4214>><<4214>>withmaximum(resp.minimum)modulus<<4215>><<4215>>$sigma_<<74>>max<<74>>(A),sigma_<<75>>min<<75>>(A)$<<4216>><<4216>>&largestandsmallestsingularvaluesof<<4217>><<4217>>$A$<<4218>><<4218>><<4219>><<4219>>$kappa_2(A)$<<4220>><<4220>>&spectralconditionnumberofmatrix<<4221>><<4221>>$A$<<4222>><<4222>><<4223>><<4223>>$calL$<<4224>><<4224>>&linearoperator<<4225>><<4225>>$overline<<76>>alpha<<76>>$<<4226>><<4226>>&complexconjugateofthescalar<<4227>><<4227>>$alpha$<<4228>><<4228>><<4229>><<4229>>$max{S}$<<4230>><<4230>>&maximumvalueinset<<4231>><<4231>>$S$<<4232>><<4232>><<4233>><<4233>>$min{S}$<<4234>><<4234>>&minimumvalueinset<<4235>><<4235>>$S$<<4236>><<4236>><<4237>><<4237>>$sum$<<4238>><<4238>>&summation<<4239>><<4239>>$O(cdot)$<<4240>><<4240>>&``big-oh''asymptoticbound&multicolumn<<77>>2<<77>><<78>>c<<78>><<79>>ConventionsUsedinthisBook<<79>>&<<4241>><<4241>>$D$<<4242>><<4242>>&diagonalmatrix<<4243>><<4243>>$L$<<4244>><<4244>>&lowertriangularmatrix<<4245>><<4245>>$U$<<4246>><<4246>>&uppertriangularmatrix<<4247>><<4247>>$Q$<<4248>><<4248>>&orthogonalmatrix<<4249>><<4249>>$M$<<4250>><<4250>>&preconditioner<<4251>><<4251>>$I,;I^<<80>>ntimesn<<80>>$<<4252>><<4252>>&<<4253>><<4253>>$ntimesn$<<4254>><<4254>>identitymatrix<<4255>><<4255>>$hat<<81>>x<<81>>$<<4256>><<4256>>&typically,theexactsolutionto<<4257>><<4257>>$Ax=b$<<4258>><<4258>><<4259>><<4259>>$h$<<4260>><<4260>>&discretizationmeshwidth{tabular}/; $cached_env_img{$key} ='

'; $key = q/{}$p^<<503>>(i)<<503>>${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}x^<<733>>(i)<<733>>in{r^<<734>>(0)<<734>>,Ar^<<735>>(0)<<735>>,ldots,A^<<736>>i-1<<736>>r^<<737>>(0)<<737>>},mbox<<738>>~~<<738>>x^<<739>>(i)<<739>>=R_ibar<<740>>y<<740>>{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$|r^<<2918>>(i)<<2918>>|_2${}/; $cached_env_img{$key} ='

'; $key = q/{}$p_i${}/; $cached_env_img{$key} ='

'; $key = q/{}$S_5${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}x^<<823>>(i)<<823>>=x^<<824>>(0)<<824>>+y_1v^<<825>>(1)<<825>>+cdots+y_iv^<<826>>(i)<<826>>,{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$A'_i${}/; $cached_env_img{$key} ='

'; $key = q/{}$jinI_i${}/; $cached_env_img{$key} ='

'; $key = q/{}$A^<<2370>>-1<<2370>>R${}/; $cached_env_img{$key} ='

'; $key = q/{}$M=P_n(A)${}/; $cached_env_img{$key} ='

'; $key = q/{tabbing}junkjunk=junk=junk=junk=junk=junkkill><<5933>><<5933>>$i=0$<<5934>><<5934>>><<2054>>bfrepeat<<2054>>>><<5935>><<5935>>$i=i+1$<<5936>><<5936>>>>Computetheapproximatesolution<<5937>><<5937>>$x^<<2055>>(i)<<2055>>$<<5938>><<5938>>.>>Computetheresidual<<5939>><<5939>>$r^<<2056>>(i)<<2056>>=Ax^<<2057>>(i)<<2057>>-b$<<5940>><<5940>>.>>Compute<<5941>><<5941>>$|r^<<2058>>(i)<<2058>>|$<<5942>><<5942>>and<<5943>><<5943>>$|x^<<2059>>(i)<<2059>>|$<<5944>><<5944>>.><<2060>>bfuntil<<2060>><<5945>><<5945>>$igeqmaxit$<<5946>><<5946>>or<<5947>><<5947>>$|r^<<2061>>(i)<<2061>>|leq<<2062>>emstop_tol<<2062>>cdot(|A|cdot|x^<<2063>>(i)<<2063>>|+|b|)$<<5948>><<5948>>.{tabbing}/; $cached_env_img{$key} ='

'; $key = q/{}$tilde<<3242>>K<<3242>>_<<3243>>BPS<<3243>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$<<2515>>ttp<<2515>>+<<2516>>ttq<<2516>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$pm10^<<2046>>-6<<2046>>|b|${}/; $cached_env_img{$key} ='

'; $key = q/{}$M^<<2847>>-1<<2847>>Av^<<2848>>(j)<<2848>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$i+d${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}M^<<2948>>-1<<2948>>Ax=M^<<2949>>-1<<2949>>f{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$|A^<<2195>>-1<<2195>>|cdot|r^<<2196>>(i)<<2196>>|${}/; $cached_env_img{$key} ='

'; $key = q/{}$|G|${}/; $cached_env_img{$key} ='

'; $key = q/{tabbing}xxxxxx=kill><<5735>><<5735>>$X_1=A_<<1897>>1,1<<1897>>$<<5736>><<5736>>><<1898>>bffor<<1898>>=<<5737>><<5737>>$igeq1$<<5738>><<5738>>>>let<<5739>><<5739>>$Y_iapproxX_iinv$<<5740>><<5740>>>><<5741>><<5741>>$X_<<1899>>i+1<<1899>>=A_<<1900>>i+1,i+1<<1900>>-A_<<1901>>i+1,i<<1901>>Y_iA_<<1902>>i,i+1<<1902>>$<<5742>><<5742>>{tabbing}/; $cached_env_img{$key} ='

'; $key = q/{}$y${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}-(tildea(x)u_x)_x-(tildeb(y)u_y)_y=f.{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$ntimesn'_i${}/; $cached_env_img{$key} ='

'; $key = q/{}$a_<<1737>>i,j<<1737>>=0${}/; $cached_env_img{$key} ='

'; $key = q/{}$Anot=LU${}/; $cached_env_img{$key} ='

'; $key = q/{}$|e^<<2349>>(i)<<2349>>|_<<2350>>infty<<2350>>leq|,|A^<<2351>>-1<<2351>>|cdot|r^<<2352>>(i)<<2352>>|,|_<<2353>>infty<<2353>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$z^<<1030>>(i-1)^T<<1030>>tilde<<1031>>r<<1031>>^<<1032>>(i-1)<<1032>>approx0${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}B_2=(D-omegaL)^<<3633>>-1<<3633>>(omegaU+(1-omega)D).{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$E_<<3131>>i<<3131>>${}/; $cached_env_img{$key} ='

'; $key = q/{}${beta_k}${}/; $cached_env_img{$key} ='

'; $key = q/{}$|r^<<2919>>(i)<<2919>>|_2${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}(I-L)^<<3676>>-1<<3676>>=cases<<1857>>I+L+L^2+L^3+cdotscr(I+L)(I+L^2)(I+L^4)cdots<<1857>>{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$(i+1,i)${}/; $cached_env_img{$key} ='

'; $key = q/{}$tilde<<3250>>A<<3250>>_i^<<3251>>-1<<3251>>A_i${}/; $cached_env_img{$key} ='

'; $key = q/{}$$K_<<3038>>as<<3038>>^<<3039>>-1<<3039>>v=R_H^TA_H^<<3040>>-1<<3040>>R_Hv+sum_<<3041>>i=1<<3041>>^pR_i^T<<3042>>A'<<3042>>_i^<<3043>>-1<<3043>>R_iv.$${}/; $cached_env_img{$key} ='

'; $key = q/{}$a_<<3430>>i,i<<3430>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$(A+deltaA)x^<<3362>>(i)<<3362>>=b+deltab${}/; $cached_env_img{$key} ='

'; $key = q/{}$|A^<<2297>>-1<<2297>>M|=|(I-G)^<<2298>>-1<<2298>>|leq1(1-|G|)${}/; $cached_env_img{$key} ='

'; $key = q/{}$ILU(0)${}/; $cached_env_img{$key} ='

'; $key = q/{}$bar<<711>>T<<711>>_i${}/; $cached_env_img{$key} ='

'; $key = q/{}$i${}/; $cached_env_img{$key} ='

'; $key = q/{}$v_B${}/; $cached_env_img{$key} ='

'; $key = q/{}$M^<<1657>>-1<<1657>>A${}/; $cached_env_img{$key} ='

'; $key = q/{}$<<2443>>ttrow_ptr(i)<<2443>>leqk<<<2444>>ttrow_ptr(i+1)<<2444>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$|A^<<2293>>-1<<2293>>M|${}/; $cached_env_img{$key} ='

'; $key = q/{}$K^<<3032>>-1<<3032>>v${}/; $cached_env_img{$key} ='

'; $key = q/{}$x^<<2041>>(i)<<2041>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$A'_i,A_H${}/; $cached_env_img{$key} ='

'; $key = q/{}$u_I${}/; $cached_env_img{$key} ='

'; $key = q/{}$r^<<2095>>(i)<<2095>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$x_<<2540>>sigma_k(i)<<2540>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$x=l_<<2701>>*1<<2701>>y_1+l_<<2702>>*2<<2702>>y_2+cdots${}/; $cached_env_img{$key} ='

'; $key = q/{}$r^<<1202>>(0)<<1202>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$(i-1,j)${}/; $cached_env_img{$key} ='

'; $key = q/{}$fequiv(f_I,f_B)^T${}/; $cached_env_img{$key} ='

'; $key = q/{}$r^<<2144>>(i)<<2144>>=Ax^<<2145>>(i)<<2145>>-b${}/; $cached_env_img{$key} ='

'; $key = q/{}$f(x)geq0${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}(1+alphaL_1)(1+alphaL_2)u^<<1999>>(m+1)<<1999>>=left[(1+betaL_1)(1+betaL_2)right]u^<<2000>>(m)<<2000>>{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$S_<<3133>>E_i<<3133>>^<<3134>>-1<<3134>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$(i,j)${}/; $cached_env_img{$key} ='

'; $key = q/{}$nnz=nnzbtimesn_b^2${}/; $cached_env_img{$key} ='

'; $key = q/{}$bar<<764>>T<<764>>_i${}/; $cached_env_img{$key} ='

'; $key = q/{}${X_i}_<<1895>>i=1..n<<1895>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$M=M_1M_2${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}<<270>>calL<<270>>u=-u_<<271>>xx<<271>>=fqquadhbox<<272>>on<<4371>><<4371>>$(0,1)$<<4372>><<4372>><<272>>,qquadu(0)=u_0,quadu(1)=u_1,{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$X=<<1911>>rmdiag<<1911>>(X_i)${}/; $cached_env_img{$key} ='

'; $key = q/{}$A_I^<<3237>>-1<<3237>>${}/; $cached_env_img{$key} ='

'; $key = q/{tabular}<<2456>>|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|<<2456>>hline<<2457>>ttval<<2457>>&10&-2&3&9&3&7&8&7&3<<6467>><<6467>>$cdots$<<6468>><<6468>>9&13&4&2&-1hline<<2458>>ttcol_ind<<2458>>&1&5&1&2&6&2&3&4&1<<6469>><<6469>>$cdots$<<6470>><<6470>>5&6&2&5&6hline{tabular}/; $cached_env_img{$key} ='

'; $key = q/{}$(A^<<3688>>-1<<3688>>)_<<3385>>i,j<<3385>>geq0${}/; $cached_env_img{$key} ='

'; $key = q/{}$kappa(M^<<3674>>-1<<3674>>A)=O(h^<<3675>>-1<<3675>>)${}/; $cached_env_img{$key} ='

'; $key = q/{}$a_<<2683>>j,j<<2683>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$x^<<502>>(i)<<502>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$Ly=x${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}m_<<1679>>i,j<<1679>>=left{<<1680>>array<<1680>><<1681>>ll<<1681>>a_<<1682>>i,j<<1682>>&mbox<<1683>>if<<5509>><<5509>>$i$<<5510>><<5510>>and<<5511>><<5511>>$j$<<5512>><<5512>>areinthesameindexsubset<<1683>>0&mbox<<1684>>otherwise<<1684>>.<<1685>>array<<1685>>right.{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$q^<<2800>>(i)<<2800>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$I${}/; $cached_env_img{$key} ='

'; $key = q/{}$r^<<2392>>(i)<<2392>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$|x|${}/; $cached_env_img{$key} ='

'; $key = q/{}$|r^<<2155>>(i)<<2155>>|letau${}/; $cached_env_img{$key} ='

'; $key = q/{}$P_i^2(A)r^<<1225>>(0)<<1225>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$bleftarrowM_1^<<3657>>-1<<3657>>b${}/; $cached_env_img{$key} ='

'; $key = q/{}$O(varepsilon)|,|A^<<2408>>-1<<2408>>|cdot(|A|cdot|x^<<2409>>(i)<<2409>>|+|b|)|${}/; $cached_env_img{$key} ='

'; $key = q/{}$(R_<<3119>>E_i<<3119>>u_B)_j=(u_B)_j${}/; $cached_env_img{$key} ='

'; $key = q/{}$O(h^<<3103>>-1<<3103>>)${}/; $cached_env_img{$key} ='

'; $key = q/{}$|deltar^<<2393>>(i)<<2393>>|leqO(varepsilon)(|A|cdot|x^<<2394>>(i)<<2394>>|+|b|)${}/; $cached_env_img{$key} ='

'; $key = q/{}$K_<<3206>>as<<3206>>,K_<<3207>>BPS<<3207>>,K_<<3208>>VS<<3208>>${}/; $cached_env_img{$key} ='

'; $key = q/{}[|e^<<2188>>(i)<<2188>>|leq|A^<<2189>>-1<<2189>>|cdot|r^<<2190>>(i)<<2190>>|leq<<2191>>emstop_tol<<2191>>cdot|A^<<2192>>-1<<2192>>|cdot|b|]{}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}M^<<1637>>-1<<1637>>Ax=M^<<1638>>-1<<1638>>b{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$<<2916>>rmspan<<2916>>{b,~Ab,~ldots,~A^<<2917>>k-1<<2917>>b}${}/; $cached_env_img{$key} ='

'; $key = q/{}${Y_i}_<<1905>>i=1..n<<1905>>${}/; $cached_env_img{$key} ='

'; $key = q/{tabular}<<2612>>|r|r|r|r|r|r|r|<<2612>>hline<<2613>>ttperm<<2613>>&5&2&3&4&1&6hline{tabular}/; $cached_env_img{$key} ='

'; $key = q/{}$r^<<2735>>(i)^T<<2735>>M^<<2736>>-1<<2736>>r^<<2737>>(i)<<2737>>${}/; $cached_env_img{$key} ='

'; $key = q/{tabular}<<56>>ll<<56>><<4141>><<4141>>$A,ldots,Z$<<4142>><<4142>>&matrices<<4143>><<4143>>$a,ldots,z$<<4144>><<4144>>&vectors<<4145>><<4145>>$alpha,beta,ldots,omega$<<4146>><<4146>>&scalars<<4147>><<4147>>$A^<<57>>T<<57>>$<<4148>><<4148>>&matrixtranspose<<4149>><<4149>>$A^<<58>>H<<58>>$<<4150>><<4150>>&conjugatetranspose(Hermitian)of<<4151>><<4151>>$A$<<4152>><<4152>><<4153>><<4153>>$A^<<59>>-1<<59>>$<<4154>><<4154>>&matrixinverse<<4155>><<4155>>$A^<<60>>-T<<60>>$<<4156>><<4156>>&theinverseof<<4157>><<4157>>$A^T$<<4158>><<4158>><<4159>><<4159>>$a_<<61>>i,j<<61>>$<<4160>><<4160>>&matrixelement<<4161>><<4161>>$a_<<62>>.,j<<62>>$<<4162>><<4162>>&<<4163>><<4163>>$j$<<4164>><<4164>>thmatrixcolumn<<4165>><<4165>>$A_<<63>>i,j<<63>>$<<4166>><<4166>>&matrixsubblock<<4167>><<4167>>$a_<<64>>i<<64>>$<<4168>><<4168>>&vectorelement<<4169>><<4169>>$u_x,u_<<65>>xx<<65>>$<<4170>><<4170>>&first,secondderivativewithrespectto~<<4171>><<4171>>$x$<<4172>><<4172>><<4173>><<4173>>$(x,y)$<<4174>><<4174>>,<<4175>><<4175>>$x^Ty$<<4176>><<4176>>&vectordotproduct(innerproduct)<<4177>><<4177>>$x^<<66>>(i)<<66>>_j$<<4178>><<4178>>&<<4179>><<4179>>$j$<<4180>><<4180>>thcomponentofvector<<4181>><<4181>>$x$<<4182>><<4182>>inthe<<4183>><<4183>>$i$<<4184>><<4184>>thiterationdiag(<<4185>><<4185>>$A$<<4186>><<4186>>)&diagonalofmatrix<<4187>><<4187>>$A$<<4188>><<4188>>diag(<<4189>><<4189>>$alpha,betaldots$<<4190>><<4190>>)&diagonalmatrixconstructedfromscalars<<4191>><<4191>>$alpha,betaldots$<<4192>><<4192>>span(<<4193>><<4193>>$a,bldots$<<4194>><<4194>>)&spanningspaceofvectors<<4195>><<4195>>$a,bldots$<<4196>><<4196>><<4197>><<4197>>$calR$<<4198>><<4198>>&setofrealnumbers<<4199>><<4199>>$<<67>>calR<<67>>^<<68>>n<<68>>$<<4200>><<4200>>&real<<4201>><<4201>>$n$<<4202>><<4202>>-space<<4203>><<4203>>$||x||_<<69>>2<<69>>$<<4204>><<4204>>&2-norm<<4205>><<4205>>$||x||_<<70>>p<<70>>$<<4206>><<4206>>&<<4207>><<4207>>$p$<<4208>><<4208>>-norm<<4209>><<4209>>$||x||_<<71>>A<<71>>$<<4210>><<4210>>&the``<<4211>><<4211>>$A$<<4212>><<4212>>-norm'',definedas<<4213>><<4213>>$(Ax,x)^<<72>>12<<72>>$<<4214>><<4214>><<4215>><<4215>>$lambda_<<73>>max<<73>>(A),lambda_<<74>>min<<74>>(A)$<<4216>><<4216>>&eigenvaluesof<<4217>><<4217>>$A$<<4218>><<4218>>withmaximum(resp.minimum)modulus<<4219>><<4219>>$sigma_<<75>>max<<75>>(A),sigma_<<76>>min<<76>>(A)$<<4220>><<4220>>&largestandsmallestsingularvaluesof<<4221>><<4221>>$A$<<4222>><<4222>><<4223>><<4223>>$kappa_2(A)$<<4224>><<4224>>&spectralconditionnumberofmatrix<<4225>><<4225>>$A$<<4226>><<4226>><<4227>><<4227>>$calL$<<4228>><<4228>>&linearoperator<<4229>><<4229>>$overline<<77>>alpha<<77>>$<<4230>><<4230>>&complexconjugateofthescalar<<4231>><<4231>>$alpha$<<4232>><<4232>><<4233>><<4233>>$max{S}$<<4234>><<4234>>&maximumvalueinset<<4235>><<4235>>$S$<<4236>><<4236>><<4237>><<4237>>$min{S}$<<4238>><<4238>>&minimumvalueinset<<4239>><<4239>>$S$<<4240>><<4240>><<4241>><<4241>>$sum$<<4242>><<4242>>&summation<<4243>><<4243>>$O(cdot)$<<4244>><<4244>>&``big-oh''asymptoticbound&multicolumn<<78>>2<<78>><<79>>c<<79>><<80>>ConventionsUsedinthisBook<<80>>&<<4245>><<4245>>$D$<<4246>><<4246>>&diagonalmatrix<<4247>><<4247>>$L$<<4248>><<4248>>&lowertriangularmatrix<<4249>><<4249>>$U$<<4250>><<4250>>&uppertriangularmatrix<<4251>><<4251>>$Q$<<4252>><<4252>>&orthogonalmatrix<<4253>><<4253>>$M$<<4254>><<4254>>&preconditioner<<4255>><<4255>>$I,;I^<<81>>ntimesn<<81>>$<<4256>><<4256>>&<<4257>><<4257>>$ntimesn$<<4258>><<4258>>identitymatrix<<4259>><<4259>>$hat<<82>>x<<82>>$<<4260>><<4260>>&typically,theexactsolutionto<<4261>><<4261>>$Ax=b$<<4262>><<4262>><<4263>><<4263>>$h$<<4264>><<4264>>&discretizationmeshwidth{tabular}/; $cached_env_img{$key} ='

'; $key = q/{}$M^<<1433>>-1<<1433>>A${}/; $cached_env_img{$key} ='

'; $key = q/{}$r^<<547>>(i)<<547>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$$S_BequivA_B-A_<<3093>>IB<<3093>>^TA_I^<<3094>>-1<<3094>>A_<<3095>>IB<<3095>>$${}/; $cached_env_img{$key} ='

'; $key = q/{}$R_<<754>>i+1<<754>>D_<<755>>i+1<<755>>^<<756>>-1<<756>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$A^TAx=A^Tb${}/; $cached_env_img{$key} ='

'; $key = q/{}$Au=f${}/; $cached_env_img{$key} ='

'; $key = q/{}$O(1H^2)${}/; $cached_env_img{$key} ='

'; $key = q/{}$r^<<543>>(i)<<543>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$r^<<1297>>(k)<<1297>>=P_k(A)r^<<1298>>(0)<<1298>>${}/; $cached_env_img{$key} ='

'; $key = q/{tabbing}junkjunk=junk=junk=junk=junk=junkkill><<2078>>bfuntil<<2078>><<5967>><<5967>>$igeqmaxit$<<5968>><<5968>>or<<5969>><<5969>>$|r^<<2079>>(i)<<2079>>|leq<<2080>>emstop_tol<<2080>>cdot|x^<<2081>>(i)<<2081>>||A^<<2082>>-1<<2082>>|;;,$<<5970>><<5970>>{tabbing}/; $cached_env_img{$key} ='

'; $key = q/{}$L^<<2741>>-T<<2741>>s${}/; $cached_env_img{$key} ='

'; $key = q/{}$x^<<2065>>(i)<<2065>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$S_<<3168>>X_k<<3168>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$X_i${}/; $cached_env_img{$key} ='

'; $key = q/{}$k+1${}/; $cached_env_img{$key} ='

'; $key = q/{}$L=(l_<<2700>>*1<<2700>>,l_<<2701>>*2<<2701>>,ldots~)${}/; $cached_env_img{$key} ='

'; $key = q/{}$L=(l_<<2699>>*1<<2699>>,l_<<2700>>*2<<2700>>,ldots~)${}/; $cached_env_img{$key} ='

'; $key = q/{}$K_<<3165>>BPS<<3165>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$|e^<<2158>>(i)<<2158>>|leq|,|A^<<2159>>-1<<2159>>|cdot|r^<<2160>>(i)<<2160>>|,|${}/; $cached_env_img{$key} ='

'; $key = q/{}$(A+deltaA)x^<<2162>>(i)<<2162>>=b+deltab${}/; $cached_env_img{$key} ='

'; $key = q/{}$y=D_B^<<3006>>12<<3006>>x_B${}/; $cached_env_img{$key} ='

'; $key = q/{}$|cdot|${}/; $cached_env_img{$key} ='

'; $key = q/{}$r^<<543>>(i-1)<<543>>${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}M_p^<<3682>>-1<<3682>>=(sum_<<1942>>i=0<<1942>>^<<1943>>p-1<<1943>>(I-M^<<3683>>-1<<3683>>A)^i)M^<<3684>>-1<<3684>>.{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$a_<<3403>>i,j<<3403>>=0${}/; $cached_env_img{$key} ='

'; $key = q/{}$x^<<302>>(k)<<302>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$|b|${}/; $cached_env_img{$key} ='

'; $key = q/{}[y_i=sum_j(A^T)_<<2650>>i,j<<2650>>x_j=sum_ja_<<2651>>j,i<<2651>>x_j,]{}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}<<2101>>array<<2101>><<2102>>lcl<<2102>>|x|_<<2103>>infty<<2103>>&equiv&max_j|x_j|;,|x|_<<2104>>1<<2104>>&equiv&sum_j|x_j|;<<2105>>rm,and<<2105>>|x|_<<2106>>2<<2106>>&equiv&(sum_j|x_j|^2)^<<2107>>12<<2107>>;;.<<2108>>array<<2108>>{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$y_1${}/; $cached_env_img{$key} ='

'; $key = q/{}$S_<<3134>>E_i<<3134>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$S_4equivmax_j(|r^<<2214>>(i)<<2214>>|_j(Ecdot|x^<<2215>>(i)<<2215>>|+f)_j)leq<<2216>>emstop_tol<<2216>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$d>0${}/; $cached_env_img{$key} ='

'; $key = q/{}$r^<<2253>>(0)<<2253>>=b${}/; $cached_env_img{$key} ='

'; $key = q/{}$(R^T_iu_i)_j=(u_i)_j${}/; $cached_env_img{$key} ='

'; $key = q/{}$lambda_<<1420>>max<<1420>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$n^2${}/; $cached_env_img{$key} ='

'; $key = q/{}$p^<<2885>>(1)<<2885>>=r^<<2886>>(0)<<2886>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$r^<<1191>>(i)<<1191>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$R_i${}/; $cached_env_img{$key} ='

'; $key = q/{}$S_<<3129>>E<<3129>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$^,${}/; $cached_env_img{$key} ='

'; $key = q/{}$|A^<<2075>>-1<<2075>>|${}/; $cached_env_img{$key} ='

'; $key = q/{}$a_<<2660>>i,i+j<<2660>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$tilde<<3242>>S<<3242>>_B${}/; $cached_env_img{$key} ='

'; $key = q/{}$s=L^<<2739>>-1<<2739>>r^<<2740>>(i)<<2740>>${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}x^<<822>>(i)<<822>>=x^<<823>>(0)<<823>>+y_1v^<<824>>(1)<<824>>+cdots+y_iv^<<825>>(i)<<825>>,{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$kappa(M_<<3531>>omega_<<1712>>rmopt<<1712>><<3531>>^<<3662>>-1<<3662>>A)=O(sqrt<<1713>>kappa(A)<<1713>>)${}/; $cached_env_img{$key} ='

'; $key = q/{}$(n^2-1)2${}/; $cached_env_img{$key} ='

'; $key = q/{}$L${}/; $cached_env_img{$key} ='

'; $key = q/{}$A^<<2322>>-1<<2322>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$X=bigcup_kX_k${}/; $cached_env_img{$key} ='

'; $key = q/{}$R_<<753>>i+1<<753>>D_<<754>>i+1<<754>>^<<755>>-1<<755>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$A=LL^T${}/; $cached_env_img{$key} ='

'; $key = q/{}$r^<<2403>>(i)<<2403>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$p^<<2732>>(i)^T<<2732>>Ap^<<2733>>(i)<<2733>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$lambda_<<633>>max<<633>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$f(x+y)leqf(x)+f(y)${}/; $cached_env_img{$key} ='

'; $key = q/{}$|r^<<2305>>(i)<<2305>>|_2${}/; $cached_env_img{$key} ='

'; $key = q/{}$|r^<<2225>>(i)<<2225>>|_<<2226>>infty<<2226>>(n|A|_<<2227>>infty<<2227>>|x^<<2228>>(i)<<2228>>|_<<2229>>infty<<2229>>+|b|_<<2230>>infty<<2230>>)${}/; $cached_env_img{$key} ='

'; $key = q/{}$A^Tp^<<2722>>(i)<<2722>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$B_1${}/; $cached_env_img{$key} ='

'; $key = q/{}$y=D_B^<<3007>>12<<3007>>x_B${}/; $cached_env_img{$key} ='

'; $key = q/{}$<<2445>>ttrow_ptr(n+1)<<2445>>=nnz+1${}/; $cached_env_img{$key} ='

'; $key = q/{}$r^<<2092>>(i)<<2092>>=Ax^<<2093>>(i)<<2093>>-b${}/; $cached_env_img{$key} ='

'; $key = q/{}$Ax=b${}/; $cached_env_img{$key} ='

'; $key = q/{}$p^<<541>>(i)<<541>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$x^<<2209>>(i)<<2209>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$X_k${}/; $cached_env_img{$key} ='

'; $key = q/{}$tilde<<3225>>A<<3225>>_H^<<3226>>-1<<3226>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$Omega_i${}/; $cached_env_img{$key} ='

'; $key = q/{}$-1${}/; $cached_env_img{$key} ='

'; $key = q/{}$hat<<3353>>x<<3353>>${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}c_1leq<<1969>>x^TAxoverx^TBx<<1969>>leqc_2quadhbox<<1970>>forall<<5805>><<5805>>$x$<<5806>><<5806>><<1970>>,{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$jrightarrowi+j${}/; $cached_env_img{$key} ='

'; $key = q/{}$|x^<<2067>>(i)<<2067>>|${}/; $cached_env_img{$key} ='

'; $key = q/{}$v^<<2851>>(j)<<2851>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$w_i${}/; $cached_env_img{$key} ='

'; $key = q/{}[|e^<<2237>>(i)<<2237>>|_<<2238>>infty<<2238>>leq|,|A^<<2239>>-1<<2239>>|cdot|r^<<2240>>(i)<<2240>>|,|leqS_4cdot|,|A^<<2241>>-1<<2241>>|(E|x^<<2242>>(i)<<2242>>|+f)|_<<2243>>infty<<2243>>]{}/; $cached_env_img{$key} ='

'; $key = q/{}$D_<<748>>i+1<<748>>equivmbox<<749>>diag<<749>>(|r^<<750>>(0)<<750>>|_2,|r^<<751>>(1)<<751>>|_2,...,|r^<<752>>(i)<<752>>|_2)${}/; $cached_env_img{$key} ='

'; $key = q/{}$alpha=(sqrt<<645>>kappa_2<<645>>-1)(sqrt<<646>>kappa_2<<646>>+1)${}/; $cached_env_img{$key} ='

'; $key = q/{}$P_n(x)=x^n${}/; $cached_env_img{$key} ='

'; $key = q/{}$R^T_H${}/; $cached_env_img{$key} ='

'; $key = q/{}$x^<<2253>>(0)<<2253>>=0${}/; $cached_env_img{$key} ='

'; $key = q/{}$<<3547>>hat<<2285>>r<<2285>><<3547>>^<<2286>>(i)<<2286>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$|deltar^<<2405>>(i)<<2405>>|leqO(varepsilon)(|A|cdot|x^<<2406>>(i)<<2406>>|+|b|)${}/; $cached_env_img{$key} ='

'; $key = q/{}$|r^<<2306>>(i)<<2306>>|_2${}/; $cached_env_img{$key} ='

'; $key = q/{}$T_iy=|r^<<766>>(0)<<766>>|_2e^<<767>>(1)<<767>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$beta_i=r^<<536>>(i)^T<<536>>r^<<537>>(i)<<537>>r^<<538>>(i-1)^T<<538>>r^<<539>>(i-1)<<539>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$<<3544>>hat<<2279>>r<<2279>><<3544>>^<<2280>>(i)<<2280>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$i-pleqjleqi+q${}/; $cached_env_img{$key} ='

'; $key = q/{}$v^<<2854>>(1)<<2854>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$tilde<<1207>>r<<1207>>^<<1208>>(i)<<1208>>=P_i(A)tilde<<1209>>r<<1209>>^<<1210>>(0)<<1210>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$i+j${}/; $cached_env_img{$key} ='

'; $key = q/{}$nge250${}/; $cached_env_img{$key} ='

'; $key = q/{}$A_I^<<3236>>-1<<3236>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$B_2${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}M=(D+L)D^<<3662>>-1<<3662>>(D+L)^T,{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$Av^<<2855>>(1)<<2855>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$sigma${}/; $cached_env_img{$key} ='

'; $key = q/{}$e^<<2398>>(i)<<2398>>=A^<<2399>>-1<<2399>>r^<<2400>>(i)<<2400>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$bleftarrowM_1^<<3659>>-1<<3659>>b${}/; $cached_env_img{$key} ='

'; $key = q/{}[e^<<2145>>(i)<<2145>>=x^<<2146>>(i)<<2146>>-x=A^<<2147>>-1<<2147>>(Ax^<<2148>>(i)<<2148>>-b)=A^<<2149>>-1<<2149>>r^<<2150>>(i)<<2150>>,]{}/; $cached_env_img{$key} ='

'; $key = q/{}$$tilde<<3217>>K<<3217>>_<<3218>>as<<3218>>^<<3219>>-1<<3219>>v=R_H^Ttilde<<3220>>A<<3220>>_H^<<3221>>-1<<3221>>R_Hv+sum_<<3222>>i=1<<3222>>^pR_i^T<<3559>>tilde<<3223>>A'<<3223>><<3559>>_i^<<3224>>-1<<3224>>R_iv.$${}/; $cached_env_img{$key} ='

'; $key = q/{}$(D_A+L_A)D_A^<<2672>>-1<<2672>>(D_A+U_A)${}/; $cached_env_img{$key} ='

'; $key = q/{}$|e^<<2256>>(i)<<2256>>|leqS_5cdot|A^<<2257>>-1<<2257>>|${}/; $cached_env_img{$key} ='

'; $key = q/{}$tilde<<3247>>A<<3247>>_H${}/; $cached_env_img{$key} ='

'; $key = q/{}$|A^<<2076>>-1<<2076>>|${}/; $cached_env_img{$key} ='

'; $key = q/{}[y_ileftarrowy_i+a_<<2657>>i,j<<2657>>x_jquadRightarrowquady_ileftarrowy_i+a_<<2658>>i,i+j<<2658>>x_<<2659>>i+j<<2659>>~.]{}/; $cached_env_img{$key} ='

'; $key = q/{tabular}<<2489>>|r|r|r|r|r|r|r|r|<<2489>>hline<<2490>>ttcol_ptr<<2490>>&1&4&8&10&13&17&20hline{tabular}/; $cached_env_img{$key} ='

'; $key = q/{}$d+c${}/; $cached_env_img{$key} ='

'; $key = q/{}$|r^<<2180>>(i)<<2180>>|leqS_2equiv<<2181>>emstop_tol<<2181>>cdot|b|${}/; $cached_env_img{$key} ='

'; $key = q/{}$<<3000>>D_B<<3000>>^<<3001>>-12<<3001>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$O(h^<<1837>>-2<<1837>>)${}/; $cached_env_img{$key} ='

'; $key = q/{}$|x|_2${}/; $cached_env_img{$key} ='

'; $key = q/{}$|r^<<2299>>(i)<<2299>>|_<<3548>>M^<<2300>>-12<<2300>>,2<<3548>>=(r^<<2301>>(i)T<<2301>>M^<<2302>>-1<<2302>>r^<<2303>>(i)<<2303>>)^<<2304>>12<<2304>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$A=D+L+U${}/; $cached_env_img{$key} ='

'; $key = q/{}$n^22${}/; $cached_env_img{$key} ='

'; $key = q/{}$r^<<555>>(0)<<555>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$bar<<710>>T<<710>>_i${}/; $cached_env_img{$key} ='

'; $key = q/{}$(i,j-1)${}/; $cached_env_img{$key} ='

'; $key = q/{}$d_<<1884>>i,i<<1884>>=1a_<<1885>>i,i<<1885>>${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}<<2585>>array<<2585>><<2586>>|r|r|r|r|r|r|r|<<2586>>hline<<2587>>ttcol_ind<<2587>>(:,1)&1&2&1&2&5&5hline<<2588>>ttcol_ind<<2588>>(:,2)&2&3&3&4&6&6hline<<2589>>ttcol_ind<<2589>>(:,3)&4&5&4&5&0&0hline<<2590>>ttcol_ind<<2590>>(:,4)&0&0&0&6&0&0hline<<2591>>array<<2591>>.{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$r^<<2098>>(i)<<2098>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$S_Bu_B=g_B${}/; $cached_env_img{$key} ='

'; $key = q/{tabular}<<2463>>|r|r|r|r|r|r|r|r|<<2463>>hline<<2464>>ttrow_ptr<<2464>>&1&3&6&9&13&17&20hline{tabular}/; $cached_env_img{$key} ='

'; $key = q/{}$O(h^<<3104>>-2<<3104>>)${}/; $cached_env_img{$key} ='

'; $key = q/{}$2nnz+n+1${}/; $cached_env_img{$key} ='

'; $key = q/{}$x^<<3361>>(i)<<3361>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$R_i^T<<3043>>A'<<3043>>_i^<<3044>>-1<<3044>>R_iv${}/; $cached_env_img{$key} ='

'; $key = q/{}$$S_BequivA_B-A_<<3092>>IB<<3092>>^TA_I^<<3093>>-1<<3093>>A_<<3094>>IB<<3094>>$${}/; $cached_env_img{$key} ='

'; $key = q/{}[A^<<3282>>(i+1)<<3282>>approxS^<<3283>>(i)<<3283>>=A^<<3284>>(i)<<3284>>_<<3285>>2,2<<3285>>-A^<<3286>>(i)<<3286>>_<<3287>>2,1<<3287>>A^<<3288>>(i)^<<3684>>-1<<3684>><<3288>>_<<3289>>1,1<<3289>>A^<<3290>>(i)<<3290>>_<<3291>>1,2<<3291>>.]{}/; $cached_env_img{$key} ='

'; $key = q/{}$a_<<3429>>i,i<<3429>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$e^<<2412>>(i)<<2412>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$r^<<2039>>(i)<<2039>>=Ax^<<2040>>(i)<<2040>>-b${}/; $cached_env_img{$key} ='

'; $key = q/{}$M^<<658>>-1<<658>>A${}/; $cached_env_img{$key} ='

'; $key = q/{}$pminfty${}/; $cached_env_img{$key} ='

'; $key = q/{}$<<2437>>ttcol_ind(k)<<2437>>=j${}/; $cached_env_img{$key} ='

'; $key = q/{}$n_d=nn_b${}/; $cached_env_img{$key} ='

'; $key = q/{}$|r^<<2300>>(i)<<2300>>|_<<3550>>M^<<2301>>-12<<2301>>,2<<3550>>=(r^<<2302>>(i)T<<2302>>M^<<2303>>-1<<2303>>r^<<2304>>(i)<<2304>>)^<<2305>>12<<2305>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$R_<<3116>>E_i<<3116>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$bar<<763>>T<<763>>_i${}/; $cached_env_img{$key} ='

'; $key = q/{}$lambda_<<1421>>max<<1421>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$x^<<240>>(k)<<240>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$Y=<<1913>>rmdiag<<1913>>(Y_i)${}/; $cached_env_img{$key} ='

'; $key = q/{}$O(varepsilon)|A^<<2401>>-1<<2401>>|cdot(|A|cdot|x^<<2402>>(i)<<2402>>|+|b|)${}/; $cached_env_img{$key} ='

'; $key = q/{}$A^<<2323>>-1<<2323>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$a_<<3383>>i,j<<3383>>leq0${}/; $cached_env_img{$key} ='

'; $key = q/{}$A^mv^<<2856>>(1)<<2856>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$<<3545>>hat<<2272>>r<<2272>><<3545>>^<<2273>>(i)<<2273>>=x^<<2274>>(i)<<2274>>-Gx^<<2275>>(i)<<2275>>-c=M^<<2276>>-1<<2276>>(Ax^<<2277>>(i)<<2277>>-b)=M^<<2278>>-1<<2278>>r^<<2279>>(i)<<2279>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$tilde<<3239>>A<<3239>>_I,tilde<<3240>>S<<3240>>_B${}/; $cached_env_img{$key} ='

'; $key = q/{}$rk${}/; $cached_env_img{$key} ='

'; $key = q/{}$|r^<<2226>>(i)<<2226>>|_<<2227>>infty<<2227>>(n|A|_<<2228>>infty<<2228>>|x^<<2229>>(i)<<2229>>|_<<2230>>infty<<2230>>+|b|_<<2231>>infty<<2231>>)${}/; $cached_env_img{$key} ='

'; $key = q/{}$r${}/; $cached_env_img{$key} ='

'; $key = q/{}$K_<<3194>>VS<<3194>>^<<3195>>-1<<3195>>S_B${}/; $cached_env_img{$key} ='

'; $key = q/{}$f_j=|b|_<<2224>>infty<<2224>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$S_<<3175>>X_k<<3175>>${}/; $cached_env_img{$key} ='

'; $key = q/{tabular}<<2457>>|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|<<2457>>hline<<2458>>ttval<<2458>>&10&-2&3&9&3&7&8&7&3<<6471>><<6471>>$cdots$<<6472>><<6472>>9&13&4&2&-1hline<<2459>>ttcol_ind<<2459>>&1&5&1&2&6&2&3&4&1<<6473>><<6473>>$cdots$<<6474>><<6474>>5&6&2&5&6hline{tabular}/; $cached_env_img{$key} ='

'; $key = q/{}$e^<<2408>>(i)<<2408>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$u^<<2006>>(m)<<2006>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$A_<<3252>>IB<<3252>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$Ap^<<542>>(i-1)<<542>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$ell${}/; $cached_env_img{$key} ='

'; $key = q/{}$<<2435>>ttval(k)<<2435>>=a_<<2436>>i,j<<2436>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$x^<<599>>(i)<<599>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$|A^<<2342>>-1<<2342>>|_<<2343>>infty<<2343>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$R^T_i,R_i,R^T_H,R_H${}/; $cached_env_img{$key} ='

'; $key = q/{}$r^<<2728>>(i+1)<<2728>>${}/; $cached_env_img{$key} ='

'; $key = q/{}[A^T=[A_1,ldots,A_m]]{}/; $cached_env_img{$key} ='

'; $key = q/{}$a_<<3377>>i,i<<3377>>>sum_<<3378>>jnot=i<<3378>>|a_<<3379>>i,j<<3379>>|${}/; $cached_env_img{$key} ='

'; $key = q/{}$f(x)=0${}/; $cached_env_img{$key} ='

'; $key = q/{}$b${}/; $cached_env_img{$key} ='

'; $key = q/{}[A^<<3273>>(i)<<3273>>=pmatrix<<3558>>A^<<3274>>(i)<<3274>>_<<3275>>1,1<<3275>>&A^<<3276>>(i)<<3276>>_<<3277>>1,2<<3277>>crA^<<3278>>(i)<<3278>>_<<3279>>2,1<<3279>>&A^<<3280>>(i)<<3280>>_<<3281>>2,2<<3281>><<3558>>]{}/; $cached_env_img{$key} ='

'; $key = q/{}$e^<<2095>>(i)<<2095>>${}/; $cached_env_img{$key} ='

'; $key = q/{}[1leq<<2663>>tti<<2663>>,<<2664>>tti+j<<2664>>leqn.]{}/; $cached_env_img{$key} ='

'; $key = q/{}$e^<<2031>>(i)<<2031>>equivx^<<2032>>(i)<<2032>>-x${}/; $cached_env_img{$key} ='

'; $key = q/{}$x^<<239>>(k-1)<<239>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$beta_<<2729>>i-1<<2729>>${}/; $cached_env_img{$key} ='

'; $key = q/{}${A^ix}_<<3372>>igeq0<<3372>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$M^<<3651>>-1<<3651>>A${}/; $cached_env_img{$key} ='

'; $key = q/{}$A=M-N${}/; $cached_env_img{$key} ='

'; $key = q/{}$Omega${}/; $cached_env_img{$key} ='

'; $key = q/{}$deltab_j${}/; $cached_env_img{$key} ='

'; $key = q/{}$lambda_<<633>>min<<633>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$K_<<3228>>BPS<<3228>>${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}p^<<2881>>(j)<<2881>>=r^<<2882>>(j-1)<<2882>>+beta_jp^<<2883>>(j-1)<<2883>>,~j=2,~3,~ldots,~k~,{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}[hbox<<2653>>forall<<6625>><<6625>>$j$<<6626>><<6626>>,doforall<<6627>><<6627>>$i$<<6628>><<6628>>:<<2653>>qquady_ileftarrowy_i+a_<<2654>>j,i<<2654>>x_j.]{}/; $cached_env_img{$key} ='

'; $key = q/{}$z^<<554>>(0)<<554>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$r^<<2387>>(i)<<2387>>=Ax^<<2388>>(i)<<2388>>-b${}/; $cached_env_img{$key} ='

'; $key = q/{}[y_i=sum_j(A^T)_<<2651>>i,j<<2651>>x_j=sum_ja_<<2652>>j,i<<2652>>x_j,]{}/; $cached_env_img{$key} ='

'; $key = q/{}$1-O(h^2)${}/; $cached_env_img{$key} ='

'; $key = q/{}$K_<<3048>>as<<3048>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$n_<<3142>>E_i<<3142>>timesn_<<3143>>E_i<<3143>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$hat<<637>>x<<637>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$R${}/; $cached_env_img{$key} ='

'; $key = q/{}$beta_i${}/; $cached_env_img{$key} ='

'; $key = q/{}$a_<<3384>>i,j<<3384>>leq0${}/; $cached_env_img{$key} ='

'; $key = q/{}$|r^<<2265>>(i)<<2265>>|${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}x=left[<<3420>>array<<3420>><<3421>>c<<3421>>x_1vdotsx_n<<3422>>array<<3422>>right]hspace<<3423>>2em<<3423>>x_iin<<3424>>calR<<3424>>.{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$<<3546>>hat<<2280>>r<<2280>><<3546>>^<<2281>>(i)<<2281>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$x=0${}/; $cached_env_img{$key} ='

'; $key = q/{}$S_<<3137>>E_i<<3137>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$beta_i=r^<<537>>(i)^T<<537>>r^<<538>>(i)<<538>>r^<<539>>(i-1)^T<<539>>r^<<540>>(i-1)<<540>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$R_iu${}/; $cached_env_img{$key} ='

'; $key = q/{}$x=l_<<2702>>*1<<2702>>y_1+l_<<2703>>*2<<2703>>y_2+cdots${}/; $cached_env_img{$key} ='

'; $key = q/{}$kappa_2(A)=O(h^<<655>>-2<<655>>)${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}<<2577>>array<<2577>><<2578>>|r|r|r|r|r|r|r|<<2578>>hline<<2579>>ttval<<2579>>(:,1)&10&9&3&6&9&5hline<<2580>>ttval<<2580>>(:,2)&-3&6&8&7&13&-1hline<<2581>>ttval<<2581>>(:,3)&1&-2&7&5&0&0hline<<2582>>ttval<<2582>>(:,4)&0&0&0&4&0&0hline<<2583>>array<<2583>>,{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$n_<<3119>>E_i<<3119>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$(A+deltaA)x^<<2163>>(i)<<2163>>=b+deltab${}/; $cached_env_img{$key} ='

'; $key = q/{}$Y_i${}/; $cached_env_img{$key} ='

'; $key = q/{}$b-Ahat<<3352>>x<<3352>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$|x|_1${}/; $cached_env_img{$key} ='

'; $key = q/{}$p^<<502>>(i)<<502>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$lambda_<<1430>>max<<1430>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$B${}/; $cached_env_img{$key} ='

'; $key = q/{}$x^<<2267>>(i)<<2267>>=M^<<2268>>-1<<2268>>Nx^<<2269>>(i-1)<<2269>>+M^<<2270>>-1<<2270>>bequivGx^<<2271>>(i-1)<<2271>>+c${}/; $cached_env_img{$key} ='

'; $key = q/{}$(cdot,cdot)_A${}/; $cached_env_img{$key} ='

'; $key = q/{}$x^<<2930>>(k)<<2930>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$beta_<<2730>>i-1<<2730>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$M^<<1431>>-1<<1431>>A${}/; $cached_env_img{$key} ='

'; $key = q/{}$A_H^<<3206>>-1<<3206>>${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}x=left[<<3421>>array<<3421>><<3422>>c<<3422>>x_1vdotsx_n<<3423>>array<<3423>>right]hspace<<3424>>2em<<3424>>x_iin<<3425>>calR<<3425>>.{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}B_1=(D-omegaU)^<<3630>>-1<<3630>>(omegaL+(1-omega)D),{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{tabular}<<2482>>|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|<<2482>>hline<<2483>>ttval<<2483>>&10&3&3&9&7&8&4&8&8<<6483>><<6483>>$cdots$<<6484>><<6484>>9&2&3&13&-1hline<<2484>>ttrow_ind<<2484>>&1&2&4&2&3&5&6&3&4<<6485>><<6485>>$cdots$<<6486>><<6486>>5&6&2&5&6hline{tabular}/; $cached_env_img{$key} ='

'; $key = q/{}$x^<<2795>>(i)<<2795>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$tilde<<3243>>K<<3243>>_<<3244>>BPS<<3244>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$|x^<<2066>>(i)<<2066>>|${}/; $cached_env_img{$key} ='

'; $key = q/{}$a_<<2819>>i,j<<2819>>${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}<<2586>>array<<2586>><<2587>>|r|r|r|r|r|r|r|<<2587>>hline<<2588>>ttcol_ind<<2588>>(:,1)&1&2&1&2&5&5hline<<2589>>ttcol_ind<<2589>>(:,2)&2&3&3&4&6&6hline<<2590>>ttcol_ind<<2590>>(:,3)&4&5&4&5&0&0hline<<2591>>ttcol_ind<<2591>>(:,4)&0&0&0&6&0&0hline<<2592>>array<<2592>>.{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$A_i^<<3110>>-1<<3110>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$D_R${}/; $cached_env_img{$key} ='

'; $key = q/{}$kappa_2(A)=O(h^<<656>>-2<<656>>)${}/; $cached_env_img{$key} ='

'; $key = q/{}$L_1+L_2=(I+L_1)(I+L_2)-I-L_1L_2${}/; $cached_env_img{$key} ='

'; $key = q/{}$(A^<<3629>>-1<<3629>>)_<<402>>i,j<<402>>geq0${}/; $cached_env_img{$key} ='

'; $key = q/{}$u${}/; $cached_env_img{$key} ='

'; $key = q/{}$a_<<2509>>i,j<<2509>>not=0${}/; $cached_env_img{$key} ='

'; $key = q/{}$2${}/; $cached_env_img{$key} ='

'; $key = q/{}$r^<<614>>(i)^T<<614>>M^<<3649>>-1<<3649>>r^<<615>>(j)<<615>>=0${}/; $cached_env_img{$key} ='

'; $key = q/{}$lambda_<<3401>>min<<3401>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$S_i${}/; $cached_env_img{$key} ='

'; $key = q/{}$x^<<2251>>(0)<<2251>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$a_<<3363>>i,j<<3363>>=0${}/; $cached_env_img{$key} ='

'; $key = q/{}$x^<<601>>(0)<<601>>+mbox<<602>>span<<602>>{r^<<603>>(0)<<603>>,ldots,A^<<604>>i-1<<604>>r^<<605>>(0)<<605>>}${}/; $cached_env_img{$key} ='

'; $key = q/{}$S_4equivmax_j(|r^<<2215>>(i)<<2215>>|_j(Ecdot|x^<<2216>>(i)<<2216>>|+f)_j)leq<<2217>>emstop_tol<<2217>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$e_<<2231>>j,k<<2231>>=|a_<<2232>>j,k<<2232>>|${}/; $cached_env_img{$key} ='

'; $key = q/{}$lambda_<<2332>>max<<2332>>(A)${}/; $cached_env_img{$key} ='

'; $key = q/{}$|A|cdot|x|gg|b|${}/; $cached_env_img{$key} ='

'; $key = q/{}$x^<<2266>>(i)<<2266>>=M^<<2267>>-1<<2267>>Nx^<<2268>>(i-1)<<2268>>+M^<<2269>>-1<<2269>>bequivGx^<<2270>>(i-1)<<2270>>+c${}/; $cached_env_img{$key} ='

'; $key = q/{}$p^<<2798>>(i)<<2798>>${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}m_<<1668>>i,j<<1668>>=left{<<1669>>array<<1669>><<1670>>ll<<1670>>a_<<1671>>i,i<<1671>>&mbox<<1672>>if<<5501>><<5501>>$i=j$<<5502>><<5502>><<1672>>0&mbox<<1673>>otherwise<<1673>>.<<1674>>array<<1674>>right.{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}left(<<2566>>array<<2566>><<2567>>rrrrrr<<2567>>10&-3&0&1&0&00&9&6&0&-2&03&0&8&7&0&00&6&0&7&5&40&0&0&0&9&130&0&0&0&5&-1<<2568>>array<<2568>>right)longrightarrowleft(<<2569>>array<<2569>><<2570>>rrrr<<2570>>10&-3&19&6&-23&8&76&7&5&49&135&-1<<2571>>array<<2571>>right){displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$|x|_<<2133>>infty<<2133>>leq1${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}A=D+L+L^T{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}|Ax^<<743>>(i)<<743>>-b|_2=|AR_ibar<<744>>y<<744>>-b|_2=|R_<<745>>i+1<<745>>bar<<746>>T<<746>>_iy-b|_2.{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$deltaA=0${}/; $cached_env_img{$key} ='

'; $key = q/{}$M_<<3146>>E_i<<3146>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$$M_<<3138>>E_i<<3138>>=alpha_<<3139>>E_i<<3139>>K^<<3140>>12<<3140>>$${}/; $cached_env_img{$key} ='

'; $key = q/{}$f:R^nrightarrowR${}/; $cached_env_img{$key} ='

'; $key = q/{tabular}<<2616>>|r|r|r|r|r|<<2616>>hline<<2617>>ttjd_ptr<<2617>>&1&7&13&17hline{tabular}/; $cached_env_img{$key} ='

'; $key = q/{}$tilde<<3248>>A<<3248>>_H^<<3249>>-1<<3249>>A_H${}/; $cached_env_img{$key} ='

'; $key = q/{}$A_<<3253>>IB<<3253>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$R_i^T<<3044>>A'<<3044>>_i^<<3045>>-1<<3045>>R_iv${}/; $cached_env_img{$key} ='

'; $key = q/{}$|A^<<2294>>-1<<2294>>M|${}/; $cached_env_img{$key} ='

'; $key = q/{}$f(alphax)=|alpha|f(x)${}/; $cached_env_img{$key} ='

'; $key = q/{}$(D_A+L_A)D_A^<<2671>>-1<<2671>>(D_A+U_A)${}/; $cached_env_img{$key} ='

'; $key = q/{}$|x|_<<2111>>B,alpha<<2111>>equiv|Bx|_<<2112>>alpha<<2112>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$(I+LD^<<3679>>-1<<3679>>)(D+U)${}/; $cached_env_img{$key} ='

'; $key = q/{}$U${}/; $cached_env_img{$key} ='

'; $key = q/{}$A_i^<<3109>>-1<<3109>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$s^Ts${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}A=left[<<3409>>array<<3409>><<3410>>ccc<<3410>>a_<<3411>>1,1<<3411>>&cdots&a_<<3412>>1,n<<3412>>vdots&&vdotsa_<<3413>>m,1<<3413>>&cdots&a_<<3414>>m,n<<3414>><<3415>>array<<3415>>right]hspace<<3416>>2em<<3416>>a_<<3417>>i,j<<3417>>in<<3418>>calR<<3418>>.{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$r^<<551>>(j)<<551>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$r^<<611>>(i)<<611>>=b-Ax^<<612>>(i)<<612>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$p^<<2731>>(i)^T<<2731>>Ap^<<2732>>(i)<<2732>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$R_H${}/; $cached_env_img{$key} ='

'; $key = q/{}$A=(a_<<2507>>i,j<<2507>>)${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}M_1^<<3655>>-1<<3655>>AM_2inv(M_2x)=M_1^<<3656>>-1<<3656>>b.{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$S_<<3170>>X_k<<3170>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$A^H${}/; $cached_env_img{$key} ='

'; $key = q/{}$mbox<<798>>span<<798>>{r^<<799>>(0)<<799>>,Ar^<<800>>(0)<<800>>,A^2r^<<801>>(0)<<801>>,ldots}${}/; $cached_env_img{$key} ='

'; $key = q/{}$|A^<<2368>>-1<<2368>>R|_<<2369>>infty<<2369>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$AA^Ty=b${}/; $cached_env_img{$key} ='

'; $key = q/{}$x_i${}/; $cached_env_img{$key} ='

'; $key = q/{}$LDL^T${}/; $cached_env_img{$key} ='

'; $key = q/{}$hat<<3354>>x<<3354>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$|deltaa_<<2219>>j,k<<2219>>|leqtolcdote_<<2220>>j,k<<2220>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$|deltaA|${}/; $cached_env_img{$key} ='

'; $key = q/{}$|e^<<2152>>(i)<<2152>>|leq|A^<<2153>>-1<<2153>>|cdot|r^<<2154>>(i)<<2154>>|${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}p^<<919>>(i)<<919>>=r^<<920>>(i-1)<<920>>+beta_<<921>>i-1<<921>>p^<<922>>(i-1)<<922>>,qquadtildep^<<923>>(i)<<923>>=tilder^<<924>>(i-1)<<924>>+beta_<<925>>i-1<<925>>tildep^<<926>>(i-1)<<926>>.{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$x^<<501>>(i)<<501>>${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}M_p^<<3680>>-1<<3680>>=(sum_<<1941>>i=0<<1941>>^<<1942>>p-1<<1942>>(I-M^<<3681>>-1<<3681>>A)^i)M^<<3682>>-1<<3682>>.{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$Ay-b${}/; $cached_env_img{$key} ='

'; $key = q/{}[i<jrightarrowsigma(i)<sigma(j).]{}/; $cached_env_img{$key} ='

'; $key = q/{}$|r^<<2361>>(i)<<2361>>|${}/; $cached_env_img{$key} ='

'; $key = q/{}[1llfrac<<2182>>|A|cdot|x|<<2182>><<2183>>|b|<<2183>>=frac<<3537>>|A|cdot|A^<<2184>>-1<<2184>>b|<<3537>><<2185>>|b|<<2185>>leq|A|cdot|A^<<2186>>-1<<2186>>|;;.]{}/; $cached_env_img{$key} ='

'; $key = q/{}$|w^<<821>>(i)<<821>>|${}/; $cached_env_img{$key} ='

'; $key = q/{}$E${}/; $cached_env_img{$key} ='

'; $key = q/{}$<<2444>>ttrow_ptr(n+1)<<2444>>=nnz+1${}/; $cached_env_img{$key} ='

'; $key = q/{}$H=D_R^<<3004>>-12<<3004>>CD_B^<<3005>>-12<<3005>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$I^<<3428>>ntimesn<<3428>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$e_<<2222>>j,k<<2222>>=|A|_<<2223>>infty<<2223>>${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}left[<<2985>>array<<2985>><<2986>>cc<<2986>>D_R&CO&D_B-C^T<<2987>>D_R<<2987>>^<<2988>>-1<<2988>>C<<2989>>array<<2989>>right]left[<<2990>>array<<2990>><<2991>>c<<2991>>x_Rx_B<<2992>>array<<2992>>right]=left[<<2993>>array<<2993>><<2994>>c<<2994>>f_Rf_B-C^T<<2995>>D_R<<2995>>^<<2996>>-1<<2996>>f_R<<2997>>array<<2997>>right].{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$K_<<3049>>as<<3049>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$r^<<1297>>(0)<<1297>>${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}x^<<734>>(i)<<734>>in{r^<<735>>(0)<<735>>,Ar^<<736>>(0)<<736>>,ldots,A^<<737>>i-1<<737>>r^<<738>>(0)<<738>>},mbox<<739>>~~<<739>>x^<<740>>(i)<<740>>=R_ibar<<741>>y<<741>>{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$(alpha_i)${}/; $cached_env_img{$key} ='

'; $key = q/{}$v^<<2850>>(2)<<2850>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$K_<<3050>>as<<3050>>${}/; $cached_env_img{$key} ='

'; $key = q/{}${A^kr^<<817>>(0)<<817>>}${}/; $cached_env_img{$key} ='

'; $key = q/{}$e^<<2090>>(i)<<2090>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$uequiv(u_I,u_B)^T${}/; $cached_env_img{$key} ='

'; $key = q/{}$S_B${}/; $cached_env_img{$key} ='

'; $key = q/{}$max_isum_j|A_<<1948>>i,j<<1948>>|${}/; $cached_env_img{$key} ='

'; $key = q/{}$S_k${}/; $cached_env_img{$key} ='

'; $key = q/{}$a_<<1830>>i,i<<1830>>${}/; $cached_env_img{$key} ='

'; $key = q/{tabular}<<2617>>|r|r|r|r|r|<<2617>>hline<<2618>>ttjd_ptr<<2618>>&1&7&13&17hline{tabular}/; $cached_env_img{$key} ='

'; $key = q/{}$x${}/; $cached_env_img{$key} ='

'; $key = q/{}$|x|_<<2387>>infty<<2387>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$|A^<<2315>>-1<<2315>>|${}/; $cached_env_img{$key} ='

'; $key = q/{}$|e^<<2348>>(i)<<2348>>|_<<2349>>infty<<2349>>leq|,|A^<<2350>>-1<<2350>>|cdot|r^<<2351>>(i)<<2351>>|,|_<<2352>>infty<<2352>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$K_<<3163>>BPS<<3163>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$r^<<2263>>(i)<<2263>>=Ax^<<2264>>(i)<<2264>>-b${}/; $cached_env_img{$key} ='

'; $key = q/{}$r^<<1224>>(0)<<1224>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$5${}/; $cached_env_img{$key} ='

'; $key = q/{}$p^<<2909>>(1)<<2909>>,~p^<<2910>>(2)<<2910>>,~ldots,~p^<<2911>>(j)<<2911>>${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}<<2114>>array<<2114>><<2115>>lcl<<2115>>|A|_<<2116>>infty<<2116>>&equiv&max_jsum_k|a_<<2117>>j,k<<2117>>|;,|A|_<<2118>>1<<2118>>&equiv&max_ksum_j|a_<<2119>>j,k<<2119>>|;<<2120>>rm,and<<2120>>|A|_<<2121>>E<<2121>>&equiv&(sum_<<2122>>jk<<2122>>|a_<<2123>>j,k<<2123>>|^2)^<<2124>>12<<2124>>;;,<<2125>>array<<2125>>{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$r^<<544>>(i-1)<<544>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$K_<<3036>>as<<3036>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$<<1997>>calL<<1997>>_2${}/; $cached_env_img{$key} ='

'; $key = q/{}$x_j${}/; $cached_env_img{$key} ='

'; $key = q/{}$j>i${}/; $cached_env_img{$key} ='

'; $key = q/{}$Ap^<<2720>>(i)<<2720>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$a_<<2682>>i,j<<2682>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$(i,sigma_k(i))${}/; $cached_env_img{$key} ='

'; $key = q/{}$a_<<2680>>j,i<<2680>>${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}A=(D+L)D^<<3675>>-1<<3675>>(D+U)=(D+L)(I+D^<<3676>>-1<<3676>>U){displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$r^<<542>>(i)<<542>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$<<2441>>ttval(k)<<2441>>=a_<<2442>>i,j<<2442>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$alpha=alpha_i=r^<<522>>(i)^T<<522>>r^<<523>>(i)<<523>>p^<<524>>(i)^T<<524>>Ap^<<525>>(i)<<525>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$A_i${}/; $cached_env_img{$key} ='

'; $key = q/{}$A^<<3681>>-1<<3681>>=sum_<<1939>>k=0<<1939>>^inftyB^k${}/; $cached_env_img{$key} ='

'; $key = q/{}$a_<<401>>i,j<<401>>leq0${}/; $cached_env_img{$key} ='

'; $key = q/{tabbing}junkjunk=junk=junk=junk=junk=junkkill><<2077>>bfuntil<<2077>><<5963>><<5963>>$igeqmaxit$<<5964>><<5964>>or<<5965>><<5965>>$|r^<<2078>>(i)<<2078>>|leq<<2079>>emstop_tol<<2079>>cdot|x^<<2080>>(i)<<2080>>||A^<<2081>>-1<<2081>>|;;,$<<5966>><<5966>>{tabbing}/; $cached_env_img{$key} ='

'; $key = q/{}$h${}/; $cached_env_img{$key} ='

'; $key = q/{}$|x|_2=(sum_j|x_j|^2)^<<2382>>12<<2382>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$Ap_i${}/; $cached_env_img{$key} ='

'; $key = q/{}$ntimesk${}/; $cached_env_img{$key} ='

'; $key = q/{}$<<2438>>ttcol_ind(k)<<2438>>=j${}/; $cached_env_img{$key} ='

'; $key = q/{}$(0,2)${}/; $cached_env_img{$key} ='

'; $key = q/{}$lambda_<<1431>>max<<1431>>${}/; $cached_env_img{$key} ='

'; $key = q/{}[y_ileftarrowy_i+a_<<2658>>i,j<<2658>>x_jquadRightarrowquady_ileftarrowy_i+a_<<2659>>i,i+j<<2659>>x_<<2660>>i+j<<2660>>~.]{}/; $cached_env_img{$key} ='

'; $key = q/{}$|,|A^<<2362>>-1<<2362>>|cdot|r^<<2363>>(i)<<2363>>|,|_<<2364>>infty<<2364>>=|A^<<2365>>-1<<2365>>R|_<<2366>>infty<<2366>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$rho_<<2796>>i+1<<2796>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$R^<<2867>>(k)<<2867>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$a_<<1828>>i,k<<1828>>a_<<1829>>k,k<<1829>>^<<3673>>-1<<3673>>a_<<1830>>k,j<<1830>>${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}left[<<2986>>array<<2986>><<2987>>cc<<2987>>D_R&CO&D_B-C^T<<2988>>D_R<<2988>>^<<2989>>-1<<2989>>C<<2990>>array<<2990>>right]left[<<2991>>array<<2991>><<2992>>c<<2992>>x_Rx_B<<2993>>array<<2993>>right]=left[<<2994>>array<<2994>><<2995>>c<<2995>>f_Rf_B-C^T<<2996>>D_R<<2996>>^<<2997>>-1<<2997>>f_R<<2998>>array<<2998>>right].{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$M^<<659>>-1<<659>>A${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}|A|_2cdot|A^<<3687>>-1<<3687>>|_2=frac<<3559>>lambda^<<3395>>12<<3395>>_<<3396>>max<<3396>>(A^TA)<<3559>><<3560>>lambda^<<3397>>12<<3397>>_<<3398>>min<<3398>>(A^TA)<<3560>>,{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$X${}/; $cached_env_img{$key} ='

'; $key = q/{}$A_I^<<3234>>-1<<3234>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$tilde<<3251>>A<<3251>>_i^<<3252>>-1<<3252>>A_i${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}p^<<920>>(i)<<920>>=r^<<921>>(i-1)<<921>>+beta_<<922>>i-1<<922>>p^<<923>>(i-1)<<923>>,qquadtildep^<<924>>(i)<<924>>=tilder^<<925>>(i-1)<<925>>+beta_<<926>>i-1<<926>>tildep^<<927>>(i-1)<<927>>.{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$A_H^<<3205>>-1<<3205>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$sqrt<<1193>>i+1<<1193>>${}/; $cached_env_img{$key} ='

'; $key = q/{}${A^kr^<<816>>(0)<<816>>}${}/; $cached_env_img{$key} ='

'; $key = q/{}$|r^<<2264>>(i)<<2264>>|${}/; $cached_env_img{$key} ='

'; $key = q/{}$hrightarrow0${}/; $cached_env_img{$key} ='

'; $key = q/{}$S_<<3128>>E_i<<3128>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$lambda_<<3402>>min<<3402>>${}/; $cached_env_img{$key} ='

'; $key = q/{}${x^<<2029>>(i)<<2029>>}${}/; $cached_env_img{$key} ='

'; $key = q/{}$imapstoQ_i(A)P_i(A)r^<<1314>>(0)<<1314>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$[lambda_<<1483>>min<<1483>>,lambda_<<1484>>max<<1484>>]${}/; $cached_env_img{$key} ='

'; $key = q/{}$L^<<1931>>-1<<1931>>AA^TL^<<1932>>-T<<1932>>y=b${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}mbox<<1661>>solve<<5493>><<5493>>$u$<<5494>><<5494>>from<<5495>><<5495>>$Mu=v$<<5496>><<5496>><<1661>>,{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$epsilon${}/; $cached_env_img{$key} ='

'; $key = q/{}$O(n^3)${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}hat<<2894>>T<<2894>>_k=R_k^TAR_k=B_k^That<<2895>>Lambda<<2895>>_kB_k{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$|A|cdot|x^<<2405>>(i)<<2405>>|+|b|${}/; $cached_env_img{$key} ='

'; $key = q/{}$H${}/; $cached_env_img{$key} ='

'; $key = q/{}$LU${}/; $cached_env_img{$key} ='

'; $key = q/{}$e^<<2088>>(i)<<2088>>=x^<<2089>>(i)<<2089>>-x${}/; $cached_env_img{$key} ='

'; $key = q/{}$kappa(M^<<3672>>-1<<3672>>A)=O(h^<<3673>>-1<<3673>>)${}/; $cached_env_img{$key} ='

'; $key = q/{}$a_<<3375>>i,j<<3375>>=a_<<3376>>j,i<<3376>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$a_<<3378>>i,i<<3378>>>sum_<<3379>>jnot=i<<3379>>|a_<<3380>>i,j<<3380>>|${}/; $cached_env_img{$key} ='

'; $key = q/{}$A^TA${}/; $cached_env_img{$key} ='

'; $key = q/{}$[lambda_<<3370>>min<<3370>>(A),lambda_<<3371>>max<<3371>>(A)]${}/; $cached_env_img{$key} ='

'; $key = q/{}$|A^<<2316>>-1<<2316>>|${}/; $cached_env_img{$key} ='

'; $key = q/{}$|r^<<2154>>(i)<<2154>>|letau${}/; $cached_env_img{$key} ='

'; $key = q/{}$(i,sigma(i))${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}M=(D+L)D^<<3660>>-1<<3660>>(D+L)^T,{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$|r^<<2256>>(0)<<2256>>|${}/; $cached_env_img{$key} ='

'; $key = q/{}[y_i=sum_ja_<<2648>>i,j<<2648>>x_j,]{}/; $cached_env_img{$key} ='

'; $key = q/{}${A^ix}_<<3373>>igeq0<<3373>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$tilde<<1035>>p<<1035>>^<<1036>>(i)^T<<1036>>q^<<1037>>(i)<<1037>>approx0${}/; $cached_env_img{$key} ='

'; $key = q/{}$A_B${}/; $cached_env_img{$key} ='

'; $key = q/{}$A_<<3108>>I<<3108>>^<<3109>>-1<<3109>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$1:n_b${}/; $cached_env_img{$key} ='

'; $key = q/{}$j>i+q${}/; $cached_env_img{$key} ='

'; $key = q/{}$a(x,y)${}/; $cached_env_img{$key} ='

'; $key = q/{}$p^<<540>>(i)<<540>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$<<3556>>tilde<<3209>>A'<<3209>><<3556>>_i^<<3210>>-1<<3210>>,tilde<<3211>>A<<3211>>_i^<<3212>>-1<<3212>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$L^T${}/; $cached_env_img{$key} ='

'; $key = q/{}$h=N^<<284>>-d<<284>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$LQ${}/; $cached_env_img{$key} ='

'; $key = q/{}$S_E${}/; $cached_env_img{$key} ='

'; $key = q/{}$a_<<3374>>i,j<<3374>>=0${}/; $cached_env_img{$key} ='

'; $key = q/{}$[lambda_<<1417>>min<<1417>>,lambda_<<1418>>max<<1418>>]${}/; $cached_env_img{$key} ='

'; $key = q/{}${A^ix}_<<623>>igeq0<<623>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$|deltaa_<<2218>>j,k<<2218>>|leqtolcdote_<<2219>>j,k<<2219>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$|deltaA|leqvarepsilon|A|${}/; $cached_env_img{$key} ='

'; $key = q/{}$|r^<<2181>>(i)<<2181>>|leqS_2equiv<<2182>>emstop_tol<<2182>>cdot|b|${}/; $cached_env_img{$key} ='

'; $key = q/{}$tilde<<3213>>A<<3213>>_H^<<3214>>-<<3214>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$a_<<2661>>i,i+j<<2661>>${}/; $cached_env_img{$key} ='

'; $key = q/{tabular}<<2552>>|r|r|r|r|r|r|r|r|r|r|r|<<2552>>hline<<2553>>ttval(:,-1)<<2553>>&0&0&3&6&0&5hline<<2554>>ttval(:,0)<<2554>>&10&9&8&7&9&-1hline<<2555>>ttval(:,+1)<<2555>>&0&-3&6&7&5&13hline<<2556>>ttval(:,+2)<<2556>>&0&1&-2&0&4&0hline{tabular}/; $cached_env_img{$key} ='

'; $key = q/{}$T_iy=|r^<<765>>(0)<<765>>|_2e^<<766>>(1)<<766>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$k${}/; $cached_env_img{$key} ='

'; $key = q/{}$1:nnzb${}/; $cached_env_img{$key} ='

'; $key = q/{}$tilde<<1208>>r<<1208>>^<<1209>>(i)<<1209>>=P_i(A)tilde<<1210>>r<<1210>>^<<1211>>(0)<<1211>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$(w^<<818>>(i)<<818>>,v^<<819>>(k)<<819>>)${}/; $cached_env_img{$key} ='

'; $key = q/{}$kp${}/; $cached_env_img{$key} ='

'; $key = q/{}$hat<<636>>x<<636>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$Q_k=R_kDelta^<<2916>>-1<<2916>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$x^<<2235>>(i)<<2235>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$|I-M^<<3683>>-1<<3683>>A|${}/; $cached_env_img{$key} ='

'; $key = q/{}$p^<<2926>>(i)^T<<2926>>p^<<2927>>(i)<<2927>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$z=A^Tcdotx${}/; $cached_env_img{$key} ='

'; $key = q/{}$A^<<2294>>-1<<2294>>M=(M-N)^<<2295>>-1<<2295>>M=(I-G)^<<2296>>-1<<2296>>${}/; $cached_env_img{$key} ='

'; $key = q/{}[y_i=sum_ja_<<2647>>i,j<<2647>>x_j,]{}/; $cached_env_img{$key} ='

'; $key = q/{}$i+j=k-1${}/; $cached_env_img{$key} ='

'; $key = q/{}$r^<<1200>>(i)<<1200>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$T_h${}/; $cached_env_img{$key} ='

'; $key = q/{}$K_<<3037>>as<<3037>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$x^<<2026>>(i)<<2026>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$kappa_2(B)=lambda_<<634>>max<<634>>(B)lambda_<<635>>min<<635>>(B))${}/; $cached_env_img{$key} ='

'; $key = q/{}$Tx${}/; $cached_env_img{$key} ='

'; $key = q/{tabular}<<2483>>|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|<<2483>>hline<<2484>>ttval<<2484>>&10&3&3&9&7&8&4&8&8<<6487>><<6487>>$cdots$<<6488>><<6488>>9&2&3&13&-1hline<<2485>>ttrow_ind<<2485>>&1&2&4&2&3&5&6&3&4<<6489>><<6489>>$cdots$<<6490>><<6490>>5&6&2&5&6hline{tabular}/; $cached_env_img{$key} ='

'; $key = q/{}$|A^<<2298>>-1<<2298>>M|=|(I-G)^<<2299>>-1<<2299>>|leq1(1-|G|)${}/; $cached_env_img{$key} ='

'; $key = q/{}$x^<<2042>>(i)<<2042>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$K^<<3171>>12<<3171>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$<<1998>>calL<<1998>>_2${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}Lu(x_i)=2u(x_i)-u(x_<<276>>i-1<<276>>)-u(x_<<277>>i+1<<277>>)=f(x_i)qquadhbox<<278>>for<<4377>><<4377>>$x_i=iN$<<4378>><<4378>>,<<4379>><<4379>>$i=1ldotsN-1$<<4380>><<4380>><<278>>.{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$(x^<<605>>(i)<<605>>-hat<<606>>x<<606>>)^TA(x^<<607>>(i)<<607>>-hat<<608>>x<<608>>)${}/; $cached_env_img{$key} ='

'; $key = q/{}$H=D_R^<<3005>>-12<<3005>>CD_B^<<3006>>-12<<3006>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$x^<<2254>>(0)<<2254>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$r^<<2096>>(i)<<2096>>${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}M^<<1636>>-1<<1636>>Ax=M^<<1637>>-1<<1637>>b{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$x^<<2929>>(k)<<2929>>${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}A=(D+L)D^<<3677>>-1<<3677>>(D+U)=(D+L)(I+D^<<3678>>-1<<3678>>U){displaymath}/; $cached_env_img{$key} ='

'; $key = q/{tabular}<<2488>>|r|r|r|r|r|r|r|r|<<2488>>hline<<2489>>ttcol_ptr<<2489>>&1&4&8&10&13&17&20hline{tabular}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}AR_i=R_<<708>>i+1<<708>>bar<<709>>T<<709>>_i,{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}x^<<2324>>(i)<<2324>>=M^<<2325>>-1<<2325>>Nx^<<2326>>(i-1)<<2326>>+M^<<2327>>-1<<2327>>b=Gx^<<2328>>(i-1)<<2328>>+c,{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$S={1,ldots,n}${}/; $cached_env_img{$key} ='

'; $key = q/{}$q^<<2798>>(i)<<2798>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$X=<<1912>>rmdiag<<1912>>(X_i)${}/; $cached_env_img{$key} ='

'; $key = q/{}[Lz=x,qquadUy=z.]{}/; $cached_env_img{$key} ='

'; $key = q/{}$|A|_<<2126>>B,alpha<<2126>>equiv|BAB^<<2127>>-1<<2127>>|_<<2128>>alpha<<2128>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$|r^<<2360>>(i)<<2360>>|${}/; $cached_env_img{$key} ='

'; $key = q/{}$M_1^<<3658>>-1<<3658>>AM_2inv${}/; $cached_env_img{$key} ='

'; $key = q/{}$Delta${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}B_k=left[<<2874>>array<<2874>><<2875>>ccccc<<2875>>1&-beta_2&&cdots&0&1&-beta_3&&vdots&ddots&ddots&ddots&vdots&&ddots&ddots&-beta_k0&cdots&&&1<<2876>>array<<2876>>right],{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$r^<<509>>(i)<<509>>=b-Ax^<<510>>(i)<<510>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$K${}/; $cached_env_img{$key} ='

'; $key = q/{}$|e^<<2156>>(i)<<2156>>|leqtaucdot|A^<<2157>>-1<<2157>>|${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}<<2113>>array<<2113>><<2114>>lcl<<2114>>|A|_<<2115>>infty<<2115>>&equiv&max_jsum_k|a_<<2116>>j,k<<2116>>|;,|A|_<<2117>>1<<2117>>&equiv&max_ksum_j|a_<<2118>>j,k<<2118>>|;<<2119>>rm,and<<2119>>|A|_<<2120>>E<<2120>>&equiv&(sum_<<2121>>jk<<2121>>|a_<<2122>>j,k<<2122>>|^2)^<<2123>>12<<2123>>;;,<<2124>>array<<2124>>{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$T_i${}/; $cached_env_img{$key} ='

'; $key = q/{}$b-Ahat<<3353>>x<<3353>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$tilde<<3232>>K<<3232>>_<<3233>>VS<<3233>>${}/; $cached_env_img{$key} ='

'; $key = q/{}[1llfrac<<2183>>|A|cdot|x|<<2183>><<2184>>|b|<<2184>>=frac<<3539>>|A|cdot|A^<<2185>>-1<<2185>>b|<<3539>><<2186>>|b|<<2186>>leq|A|cdot|A^<<2187>>-1<<2187>>|;;.]{}/; $cached_env_img{$key} ='

'; $key = q/{}$lambda_<<1432>>max<<1432>>${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}AR_i=R_<<707>>i+1<<707>>bar<<708>>T<<708>>_i,{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$M^<<1658>>-1<<1658>>A${}/; $cached_env_img{$key} ='

'; $key = q/{}$p^<<2728>>(i)<<2728>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$q^<<2801>>(i)<<2801>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$e_<<2223>>j,k<<2223>>=|A|_<<2224>>infty<<2224>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$[lambda_<<1484>>min<<1484>>,lambda_<<1485>>max<<1485>>]${}/; $cached_env_img{$key} ='

'; $key = q/{}$lambda_<<2332>>min<<2332>>(A)${}/; $cached_env_img{$key} ='

'; $key = q/{}$L^<<2733>>-1<<2733>>r^<<2734>>(i)<<2734>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$|r^<<1190>>(i)<<1190>>|${}/; $cached_env_img{$key} ='

'; $key = q/{}$r^<<1298>>(k)<<1298>>=P_k(A)r^<<1299>>(0)<<1299>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$bar<<768>>T<<768>>_i${}/; $cached_env_img{$key} ='

'; $key = q/{}$M^<<1639>>-1<<1639>>A${}/; $cached_env_img{$key} ='

'; $key = q/{}$x^Tx=0${}/; $cached_env_img{$key} ='

'; $key = q/{}$S_Bv_B${}/; $cached_env_img{$key} ='

'; $key = q/{}$|deltab|leq<<2169>>emstop_tol<<2169>>cdot|b|${}/; $cached_env_img{$key} ='

'; $key = q/{}$|A+B|leq|A|+|B|${}/; $cached_env_img{$key} ='

'; $key = q/{}$2times2${}/; $cached_env_img{$key} ='

'; $key = q/{}$u^<<2004>>(m)<<2004>>${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}|A|_2cdot|A^<<3689>>-1<<3689>>|_2=frac<<3561>>lambda^<<3396>>12<<3396>>_<<3397>>max<<3397>>(A^TA)<<3561>><<3562>>lambda^<<3398>>12<<3398>>_<<3399>>min<<3399>>(A^TA)<<3562>>,{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}|Ax|leq|A|,|x|.{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$max_isum_j|A_<<1947>>i,j<<1947>>|${}/; $cached_env_img{$key} ='

'; $key = q/{}$r^<<2260>>(i)<<2260>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$A^<<2295>>-1<<2295>>M=(M-N)^<<2296>>-1<<2296>>M=(I-G)^<<2297>>-1<<2297>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$M^<<1640>>-1<<1640>>A${}/; $cached_env_img{$key} ='

'; $key = q/{}$ntimesn${}/; $cached_env_img{$key} ='

'; $key = q/{}$M^<<631>>-1<<631>>A${}/; $cached_env_img{$key} ='

'; $key = q/{}${r^<<2878>>(k)<<2878>>}${}/; $cached_env_img{$key} ='

'; $key = q/{}$|A|cdot|x^<<2404>>(i)<<2404>>|+|b|${}/; $cached_env_img{$key} ='

'; $key = q/{tabular}<<2606>>|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|<<2606>>hline<<2607>>ttjdiag<<2607>>&1&3&7&8&10&2;&9&9&8<<6593>><<6593>>$cdots$<<6594>><<6594>>-1;&9&6&7&5;&13hline<<2608>>ttcol_ind<<2608>>&1&1&2&3&1&5;&4&2&3<<6595>><<6595>>$cdots$<<6596>><<6596>>6;&5&3&4&5;&6hline{tabular}/; $cached_env_img{$key} ='

'; $key = q/{}$min_i{a_<<3381>>i,i<<3381>>-sum_<<3382>>jnot=i<<3382>>|a_<<3383>>i,j<<3383>>|}${}/; $cached_env_img{$key} ='

'; $key = q/{}$|r^<<2307>>(i)<<2307>>|_<<3551>>M^<<2308>>-12<<2308>>,2<<3551>>|r^<<2309>>(0)<<2309>>|_<<3552>>M^<<2310>>-12<<2310>>,2<<3552>>leqtol${}/; $cached_env_img{$key} ='

'; $key = q/{}$A_I^<<3233>>-1<<3233>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$r^<<1224>>(i)<<1224>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$n${}/; $cached_env_img{$key} ='

'; $key = q/{}$Ap^<<1047>>(k)<<1047>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$(A+deltaA)x^<<2142>>(i)<<2142>>=(b+deltab)${}/; $cached_env_img{$key} ='

'; $key = q/{}$ncofm${}/; $cached_env_img{$key} ='

'; $key = q/{}$A_H^<<3225>>-1<<3225>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$|x+y|leq|x|+|y|${}/; $cached_env_img{$key} ='

'; $key = q/{}$|e^<<2072>>(i)<<2072>>|leq|A^<<2073>>-1<<2073>>|cdot|r^<<2074>>(i)<<2074>>|${}/; $cached_env_img{$key} ='

'; $key = q/{}$|A^<<2354>>-1<<2354>>|_<<2355>>infty<<2355>>cdot|r^<<2356>>(i)<<2356>>|_<<2357>>infty<<2357>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$x^<<2066>>(i)<<2066>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$varepsilon=2^<<2051>>-53<<2051>>approx10^<<2052>>-16<<2052>>${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}m_<<1678>>i,j<<1678>>=left{<<1679>>array<<1679>><<1680>>ll<<1680>>a_<<1681>>i,j<<1681>>&mbox<<1682>>if<<5505>><<5505>>$i$<<5506>><<5506>>and<<5507>><<5507>>$j$<<5508>><<5508>>areinthesameindexsubset<<1682>>0&mbox<<1683>>otherwise<<1683>>.<<1684>>array<<1684>>right.{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$Ax=f${}/; $cached_env_img{$key} ='

'; $key = q/{}$sqrt<<2136>>n<<2136>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$[lambda_<<1418>>min<<1418>>,lambda_<<1419>>max<<1419>>]${}/; $cached_env_img{$key} ='

'; $key = q/{}$a_<<397>>ii<<397>>>sum_<<398>>jnot=i<<398>>|a_<<399>>ij<<399>>|${}/; $cached_env_img{$key} ='

'; $key = q/{}$y_i${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}r^<<910>>(i)<<910>>=r^<<911>>(i-1)<<911>>-alpha_iAp^<<912>>(i)<<912>>,qquadtilder^<<913>>(i)<<913>>=tilder^<<914>>(i-1)<<914>>-alpha_iA^Ttildep^<<915>>(i)<<915>>,{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$K_<<3166>>BPS<<3166>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$r^<<2736>>(i)^T<<2736>>M^<<2737>>-1<<2737>>r^<<2738>>(i)<<2738>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$Ax^<<2390>>(i)<<2390>>${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}B_k=left[<<2873>>array<<2873>><<2874>>ccccc<<2874>>1&-beta_2&&cdots&0&1&-beta_3&&vdots&ddots&ddots&ddots&vdots&&ddots&ddots&-beta_k0&cdots&&&1<<2875>>array<<2875>>right],{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$nnz${}/; $cached_env_img{$key} ='

'; $key = q/{}$A^<<2244>>-1<<2244>>${}/; $cached_env_img{$key} ='

'; $key = q/{}$A_H,R_H${}/; $cached_env_img{$key} ='

'; $key = q/{displaymath}B_1=(D-omegaU)^<<3632>>-1<<3632>>(omegaL+(1-omega)D),{displaymath}/; $cached_env_img{$key} ='

'; $key = q/{}$V_H=bigcup_kV_k${}/; $cached_env_img{$key} ='

'; $key = q/{}$|(I-G)^<<2330>>-1<<2330>>|leq1(1-|G|)${}/; $cached_env_img{$key} ='

'; $key = q/{}$K_<<3158>>BPS<<3158>>^<<3159>>-1<<3159>>S_B${}/; $cached_env_img{$key} ='

'; 1;