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# LaTeX2HTML 99.2beta6 (1.42) # Associate images original text with physical files.
$key = q/{displaymath}hat{f}(x)-f(x).{displaymath};MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|78#78|;
$key = q/{displaymath}left|left|left[array{{r}-1.61.2array{right]right|right|_1=2.8,qquady{{r}-1.61.2array{right]right|right|_{infty}=1.6{displaymath};MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|614#614|;
$key = q/x+Δx;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|1632#1632|;
$key = q/x^*;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|1052#1052|;
$key = q/{multicols}%latex2htmlidmarker16921{2}theorem_type{[askhseis][askhseis][chapter]λλέςμαθηματικέςβιβλιοθήκεςλογισμικού.theorem_type{{multicols};MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|1749#1749|;
$key = q/a;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|431#431|;
$key = q/{displaymath}nablaf(x)=left[array{{cc}x_15x_2array{right].{displaymath};MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|1290#1290|;
$key = q/"toolboxes";MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|301#301|;
$key = q/{displaymath}x_{k+1}=x_k-f(x_k)slashf'(x_k){displaymath};MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|1123#1123|;
$key = q/LDL^{T};MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|1211#1211|;
$key = q/mbox{τ}=(sqrt{5}-1)slash2approx0.618;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|1238#1238|;
$key = q/front;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|1624#1624|;
$key = q/http:slashslashwww.rl.ac.ukslashdepartmentsslashccdslashnumericalslashhslslashhsl.html;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|244#244|;
$key = q/(t,y);MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|1539#1539|;
$key = q/Az=u;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|585#585|;
$key = q/g(x);MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|1557#1557|;
$key = q/{displaymath}y'=-100y+100t+101{displaymath};MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|1513#1513|;
$key = q/Vandermonde;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|1672#1672|;
$key = q/{displaymath}u_j^{k+1}=u_j^k+cfrac{Δt}{(Δx)^2}(u_{j+1}^k-2u_j^k+u_{j-1}^k),j=1,ldots,n.{displaymath};MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|1595#1595|;
$key = q/{displaymath}y(t+h)=y(t)+y'(t)h+frac{y''(t)h}{2}h^2+cdots=y(t)+f(t,y(t))h+frac{y''(t)}{2}h^2+cdots.{displaymath};MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|1419#1419|;
$key = q/h=0.5;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|1432#1432|;
$key = q/P_{4};MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|1687#1687|;
$key = q/{displaymath}left[array{{cc}alphaI&AA^T&0array{right]left[array{{c}rslashalphaxarray{right]=left[array{{c}b0array{right]{displaymath};MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|750#750|;
$key = q/{displaymath}s=frac{alphaa_2}{a_1^2+a_2^2},c=frac{alphaa_1}{a_1^2+a_2^2}.{displaymath};MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|808#808|;
$key = q/Scratchpad;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|319#319|;
$key = q/Axiom;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|313#313|;
$key = q/{displaymath}Μ_1P_1Ax=left[array{{rrr}4&9&-30&-frac{1}{2}&-frac{1}{2}0&frac{3}{2{right]=left[array{{r}8-214array{right]=M_1P_1b.{displaymath};MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|514#514|;
$key = q/Lagrange;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|1160#1160|;
$key = q/HOUSEHOLDER;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|765#765|;
$key = q/{displaymath}y'=f(t,y,y),aleqtleqb,{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1546#1546|;
$key = q/prightarrowinfty;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|611#611|;
$key = q/{displaymath}sum_{j=1}^{n}a'_{j}(t)phi_{j}(x_{i})=csum_{j=1}{n}a_{j}(t)phi^{''}_{j}(x_{i}),i=1,...n,{displaymath};MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|1581#1581|;
$key = q/{displaymath}left[array{{rrrr}0.141&0.825&-0.420&-0.3510.344&0.426&0.298&0.7820.0.761&-0.057&0.6460.408&-0.816&0.408array{right]{displaymath};MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|977#977|;
$key = q/AppliedNumericalLinearAlgebra;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|263#263|;
$key = q/b^{2};MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|193#193|;
$key = q/y_2;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|1393#1393|;
$key = q/w^{0}=1;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|1681#1681|;
$key = q/frac{f(x+h)-f(x)}{f(x)}approxhfrac{f^prime(x)}{f(x)},;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|55#55|;
$key = q/epsilon_{mach}=0.25;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|145#145|;
$key = q/2epsilon;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|184#184|;
$key = q/Α,;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|543#543|;
$key = q/{displaymath}(1-λh)y_{k+1}=y_{k},{displaymath};MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|1497#1497|;
$key = q/{displaymath}x_{k}=x_{k-17}-x_{k-5}.{displaymath};MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|1757#1757|;
$key = q/b^{2}-4ac;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|1550#1550|;
$key = q/{displaymath}%latex2htmlidmarker4923parallelx-yparallelgeqleft|parallelxparallel-parallelyparallelright|.{displaymath};MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|622#622|;
$key = q/{displaymath}E_{k}=y_{k}-u_{0}(t_{k})=y_{k}-y(t_{k}).{displaymath};MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|1450#1450|;
$key = q/a_{ij}=a_{ij}-(a_{ik}slasha_{kk})a_{kj};MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|527#527|;
$key = q/{displaymath}c=frac{a_1}{sqrt{a_1^2+a_2^2}},s=frac{a_2}{sqrt{a_1^2+a_2^2}}.{displaymath};MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|809#809|;
$key = q/fell(f(x+h))=fell(f(x));MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|207#207|;
$key = q/J<0;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|1503#1503|;
$key = q/{displaymath}%latex2htmlidmarker7398parallelrparallel_2^2=r^Tr=(b-Ax)^T(b-Ax)=b^Tb-2x^TA^Tb+x^TA^TAx{displaymath};MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|731#731|;
$key = q/{displaymath}mbox{φ}=frac{x-u}{vx-w}.{displaymath};MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|1168#1168|;
$key = q/{displaymath}k_{1}=f(t_{k},y_{k})h_{k},k_{2}=f(t_{k}+h_{k},y_{k}+k_{1})H_{K}.{displaymath};MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|1544#1544|;
$key = q/{displaymath}x=(A-uv^{T})^{-1}b=A^{-1}b+A^{-1}u(1-v^{T}A^{-1}u)^{-1}v^{T}A^{-1}b{displaymath};MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|584#584|;
$key = q/t_{1};MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|899#899|;
$key = q/{displaymath}lim_{krightarrowinfty}frac{|e_{k+1|}}{|e_k|cdot|e_{k-1}|}=c{displaymath};MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|1145#1145|;
$key = q/{displaymath}cos(1,57078)=1.63267949times10^{-5},{displaymath};MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|69#69|;
$key = q/f:{mathbb{R}{^nlongrightarrow{mathbb{R}{;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|1202#1202|;
$key = q/M^{-1};MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|490#490|;
$key = q/s_{k+1}=-g_{k+1}+mbox{β}_{k+1}s_k.;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|1338#1338|;
$key = q/y=F(x);MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|1762#1762|;
$key = q/{displaymath}%latex2htmlidmarker4929frac{parallelzparallel}{parallelyparallel}leqparallelA^{-1}parallel{displaymath};MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|651#651|;
$key = q/alpha_1=0.6;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|1350#1350|;
$key = q/{displaymath}x=-frac{1}{nu_{n+1,n+1}}left[array{{c}nu_{1,n+1}vdotsnu_{n,n+1}array{right]{displaymath};MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|998#998|;
$key = q/C=1-(1slashm);MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|1134#1134|;
$key = q/{displaymath}x=(AD)^{-1}b=D^{-1}A^{-1}b,{displaymath};MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|411#411|;
$key = q/f_2=f_1;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|1252#1252|;
$key = q/h>-2slashJ;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|1471#1471|;
$key = q/Teukolsky;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|278#278|;
$key = q/Μ_2Μ_1Αx=M_2M_1b;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|461#461|;
$key = q/cond(gammaA)=cond(A);MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|645#645|;
$key = q/{displaymath}r=b-Ax=left[array{{l}-0.000622-0.000202array{right],{displaymath};MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|671#671|;
$key = q/x_{k+1}=g(x_k);MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|1104#1104|;
$key = q/mbox{β}_{κ+1};MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|1339#1339|;
$key = q/{displaymath}X_{1}^{2}-x_{2}+0.25=0,{displaymath};MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|1033#1033|;
$key = q/http:slashslashwww.mathworks.com;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|299#299|;
$key = q/R;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|777#777|;
$key = q/log_beta(C)mbox{βάσης}-beta;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|1059#1059|;
$key = q/Dekker;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|1179#1179|;
$key = q/u_{j}^{k};MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|1589#1589|;
$key = q/4ac;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|194#194|;
$key = q/{figure}{centering{
$key = q/x_0=0,x_1=0.6;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|1265#1265|;
$key = q/i,j=1,ldots,n;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|1650#1650|;
$key = q/xpr;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|335#335|;
$key = q/ntimesk;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|591#591|;
$key = q/Q^{T};MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|840#840|;
$key = q/{displaymath}1+frac{s}{c}frac{(a-d)}{b}-frac{s^2}{c^2}=0{displaymath};MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|928#928|;
$key = q/u_{0}^{k}=α;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|1596#1596|;
$key = q/{displaymath}x_2=H^{-1}left[array{{c}alphay_2array{right],qquadmbox{where}qquadalpha=frac{b^Ty_2}{lambda_2-lambda_1}{displaymath};MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|950#950|;
$key = q/bracket;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|1061#1061|;
$key = q/mtimesn;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|364#364|;
$key = q/{displaymath}f(x)=e^x=1+x+frac{x^2}{2!}+frac{x^3}{3!}+cdots,{displaymath};MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|76#76|;
$key = q/{displaymath}%latex2htmlidmarker7418parallelb-Axparallel_2=parallelb-Qleft[arrayllelQ^Tb-left[array{{c}R0array{right]xparallel_2{displaymath};MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|758#758|;
$key = q/M^{T},;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|401#401|;
$key = q/Minitab;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|893#893|;
$key = q/16;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|150#150|;
$key = q/f_b;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|1158#1158|;
$key = q/c=y_0;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|1377#1377|;
$key = q/mathit{j};MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|529#529|;
$key = q/f(x^{}+h)approxf(x^{})+ch^{2};MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|1220#1220|;
$key = q/det(Α)=0.;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|375#375|;
$key = q/{displaymath}A=left[right],b=left[array{{c}1.00.50.00.52.0array{right].{displaymath};MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|785#785|;
$key = q/{displaymath}%latex2htmlidmarker10455x_k=y_kslashparallely_kparallel_infty{displaymath};MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|952#952|;
$key = q/alpha=x_1;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|1247#1247|;
$key = q/M;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|213#213|;
$key = q/{displaymath}Μ_2=left[array{{rrr}1&0&00&1&00&frac{1}{3}&1array{right],{displaymath};MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|518#518|;
$key = q/100;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|41#41|;
$key = q/DFTs;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|1683#1683|;
$key = q/{displaymath}%latex2htmlidmarker4919parallelxparallel_1=sum_{i=1}^n|x_i|,{displaymath};MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|606#606|;
$key = q/e^{-x}-x=0;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|1037#1037|;
$key = q/Broyden;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|1319#1319|;
$key = q/{displaymath}left[array{{rrr}frac{1}{2}&0&00&1&00&0&frac{1}{4}array{right]left[a1&frac{3}{4}&-frac{1}{4}&frac{1}{4}array{right],{displaymath};MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|565#565|;
$key = q/mathbf{strtod},;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|137#137|;
$key = q/{displaymath}f(x)=e^x=1+x+frac{x^2}{2!}+frac{x^3}{3!}+cdots{displaymath};MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|185#185|;
$key = q/c=5.015;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|199#199|;
$key = q/{displaymath}Q^{T}AP=left[matrix{R&Scr0&0}right],{displaymath};MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|876#876|;
$key = q/f(x_k);MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|1124#1124|;
$key = q/[a,x_1);MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|1231#1231|;
$key = q/{displaymath}hat{f}(x)=1+x+frac{x^2}{2}+frac{x^3}{6}.{displaymath};MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|77#77|;
$key = q/Cramer;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|555#555|;
$key = q/randomness;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|1751#1751|;
$key = q/hat{f};MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|36#36|;
$key = q/6%;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|189#189|;
$key = q/|(1+λhslash2)slash(1-λhslash2)|<1;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|1509#1509|;
$key = q/{displaymath}au_{xx}+bu_{xy}+cu_{yy}+du_{x}+eu_{y}+fu+g=0,b^{2}-4ac>0:mbox{υπερβ0:mbox{παραβολική},b^{2}-4ac<0:mbox{έλλειπτική}.{displaymath};MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|1551#1551|;
$key = q/{displaymath}sum_{i=1,ineqj}^{n}|a_{ij}|<|a_{jj}|,j=1,...,n,{displaymath};MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|526#526|;
$key = q/{displaymath}b^{2}-4ac=9757-1.005=9756,{displaymath};MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|201#201|;
$key = q/y'=a;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|1402#1402|;
$key = q/Jordan;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|556#556|;
$key = q/SystemsOptimizationLaboratory;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|292#292|;
$key = q/NAGInc.;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|258#258|;
$key = q/secant;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|16#16|;
$key = q/{displaymath}Ax=lambdax{displaymath};MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|905#905|;
$key = q/{displaymath}frac{1}{n}left[matrix{1&1&1&1cr1&w^{-1}&w^{-2}&w^{-3}cr1&w^{-2}&w^{y_3}right]=left[matrix{x_0crx_1crx_2crx_3}right]{displaymath};MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|1668#1668|;
$key = q/Chebyshev;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|1359#1359|;
$key = q/(2,2);MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|1101#1101|;
$key = q/[10]^{T};MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|704#704|;
$key = q/{displaymath}P_2=left[array{{ccc}1&0&00&0&10&1&0array{right],{displaymath};MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|516#516|;
$key = q/%latex2htmlidmarker6006parallelzparallelslashparallelyparallel;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|652#652|;
$key = q/0leqxleq1,0leqyleq1;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|1638#1638|;
$key = q/{displaymath}left[array{{c}R0array{right]xapproxleft[array{{c}b_1b_2array{right]{displaymath};MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|752#752|;
$key = q/beta=2;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|95#95|;
$key = q/z=left[-frac{3}{2}frac{1}{2}-frac{1}{2}right]^{T};MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|599#599|;
$key = q/hat{v};MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|1724#1724|;
$key = q/SVD;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|12#12|;
$key = q/D_{2};MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|1689#1689|;
$key = q/Gauss-Jordan;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|3#3|;
$key = q/GIVENSQR;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|829#829|;
$key = q/beta^{U+1}(1-beta^{-t}),;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|124#124|;
$key = q/p(0);MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|1156#1156|;
$key = q/Jenkins;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|1187#1187|;
$key = q/{displaymath}f(t)=lambdae^{-lambdat},t>0.{displaymath};MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|1760#1760|;
$key = q/xwd;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|334#334|;
$key = q/4x_3=8;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|422#422|;
$key = q/mathbf{else};MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|1072#1072|;
$key = q/MonteCarlo;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|187#187|;
$key = q/nxn;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|1409#1409|;
$key = q/|f''(x)|;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|214#214|;
$key = q/{displaymath}A=left[matrix{1&-1.0&1.0cr1&-0.5&0.25cr1&0.0&0.0cr1&0.5&0.25cr1&1.0t],b=left[matrix{1.0cr0.5cr0.0cr0.5cr2.0}right].{displaymath};MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|830#830|;
$key = q/{displaymath}P_4=left[matrix{1&0&0&0cr0&0&1&0cr0&1&0&0cr0&0&0&1}right],{displaymath};MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|1688#1688|;
$key = q/mathbf{NAG:};MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|256#256|;
$key = q/{displaymath}Α^{-1}frac{1}{4}left[array{{rrr}27&-11&3-11&5&-13&-1&1array{right].{displaymath};MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|566#566|;
$key = q/Μ_1Α;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|460#460|;
$key = q/{displaymath}J_0=left[array{{ccc}0.707&0&-0.7070&1&00.707&0&0.707array{right]mboccc}3&0.707&00.707&2&0.7070&0.707&-1array{right]{displaymath};MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|931#931|;
$key = q/approx-htan(x)approxinfty;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|66#66|;
$key = q/r=2;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|1057#1057|;
$key = q/h=pslashq;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|1167#1167|;
$key = q/EYLER;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|1462#1462|;
$key = q/C;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|250#250|;
$key = q/DownersGrove;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|259#259|;
$key = q/Uz=y;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|496#496|;
$key = q/{displaymath}f(t+h,y+hf)=f+hf_{1}+hff_{y}+O(h^{2}),{displaymath};MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|1540#1540|;
$key = q/mathbf{NaN};MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|157#157|;
$key = q/[a,b];MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|1060#1060|;
$key = q/{displaymath}x_1=x_0+s_0=left[array{{c}51array{right]+left[array{{c}-5-5array{right]=left[array{{c}0-4array{right].{displaymath};MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|1331#1331|;
$key = q/(x_2,b);MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|1230#1230|;
$key = q/{displaymath}Rx=b_1{displaymath};MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|754#754|;
$key = q/fell(x+h)slashneqfell(x);MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|206#206|;
$key = q/%latex2htmlidmarker5912parallelxparallel>0;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|618#618|;
$key = q/A,;MSF=1.6;LFS=11;AAT/; $cached_env_img{$key} = q|597#597|;
$key = q/x_1 $key = q/t_{2}=0.5;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1531#1531|; $key = q/{displaymath}A^TAx=A^Tb{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|733#733|; $key = q/{displaymath}sigma_k=x_k^TAx_kslashx_k^Tx_k{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|959#959|; $key = q/i>j;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|415#415|; $key = q/h=-f(x)slashf'(x);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1121#1121|; $key = q/LAPLACE;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1643#1643|; $key = q/x_1=alpha+(1-tau)(b-alpha);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1241#1241|; $key = q/(y''slash2)h^{2};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1484#1484|; $key = q/{displaymath}beta_0=(g_1^Tg_1)slash(g_0^Tg_0)=0.444,{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1348#1348|; $key = q/[α,χ_1);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1234#1234|; $key = q/{displaymath}left[matrix{u_{1,1}cru_{2,1}cru_{2,1}cru_{2,2}}right]=left[matrix{0.125cr0.125cr0.375cr0.375}right]{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1653#1653|; $key = q/L_{k},;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|541#541|; $key = q/y(t)=ce^{-t};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1400#1400|; $key = q/IEEESP;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|109#109|; $key = q/EISPACK;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1023#1023|; $key = q/cos(x+h);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|64#64|; $key = q/x=[-122]^Τ;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|477#477|; $key = q/{displaymath}x_{k}=-frac{log(y_k)}{lambda},{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1764#1764|; $key = q/A^{T}y=c;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|654#654|; $key = q/x_1=0.764;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1256#1256|; $key = q/{displaymath}x_k+1=x_k-a_kB_k^-1nablaf(x_k){displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1315#1315|; $key = q/mathbf{naN};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|161#161|; $key = q/x_0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1092#1092|; $key = q/{displaymath}frac{u_j^{k+1}-u_j^k}{Δt}=cfrac{u_{j+1}^k-2u_j^k+u_{-1}^k}{(Δx)^2},j=1,ldots,n,{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1594#1594|; $key = q/alpha_{k};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|436#436|; $key = q/y_2(t)=u';MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1382#1382|; $key = q/m_{i}=a_{i}slasha_{k};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|558#558|; $key = q/Crout;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|532#532|; $key = q/napack;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1011#1011|; $key = q/{displaymath}e_{k+1}=x_{k+1}-x^=g(x_k)-g(x^){displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1109#1109|; $key = q/roundingerror;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|132#132|; $key = q/f:{mathbb{R}{^{n+1}rightarrow{mathbb{R}{^n;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1363#1363|; $key = q/cond(D)=(max|d_{i}|)slash(min|d_{i}|);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|647#647|; $key = q/Householdtransformations;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|828#828|; $key = q/y=1slashv;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1171#1171|; $key = q/{displaymath}y'=f(t,y)=-2ty^{2},{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1525#1525|; $key = q/{displaymath}E_{k+1}=(1+h_{k}J)E_{k}+L_{k+1}.{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1465#1465|; $key = q/x(t);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|691#691|; $key = q/f(x_1)>f(x_2);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1232#1232|; $key = q/{displaymath}y(a)=alpha,y(b)=beta.{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1547#1547|; $key = q/y_{m};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1669#1669|; $key = q/{displaymath}(A+E)hat{x}=b{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|660#660|; $key = q/{eqnarraystar}J^TAJ&=&left[array{{cr}c&-ss&carray{right]left[array{{cc}a&bb&darr+cs(a-d)-s^2b&c^2d+2csb+s^2aarray{right]{eqnarraystar};MSF=1.6;LFS=11;TAGS=R;AAT/;
$cached_env_img{$key} = q|926#926|; $key = q/DTF;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1682#1682|; $key = q/seculartrends;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1718#1718|; $key = q/y_1(t)=u;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1381#1381|; $key = q/{displaymath}(A-uv^{T})^{-1}=A^{-1}+A^{-1}u(1-v^{T}A^{-1}u)^{-1}v^{T}A^{-1}{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|581#581|; $key = q/Unix;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|329#329|; $key = q/u_{x};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1641#1641|; $key = q/{displaymath}F_{1}=1,F_{2}=left[matrix{171cr1&-1}right],F_{4}=left[matrix{1&1&1&1cr1&i&-1&-icr1&-1&1&-1cr1&-i&-1&icr}right].{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1686#1686|; $key = q/Fixed-point;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1103#1103|; $key = q/{displaymath}s_k=|e_{k+1}slash|e_k|^r,{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1146#1146|; $key = q/infty;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|609#609|; $key = q/M-1;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1756#1756|; $key = q/m=[0,ldots,0,m_{k+1},ldots,m_{n}]^{T};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|440#440|; $key = q/PC;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|340#340|; $key = q/PA=LDU;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|540#540|; $key = q/{displaymath}mbox{Normal}:qquadleft[array{{ccc}1&2&00&1&22&0&1array{right],qquadonnormal}:qquadleft[array{{cc}1&10&1array{right]{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|919#919|; $key = q/4;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|510#510|; $key = q/{displaymath}1leqm $key = q/x_2=1.2;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1266#1266|; $key = q/PAx=Pb;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|404#404|; $key = q/{multicols}{2}{par{textbf{4.1.}Σωστόήλάθος:Οιιδιοτιμέςενόςπίνακαδενείναιαπαραίτηικέςεξισώσεις.item[(γ)]Ανάλυσηιδιάζουσας.description{{par{{multicols};LFS=11;AAT/;
$cached_env_img{$key} = q|1025#1025|; $key = q/Dirichlet;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1639#1639|; $key = q/{displaymath}frac{t_k-t_0}{h}O(h^{p+1})=O(h^p),{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1453#1453|; $key = q/{calO}(n^{2});MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|567#567|; $key = q/r_{jk}=q^{T}_{j}alpha_{k};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|848#848|; $key = q/t_1 $key = q/A^{+}b;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|986#986|; $key = q/1.00ldots;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|152#152|; $key = q/{displaymath}alpha=left[array{{c}212array{right].{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|766#766|; $key = q/2times2;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|387#387|; $key = q/y_{1}=exp(1)~=2.718;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1441#1441|; $key = q/rank(Α) $key = q/f_1 $key = q/gnuplot;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|336#336|; $key = q/y'=f(t,y);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1367#1367|; $key = q/{displaymath}u(t,0)=α,u(t,1)=beta.{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1603#1603|; $key = q/rapprox1.839;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1163#1163|; $key = q/f(x^{*});MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1217#1217|; $key = q/O(nlog_2n);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1675#1675|; $key = q/alpha=m;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1071#1071|; $key = q/{displaymath}left[array{{cc}2&34&6array{right]left[array{{c}x_1x_2array{right]=left[array{{c}b_1b_2array{right]{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|392#392|; $key = q/(FFT);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1674#1674|; $key = q/{displaymath}mbox{Symmetric}:qquadleft[array{{cc}1&22&3array{right],qquadmbox{noymmetric}:qquadleft[array{{cc}1&32&4array{right]{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|915#915|; $key = q/Axapproxb;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|878#878|; $key = q/1-mbox{τ}approx0.382;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1239#1239|; $key = q/{figure}{centering{ $key = q/Αx=b;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|379#379|; $key = q/y;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|170#170|; $key = q/nablaf(x)=0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1204#1204|; $key = q/{mathbb{R}{^n;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1283#1283|; $key = q/{displaymath}M=left[array{{cc}1&0-epsilon&1array{right],{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|507#507|; $key = q/p>0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|604#604|; $key = q/mathbf{InfslashInf};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|160#160|; $key = q/{displaymath}x_i=(b_i-sum_{j=i+1}^na_{ij}x_j)slasha_{ii},qquadi=n-1,ldots,1.{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|418#418|; $key = q/t=0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|695#695|; $key = q/x_k;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1051#1051|; $key = q/{displaymath}sum_{n=1}^{infty}frac{1}{n}{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|173#173|; $key = q/f'(x^{*})=0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1218#1218|; $key = q/x<0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|186#186|; $key = q/E_{i};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|989#989|; $key = q/f(x^{*}+h)approxch;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1215#1215|; $key = q/{displaymath}x_{k+1}=g(x_k),{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1088#1088|; $key = q/interpretedmode;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|345#345|; $key = q/{displaymath}x_{k+1}=g(x_k).{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1192#1192|; $key = q/Sherman;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|579#579|; $key = q/{displaymath}J_1=left[array{{ccc}0.888&-0.460&00.460&0.888&00&0&1array{right]mbo6&0&0.3250&1.634&0.6280.325&0.628&-1array{right]{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|932#932|; $key = q/{displaymath}left[array{{rr}0.641&0.2420.321&0.121array{right]left[array{{c}x_1xray{right]=left[array{{c}0.8830.442array{right],{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|668#668|; $key = q/{displaymath}x_{k+1}=x_k-H_f^-1(x_k)nablaf(x_k),{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1297#1297|; $key = q/{displaymath}y'_{i}=cslash(Δx)^{2}[y_{i+1}(t)-2y_{i}(t)+y_{i-1}(t)],i=1,ldots,n,{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1566#1566|; $key = q/t;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|49#49|; $key = q/{displaymath}Ga=left[matrix{0.8&0.6cr-0.6&0.8}right]left[matrix{4cr3}right]=left[matrix{5cr0}right]{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|822#822|; $key = q/p(I+hJ)<1;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1473#1473|; $key = q/e-mail;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|268#268|; $key = q/O(n^{2});MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1658#1658|; $key = q/{displaymath}F(x)=int^x_0f(t)dt=1-e^{-lambdax}.{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1761#1761|; $key = q/dino.ph.utexas.eduslashplplot;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|339#339|; $key = q/H_{n}cdotsH_{1};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|778#778|; $key = q/{displaymath}|e_{k+1}|leqC{e_k}leqcdotsC^k|e_0|,{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1115#1115|; $key = q/resulting;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1140#1140|; $key = q/dfslashdy=-1;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1401#1401|; $key = q/Fibonacci;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|25#25|; $key = q/y(t)=at+c;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1403#1403|; $key = q/x_{0};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1716#1716|; $key = q/approxh;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|59#59|; $key = q/mathbf{while}((b-alpha)>mathit{tol})mathbf{do};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1245#1245|; $key = q/{displaymath}array{{rrr}1&-4&00&1&00&-1&1array{left[array{{rrr|rrr}2&4&-2&1&0&000&-6&9&-4&00&1&1&-2&1&00&0&4&3&-1&0array{right],{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|563#563|; $key = q/{displaymath}u(t,0)=a,u(t,L)=beta,{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1556#1556|; $key = q/f'(x^{*})neq0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1214#1214|; $key = q/x_{k+1}=x_k-(2x_k^2-1)slash(2x_k(3-2x_k^2));MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1279#1279|; $key = q/{displaymath}x=left[array{{r}y-10array{right]{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|923#923|; $key = q/ADeltax=Deltab;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|679#679|; $key = q/u_{t}=g(x);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1610#1610|; $key = q/{displaymath}H_f(x_k)S_k=-nablaf(x_k){displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1299#1299|; $key = q/{displaymath}y_{k+1}=y_{k}+f(t_{k+1},y_{k+1})h_{k}.{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1489#1489|; $key = q/f(x);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|34#34|; $key = q/{multicols}%latex2htmlidmarker1466{2}theorem_type{[askhseis][askhseis][chapter][περασμάτωνπουδόθηκεστοΠαράδειγμα1.12.theorem_type{{multicols};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|360#360|; $key = q/{displaymath}f(t,x)=x_1phi_1(t)+x_2phi_2(t)+cdots+x_nphi_n(t){displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|717#717|; $key = q/{displaymath}mbox{Transpose}:qquadleft[array{{cc}1&23&4array{right]^T=left[array{{cc}1&32&4array{right]{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|913#913|; $key = q/approxhe^{x};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|58#58|; $key = q/IMSL;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|321#321|; $key = q/P_{n};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1694#1694|; $key = q/a_{ij}=0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|414#414|; $key = q/u_i=1,...,n;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1412#1412|; $key = q/b=k2^{j};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1739#1739|; $key = q/computeralgebra;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|312#312|; $key = q/Μ_2;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|459#459|; $key = q/{displaymath}y_0=sum^n_{i=1}alpha_iu_i.{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1413#1413|; $key = q/{displaymath}T^{-1}AT=left[array{{rr}0.5&0.50.5&-0.5array{right]left[array{{rr}3y{right]left[array{{rr}2&00&4array{right]=Lambda{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|921#921|; $key = q/log_{10}(cond(A));MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|698#698|; $key = q/times;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|50#50|; $key = q/{displaymath}%latex2htmlidmarker7396parallelrparallel_2^2=r^Tr{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|729#729|; $key = q/c=0.707,s=0.707.;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|831#831|; $key = q/B_kv;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1353#1353|; $key = q/ax=b,;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|224#224|; $key = q/epsilon_{mach}=2^{-53}approx10^{-16};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|148#148|; $key = q/SPSS;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|896#896|; $key = q/j;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|573#573|; $key = q/phi(x);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1734#1734|; $key = q/{displaymath}y''=f_{t}+ff_{y}{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1538#1538|; $key = q/{displaymath}A^+=VSigma^+U^T{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|985#985|; $key = q/y_{0}=1;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1493#1493|; $key = q/h^{2};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1222#1222|; $key = q/alpha_k;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1287#1287|; $key = q/y_n(t)=u^{(n-1)};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1383#1383|; $key = q/%latex2htmlidmarker5920parallelx+yparallelleqparallelxparallel+parallelyparallel;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|621#621|; $key = q/df(x)slashdx_i;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1207#1207|; $key = q/mathbf{if}sign(f(alpha))=sign(f(m))mathbf{then};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1070#1070|; $key = q/Rayleigh;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|9#9|; $key = q/t_{k};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1421#1421|; $key = q/TOMS(#735);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1745#1745|; $key = q/{displaymath}e^{mbox{λ}h}=1+λh+frac{(mbox{λ}h)^2}{2}+frac{(mbox{λ}h)^3}{2}+cdots{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1483#1483|; $key = q/{displaymath}A^{T}A={displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|739#739|; $key = q/ComputerMethodsforMathematicalComputations;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|237#237|; $key = q/1,ldots,n;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|544#544|; $key = q/fell(x+h);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|204#204|; $key = q/{displaymath}p(w^{m})=sum{k=0}^{n-1}x_{k}(w^{m})^{k}=sum{k=0}^{n-1}x_{k}w^{mk}=y_{m}.{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1732#1732|; $key = q/mathbf{NUMAL:};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|281#281|; $key = q/functionaliteration;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1091#1091|; $key = q/e;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|93#93|; $key = q/k;;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|483#483|; $key = q/H_f(x_k));MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1311#1311|; $key = q/mathbf{EISPACK};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|999#999|; $key = q/(A-uv^{T})x=b;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|583#583|; $key = q/|alpha|<1;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|640#640|; $key = q/nu_{1};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|787#787|; $key = q/Μ;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|372#372|; $key = q/{displaymath}array{{c|rrrrr}hlinehlinet&-1.0&-0.5&0.0&0.5&1.0y&1.0&0.5&0.0&0.5&2.0hlinearray{{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|725#725|; $key = q/y=x;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1100#1100|; $key = q/{displaymath}L=left[array{{cc}1&01slashepsilon&1array{right]kappaalphaiotaU=left[array{{cc}epsilon&10&-1slashepsilonarray{right]{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|504#504|; $key = q/{displaymath}L=M^{-1}=(M_2P_2M_1P_1)^{-1}=P_1^TL_1P_2^TL_2={displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|520#520|; $key = q/H_{1};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|789#789|; $key = q/Az=y;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|650#650|; $key = q/mathbf{NAPACK:};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|262#262|; $key = q/f'(x^*)neq0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1133#1133|; $key = q/C=0.5;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1082#1082|; $key = q/x^*geqx;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1226#1226|; $key = q/y'=y;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|17#17|; $key = q/{figure}{centering{ $key = q/q(t);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1727#1727|; $key = q/{displaymath}G=left[matrix{c&scr-s&c}right]=left[matrix{0.8&0.6cr-0.6&0.8}right].{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|821#821|; $key = q/{displaymath}|e_{k+1}|=s_k|e_k|^r=s_k(s_{k-1}|e_{k-1}|^r)^r=s_ks_k^r|e_{k-1}|^{r^{2}},{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1147#1147|; $key = q/Ax=b;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|383#383|; $key = q/{table}{centering{{{{table};LFS=11;AAT/;
$cached_env_img{$key} = q|886#886|; $key = q/{displaymath}y_{1}=y_{0}+y_{0}'h=y_{0}+y_{0}h=y_{0}(1+h).{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1428#1428|; $key = q/mathbf{STELIOSNOTES:}table;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1713#1713|; $key = q/M_{j},j>k,;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|446#446|; $key = q/{displaymath}array{{l}detleft(left[array{{rr}3&-1-1&3array{right]-lambdaleft[arra)(3-lambda)-(-1)(-1)=lambda^2-6lambda+8=0array{{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|911#911|; $key = q/{displaymath}y_{1}=y_{0}+f(t_{1},y_{1})h=1-0.5y_{1}^{3}{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1492#1492|; $key = q/beta=10;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|100#100|; $key = q/Fortran;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|233#233|; $key = q/Amsterdam;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|283#283|; $key = q/mathbf{for}j=1mathbf{to}k-1;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|847#847|; $key = q/e_{k}^{T}t=0.;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|448#448|; $key = q/cond(A)geq1;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|642#642|; $key = q/{displaymath}happroxsqrt{epsilon_{mach}}cdot|x|.{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|218#218|; $key = q/MPFUN;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|353#353|; $key = q/Francis;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1017#1017|; $key = q/Η;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|773#773|; $key = q/{displaymath}HAH^{-1}=left[array{{cc}lambda_1&b^T0&Barray{right]{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|949#949|; $key = q/1slash(1-hJ);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1502#1502|; $key = q/laso;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1007#1007|; $key = q/{displaymath}r_{1,2}=q_1^Ta_2=0,r_{1,3}-q_1^Ta_3=1.118,{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|862#862|; $key = q/hJ;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1468#1468|; $key = q/λ_i;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1411#1411|; $key = q/|1slash(1-λh)|<1;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1499#1499|; $key = q/u_{tt};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1604#1604|; $key = q/mbox{det}(alphaI_{n})=alpha^{n};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|639#639|; $key = q/{displaymath}c=1slashsqrt{1+t^{2}},s=ccdottau.{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|812#812|; $key = q/b=[48]^{T},;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|394#394|; $key = q/{displaymath}%latex2htmlidmarker4931parallelrparallel=parallelb-Ahat{x}parallel=elleqparallelEparallelcdotparallelhat{x}parallel{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|661#661|; $key = q/{displaymath}y(t+h)=y(t)+y'(t)h+O(h^{2})=y(t)+f(t,y(t))h+O(h^{2}).{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1454#1454|; $key = q/{displaymath}A=USigmaV^T{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|974#974|; $key = q/{displaymath}g(x_k)-g(x^*)=g'(Θ_k)(x_k-x^*){displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1112#1112|; $key = q/y_{2}=y_{1}+y_{1}h=1.5+(1.5)(0.5)=2.25;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1438#1438|; $key = q/beta^{L};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|153#153|; $key = q/bit;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|118#118|; $key = q/{figure}{centering{ $key = q/%latex2htmlidmarker10637parallely_kparallel_infty;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|957#957|; $key = q/f(x_2);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1229#1229|; $key = q/LU;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|2#2|; $key = q/a_{j}=a_{j}-r_{kl}q_{k};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|859#859|; $key = q/{displaymath}x''=Fslashm{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1387#1387|; $key = q/Β;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|628#628|; $key = q/H_{2}H_{1}A;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|796#796|; $key = q/pointwise;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1720#1720|; $key = q/Vetterling;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|279#279|; $key = q/b+pslashq;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1162#1162|; $key = q/y_1;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1392#1392|; $key = q/approx|x|;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|60#60|; $key = q/{displaymath}A^{T}Ax=A^{T}b,{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|735#735|; $key = q/{displaymath}f(x)=0.5x_1^2+2.5x_2^2,{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1289#1289|; $key = q/{multicols}%latex2htmlidmarker6990{2}theorem_type{[ervthseis][ervthseis][chapterνΕυκλείδιανόρματουδιανύσματος{x{theorem_type{{par{{multicols};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|901#901|; $key = q/{displaymath}A=Qleft[array{{c}R0array{right]{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|757#757|; $key = q/f(x)=x^2-x-2=0,;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1096#1096|; $key = q/WilliamKahan;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|350#350|; $key = q/P_{k};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|542#542|; $key = q/delta_{1}b;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|230#230|; $key = q/{figure}{centering{ $key = q/V;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|592#592|; $key = q/epsilon_{mach}=2^{-24}approx10^{-7};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|147#147|; $key = q/k,;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|482#482|; $key = q/q_{k}=alpha_{k};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|846#846|; $key = q/Wielandt;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1016#1016|; $key = q/{displaymath}nablaf(x)=left[array{{c}x_15x_2array{right]mbox{και}H_f(x)=left[array{{cc}1&00&5array{right].{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1303#1303|; $key = q/[0-epsilon]^{T};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|702#702|; $key = q/{displaymath}left[matrix{hat{z}_0crhat{z}_1crvdotshat{z}_{n-2}crhat{z}_{n-1}}rignu}_1crvdotshat{nu}_{n-2}crhat{nu}_{n-1}}right],{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1725#1725|; $key = q/beta;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|86#86|; $key = q/Manhattan;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|607#607|; $key = q/{displaymath}xlambdaapproxAx{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|954#954|; $key = q/{displaymath}y_{k+1}=y_{k}+λy_{k}h=(1+λh)y_{k},{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1477#1477|; $key = q/stiff;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1576#1576|; $key = q/{displaymath}A=left[array{{rr}3&-1-1&3array{right]:lambda=2,x=left[array{{c}11arbox{and}lambda=4,x=left[array{{r}1-1array{right]{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|908#908|; $key = q/netlib;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|138#138|; $key = q/magnitude;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1081#1081|; $key = q/y'=-y;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1399#1399|; $key = q/{displaymath}y_{k+1}-y(t_{k+1})=[y_{k}-y(t_{k})+[f(t_{k},y_{k})-f(t_{k},y(t_{k}))]h_{k}-O(h_{k}^{2}).{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1458#1458|; $key = q/{displaymath}c^2b+cs(a-d)-s^2b=0{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|927#927|; $key = q/64-bits;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|101#101|; $key = q/U-L+1;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|122#122|; $key = q/Poisson;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1636#1636|; $key = q/x-z=1.28000times10^{-1};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|181#181|; $key = q/{displaymath}f(t,x)=x_1e^{x_{2}t}+cdots+x_{n-1}e^{x_{n}t}{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|720#720|; $key = q/mathbf{if}n=1mathbf{then};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1700#1700|; $key = q/Strang;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|358#358|; $key = q/c<0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1621#1621|; $key = q/Q;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|495#495|; $key = q/Aneq0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|630#630|; $key = q/Plank;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|26#26|; $key = q/NAG;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|302#302|; $key = q/alpha_k=q_k^Tr_k;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|967#967|; $key = q/cond(A);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|981#981|; $key = q/E_{i}=u_{i}v_{i}^{T};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|988#988|; $key = q/{displaymath}u_{t}=-cu_{x},tgeq0,{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1617#1617|; $key = q/NEWTON;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1274#1274|; $key = q/y'=1slash2;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1406#1406|; $key = q/Derive;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|314#314|; $key = q/{displaymath}left[array{{rrr}2&4&-24&9&-3-2&-3&7array{right]left[array{{c}x_1x_2x_3array{right]=left[array{{r}2810array{right].{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|473#473|; $key = q/mantissa;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|92#92|; $key = q/n^{3}slash3;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|545#545|; $key = q/workstation;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|347#347|; $key = q/f_a;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1157#1157|; $key = q/x=1;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|75#75|; $key = q/1+mathbf{Inf}=mathbf{Inf};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|163#163|; $key = q/{displaymath}rule{25em}{3pt}{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|27#27|; $key = q/6.37times10^{-6};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|72#72|; $key = q/phi_{j}(x);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1578#1578|; $key = q/eigenvalueroutine;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1186#1186|; $key = q/L;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|107#107|; $key = q/r=b-ahat{x};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|225#225|; $key = q/zneq0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|378#378|; $key = q/alpha_{2};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|842#842|; $key = q/F=ma;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1386#1386|; $key = q/{displaymath}(L^{-1}AL^{-T})y=lambday{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|973#973|; $key = q/{displaymath}Δtleqfrac{Δx}{sqrt{c}}{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1611#1611|; $key = q/RUNGE-KUTTA;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1536#1536|; $key = q/robustness;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1318#1318|; $key = q/beta-1;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|120#120|; $key = q/{displaymath}y'=-y^{3}{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1490#1490|; $key = q/tilde{alpha}_{jk};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|575#575|; $key = q/{displaymath}x_n=b_nslasha_{nn},{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|417#417|; $key = q/alpha=alpha_{jk}-tilde{alpha}_{jk};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|578#578|; $key = q/{displaymath}B_1left[array{{cc}0.667&0.3330.333&0.667array{right].{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1332#1332|; $key = q/x^{2}-2x+1=0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1045#1045|; $key = q/%latex2htmlidmarker10661beta_k=parallelr_kparallel_2;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|969#969|; $key = q/10;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|43#43|; $key = q/netlib2.cs.utk.edu;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|274#274|; $key = q/{displaymath}x=y+frac{nu^Ty}{1-nu^Tz}z=left[array{{r}-122array{right]+frac{2}{1-1}{2}array{right]=left[array{{r}-740array{right]{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|602#602|; $key = q/y:{mathbb{R}{rightarrow{mathbb{R}{^n;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1362#1362|; $key = q/QRHOUSEHOLDER;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|782#782|; $key = q/{displaymath}J_2=left[array{{ccc}1&0&00&0.975&-0.2210&0.221&0.975array{right]mbo072&0.3170.072&1.776&00.317&0&-1.142array{right]{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|933#933|; $key = q/hat{f}(x)-f(x)=-0.051615,;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|82#82|; $key = q/{displaymath}Q^{T}=H_{n}cdotsH_{1}mbox{ήισοδύναμα,}Q=H_{n}^{T}cdotsH_{1}^{T},{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|779#779|; $key = q/beta^{L},;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|123#123|; $key = q/mbox{cond}(A)=infty;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|636#636|; $key = q/{displaymath}y_{k+1}=y_{k}+f(t_{k},y_{k})h_{k}+frac{f(t_k+h_k,y_k+h_kf(t_k,y_k)){f(t_k,y_k)+f(t_k+h_k,y_k+y_kf(t_k,y_k))}{2}h_k,{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1542#1542|; $key = q/M_{k}^{-1}=I+me_{k}^{T};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|442#442|; $key = q/rapprox6370km;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|30#30|; $key = q/{displaymath}|frac{fell(x)-x}{x}|leqepsilon_{mach}.{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|142#142|; $key = q/m=n;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|368#368|; $key = q/10^{-t};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|48#48|; $key = q/{displaymath}(A-UV^{T})^{-1}=A^{-1}+A^{-1}U(I-V^{T}A^{-1}U)^{-1}V^{T}A^{-1}{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|593#593|; $key = q/mathbf{MATLAB}[V,D]=eig(A);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1003#1003|; $key = q/x=18slash3=6;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|551#551|; $key = q/r_{kj}=q_{k}^{T}alpha_{j};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|858#858|; $key = q/r=1;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1054#1054|; $key = q/seed;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1755#1755|; $key = q/Lau;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|286#286|; $key = q/B;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|554#554|; $key = q/D=diag(d_{i};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|646#646|; $key = q/Kahan;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1021#1021|; $key = q/{displaymath}f:{mathbb{R}{^{n}rightarrow{mathbb{R}{^{2}(mbox{διανυσματική}).{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1030#1030|; $key = q/{displaymath}frac{1}{1-mbox{λ}h}=1+λh+(λh)^{2}+...{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1500#1500|; $key = q/(-infty,0);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1512#1512|; $key = q/[L,U];MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|87#87|; $key = q/A{z}=u;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|598#598|; $key = q/{displaymath}f(x)=0.5-x^{-x^{2}}.{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1264#1264|; $key = q/J=-100;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1518#1518|; $key = q/e_{k};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|441#441|; $key = q/{displaymath}y_k=Ax_{k-1}{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|942#942|; $key = q/A^{T}A;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|627#627|; $key = q/{displaymath}J_3=left[array{{ccc}0.998&0&-0.0700&1&00.070&0&0.998array{right]mbo072&00.072&1.776&0.0050&0.005&-1.164array{right]{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|934#934|; $key = q/u_{x}=duslashdx,u_{xy}=d^{2}uslashdxdy;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1549#1549|; $key = q/x=1.9;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1032#1032|; $key = q/y'=dyslashdt;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1364#1364|; $key = q/CholeskyL;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1664#1664|; $key = q/alpha_{ij};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|827#827|; $key = q/A{y}=b;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|600#600|; $key = q/cache;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|342#342|; $key = q/{figure}{centering{ $key = q/x_0=[51]^T;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1291#1291|; $key = q/{displaymath}%latex2htmlidmarker4946parallelDeltaxparallelleqparallelA^{-1}paralcdotparallelEparallelcdotparallelhat{x}parallel,{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|687#687|; $key = q/t=cslashs=alpha_{2}slashalpha_{1};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|811#811|; $key = q/y_{k}+k_{1};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1545#1545|; $key = q/%latex2htmlidmarker5946parallelA+BparallelleqparallelAparallel+parallelBparallel;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|632#632|; $key = q/UFL=(1.00)_{2}times2^{-1}=(0.5)_{10};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|129#129|; $key = q/{displaymath}left[array{{c}ROarray{right]xapproxQ^{T}b.{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|781#781|; $key = q/quadratic;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1058#1058|; $key = q/f'(x_k);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1135#1135|; $key = q/Mhslash2;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|212#212|; $key = q/mathbf{FMM:};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|236#236|; $key = q/Schur;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1004#1004|; $key = q/{displaymath}Α=left[array{{cc}epsilon&11&1array{right],{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|501#501|; $key = q/{displaymath}array{{c}M_1A=left[array{{rcc}1&0&0-2&1&01&0&1array{right]left[arraray{right]=left[array{{r}2412array{right].array{{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|474#474|; $key = q/Jacobi;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|7#7|; $key = q/Nash;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|255#255|; $key = q/NumericalRecipes;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|276#276|; $key = q/Fletcher;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1340#1340|; $key = q/{displaymath}A=left[array{{cc}0&11&0array{right]{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|499#499|; $key = q/{displaymath}2A^TAx-2A^Tb=0{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|732#732|; $key = q/{displaymath}1+tfrac{(a-d)}{b}-t^2=0{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|929#929|; $key = q/{displaymath}mbox{Orthogonal}:qquadleft[array{{cc}0&11&0array{right],left[array{thogonal}:qquadleft[array{{cc}1&11&2array{right]{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|917#917|; $key = q/{displaymath}A_0=left[array{{ccc}1&0&20&2&12&1&1array{right]{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|930#930|; $key = q/{displaymath}left[matrix{a_1&a_2cr0&-a_1-a_2^2slasha_1}right]left[matrix{ccrs}right]=left[matrix{a&-alphaa_2slasha_1}right].{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|807#807|; $key = q/{displaymath}L_{k}=O(h_{k}^{p+1}).{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1451#1451|; $key = q/{displaymath}u_{x}approxfrac{u(t,x)-u(t,x-Δx)}{Δx}.{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1631#1631|; $key = q/m_{i}=alpha_{i}slashalpha_{k},i=k+1,ldots,n;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|435#435|; $key = q/{displaymath}left[array{{cc}1&0-a_2slasha_1&1array{right]left[array{{c}a_1a_2array{right]=left[array{{c}a_10array{right].{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|430#430|; $key = q/TAYLOR;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1524#1524|; $key = q/Omitab;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|894#894|; $key = q/y[0]=x[0];MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1701#1701|; $key = q/x';MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1394#1394|; $key = q/{displaymath}%latex2htmlidmarker4928parallelAparallelcdotparallelA^{-1}parallel=rallelAxparallel}{parallelxparallel}right)^{-1},{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|637#637|; $key = q/alpha_0=frac{1}{3};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1346#1346|; $key = q/{displaymath}A=left[array{{rrr}1&2&34&5&67&8&910&11&12array{right]{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|976#976|; $key = q/f:{mathbb{R}{^{n+1}rightarrow{mathbb{R}{;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|713#713|; $key = q/{displaymath}epsilon_{mach}=beta^{1-t},{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|140#140|; $key = q/alpha=pm||a||_{2}=pm3;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|768#768|; $key = q/{displaymath}%latex2htmlidmarker4950frac{parallelx^{prime}parallel}{parallelxparllelb^{prime}parallel}{parallelbparallel}right).{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|696#696|; $key = q/3times3;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1661#1661|; $key = q/Household;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|786#786|; $key = q/hat{z};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1722#1722|; $key = q/Mathematics;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|317#317|; $key = q/{displaymath}b-{fl}(atimesfl(hat{x}))=[b-ahat{x}(1+delta_{1})](1+delta_{2})=[r-d1}delta_{2}ahat{x}approxr+delta_{2}r-delta_{1}b,{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|228#228|; $key = q/MATLAB;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|162#162|; $key = q/f''(x)neq0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1219#1219|; $key = q/graph;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|330#330|; $key = q/J=λ;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1481#1481|; $key = q/m=α+(b-α)slash2=2;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1077#1077|; $key = q/100%;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|46#46|; $key = q/{displaymath}LU=left[array{{cc}1&0epsilon&1array{right]left[array{{cc}1&10&1arra{right]=left[array{{cc}1&1epsilon&1array{right],{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|509#509|; $key = q/x^{3}+6x^{2}+11x-6=0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1039#1039|; $key = q/Polak;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1342#1342|; $key = q/ΙΕΕΕ;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|135#135|; $key = q/{displaymath}A-lambdaI=left[array{{ccc}U_{11}&u&U_{13}0&0&nu^T0&0&U_{33}array{right]{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|922#922|; $key = q/{displaymath}D_2=left[matrix{1&0cr0&i}right].{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1690#1690|; $key = q/Μb,;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|471#471|; $key = q/D_{nslash2}=diag(1,ldots,,w^{nslash2});MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1696#1696|; $key = q/y_{1};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|898#898|; $key = q/StanfordUniversity;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|293#293|; $key = q/A-uv^{T};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|596#596|; $key = q/{displaymath}u_j^{k+1}=2u_j^k-u_j^{k-1}+cfrac{(Δt)^2}{(Δx)^2}(u_{j+1}^k-2u_j^k+u_{j-1}^k).{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1606#1606|; $key = q/2epsilon|f(x)|slashh.;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|209#209|; $key = q/{displaymath}%latex2htmlidmarker4943frac{parallelDeltaxparallel}{parallelxparallfrac{parallelDeltabparallel}{parallelbparallel}.{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|684#684|; $key = q/{displaymath}u(0,x)=f(x),u(t,0)=α,u(t,1)=β.{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1588#1588|; $key = q/{displaymath}[]=[],{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1174#1174|; $key = q/{displaymath}%latex2htmlidmarker4920parallelxparallel_2=left(sum_{j=1}^n|x_i|^2right)^{1slash2},{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|608#608|; $key = q/n+1;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1282#1282|; $key = q/Frobenius;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|992#992|; $key = q/B-splines;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1584#1584|; $key = q/{displaymath}u_{x}approxfrac{u(t,x+Δx)-u(t,x-Δx)}{2Δx},{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1630#1630|; $key = q/G_{1};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|832#832|; $key = q/y_k=nablaf(x_{k+1})-nablaf(x_k);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1324#1324|; $key = q/g_0=nablaf(x_0);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1333#1333|; $key = q/netlib@οrnl.gov;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|270#270|; $key = q/n^{2};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|546#546|; $key = q/x;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|33#33|; $key = q/f(x+h)-f(x)approxhf^prime(x).;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|54#54|; $key = q/Malcom;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|239#239|; $key = q/{displaymath}left{H_f(x)right}_{ij}=frac{d^{2}f(x)}{dx_{i}dx_{j}},{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1209#1209|; $key = q/Nelder;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1280#1280|; $key = q/{displaymath}PAQ=LU,{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|494#494|; $key = q/{displaymath}%latex2htmlidmarker4924parallelAparallel=max_{xneq0}frac{parallelAxparallel}{parallelxparallel}.{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|623#623|; $key = q/r_{ij};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|854#854|; $key = q/{displaymath}nu_{1}=left[array{{c}11111array{right]-left[array{{c}-2.2360000array{right]=left[array{{c}3.2361111array{right].{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|788#788|; $key = q/{displaymath}left[array{{ccc}0&1&01&0&00&0&1array{right]left[array{{rrr}1&0&0fra}&-frac{1}{3}&11&0&0-frac{1}{2}&1&0array{right],{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|521#521|; $key = q/{displaymath}x=left[array{{c}0.0860.4001.429array{right]{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|743#743|; $key = q/{displaymath}nablaf(x_k)^Ts_k<0{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1309#1309|; $key = q/{displaymath}sigma=left[frac{1}{n-1}left(sum_{i=1}^nx_i^2-nbar{x}^2right)right]^{1slash2}{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|223#223|; $key = q/f(hat{x})=hat{f}(x);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|74#74|; $key = q/{displaymath}fl(A^TA)=left[array{{cc}1&11&1array{right]{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|746#746|; $key = q/k+1;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1464#1464|; $key = q/{displaymath}Ay_k=x_{k-1}{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|951#951|; $key = q/HouseholdQRfactorization;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|783#783|; $key = q/Galerkin;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1656#1656|; $key = q/{displaymath}g(x)=x-f(x)slashf'(x).{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1130#1130|; $key = q/Muller;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1153#1153|; $key = q/B_0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1321#1321|; $key = q/{displaymath}f(x)=2.718282,qquadhat{f}(x)=2.666667,{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|80#80|; $key = q/s;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|802#802|; $key = q/neq0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|983#983|; $key = q/{displaymath}left[array{{cc}1&00&5array{right]s_0=left[array{{c}-5-5array{right],{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1304#1304|; $key = q/p(t);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1726#1726|; $key = q/http:slashslashwww.netlib.org;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|267#267|; $key = q/mathbf{MATLABqr[Q,R]=qr(A)};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|891#891|; $key = q/{displaymath}%latex2htmlidmarker4947frac{parallelDeltaxparallel}{parallelhat{x}pd}(A)frac{parallelEparallel}{parallelAparallel}.{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|688#688|; $key = q/a_0=frac{1}{3};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1294#1294|; $key = q/steepfronts;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1622#1622|; $key = q/{displaymath}min_xsum_{i=1}^m(y_i-f(t_i,x))^2{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|714#714|; $key = q/n=4;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1667#1667|; $key = q/{eqnarraystar}2x_1+3x_2=b_1,5x_1+4x_2=b_2,{eqnarraystar};MSF=1.6;LFS=11;TAGS=R;AAT/;
$cached_env_img{$key} = q|388#388|; $key = q/x_2=1.236;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1257#1257|; $key = q/{displaymath}Ax=lambdaBx{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|971#971|; $key = q/%latex2htmlidmarker6040parallelEparallel;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|663#663|; $key = q/dfslashdy;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1398#1398|; $key = q/{displaymath}frac{y_{k+1}-y_k}{h_k}=f(t_k,y_k),{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1423#1423|; $key = q/{displaymath}%latex2htmlidmarker7458r_{2,2}=parallela_2parallel_2=1.581,q_2=a_2st[matrix{-0.632cr-0.316cr0cr0.316cr0.632}right].{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|864#864|; $key = q/{displaymath}H_{n}cdotsH_{1}A=left[array{{c}ROarray{right],{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|776#776|; $key = q/{displaymath}%latex2htmlidmarker5888parallelxparallel_{infty}=max_i|x_i|,{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|610#610|; $key = q/%latex2htmlidmarker5942parallelgammaAparallel=|gamma|cdotparallelAparallel;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|631#631|; $key = q/{displaymath}x=(AP)^{-1}b=P^{-1}A^{-1}b=P^T(A^{-1}b).{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|405#405|; $key = q/mathit{k};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|530#530|; $key = q/n;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|366#366|; $key = q/A_{t}=A+tE;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|689#689|; $key = q/Gauss,;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|480#480|; $key = q/Mathcad;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|310#310|; $key = q/mbox{τ}=(sqrt{5}-1)slash2;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1240#1240|; $key = q/{displaymath}left[array{{cc}I&AA^T&0array{right]left[array{{c}rxarray{right]=left[array{{c}b0array{right]{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|749#749|; $key = q/{displaymath}u^{(n)}=f(t,u,u',...,u^{(n-1)}){displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1380#1380|; $key = q/{displaymath}A=left[matrix{0.447&-0.632&0.535cr0.447&-0.316&0.267cr0.447&0&-0.53x{2.236&0&1.118cr0&1.581&0cr0&0&0.935}right]=QR.{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|867#867|; $key = q/y(0)=y_{0};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1475#1475|; $key = q/{displaymath}y_{k+1}=y_{k}+1slash2(k_{1}+k_{2}),{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1543#1543|; $key = q/{displaymath}epsilon_{mach}=frac{1}{2}beta^{1-t}.{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|141#141|; $key = q/Δt;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1600#1600|; $key = q/x_{0}=x_{n};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1714#1714|; $key = q/t_{k+1}=t_{k}+h_{k};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1422#1422|; $key = q/ODEy'=-y;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1444#1444|; $key = q/Μ_1;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|456#456|; $key = q/1slash0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|156#156|; $key = q/Deltax=A^{-1}Deltab;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|680#680|; $key = q/{multicols}%latex2htmlidmarker1549{2}theorem_type{[provlhmata][provlhmata][chaptοτικότηταηδυναμικήρουτίνασαςθυσιάζει;theorem_type{{multicols};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|361#361|; $key = q/nu;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|774#774|; $key = q/(0.01)_2times2^{-1}=(0.125)_{10};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|154#154|; $key = q/f(t,y)=y;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1373#1373|; $key = q/{displaymath}Halpha=.{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|761#761|; $key = q/i;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|824#824|; $key = q/{displaymath}A^{prime}(t)x(t)+A(t)x^{prime}(t)=b^{prime(t)},{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|693#693|; $key = q/{displaymath}u_{1}=cu_{xx},0leqxleq1,tgeq0,{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1560#1560|; $key = q/1.58;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|70#70|; $key = q/H_f;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1307#1307|; $key = q/%latex2htmlidmarker6088mbox{cond}(A)=parallelAparallelcdotparallelA^{-1}parallel,;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|683#683|; $key = q/{multicols}{2}{par{textbf{4.1.}description{item[(α)]Εφαρμόστετηνεκθετικήμέθοδογιτρειςευθείεςσεέναμοναδικόγράφημα.description{{par{{multicols};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1027#1027|; $key = q/Ly=Pb;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|493#493|; $key = q/Runge-Kutta;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|20#20|; $key = q/f(t_i,x);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|715#715|; $key = q/f(x^*);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1224#1224|; $key = q/y_{i}(0)=f(x_{i}),i=1,...,n;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1570#1570|; $key = q/Kahaner;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|254#254|; $key = q/{displaymath}f(x)=0.{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1028#1028|; $key = q/Νewton;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1385#1385|; $key = q/{displaymath}y''=f_{t}(t,y)+f(t,y)f_{y}(t,y)=-2y^{2}+(-2ty^{2})(-4t)=2y^{2}(4t^{2}-1).{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1526#1526|; $key = q/f(x+h)-f(x);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|208#208|; $key = q/0.01;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|44#44|; $key = q/U=MA;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|466#466|; $key = q/Gram-Schmidt;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|6#6|; $key = q/{displaymath}Q^{T}b=left[matrix{0.447&0.447&0.447&0.447&0.447cr-0.632&-0.316&0&0.0}right]=left[matrix{1.789cr0.632cr1.336}right]{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|868#868|; $key = q/gamma;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|385#385|; $key = q/QR;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|8#8|; $key = q/IEEEDP;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|110#110|; $key = q/ea15;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1008#1008|; $key = q/B_k;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1326#1326|; $key = q/mathbf{dtoa};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|136#136|; $key = q/{displaymath}mbox{Hermitian}:qquadleft[array{{cc}1&1+i1-i&2array{right],qquadmbotian}:qquadleft[array{{cc}1&1+i1+i&2array{right]{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|916#916|; $key = q/f(2)=0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1102#1102|; $key = q/(-οο,0);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1504#1504|; $key = q/p[k]=x[2k]mbox{{διαχωρισμόςσεάρτιεςκαι};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1703#1703|; $key = q/{displaymath}nu_{2}={displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|792#792|; $key = q/RANDOMNESS;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|24#24|; $key = q/{multicols}%latex2htmlidmarker7069{2}{par{theorem_type{[askhseis][askhseis][chapγιαναδείξτετουςκαθρέφτες.description{theorem_type{{multicols};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|903#903|; $key = q/delta_2:0leqdelta_2leqepsilon_{mach}.;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|229#229|; $key = q/http:slashslashwww.vni.com;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|251#251|; $key = q/E_{i}x;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|991#991|; $key = q/{displaymath}left[matrix{z_0crz_1crvdotsz_{n-2}crz_{n-1}}right]=left[matrix{u_0&trix{nu_0crnu_1crvdotsnu_{n-2}crnu_{n-1}}right],{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1721#1721|; $key = q/{displaymath}frac{2epsilon|f(x)|}{h}+frac{Mh}{2},{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|215#215|; $key = q/s_0=-g_0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1334#1334|; $key = q/mathbf{for}k=1mathbf{to}n;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|845#845|; $key = q/IBMmainframe;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|112#112|; $key = q/interactivescientificcomputingenvironment;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|327#327|; $key = q/{displaymath}I(f)=int_{alpha}^{b}f(x)dx=sum^{n}_{i=1}w_if(x_i)+R_n.{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1360#1360|; $key = q/φ_{jk}(x);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1742#1742|; $key = q/y=A^{-1}b;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|588#588|; $key = q/{displaymath}M_2P_2Μ_1P_1Ax=left[array{{rrr}4&9&-30&frac{3}{2}&frac{11}{2}0&0&frrray{{r}814frac{8}{3}array{right]=M_2P_2M_1P_1b.{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|519#519|; $key = q/%latex2htmlidmarker12199MeanValueTheorem;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1110#1110|; $key = q/F_{4};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1692#1692|; $key = q/stencil;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1655#1655|; $key = q/{displaymath}left[array{{rrr}1&0&frac{3}{2}0&1&-frac{1}{4}0&0&1array{right]left[&frac{5}{4}&-frac{1}{4}0&0&4&3&-1&1array{right],{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|564#564|; $key = q/mathbf{[U,S,V]=svd(A)};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1006#1006|; $key = q/{displaymath}%latex2htmlidmarker10470parallelAparallel_2=max_{xneq0}frac{parallelAxparallel_2}{parallelxparallel_2}=sigma_{max}{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|979#979|; $key = q/{displaymath}s_1=-g_1+beta_0s_0=left[array{{c}3.333-3.333array{right]+0.444left[ray{right]=left[array{{c}-5.5561.111array{right]{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1349#1349|; $key = q/{displaymath}%latex2htmlidmarker4941b=AxRightarrowparallelbparallelleqparallelAparallelcdotparallelxparallel,{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|681#681|; $key = q/{displaymath}nu=alpha-alphae_{1}=left[array{{c}212array{right]-alphaleft[array{{2array{right]-left[array{{c}alpha12array{right],{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|767#767|; $key = q/{displaymath}A=left[array{{cc}1&00&2array{right]:lambda=1,x=left[array{{c}10arrambox{and}lambda=2,x=left[array{{c}01array{right]{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|906#906|; $key = q/{displaymath}lambda=frac{x^TAx}{x^Tx}{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|956#956|; $key = q/Α;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|369#369|; $key = q/{displaymath}x=pmleft(d_0+frac{d_1}{beta}+frac{d_2}{beta^2}+cdots+frac{d_{t-1}}{beta^{t-1}}right)beta^{e},{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|88#88|; $key = q/mathbf{netlib:};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|266#266|; $key = q/Ux=y;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|470#470|; $key = q/SP;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|104#104|; $key = q/{displaymath}array{{c}M_2M_1A=left[array{{crc}1&0&00&1&00&-1&1array{right]left[array{right]=left[array{{r}248array{right].array{{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|475#475|; $key = q/{displaymath}u_{0}(x)=left{array{{ll}1ifxleq00ifx>0array{right..{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1628#1628|; $key = q/u_{y};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1642#1642|; $key = q/pislash2;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|61#61|; $key = q/{displaymath}R=left[matrix{1.1997&0.4527cr0&0.0002}right]{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|874#874|; $key = q/y_0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1369#1369|; $key = q/t_{0}=0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1427#1427|; $key = q/FRONT;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1626#1626|; $key = q/{multicols}%latex2htmlidmarker16887{2}{par{theorem_type{[ervthseis][ervthseis][cισμούπουεφαρμόζεταιστακυματάκια;theorem_type{{par{{multicols};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1748#1748|; $key = q/{displaymath}x_{k+1}=x_k+S_k{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1301#1301|; $key = q/Algol;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|284#284|; $key = q/vanWinjgaarden;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1180#1180|; $key = q/(RAM);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|343#343|; $key = q/1slashepsilon,;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|701#701|; $key = q/subscripts;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1193#1193|; $key = q/y(t_0)=y_0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1375#1375|; $key = q/{displaymath}x^Txlambda=x^TAx{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|955#955|; $key = q/U;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|108#108|; $key = q/y(t)=ce^t;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1374#1374|; $key = q/{displaymath}nu=(alpha-alphae_{1})frac{nu^{T}nu}{2nu^{T}alpha}.{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|763#763|; $key = q/LUx=b;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|468#468|; $key = q/x_2=2;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|424#424|; $key = q/x^{*};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1048#1048|; $key = q/pi;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|31#31|; $key = q/{displaymath}nu_{3}={displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|797#797|; $key = q/fixed-points;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1098#1098|; $key = q/{displaymath}A=LU,{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|467#467|; $key = q/1+(epsilon+epsilon)>1;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|179#179|; $key = q/{displaymath}A=4pir^2{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|28#28|; $key = q/{displaymath}H_{1}A=left[right],H_{1}B=left[array{{c}-1.789-0.362-0.862-0.3621.128array{right].{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|790#790|; $key = q/ntimesn;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|371#371|; $key = q/{displaymath}G_{1}b=left[matrix{1&0&0&0&0cr0&1&0&0&0cr0&0&1&0&0cr0&0&0&0.707&0.7=left[matrix{1.0cr0.5cr0.0cr1.768cr1.061}right].{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|834#834|; $key = q/{displaymath}%latex2htmlidmarker7455r_{1,1}=parallela_1parallel_2=2.236,q_1=a_1smatrix{0.477cr0.477cr0.477cr0.477cr0.477}right].{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|861#861|; $key = q/mainframes;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|97#97|; $key = q/[11]^{T};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|703#703|; $key = q/{displaymath}%latex2htmlidmarker10449parallely_kparallel_inftyrightarrow|lambda_~~x_krightarrowu_1slashparallelu_1parallel_infty{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|944#944|; $key = q/{displaymath}f(x)=x^2-4sin(x)=0{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1141#1141|; $key = q/rounding;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|131#131|; $key = q/interpreted;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1139#1139|; $key = q/y(t)=1+t+ce^{-100t};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1514#1514|; $key = q/L_kL_j;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|463#463|; $key = q/{displaymath}x=frac{-bpmsqrt{b^{2}-4ac}}{2a}.{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|192#192|; $key = q/h=h_{k};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1456#1456|; $key = q/P;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|398#398|; $key = q/{displaymath}left[matrix{1&0&0&0&0cr0&c&0&s&0cr0&0&1&0&0cr0&-s&0&c&0cr0&0&0&0&1}ight]=left[matrix{a_1cralphacra_3cr0cra_5}right]{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|826#826|; $key = q/MATrixLABoratory;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|300#300|; $key = q/9times9;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1662#1662|; $key = q/{displaymath}A=USigmaV^T=sigma_1E_1+sigma_2E_2+cdots+sigma_nE_n{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|987#987|; $key = q/Η_f(x);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1210#1210|; $key = q/mathbf{STELIOSNOTES:}pinakas;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1572#1572|; $key = q/roundtowardzero;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|134#134|; $key = q/q_{1};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|843#843|; $key = q/%latex2htmlidmarker6032parallelhat{x}-xparallel=0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|658#658|; $key = q/{displaymath}A=Qleft[array{{c}ROarray{right].{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|780#780|; $key = q/{calO}(n^{3});MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|582#582|; $key = q/NR;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|323#323|; $key = q/A^{+};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|984#984|; $key = q/n^{3}slash2;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|560#560|; $key = q/{displaymath}Α=left[array{{rrr}frac{1}{2}&-frac{1}{3}&11&0&0-frac{1}{2}&1&0array{3}{2}&frac{11}{2}0&0&frac{4}{3}array{right]=LU.{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|522#522|; $key = q/%latex2htmlidmarker5996parallelA^{-1}parallel;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|649#649|; $key = q/{displaymath}f(x+h)=f(x)+f'(x)h+frac{f''(x)}{2}h^{2}+O(h^{3}),{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1212#1212|; $key = q/x=0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1629#1629|; $key = q/SAS;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|895#895|; $key = q/t=t_{k};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1455#1455|; $key = q/phi_{jk}(x)=phi(2^{j}x-k);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1740#1740|; $key = q/{displaymath}0 $key = q/b=x_2;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1250#1250|; $key = q/y(0.5)=exp(0.5)~=1.649;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1434#1434|; $key = q/{displaymath}f(t_{k},y_{k})-f(t_{k},y(t_{k}))=J(ξ)(y_{k}-y(t_{k})){displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1463#1463|; $key = q/left[Abright]USigmaV^{T};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|995#995|; $key = q/|alpha_{1}|>|alpha_{2}|;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|810#810|; $key = q/loops;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|328#328|; $key = q/i $key = q/frac{3}{2};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|515#515|; $key = q/{displaymath}DeltaK=frac{|Sigmachivarepsilontauiotakappaacute{eta}alphalambdalam[f(hat{x})-f(x)]slashf(x)|}{|(hat{x}-x)slashx|},{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|52#52|; $key = q/%latex2htmlidmarker5948parallelABparallelleqparallelAparallelcdotparallelΒparallel;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|633#633|; $key = q/{displaymath}Ax=b,{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|362#362|; $key = q/Α^{-1};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|380#380|; $key = q/%latex2htmlidmarker6008parallelA^{-1}parallel;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|653#653|; $key = q/s[k]=x[2k+1]mbox{περιττέςυπακολουθίες}};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1704#1704|; $key = q/log_{2}n;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1709#1709|; $key = q/e^{x}+1=0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1036#1036|; $key = q/f(x^{*})=0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1213#1213|; $key = q/{displaymath}s=1slashsqrt{1+tau^{2}},c=scdottau.{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|815#815|; $key = q/{displaymath}nablaf(x)=left[array{{c}x_15x_2array{right].{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1329#1329|; $key = q/%latex2htmlidmarker6034parallelrparallel=0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|659#659|; $key = q/x_{1};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|675#675|; $key = q/Ribiere;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1343#1343|; $key = q/{displaymath}left[array{{ccccccc}1&cdots&0&-m_1&0&cdots&0vdots&ddots&vdots&vdotsht]=left[array{{c}0vdots0a_k0vdots0array{right],{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|557#557|; $key = q/{displaymath}nu=alpha-alphae_{1}.{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|764#764|; $key = q/F;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1388#1388|; $key = q/FIXED-POINT;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1095#1095|; $key = q/alpha=[24-1]^{T};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|450#450|; $key = q/mggn;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|884#884|; $key = q/x=u;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1169#1169|; $key = q/ODES;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|19#19|; $key = q/Β_κ;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1316#1316|; $key = q/Α=USigmaV^T;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|978#978|; $key = q/xmathbf{flop}y=(xmathbf{op}y);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|176#176|; $key = q/f'';MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|217#217|; $key = q/epsilon_{mach},;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|182#182|; $key = q/InternationalMathematicalandStatisticalLibraries;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|246#246|; $key = q/(b-α)slash2^k;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1083#1083|; $key = q/O(n^{3});MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1659#1659|; $key = q/{displaymath}x=-frac{log(1-y)}{lambda}.{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1763#1763|; $key = q/y=6.35782times10^{-1};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|168#168|; $key = q/fixedpoint;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1086#1086|; $key = q/{displaymath}y_{0}=(x_{0}+w^{0}x_{2})+w^{0}(x_{1}+w^{0}x^{3}),y_{1}=(x_{0}-w^{0}_{3}=(x_{0}-w^{0}x_{2})+w^{3}(x_{1}-w^{0}x^{3}).{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1679#1679|; $key = q/mathbf{MATLAB};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1002#1002|; $key = q/y_{k};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1446#1446|; $key = q/F(x)=0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1041#1041|; $key = q/b(t)=b+tDeltambox{b};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|690#690|; $key = q/{displaymath}y_{1}=y_{o}+y'_{o}h+frac{y''_0}{2}h^2=1+0-0.0625=0.9375.{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1530#1530|; $key = q/A;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|363#363|; $key = q/z=A^{-1}u;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|586#586|; $key = q/GRAM-SCHMIDTQR;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|860#860|; $key = q/{displaymath}f_{t}+ff_{y}=frac{f(t+h,y+hf)-f(t,y)}{h}+O(h^2).{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1541#1541|; $key = q/{displaymath}AT=left[array{{rr}3&-1-1&3array{right]left[array{{cr}1&11&-1array{r{right]left[array{{cc}2&00&4array{right]=TLambda{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|920#920|; $key = q/Deltax=1slash(n+1);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1564#1564|; $key = q/-epsilon;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|506#506|; $key = q/qrdcmp;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|888#888|; $key = q/x_3;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|425#425|; $key = q/x=A^{-1}b;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|381#381|; $key = q/{displaymath}x=sum_{sigma_{i}neq0}^{}frac{u_i^Tb}{sigma_i}nu_i{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|982#982|; $key = q/M_{1}Ax=M_{1}b,;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|458#458|; $key = q/HarwellLaboratory;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|243#243|; $key = q/f_1=f(x_1);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1242#1242|; $key = q/M_{1};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|457#457|; $key = q/{displaymath}P=P_{n-1}ldotsP_1{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|491#491|; $key = q/{displaymath}x=left[array{{c}gamma(4-2gamma)slash3)array{right]{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|395#395|; $key = q/1%;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|45#45|; $key = q/PA=LU=hat{L}hat{U};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|534#534|; $key = q/Lanczos;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|11#11|; $key = q/{figure}{centering{ $key = q/f(x)=0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1043#1043|; $key = q/s_k;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1300#1300|; $key = q/{displaymath}Hx=Biggl(I-2frac{nunu^{T}}{nu^{T}nu}Biggr)x=x-Biggl(2frac{nu^{T}x}{nu^{T}nu}Biggr)nu,{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|775#775|; $key = q/H_f(x_k);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1308#1308|; $key = q/{displaymath}x_{k+1}=x_k-a_knablaf(x_k),{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1285#1285|; $key = q/{displaymath}beta_{k+1}=((g_{k+1}-g_k)^Tg_{k+1})slash(g_{k}^{T}g_k).{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1344#1344|; $key = q/VisualNumericsInc.;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|247#247|; $key = q/{displaymath}y_k=left(frac{1}{1-mbox{λ}h}right)^ky_0.{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1498#1498|; $key = q/{displaymath}Ax=left[array{{crl}1&-1.0&1.01&-0.5&0.251&0.0&0.01&0.5&0.251&1.0&1.pproxleft[array{{l}1.00.50.00.52.0array{right]=b{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|726#726|; $key = q/{displaymath}r=b-Ax{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|730#730|; $key = q/rank(A)=n;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|871#871|; $key = q/{displaymath}r=b-Ahat{x}.{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|657#657|; $key = q/x_{k+1}=x_k+alpha+alpha_ks_k,mbox{όπουτο}alpha_kmbox{καθορίζεταιαπόμίααναζήτησηευθείας}.;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1335#1335|; $key = q/f_1=0.074;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1258#1258|; $key = q/nu_{2};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|791#791|; $key = q/1+λh;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1482#1482|; $key = q/datafitting;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|784#784|; $key = q/{displaymath}x=left[array{{c}0.0860.4001.429array{right].{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|870#870|; $key = q/userinterface;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|232#232|; $key = q/X-Windows;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|333#333|; $key = q/H_{2};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|793#793|; $key = q/7;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|149#149|; $key = q/{displaymath}%latex2htmlidmarker4951frac{parallelhat{x}-xparallel}{parallelxparallel}approx~mbox{cond}(A)epsilon_{mach}.{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|697#697|; $key = q/2x2;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1654#1654|; $key = q/http:slashslashgams.nist.gov;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|235#235|; $key = q/mneqn;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|370#370|; $key = q/{displaymath}cos(1.57079)=0.63267949times10^{-5},{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|68#68|; $key = q/alpha_{1}neq0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|429#429|; $key = q/x=g(x);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1094#1094|; $key = q/{displaymath}u_{j}^{k+1}=u_{j}^{k}+cfrac{Δt}{2(Δx)^2}(u_{j+1}^{k+1}-2u_{j}^{k+1}{j-1}^{k+1}+u_{j+1}^{k}-2u_{j}^{k}+u_{j-1}^{k}),{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1615#1615|; $key = q/3times2;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|872#872|; $key = q/{displaymath}x_{k+1}-frac{x^2_k-4sin(x_k)}{2x_k-4cos(x_k)}.{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1126#1126|; $key = q/rank;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|572#572|; $key = q/{displaymath}u_{xx}+u_{yy}=0,{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1644#1644|; $key = q/Cray;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|102#102|; $key = q/C<1;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1055#1055|; $key = q/"symbolicmath";MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|320#320|; $key = q/2;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|612#612|; $key = q/{displaymath}left[array{{cc}1&00&epsilonarray{right]left[array{{c}x_1x_2array{right]=left[array{{c}1epsilonarray{right]{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|700#700|; $key = q/A^{-1};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|547#547|; $key = q/IDL;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|309#309|; $key = q/Ay=b;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|587#587|; $key = q/{displaymath}%latex2htmlidmarker4921parallelxparallel_1geqparallelxparallel_2geqparallelxparallel_{infty}.{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|616#616|; $key = q/{displaymath}left[array{{ccc}0&0&11&0&00&1&0array{right]left[array{{c}b_1b_2b_3array{right]=left[array{{c}b_3b_1b_2array{right].{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|399#399|; $key = q/{displaymath}Az_1=r_1{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|706#706|; $key = q/Safeguarded;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|13#13|; $key = q/{displaymath}u_{t}=-u_{x}{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1627#1627|; $key = q/γ;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1190#1190|; $key = q/fit;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1152#1152|; $key = q/(frac{Apiacute{o}lambdaupsilontauosigmaphiacute{alpha}lambdamualpha}{Αlambdaetathetaacute{eta}sigmatauiotamuacute{eta}});MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|40#40|; $key = q/i=1,ldots,n;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|559#559|; $key = q/companion;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1185#1185|; $key = q/Legendre;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1585#1585|; $key = q/{displaymath}y(t)=sum^n_{i=1}alpha_iu_ie^{l_it}{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1414#1414|; $key = q/Givens;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|4#4|; $key = q/epsilon_{mach}b;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|231#231|; $key = q/xneq0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|619#619|; $key = q/m_{i};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|437#437|; $key = q/{displaymath}y_{k+1}=y_{k}+frac{f(t_k,y_k)+f(t_{k+1}y_{k+1}}{2}h_k.{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1506#1506|; $key = q/{displaymath}alpha=max_{i,j}|a_{ij}|slash1000{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|751#751|; $key = q/{displaymath}1971.605916και0.05077069387.{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|200#200|; $key = q/y(0.5)=0.8;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1535#1535|; $key = q/w;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1175#1175|; $key = q/%latex2htmlidmarker8209cross-correlations;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|900#900|; $key = q/b=-98.78;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|198#198|; $key = q/(x_2,b];MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1233#1233|; $key = q/ΤαχείςΜετασχηματισμούςFourier;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|265#265|; $key = q/man;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|332#332|; $key = q/Μ_k=Ι;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|484#484|; $key = q/a=2^{-j};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1738#1738|; $key = q/MP;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|351#351|; $key = q/Δx=1slash(n+1);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1593#1593|; $key = q/x_i;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1203#1203|; $key = q/FFT;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|22#22|; $key = q/{displaymath}f'(x)approxfrac{f(x+h)-f(x)}{h}.{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|203#203|; $key = q/v_{n+1,n+1}neq0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|997#997|; $key = q/2π;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1666#1666|; $key = q/E_{k+1};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1459#1459|; $key = q/delta_{1},0leqdeltaleqepsilon_{mach};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|227#227|; $key = q/{displaymath}LleqeleqU.{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|90#90|; $key = q/{displaymath}f(x)=0.5-xe^{-x^{2}}.{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1254#1254|; $key = q/Cooley;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1746#1746|; $key = q/n=5,i=2,j=4;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|825#825|; $key = q/%latex2htmlidmarker5994parallelAparallel;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|648#648|; $key = q/{displaymath}c=frac{a_1}{sqrt{a_1^2+a_2^2}}=frac{4}{5}=0.8mbox{και}s=frac{a_2}{sqrt{a_1^2+a_2^2}}=frac{3}{5}=0.6,{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|818#818|; $key = q/ΜΑx=Mb;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|396#396|; $key = q/p(x)slash(x-x_1);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1184#1184|; $key = q/β=3;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1076#1076|; $key = q/r;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|29#29|; $key = q/{displaymath}%latex2htmlidmarker11959parallelf(hat{x})parallelapprox0~{rm{{ή}~parallelhat{x}-x^{*}parallelapprox=0{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1047#1047|; $key = q/{displaymath}{ustarv}_{m}=sum_{k=0}^{n-1}v_{k}u_{m-k},m=0,1,ldots,n-1,{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1719#1719|; $key = q/u_{n+1}^{k}=β;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1597#1597|; $key = q/Β_0=Ι;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1327#1327|; $key = q/f''(x)=2x(3-2x^2)e^{-x^{2}};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1277#1277|; $key = q/y_k=f(x_k);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1154#1154|; $key = q/mathbf{for}j=k+1mathbf{to}n;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|857#857|; $key = q/Δx;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1575#1575|; $key = q/%latex2htmlidmarker7975r_{kk}=parallelalpha_{k}parallel_{2};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|855#855|; $key = q/x=wslashv;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1170#1170|; $key = q/b=[813]^{T};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|390#390|; $key = q/mathbf{Procedurefft}(x,y,n,w);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1699#1699|; $key = q/{displaymath}Q_0=left[array{{cr}0.832&0.5550.555&-0.832array{right]qquadmbox{andadR_0=left[array{{cc}3.61&1.660&1.11array{right]{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|937#937|; $key = q/{displaymath}F_4P_4=left[matrix{1&1&1&1cr1&-1&i&-icr1&1&-1&-icr1&-1&-i&i}right]=left[matrix{F_2&D_2F_2crF_2&-D_2F_2}right],{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1691#1691|; $key = q/y(0)=y_0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1410#1410|; $key = q/{displaymath}%latex2htmlidmarker4942Deltax=A^{-1}DeltabRightarrowparallelDeltaxparallelA^{-1}parallelcdotparallelDeltabparallel,{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|682#682|; $key = q/1slash4;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1435#1435|; $key = q/Q(n^2);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1313#1313|; $key = q/y=left[-122right]^{T};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|601#601|; $key = q/x_3=0.754;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1268#1268|; $key = q/phi((x-b)slasha);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1735#1735|; $key = q/x=left[0.50.5right]^{T};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1035#1035|; $key = q/{displaymath}left[array{{ccr}2&4&-20&1&10&0&4array{right]left[array{{c}x_1x_2x_3array{right]=left[array{{c}248array{right],{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|476#476|; $key = q/25;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|125#125|; $key = q/qrsolv;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|889#889|; $key = q/{displaymath}frac{u_{i+1,j}-2u_{i,j}+u_{i-1,j}}{h^2}+frac{u_{i.j+1}-2u_{i,j}+u_{i,j-1}}{h^2}=0,{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1647#1647|; $key = q/{displaymath}left[array{{ccr}2&4&-20&1&10&0&4array{right]left[array{{c}x_1x_2x_3array{right]=left[array{{c}248array{right].{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|421#421|; $key = q/F_{n};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1695#1695|; $key = q/{displaymath}M_{k}M_{j}=I-me_{k}^{T}-te_{j}^{T}-me_{k}^{T}te_{j}^{T}=I-me_{k}^{T}-te_{j}^{T},{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|447#447|; $key = q/m;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|114#114|; $key = q/Press;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|277#277|; $key = q/{displaymath}----------------Stelios:Toshmeioaφtodeneinaikaθoloυsafes...Mallonkatileipei.----------------{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|885#885|; $key = q/{displaymath}mbox{Conjugate~transpose}:qquadleft[array{{ll}1+i&1+2i2-i&2-2iarrayt]^H=left[array{{ll}1-i&2+i1-2i&2+2iarray{right]{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|914#914|; $key = q/e^{λh};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1501#1501|; $key = q/mbox{τ}^2=1-mbox{τ};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1237#1237|; $key = q/meanfreepath;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1752#1752|; $key = q/sign(f(a))!=sign(f(b));MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1066#1066|; $key = q/{displaymath}A=left[array{{cc}1.5&0.50.5&1.5array{right]:lambda=2,x=left[array{{box{and}lambda=1,x=left[array{{r}-11array{right]{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|909#909|; $key = q/{displaymath}y_{0}=x_{0}w^{0}+x_{1}w^{0}+x_{2}w^{0}+x_{3}w^{0},y_{1}=x_{0}w^{0}+{3}=x_{0}w^{0}+x_{1}w^{3}+x_{2}w^{6}+x_{3}w^{9}.{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1677#1677|; $key = q/c=u;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1176#1176|; $key = q/x_0=3;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1127#1127|; $key = q/IEEE;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|103#103|; $key = q/%latex2htmlidmarker2851CollectedAlgorithmsoftheACM;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|297#297|; $key = q/nODEs;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1565#1565|; $key = q/MathWorksInc;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|298#298|; $key = q/-nablaf(x);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1284#1284|; $key = q/sigma_k;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|962#962|; $key = q/b:;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|382#382|; $key = q/Fourier;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|21#21|; $key = q/u_{j}^{0};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1607#1607|; $key = q/u_{0};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1620#1620|; $key = q/ΑΜ=ΜΑ=Ι;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|373#373|; $key = q/Moler;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|240#240|; $key = q/robust;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1317#1317|; $key = q/{displaymath}(B^{-1}A)x=lambdax~~mbox{or}~~(A^{-1}B)x=(1slashlambda)x{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|972#972|; $key = q/Ν=Μ^{Τ};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|402#402|; $key = q/{displaymath}f(x)=0.5x_1^2+2.5x_2^2{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1302#1302|; $key = q/regression;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|994#994|; $key = q/7.396times10^{-4};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|85#85|; $key = q/{displaymath}left[array{{c}y'_1y'_2vdotsy'_{n-1}y'_narray{right]=left[array{{c}y_2y_3vdotsy_nf(t,y_1,y_2,ldots,y_n)array{right].{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1384#1384|; $key = q/Crank-Nicolson;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1614#1614|; $key = q/t_1;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1429#1429|; $key = q/h;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|63#63|; $key = q/{displaymath}a_{ij}=φ_{j}(x_{i}),b_{ij}=φ''_{j}(x_{i}).{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1582#1582|; $key = q/{displaymath}A=left[array{{rc}0&1-1&0array{right]:lambda=i,x=left[array{{c}1iarrft[array{{r}i1array{right],mbox{where}i=sqrt{-1}{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|910#910|; $key = q/a_k;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1286#1286|; $key = q/{displaymath}u=left[array{{r}00-2array{right]kappaalphaiotaleft[array{{c}010array{right],{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|595#595|; $key = q/fixedpoints;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1089#1089|; $key = q/beta=2,t=3,L=-1;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|126#126|; $key = q/r_k=u_k-beta_{k-1}q_{k-1};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|966#966|; $key = q/{displaymath}%latex2htmlidmarker7416parallelQxparallel_2^2=(Qx)^TQx=x^TQ^TQx=x^Tx=parallelxparallel_2^2{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|756#756|; $key = q/Forsythe;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|238#238|; $key = q/u_k=Aq_k;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|965#965|; $key = q/{figure}{centering{ $key = q/m>n;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|709#709|; $key = q/{displaymath}Ax_2=A(x_1+z_1)=Ax_1+Az_1=(b-r_1)+r_1=b.{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|708#708|; $key = q/ftp.che.utexas.eduslashpubslashoctave;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|304#304|; $key = q/Flannery;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|280#280|; $key = q/r>1;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1056#1056|; $key = q/x=y+left[(v^{T}y)slash(1-v{^T}z)right]z;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|589#589|; $key = q/%latex2htmlidmarker12597r(A)leqparallelAparallel;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1200#1200|; $key = q/x_2=alpha+tau(b-a);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1243#1243|; $key = q/c;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|655#655|; $key = q/{displaymath}y_{m}=sum_{k=0}^3x_{k}w^{mk},m=0,ldots,3.{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1676#1676|; $key = q/Αx=b,;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|413#413|; $key = q/{displaymath}atimes{fl}(hat{x})=ahat{x}(1+delta_{1}){displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|226#226|; $key = q/System;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|106#106|; $key = q/{displaymath}mbox{cond}(A)=sigma_{max}slashsigma_{min}{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|980#980|; $key = q/36.87;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|823#823|; $key = q/netlib.bell-labs.com;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|273#273|; $key = q/250000;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|71#71|; $key = q/{displaymath}f'(x)=2x-4cos(x),{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1125#1125|; $key = q/{displaymath}u_{t}=cu_{xx},0leqxleqL,tgeq0,{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1552#1552|; $key = q/50%;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|561#561|; $key = q/{displaymath}PA=LU,{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|492#492|; $key = q/nablaf(x_0)=[55]^T;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1292#1292|; $key = q/NumericalMethodsandSoftware;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|253#253|; $key = q/q_{k};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|853#853|; $key = q/x_k^T;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|941#941|; $key = q/Morrison;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|580#580|; $key = q/{displaymath}sqrt{b^{2}-4ac}=98.77.{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|202#202|; $key = q/p(x);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1182#1182|; $key = q/Q^{T}b;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|880#880|; $key = q/C=0.618;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1253#1253|; $key = q/{displaymath}A^{T}Ax=A^{T}b{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|737#737|; $key = q/array;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|164#164|; $key = q/mathbf{NOTE};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|887#887|; $key = q/{displaymath}y_{k+1}=y_{k}+λy_{k+1}h,{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1496#1496|; $key = q/{displaymath}A=left[array{{cc}1&10&2array{right]:lambda=1,x=left[array{{c}10arrambox{and}lambda=2,x=left[array{{c}11array{right]{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|907#907|; $key = q/x^{3}-3x^{2}+3x-1=0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1046#1046|; $key = q/{displaymath}Q_0=left[array{{cr}0.962&-0.2750.275&0.962array{right]qquadmbox{andadR_0=left[array{{cc}7.28&3.020&3.30array{right]{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|936#936|; $key = q/{displaymath}G_{2}G_{1}A=left[matrix{1&0&0&0&0cr0&1&0&0&0cr0&0&0.577&0.816&0cr0&0.866&0.722cr0&0.612&0.510cr0&0.354&0.530}right]{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|837#837|; $key = q/t_{o}=0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1527#1527|; $key = q/OSE;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1574#1574|; $key = q/{displaymath}u(0,x)=u_{0}(x),xgeq0.{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1618#1618|; $key = q/O(n);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1673#1673|; $key = q/B_ks_k=-nablaf(x_k)mbox{για}s_k.;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1322#1322|; $key = q/{displaymath}Axapproxb,{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|721#721|; $key = q/(t_i,y_i;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|711#711|; $key = q/{displaymath}H=I-2frac{nunu^{T}}{nu^{T}nu},{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|760#760|; $key = q/Μ_k;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|486#486|; $key = q/Jacobian;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1197#1197|; $key = q/d_{ii},i=1,2,ldots,n;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|409#409|; $key = q/x_{k};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1670#1670|; $key = q/|g'(Θ_k)|leqC<1;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1113#1113|; $key = q/Dx;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|412#412|; $key = q/ADx=b;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|410#410|; $key = q/u(t,x);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1571#1571|; $key = q/y';MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1548#1548|; $key = q/4000;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|673#673|; $key = q/JamesH.Wilkinson;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|348#348|; $key = q/memorymanagementoverhead;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|346#346|; $key = q/alpha=sqrt{alpha_{1}^{2}+alpha_2^{2}};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|805#805|; $key = q/{displaymath}left[array{{cl}0.641&0.2420&0.000242array{right]left[array{{c}x_1x_right]=left[array{{c}0.883-0.000383array{right],{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|669#669|; $key = q/BellLaboratories;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|288#288|; $key = q/fixed-point;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1093#1093|; $key = q/t_{1}=t_{0}+h;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1418#1418|; $key = q/Macsyma;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|315#315|; $key = q/{displaymath}(x_{i},y_{j})=(ih,jh),i,j=1,ldots,n,{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1645#1645|; $key = q/%latex2htmlidmarker2741NumericalAlgorithmsGroup;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|257#257|; $key = q/g:{mathbb{R}{^nlongrightarrow{mathbb{R}{^n;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1191#1191|; $key = q/{displaymath}u_{t}=cu_{xx},0leqxleq1,tgeq0,{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1587#1587|; $key = q/tol;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1067#1067|; $key = q/α=1;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1075#1075|; $key = q/{displaymath}y(t_0)=y_0.{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1371#1371|; $key = q/{displaymath}G_{1}A=left[matrix{1&0&0&0&0cr0&1&0&0&0cr0&0&1&0&0cr0&0&0&0.707&0.70.0&0.0cr1.414&1.061&0.884cr0&0.354&0.530}right]{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|833#833|; $key = q/{displaymath}x^{2}-4sin{x}=0{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1031#1031|; $key = q/{displaymath}c=frac{1}{sqrt{1+(0.75)^2}}=frac{1}{1.15}=0.8mbox{και}s=ccdott=(0.8)(0.75)=0.6.{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|820#820|; $key = q/{displaymath}A=left[array{{rrr}2&4&-24&9&-3-2&-3&7array{right]=left[array{{rcc}1left[array{{ccr}2&4&-20&1&10&0&4array{right]=LU.{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|479#479|; $key = q/3x3;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1173#1173|; $key = q/{displaymath}u(0,x)=f(x),u_{t}(0,x)=g(x),{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1555#1555|; $key = q/mathbf{fft}(s,t,nslash2,w^{2});MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1706#1706|; $key = q/{displaymath}%latex2htmlidmarker7413parallelrparallel_2^2=parallelb-Axparallel_2rallelb_1-Rxparallel_2^2+parallelb_2parallel_2^2{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|753#753|; $key = q/y_{2};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1439#1439|; $key = q/{displaymath}P_2Μ_1P_1Ax=left[array{{rrr}4&9&-30&frac{3}{2}&frac{11}{2}0&-frac{1ght]=left[array{{r}814-2array{right]=P_2M_1P_1b.{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|517#517|; $key = q/Texas;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|249#249|; $key = q/m+n;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|990#990|; $key = q/mathbf{Inf};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|155#155|; $key = q/{displaymath}min_{alpha}f(x_0-anablaf(x_0)).{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1293#1293|; $key = q/lags;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1758#1758|; $key = q/{displaymath}x=left[array{{l}1.001.00array{right],{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|672#672|; $key = q/y(0)=1;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1491#1491|; $key = q/{displaymath}A=left[array{{cc}1&1epsilon&00&epsilonarray{right]{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|744#744|; $key = q/c=f''(x^{*})slash2;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1221#1221|; $key = q/Reeves;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1341#1341|; $key = q/|alpha_{2}|>|alpha_{1}|;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|813#813|; $key = q/{displaymath}%latex2htmlidmarker4918parallelxparallel_p=left(sum_{i=1}^n|x_i|^pright)^{1slashp}.{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|605#605|; $key = q/n_{ij}=m_{ji};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|403#403|; $key = q/pm1;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|656#656|; $key = q/[x_1,b];MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1236#1236|; $key = q/M_{k};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|445#445|; $key = q/G_{2};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|836#836|; $key = q/AΝumericalLibraryinCforScientistsandEngineers;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|285#285|; $key = q/y_{1}=1.5;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1433#1433|; $key = q/n^{3};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|548#548|; $key = q/x_1=[3.333-0.667]^T;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1295#1295|; $key = q/{displaymath}%latex2htmlidmarker7460r_{3,3}=parallela_3parallel_2=0.935,q_3=a_3srix{0.535cr-0.267cr-0.535cr-0.267cr0.535}right].{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|866#866|; $key = q/J>0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1469#1469|; $key = q/{displaymath}frac{u_j^{k+1}-2u_j^k+u_j^{k-1}}{(Δt)^2}=cfrac{u_{j+1}^k-2u_j^k+u_{j-1}^k}{(Δx)^2},{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1605#1605|; $key = q/h=-f'(x)slashf''(x);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1272#1272|; $key = q/{displaymath}u_{j}^{k+1}=u_{j}^{k}+cfrac{Δt}{(Δx)^2}(u_{j+1}^{k+1}-2u_{j}^{k+1}+u_{j-1}^{k+1}),{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1612#1612|; $key = q/mathbf{end};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|849#849|; $key = q/Cholesky;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|883#883|; $key = q/HPcalculator;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|111#111|; $key = q/{displaymath}p(t)=sum_{k=0}^{n-1}x_{k}t^{k},{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1731#1731|; $key = q/{displaymath}f(t_1)=y_i,i=1,ldots,n,{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1357#1357|; $key = q/{displaymath}nu=left[array{{c}212array{right]-left[array{{c}-300array{right]=left[array{{c}512array{right].{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|770#770|; $key = q/y_{0}(t);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1568#1568|; $key = q/11.2d;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1616#1616|; $key = q/{displaymath}array{{rclclcl}2x_1&+&4x_2&-&2x_3&=&2,4x_1&+&9x_2&-&3x_3&=&8,-2x_1&-&3x_2&+&7x_3&=&10,array{{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|472#472|; $key = q/{displaymath}y'=Ay{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1408#1408|; $key = q/b_i=y_i;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|723#723|; $key = q/J;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1396#1396|; $key = q/{displaymath}A=Qleft[array{{c}R0array{right]=[Q_1Q_2]left[array{{c}R0array{right]=Q_1R{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|759#759|; $key = q/svdpack;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1014#1014|; $key = q/{displaymath}L=left[array{{cc}1&0epsilon&1array{right]kappaalphaiotaU=left[arrayonarray{right]=left[array{{cc}1&10&1array{right]{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|508#508|; $key = q/Doolittle;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|533#533|; $key = q/Α(x+γz)=b;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|384#384|; $key = q/{displaymath}ax^{2}+bx+c=0{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|191#191|; $key = q/{displaymath}h<=(2tolslash|y''|)^{1slash2}{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1485#1485|; $key = q/{displaymath}f(x)=x^2-4sin(x)=0.{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1074#1074|; $key = q/HarwellSubroutineLibrary;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|242#242|; $key = q/{displaymath}{G(x)}_{ij}=frac{dg_i(x)}{dx_j},{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1198#1198|; $key = q/Golub;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|897#897|; $key = q/mathbf{IMSL:};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|245#245|; $key = q/U=hat{U};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|538#538|; $key = q/Householder;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|771#771|; $key = q/"_";MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|325#325|; $key = q/GAUSS;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|307#307|; $key = q/f'(x)=(2x^2-1)e^{-x^{2}};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1276#1276|; $key = q/Lax;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1586#1586|; $key = q/{displaymath}x-hat{x}=A^{-1}(b-Ahat{x})=A^{-1}Ehat{x}{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|686#686|; $key = q/{displaymath}%latex2htmlidmarker4925parallelAparallel_1=max_jsum_{i=1}^n|a_{ij}|{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|624#624|; $key = q/PV-WAVE;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|311#311|; $key = q/http:slashslashwww.nag.com;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|261#261|; $key = q/FourierF_{n};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1684#1684|; $key = q/Real^{n};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|615#615|; $key = q/{displaymath}frac{1+mbox{λ}hslash2}{1-mbox{λ}hslash2}=left(1+frac{mbox{λ}h}{2}rirac{(mbox{λ}h)^2}{2}+frac{(mbox{λ}h)^3}{4}+cdots{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1510#1510|; $key = q/{displaymath}A_0=left[array{{cc}7&22&4array{right]{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|935#935|; $key = q/Ry=c;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|879#879|; $key = q/f';MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1278#1278|; $key = q/{multicols}%latex2htmlidmarker16960{2}theorem_type{[provlhmata][provlhmata][chapβλημαχρησιμοποιώνταςδισδιάστατο{FFT{.theorem_type{{multicols};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1750#1750|; $key = q/{displaymath}f'(x)approxfrac{f(x+h)-(f(x-h)}{2h}.{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|219#219|; $key = q/y_{1}approx0.7709;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1495#1495|; $key = q/Laplace;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1637#1637|; $key = q/{displaymath}mbox{Orthogonal}:qquadleft[array{{rr}sqrt{2}slash2&sqrt{2}slash2-sqslash2-sqrt{2}slash2&-isqrt{2}slash2array{right]{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|918#918|; $key = q/f(t_{k},y_{k});MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1425#1425|; $key = q/{figure}{centering{ $key = q/B_0=I;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1330#1330|; $key = q/Reduce;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|318#318|; $key = q/{displaymath}Μ_1=left[array{{rrr}1&0&0-frac{1}{2}&1&0frac{1}{2}&0&1array{right],{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|513#513|; $key = q/epsilon;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|143#143|; $key = q/x_{n-1};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1717#1717|; $key = q/hat;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1657#1657|; $key = q/{displaymath}lambda=frac{6pmsqrt{36-32}}{2}=frac{6pm2}{2}=2mbox{and}4{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|912#912|; $key = q/{displaymath}y''approxfrac{y'_k-y'_{k-1}}{t_k-t_{k-1}}.{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1487#1487|; $key = q/LDU;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|539#539|; $key = q/B_{k+1}=B+(y_ky_k^T)slash(y_k^Ts_k)-(B_ks_ks_k^TB_k)slash(s^T_kB_ks_k).;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1325#1325|; $key = q/[0,2];MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1255#1255|; $key = q/{displaymath}B_ks_k=-nablaf(x_k),{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1352#1352|; $key = q/y_2=x';MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1390#1390|; $key = q/{displaymath}x_1=b_1slasha_{11},{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|419#419|; $key = q/Arnoldi;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1012#1012|; $key = q/a_{1};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|769#769|; $key = q/x_2;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|426#426|; $key = q/L=hat{L};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|537#537|; $key = q/{displaymath}H_{2}H_{1}A=left[right],H_{2}H_{1}b=left[right].{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|794#794|; $key = q/shocks;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1623#1623|; $key = q/{displaymath}%latex2htmlidmarker10448x_k=y_kslashparallely_kparallel_infty{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|943#943|; $key = q/hat{x};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|47#47|; $key = q/{displaymath}x=frac{2a}{-bmpsqrt{b^{2}-4ac}},{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|196#196|; $key = q/mathbf{if}(f_1>f_2)mathbf{then};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1246#1246|; $key = q/a=1;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1164#1164|; $key = q/D_{4};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1743#1743|; $key = q/{displaymath}%latex2htmlidmarker4933frac{parallelEparallel}{parallelAparallel}leqrhonepsilon_{mach},{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|665#665|; $key = q/x(t+p)=x(t);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1665#1665|; $key = q/Acton;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|357#357|; $key = q/dtls;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1015#1015|; $key = q/{displaymath}y_{k}=(1+λh)^{k}y_{0}.{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1478#1478|; $key = q/IBM;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|98#98|; $key = q/xmathbf{op}y;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|177#177|; $key = q/L_{k}L_{j};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|449#449|; $key = q/epsilon_{mach}=0.125;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|146#146|; $key = q/EULER;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1426#1426|; $key = q/{displaymath}u_{tt}=cu_{xx},0leqxleqL,tgeq0,{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1554#1554|; $key = q/k=0,1,ldots,.;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1114#1114|; $key = q/x*;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1107#1107|; $key = q/x>=0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1063#1063|; $key = q/Ahat{x}=b+Deltab;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|676#676|; $key = q/{displaymath}%latex2htmlidmarker4926parallelAparallel_{infty}=max_isum_{j=1}^n|a_{ij}|.{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|625#625|; $key = q/{displaymath}0=A^{T}r=A^{T}(b-Ax),{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|734#734|; $key = q/3!=6;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|531#531|; $key = q/evans.ee.adfa.oz.auslashpubslashRLaB;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|306#306|; $key = q/f_2=0.232;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1259#1259|; $key = q/h=0.25;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1529#1529|; $key = q/x_{k}=x_{n+k};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1715#1715|; $key = q/x_{i},i=1,ldots,n,;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|220#220|; $key = q/f_2=f(x_2);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1244#1244|; $key = q/c=f'(x^{*});MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1216#1216|; $key = q/%latex2htmlidmarker5950parallelAxparallelleqparallelAparallelcdotparallelxparallel;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|634#634|; $key = q/x=3^{-1}times18=0.333times18=5.99;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|552#552|; $key = q/{displaymath}x^{prime}(t)=-A^{-1}(t)A^{prime}(t)x(t)+A^{-1}(t)b^{prime}(t){displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|694#694|; $key = q/ODEy'=λy;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1474#1474|; $key = q/a_{11};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|455#455|; $key = q/portdirectory;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|289#289|; $key = q/{displaymath}-x_{1}+x_{2}^{2}+0.25=0,{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1034#1034|; $key = q/compiler;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|344#344|; $key = q/t_{0};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1416#1416|; $key = q/t_{1}=0.25;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1528#1528|; $key = q/d_{1}neq0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|116#116|; $key = q/4^{2}=16;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1680#1680|; $key = q/d_{0}d_{1}cdotsd_{t-1};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|91#91|; $key = q/R^{mm};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|993#993|; $key = q/{displaymath}%latex2htmlidmarker4927mbox{cond}(A)=parallelAparallelcdotparallelA^{-1}parallel.{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|635#635|; $key = q/mathbf{for}k=0mathbf{to}n-1;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1707#1707|; $key = q/f(m);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1078#1078|; $key = q/y'';MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1486#1486|; $key = q/x=[12]^{T}.;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|391#391|; $key = q/phi_i;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|718#718|; $key = q/{displaymath}(A+E)hat{x}=b,{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|685#685|; $key = q/{displaymath}%latex2htmlidmarker11985lim_{krightarrowinfty}frac{parallele_{k+1}parallel}{parallele_kparallel^r}=C{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1053#1053|; $key = q/0*mathbf{Inf},;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|159#159|; $key = q/Μ_{k};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|438#438|; $key = q/Bigl(hat{f}bigl(hat{x}bigr)-fbigl(xbigr)Bigr)+Bigl(fbigl(hat{x}bigr)-fbigl(xbigr)Bigr);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|38#38|; $key = q/MAC;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|341#341|; $key = q/1;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|39#39|; $key = q/Neumann;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1640#1640|; $key = q/DP;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|105#105|; $key = q/{displaymath}x_1=x_0+s_0=left[array{{c}51array{right]+left[array{{c}-5-5array{right]=left[array{{c}00array{right],{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1305#1305|; $key = q/y(t)=y_{0}e^{λt};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1476#1476|; $key = q/{displaymath}s_0=-g_0=-nablaf(x_0)=left[array{{c}-5-5array{right].{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1345#1345|; $key = q/F_{nslash2};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1697#1697|; $key = q/xleqx^*;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1225#1225|; $key = q/x_{i}=jΔx;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1590#1590|; $key = q/NUMAL;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|324#324|; $key = q/c^{2}+s^{2}=1;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|804#804|; $key = q/(1+h_{k}J);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1466#1466|; $key = q/octave;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|303#303|; $key = q/{displaymath}f:{mathbb{R}{rightarrow{mathbb{R}{(mbox{βαθμωτή}),{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1029#1029|; $key = q/{displaymath}u_{tt}=cu_{xx},0leqxleq1,tgeq0,{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1602#1602|; $key = q/Gauss;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1#1|; $key = q/f'(x)=0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1042#1042|; $key = q/mathit{i};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|528#528|; $key = q/h=1slash(n+1)=1slash3;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1646#1646|; $key = q/10^{3};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|699#699|; $key = q/epsilon_{mach};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|139#139|; $key = q/{displaymath}A_1=R_0Q_0+sigma_0I=left[array{{ll}7.92&0.6150.615&3.08array{right]{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|938#938|; $key = q/{displaymath}x_0=left[array{{c}01array{right]=1left[array{{c}11array{right]+1left[array{{r}-11array{right]{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|946#946|; $key = q/t_0=0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1376#1376|; $key = q/|1+λh|<1;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1480#1480|; $key = q/C^klongrightarrow0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1116#1116|; $key = q/t_{2}=t_{1}+h=1.0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1437#1437|; $key = q/v;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|571#571|; $key = q/|g'(x^*)|;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1194#1194|; $key = q/r=(1+sqrt{5})slash2approx1.618;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1151#1151|; $key = q/{displaymath}A^Tb=left[array{{lllll},1&,1&1&1&1-1.0&-0.5&0.0&0.5&1.0,1.0&,0.25&0rray{right]=left[array{{l}4.01.03.25array{right]{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|740#740|; $key = q/{displaymath}b+Deltab=Ahat{x}=A(x+Deltax)=Ax+ADeltax.{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|678#678|; $key = q/a=1slash2;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1404#1404|; $key = q/{displaymath}f(x+h)approxf(x)+f'(x)h+frac{f''(x)}{2}h^2{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1269#1269|; $key = q/BMDP;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|892#892|; $key = q/U=1;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|127#127|; $key = q/χ<0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1065#1065|; $key = q/"sendindex";MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|269#269|; $key = q/mathbf{TOMS:};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|295#295|; $key = q/{displaymath}left[array{{c}y'_1y'_2vdotsy'_narray{right]=left[array{{c}dy_1slashdtdy_2slashdtvdotsdy_nslashdtarray{right].{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1365#1365|; $key = q/^{2};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1766#1766|; $key = q/f'(x)neq0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1122#1122|; $key = q/hat{u};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1723#1723|; $key = q/mathbf{HSL:};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|241#241|; $key = q/{multicols}{2}{par{textbf{4.1.}description{item[(α)]Αποδείξτεότιτο5είναιιδιοτιμήυδοαντιστρόφουενόςδοσμένουπίνακα;description{{par{{multicols};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1026#1026|; $key = q/LAPACK;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1024#1024|; $key = q/cos(x);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|67#67|; $key = q/{displaymath}u(t,x)approxsum_{j=1}^{n}a_{j}(t)phi_{j}(x),{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1577#1577|; $key = q/Houston;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|248#248|; $key = q/{displaymath}A=left[array{{cc}1&11&1array{right]{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|500#500|; $key = q/q;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1730#1730|; $key = q/cond(P)=1;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|644#644|; $key = q/a_{ij}neq0Leftrightarrow;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1663#1663|; $key = q/{displaymath}f(x+h)=f(x)+f'(x)h+f''(x)h^{2}slash2+cdots{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|211#211|; $key = q/{displaymath}u_{xx}+u_{yy}+lambdau=f(x,y).{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1635#1635|; $key = q/aneq0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1736#1736|; $key = q/{displaymath}y''=f_{t}(t,y)+f(t,y)f_{y}(t,y),{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1523#1523|; $key = q/x_3=2;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|423#423|; $key = q/{displaymath}{H_f(x)}_{ij}=frac{d^2f(x)}{dx_{i}dx_{j}},{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1298#1298|; $key = q/q_{2};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|844#844|; $key = q/%latex2htmlidmarker10623parallely_kparallel_infty;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|945#945|; $key = q/f2c;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|326#326|; $key = q/Brent;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1181#1181|; $key = q/Euler;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|18#18|; $key = q/M_1;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|454#454|; $key = q/n=2;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1351#1351|; $key = q/Traub;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1188#1188|; $key = q/a_{ij}=phi_j(t_i);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|722#722|; $key = q/x*y=1.22326times10^{2};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|171#171|; $key = q/HiQ;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|308#308|; $key = q/{displaymath}0leqd_ileqbeta-1,qquadi=0,ldots,t-1,{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|89#89|; $key = q/{displaymath}f(x+h)approxf(x)+f'(x)h{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1120#1120|; $key = q/{displaymath}x_1^2-x_2+gamma=0,-x_1+x_2^2+gamma=0,{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1189#1189|; $key = q/y'(t);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1368#1368|; $key = q/E_{k};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1449#1449|; $key = q/a_{j}(t);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1579#1579|; $key = q/Internet;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|322#322|; $key = q/fixedpointiteration;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1090#1090|; $key = q/Quasi-Newton;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|15#15|; $key = q/mathbf{hess}mbox{και}mathbf{schur};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1005#1005|; $key = q/O(h_{k}^{2});MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1461#1461|; $key = q/-1slashepsilon;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|502#502|; $key = q/{displaymath}left[array{{c}y'_1y'_2array{right]=left[array{{c}y_2Fslashmarray{right].{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1391#1391|; $key = q/f:{mathbb{R}{longrightarrow{mathbb{R}{;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|32#32|; $key = q/{displaymath}hat{x}=logBigl(hat{f}bigl(xbigr)Bigr){displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|79#79|; $key = q/{displaymath}left[matrix{0.477&-1.0&0.50cr0.447&-0.5&-0.25cr0.447&0.0&-0.50cr0.447&0.5&-0.25cr0.447&1.0&0.50}right].{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|863#863|; $key = q/(I+hJ);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1472#1472|; $key = q/xapproxpislash2;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|62#62|; $key = q/derivativeinformation;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1137#1137|; $key = q/{displaymath}frac{|e_{k+1}|}{|e_{k}|cdot|e_{k-1}|}=frac{s_ks_{k-1}^r|e_{k-1}|^{rr|e_{k-1}|}=s_ks_{k-2}^{r-1}|e_{k-1}|^{r^2-r-1}.{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1148#1148|; $key = q/fell(x).;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|205#205|; $key = q/u_{xx};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1562#1562|; $key = q/y_1=x;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1389#1389|; $key = q/y_{n+1}(t);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1569#1569|; $key = q/fhat(x)neq0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1044#1044|; $key = q/x_{2};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|674#674|; $key = q/%latex2htmlidmarker5898{x:parallelxparallel=1};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|613#613|; $key = q/BFGS;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1320#1320|; $key = q/{displaymath}u(t,x)=u_{0}(x-ct).{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1619#1619|; $key = q/mirrorsites;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|272#272|; $key = q/M_{k}^{-1};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|443#443|; $key = q/t_0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1370#1370|; $key = q/{displaymath}%latex2htmlidmarker4932frac{parallelrparallel}{parallelAparallelcdol}leqfrac{parallelEparallel}{parallelAparallel},{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|662#662|; $key = q/dissipative;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1633#1633|; $key = q/{displaymath}r_{2,3}=q_2^Ta_3=0,{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|865#865|; $key = q/g;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1087#1087|; $key = q/{displaymath}left|frac{lambda_2-sigma}{lambda_1-sigma}right| $key = q/{displaymath}A=left[array{{cc}1.5&0.50.5&1.5array{right]qquadmbox{and}qquadx_0=left[array{{c}01array{right]{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|940#940|; $key = q/{displaymath}left[matrix{a_1&a_2cra_2&-a_1}right]left[matrix{ccrs}right]=left[matrix{acr0}right].{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|806#806|; $key = q/ftp;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|271#271|; $key = q/quasi-Newton;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1314#1314|; $key = q/L_{k};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|444#444|; $key = q/dfslashdy=0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1405#1405|; $key = q/F_{2};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1693#1693|; $key = q/rank(A)=k $key = q/|e_k|longrightarrow0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1117#1117|; $key = q/{displaymath}%latex2htmlidmarker7415parallelrparallel_2^2=parallelb_2parallel_2^2{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|755#755|; $key = q/(pq)(t)=p(t)q(t);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1729#1729|; $key = q/uv^{T};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|569#569|; $key = q/hat{f}(hat{x})-f(x);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|37#37|; $key = q/%latex2htmlidmarker5916parallelgammaxparallel=|gamma|cdotparallelxparallel;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|620#620|; $key = q/{displaymath}w^{0}=w^{4}=1,w^{2}=w^{6}=-1,w^{9}=w^{1},{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1678#1678|; $key = q/q_{k}=q_{k}slashr_{kk};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|851#851|; $key = q/{displaymath}h=frac{(a-c)(b-c)(f_a-f_b)f_c}{(a-c)(f_c-f_b)f_a-(b-c)(f_c-f_a)f_b}.{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1177#1177|; $key = q/mathbf{SLATEC:};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|290#290|; $key = q/ntimes1;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|626#626|; $key = q/x^{2}-4sin(x)=0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1038#1038|; $key = q/b;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|365#365|; $key = q/{displaymath}x_{k+1}=x_k-f(x_k)frac{x_k-x{k-1}}{f(x_k)-f(x_{k-1})}.{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1138#1138|; $key = q/d_{1}d_{2}cdotsd_{t-1};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|94#94|; $key = q/2t;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|165#165|; $key = q/Helmoholtz;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1634#1634|; $key = q/Α_{0};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1505#1505|; $key = q/mathbf{PORT:};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|287#287|; $key = q/Ι;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|374#374|; $key = q/u_{i,j};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1648#1648|; $key = q/Dr.RobertPanoff;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|188#188|; $key = q/{displaymath}g'(x)=f(x)f' $key = q/x_{k}+m;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1767#1767|; $key = q/{displaymath}M_1M_2=left[array{{rcc}1&0&0-2&1&01&frac{1}{2}&1array{right],L_1L_2ray{{rrc}1&0&02&1&0-1&-frac{1}{2}&1array{right].{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|453#453|; $key = q/{displaymath}y_{k+1}=y{k}+frac{mbox{λ}y_k+mbox{λ}y_{k+1}}{2}h,{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1507#1507|; $key = q/|1+h_{k}J|<1;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1467#1467|; $key = q/λ<0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1479#1479|; $key = q/{displaymath}P_1Ax=left[array{{rrr}4&9&-32&4&-2-2&-3&7array{right]left[array{{c}rray{right]=left[array{{r}8210array{right]=P_1b.{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|512#512|; $key = q/1slashn;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|174#174|; $key = q/y(0.25)=0.9412;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1534#1534|; $key = q/O(n^2);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1712#1712|; $key = q/Rutishauser;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1020#1020|; $key = q/%latex2htmlidmarker7931r_{kk}=parallelq_{k}parallel_{2};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|850#850|; $key = q/mbox{β}=(g_{k+1}^Tg_{k+1})slash(g_{k}^Tg_k).;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1337#1337|; $key = q/{displaymath}{J}_ij=df_islashdy_j.{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1397#1397|; $key = q/TOMS;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|354#354|; $key = q/Ly=b;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|469#469|; $key = q/|g'(x^*)|<1;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1108#1108|; $key = q/{displaymath}r(G(x^*))<1,{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1195#1195|; $key = q/Hager;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|264#264|; $key = q/(b-a)x_{k}+a;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1759#1759|; $key = q/alpha_{k}neq0,;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|433#433|; $key = q/u_{j}^{0}=f(x_{i});MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1598#1598|; $key = q/{displaymath}intphi_{jk}phi_{mn}=0{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1741#1741|; $key = q/2.347times10^{3};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|84#84|; $key = q/dartmouth.eduslashpubslashgnuplot;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|337#337|; $key = q/nu_{3};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|795#795|; $key = q/x=Qz.;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|497#497|; $key = q/{displaymath}M=M_{n-1}P_{n-1}ldotsM_{1}P_{1}{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|489#489|; $key = q/b=2;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1165#1165|; $key = q/H_{3};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|798#798|; $key = q/θ_k;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1111#1111|; $key = q/h=0.1;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1516#1516|; $key = q/hat{x}=x+h;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|53#53|; $key = q/D_{nslash2};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1698#1698|; $key = q/lancz;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1010#1010|; $key = q/{displaymath}G_{2}G_{1}b=left[matrix{1&0&0&0&0cr0&1&0&0&0cr0&0&0.577&0.816&0cr0&eft[matrix{1.0cr0.5cr1.443cr1.020cr1.061}right].{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|838#838|; $key = q/b=[47]^{T},;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|393#393|; $key = q/X;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1205#1205|; $key = q/Newton;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|14#14|; $key = q/c=577,s=0.816;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|835#835|; $key = q/LR;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1019#1019|; $key = q/f(x)=e^{x};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|57#57|; $key = q/|h|;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1128#1128|; $key = q/Illinois;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|260#260|; $key = q/{displaymath}%latex2htmlidmarker10461x_{k+1}=y_{k+1}slashparallely_{k+1}parallel_infty{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|961#961|; $key = q/h<0.02;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1519#1519|; $key = q/Malcolm;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|355#355|; $key = q/{displaymath}(t_i,y_i),i=1,ldots,n,{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1355#1355|; $key = q/{displaymath}frac{1}{n}left[matrix{1&1&1&1cr1&w^{-1}&w^{-2}&w^{-3}cr1&w^{-2}&w^{atrix{1&0&0&0cr0&1&0&0cr0&0&1&0cr0&0&0&1}right].{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1671#1671|; $key = q/table;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1080#1080|; $key = q/{displaymath}M=left[array{{cc}1&0-1slashepsilon&1array{right],{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|503#503|; $key = q/{displaymath}L_{k}=y_{k}-u_{k-1}(t_{k}),{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1445#1445|; $key = q/[a,x_2];MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1261#1261|; $key = q/Harwell;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1009#1009|; $key = q/OFL=(1.11)_{2}times2^{1}=(3.5)_{10};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|128#128|; $key = q/|x|;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1129#1129|; $key = q/arpack;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1013#1013|; $key = q/mathbf{SOL:};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|291#291|; $key = q/S;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|882#882|; $key = q/y_{1}=slash=y(t_{1};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1430#1430|; $key = q/mathbf{LAPACK};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1000#1000|; $key = q/stepsize;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1136#1136|; $key = q/y_{1}=y_{0}-0.5y_{0}^{3}=0.5;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1494#1494|; $key = q/{figure}{centering{{{{figure};LFS=11;FSF=1.6;AAT/;
$cached_env_img{$key} = q|73#73|; $key = q/y(t)=y_0e^t;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1378#1378|; $key = q/{displaymath}Halpha=alpha-2frac{nu^{T}alpha}{nu^{T}nu}nu=left[array{{c}212array{}512array{right]=left[array{{c}-300array{right],{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|772#772|; $key = q/areanchored;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1558#1558|; $key = q/fronts;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1625#1625|; $key = q/alpha_k=1;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1306#1306|; $key = q/sqrt(c);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1559#1559|; $key = q/sin(x)=0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1040#1040|; $key = q/x=1.92403times10^{2};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|167#167|; $key = q/RLaB;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|305#305|; $key = q/{eqnarraystar}r+Ax=b,A^Tr=0,{eqnarraystar};MSF=1.6;LFS=11;TAGS=R;AAT/;
$cached_env_img{$key} = q|748#748|; $key = q/t_{0}=0,y_{0}=1;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1431#1431|; $key = q/{displaymath}%latex2htmlidmarker4934frac{parallelEparallel}{parallelAparallel}approxnepsilon_{mach}.{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|667#667|; $key = q/{displaymath}y_{k}=left(frac{1+mbox{λ}hslash2}{1-mbox{λ}hslash2}right)^ky_0.{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1508#1508|; $key = q/Hessenberg;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1001#1001|; $key = q/Wooldbury;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|590#590|; $key = q/{calO}(n^3);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|568#568|; $key = q/{displaymath}2(beta-1)beta^{t-1}(U-L+1)+1{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|119#119|; $key = q/f_c;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1159#1159|; $key = q/c=slash=0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1517#1517|; $key = q/t_{1},y_{1};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1442#1442|; $key = q/mbox{det}(A)=0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|638#638|; $key = q/cond(I)=1;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|643#643|; $key = q/beta^{-1}leqm<1;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|117#117|; $key = q/g'(x^*)=0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1119#1119|; $key = q/tau=cslashs=alpha_{1}slashalpha_{2};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|814#814|; $key = q/{displaymath}h=sqrt{epsilon|f(x)|slashM}.{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|216#216|; $key = q/t=alpha_{2}slashalpha_{1}=3slash4=0.75;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|819#819|; $key = q/3x=18;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|550#550|; $key = q/(t-1)o;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|133#133|; $key = q/URL;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|234#234|; $key = q/{displaymath}M_ka=left[array{{cccccc}1&cdots&0&0&cdots&0vdots&ddots&vdots&vdots&t]=left[array{{c}a_1vdotsa_k0vdots0array{right],{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|434#434|; $key = q/A(t)x(t)=b(t).;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|692#692|; $key = q/{displaymath}P=P_2P_1=left[array{{ccc}1&0&00&0&10&1&0array{right]left[array{{cccght]=left[array{{ccc}0&1&00&0&11&0&0array{right]{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|523#523|; $key = q/A-alphae_{j}e_{k}^{T},;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|576#576|; $key = q/{displaymath}L=left[array{{rrr}1&0&0-frac{1}{2}&1&0frac{1}{2}&-frac{1}{3}&1array{right],{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|524#524|; $key = q/C=|g'(x^*)|;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1118#1118|; $key = q/zerofinder;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1178#1178|; $key = q/ODΕ;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1520#1520|; $key = q/mathbf{fft}(p,q,nslash2,w^{2});MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1705#1705|; $key = q/plplot;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|338#338|; $key = q/{displaymath}left[array{{cc}2&35&4array{right]left[array{{c}x_1x_2array{right]=left[array{{c}b_1b_2array{right],{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|389#389|; $key = q/y(t)=1+t(δηλ.,c=0);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1515#1515|; $key = q/NP;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1660#1660|; $key = q/{displaymath}y_{k+1}=y_{k}+y'_{k}h_{k}+frac{y''_k}{2}h^2_k.{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1522#1522|; $key = q/u_{t};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1592#1592|; $key = q/q_k=r_{k-1}slashbeta_{k-1};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|964#964|; $key = q/G(x^*)=0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1201#1201|; $key = q/11.2c;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1613#1613|; $key = q/{displaymath}P_1=left[array{{ccc}0&1&01&0&00&0&1array{right],{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|511#511|; $key = q/sign(x)=1;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1062#1062|; $key = q/{displaymath}y_{2}=y_{1}+y'_{1}h+frac{y''_1}{2}h^2=0.9375-0.1099-0.0412=0.7846.{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1532#1532|; $key = q/Graham-Schmidt;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|5#5|; $key = q/{displaymath}y'=f(t,y),{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1361#1361|; $key = q/z=1.92275times10^{2};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|180#180|; $key = q/alpha=left[alpha_{1}alpha_{2}right]^{T};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|801#801|; $key = q/b=m;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1073#1073|; $key = q/sqrt{epsilon_{mach}};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1223#1223|; $key = q/Sterbenz;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|349#349|; $key = q/nablaf(x);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1206#1206|; $key = q/[α,χ_2];MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1235#1235|; $key = q/{displaymath}left[array{{rrr}1&0&0-2&1&01&0&1array{right]left[array{{rrr|rrr}2&42&4&-2&1&0&00&1&1&-2&1&00&1&5&1&0&1array{right],{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|562#562|; $key = q/{displaymath}PA=left[array{{ccc}0&1&00&0&11&0&0array{right]left[array{{rrr}2&4&-{3}{2}&frac{11}{2}0&0&frac{4}{3}array{right]=LU,{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|525#525|; $key = q/{displaymath}%latex2htmlidmarker4922parallelxparallel_1leqsqrt{n}parallelxparallarallelxparallel_1leqnparallelxparallel_{infty}.{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|617#617|; $key = q/{displaymath}|g'(x^*)|<1{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1106#1106|; $key = q/ktimesk;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|877#877|; $key = q/t_{k-1},y_{k-1};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1448#1448|; $key = q/g_{k+1}=nablaf(x_{k+1}).;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1336#1336|; $key = q/f(t_{0},y_{0});MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1417#1417|; $key = q/y[k]=q[kmod(nslash2)]+w^{k}t[kmod(nslash2)];MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1708#1708|; $key = q/mathbf{for}k=0mathbf{to}(nslash2)-1;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1702#1702|; $key = q/{displaymath}H_{3}H_{2}H_{1}A={displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|799#799|; $key = q/y_i;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|716#716|; $key = q/Q(n^3);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1312#1312|; $key = q/D;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|406#406|; $key = q/RAND;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1753#1753|; $key = q/{displaymath}LEIPOYNEOISELIDES106kai107...Stelios{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|902#902|; $key = q/{displaymath}A^{T}A=LL^{T},{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|738#738|; $key = q/{displaymath}LU=left[array{{cc}1&01slashepsilon&1array{right]left[array{{cc}epsiht]=left[array{{cc}epsilon&11&0array{right]neqA.{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|505#505|; $key = q/{displaymath}left[array{{ccccc}0&1&0&cdots&00&0&1&cdots&0vdots&vdots&vdots&ddotscdots&1-a_0&-a_1&-a_2&cdots&-a_{n-1}array{right]{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|924#924|; $key = q/-b;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|195#195|; $key = q/a=0.05010;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|197#197|; $key = q/{displaymath}lceillog_2(frac{b-alpha}{tol}rceil{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1084#1084|; $key = q/H_f(x);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1208#1208|; $key = q/ineqj;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|408#408|; $key = q/y_{k+1};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1424#1424|; $key = q/Taylor;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|210#210|; $key = q/d_{ij}=0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|407#407|; $key = q/ACMTransactionsonMathematicalSoftware;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|296#296|; $key = q/{displaymath}Δtleq(Δx)^{2}slash2c.{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1601#1601|; $key = q/u_{j}^{1};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1608#1608|; $key = q/144%;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|190#190|; $key = q/DFT;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|23#23|; $key = q/Rx=y;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|800#800|; $key = q/MA=U;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|488#488|; $key = q/|e_k|rightarrow0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1149#1149|; $key = q/%latex2htmlidmarker10629parallely_kparallel_infty;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|953#953|; $key = q/mbox{σ}_{n+1};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|996#996|; $key = q/alpha=[alpha_{1}alpha_{2}]^{T};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|428#428|; $key = q/{displaymath}y(t+h)=y(t)+y'(t)h+frac{y''(t)}{2}h^2+frac{y'''(t)}{6}h^3+cdots{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1521#1521|; $key = q/mathbf{NR:};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|275#275|; $key = q/u(x_{i},y_{j});MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1649#1649|; $key = q/x.;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|367#367|; $key = q/y=g(x);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1099#1099|; $key = q/x_1;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1183#1183|; $key = q/J=A;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1415#1415|; $key = q/{displaymath}DeltaΚapproxleft|frac{hf^prime(x)slashf(x)}{hslashx}right|=left|xfrac{f^prime(x)}{f(x)}right|.{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|56#56|; $key = q/1slash10;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|172#172|; $key = q/cos(x+h)-cos(x)approx-hsin(x)approx-h;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|65#65|; $key = q/{displaymath}u_{xx}approxfrac{u(t,x+Deltax)-2u(t,x)+u(t,x-Δx)}{(Δx)^{2}},{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1563#1563|; $key = q/0.1;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|42#42|; $key = q/Maple;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|316#316|; $key = q/e_k;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1050#1050|; $key = q/q_{k}=alpha_{k}slashr_{kk};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|856#856|; $key = q/bits;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|99#99|; $key = q/{displaymath}cond(A^TA)=[cond(A)]^2{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|747#747|; $key = q/{displaymath}r_1=b-Ax_1.{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|705#705|; $key = q/{displaymath}x=g(x){displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1085#1085|; $key = q/{displaymath}alpha=left[array{{c}43array{right].{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|817#817|; $key = q/f(α);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1079#1079|; $key = q/x_1=3;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1143#1143|; $key = q/O(Δt)+O((Δx)^{2});MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1599#1599|; $key = q/{displaymath}left[matrix{c&scr-s&c}right]left[matrix{a_1cra_2}right]=left[matrix{acr0}right],{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|803#803|; $key = q/{displaymath}x=Pleft[array{{c}y0array{right].{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|881#881|; $key = q/{displaymath}hat{x}=log(2.666667)=0.980829.{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|81#81|; $key = q/f(x_1);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1228#1228|; $key = q/Rx=Q^{T}b;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|869#869|; $key = q/{displaymath}(A-sigma_kI)y_{k+1}=x_k{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|960#960|; $key = q/f(t);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1358#1358|; $key = q/fell(x);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|130#130|; $key = q/f(x)=0.5-xe^{-x^{2}};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1275#1275|; $key = q/r_k=r_k-alpha_kq_k;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|968#968|; $key = q/{displaymath}M_1a=left[array{{rcc}1&0&0-2&1&01&0&1array{right]left[array{{r}24-2}24-2array{right]=left[array{{c}240array{right].{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|451#451|; $key = q/{displaymath}min_{alpha}f(x_k-anablaf(x_k)){displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1288#1288|; $key = q/hat{L}^{-1}L=hat{U}U^{-1}=D;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|535#535|; $key = q/Tukey;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1747#1747|; $key = q/{displaymath}x_2=x_1+z_1{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|707#707|; $key = q/{displaymath}g(x)=x^2-2,g(x)=sqrt{x+2}g(x)=1+2slashx,g(x)=frac{x^2+2}{2x-1}{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1097#1097|; $key = q/-4cslash(Δx)^{2};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1573#1573|; $key = q/{F_{n}}_{mk}=w^{mk};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1685#1685|; $key = q/sing(x)=-1;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1064#1064|; $key = q/{displaymath}left[array{{rc}c&s-s&carray{right]{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|925#925|; $key = q/ODE;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1395#1395|; $key = q/w^{k};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1711#1711|; $key = q/p(y);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1155#1155|; $key = q/Kublanovskaya;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1018#1018|; $key = q/%latex2htmlidmarker5938parallelAparallel>0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|629#629|; $key = q/{eqnarraystar}x_k&=&Ax_{k-1}=A^2x_{k-2}=cdots=A^kx_0&=&A^ksum_{i=1}^nalpha_iu_i=2}^n(lambda_islashlambda_1)^kalpha_iu_i){eqnarraystar};MSF=1.6;LFS=11;TAGS=R;AAT/;
$cached_env_img{$key} = q|939#939|; $key = q/Wilkinson;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|664#664|; $key = q/{multicols}%latex2htmlidmarker7148{2}theorem_type{[provlhmata][provlhmata][chaptοίησηεφαρμόζεταιδύοφορές)description{theorem_type{{multicols};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|904#904|; $key = q/hat{L};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|536#536|; $key = q/spectralradius;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1199#1199|; $key = q/{displaymath}α'(t)=cA^{-1}Bα(t),{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1583#1583|; $key = q/t=6;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|166#166|; $key = q/Gibbs;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1733#1733|; $key = q/Αz=0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|377#377|; $key = q/{displaymath}sigma_{ij}=left{array{{lll}0&mbox{for}&ineqjsigma_igeq0&mbox{for}&i=jarray{right.{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|975#975|; $key = q/mathbf{KMN:};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|252#252|; $key = q/{figure}{centering{ $key = q/{displaymath}alphae=Halpha=(I-2frac{nunu^{T}}{nu^{T}nu})alpha=alpha-nufrac{2nu^{T}alpha}{nu^{T}nu},{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|762#762|; $key = q/{multicols}%latex2htmlidmarker1340{2}theorem_type{[ervthseis][ervthseis][chapterολίεςσεαριθμητικήκινητήςυποδιαστολής.theorem_type{{multicols};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|359#359|; $key = q/x_1=x_2;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1248#1248|; $key = q/NR[205];MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1744#1744|; $key = q/{displaymath}left[array{{rrr}2&4&24&9&-3-2&-1&7array{right]left[array{{c}x_1x_2x_3array{right]=left[array{{r}2810array{right].{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|594#594|; $key = q/0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1172#1172|; $key = q/alpha_{jk};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|574#574|; $key = q/t-1;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|121#121|; $key = q/t_{k+1};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1488#1488|; $key = q/2^{n-1};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|666#666|; $key = q/{displaymath}fell(1+epsilon)>1,{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|144#144|; $key = q/m_i;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|481#481|; $key = q/(1+hJslash2)slash(1-hJslash2);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1511#1511|; $key = q/(1+epsilon)-(1-epsilon)=1-1=0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|183#183|; $key = q/t_{k}=kΔt;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1591#1591|; $key = q/{displaymath}x=left[array{{l}0.7821.58array{right].{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|670#670|; $key = q/{displaymath}x,Ax,A^2x,ldots,A^{m-1}x{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|963#963|; $key = q/z;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|386#386|; $key = q/{displaymath}L=M^{-1}=(M_{n-1}cdotsM_1)^{-1}=M_1^{-1}cdotsM_{n-1}^{-1}=L_1cdotsL_{n-1}{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|465#465|; $key = q/plot;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|331#331|; $key = q/{displaymath}x_{k+1}=x_k-f'(x_k)slashf''(x_k),{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1273#1273|; $key = q/mean;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1765#1765|; $key = q/MathematischCentrum,;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|282#282|; $key = q/t=1;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1440#1440|; $key = q/U.;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|485#485|; $key = q/1times1;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|549#549|; $key = q/mathbf{while}((b-alpha)>tol)mathbf{do};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1068#1068|; $key = q/alpha;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|641#641|; $key = q/O(nlog_{2}n);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1710#1710|; $key = q/{displaymath}x=(MA)^{-1}Mb=A^{-1}M^{-1}Mb=A^{-1}b.{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|397#397|; $key = q/x_{k+1}=x_k+s_k.;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1323#1323|; $key = q/alpha_{1};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|841#841|; $key = q/1slashn $key = q/{displaymath}Ax=left[array{{ccc}1&t_1&t_1^21&t_2&t_2^21&t_3&t_3^21&t_4&t_4^21&t_pproxleft[array{{c}y_1y_2y_3y_4y_5array{right]=b{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|724#724|; $key = q/y(t)=1slash(1+t^{2});MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1533#1533|; $key = q/c=3;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1166#1166|; $key = q/{displaymath}y_{k+1}=y_{k}+f(t_{k},y_{k})h_{k},{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1420#1420|; $key = q/u;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|570#570|; $key = q/{displaymath}B_knuapproxfrac{nablaf(x_k+hnu)-nablaf(x_k}{h}{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1354#1354|; $key = q/Mead;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1281#1281|; $key = q/{displaymath}table{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1144#1144|; $key = q/{displaymath}u(0,x)=f(x),u(t,0)=a,u(t,L)=beta,{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1553#1553|; $key = q/{figure}{centering{ $key = q/{displaymath}u=f_bslashf_c,v=f_bslashf_s,w=f_a|f_c,p=v(w(u-w)(c-b)-(1-u)(b-a)),q=(w-1)(u-1)(v-1).{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1161#1161|; $key = q/y(t);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1372#1372|; $key = q/Businger;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1022#1022|; $key = q/{displaymath}u_{j}^{0}=f(x_{j}),u_{j}^{1}=f(x_{j})+(Δt)g(x_{j}),{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1609#1609|; $key = q/y_{i}(t)approxu(t,iΔx);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1567#1567|; $key = q/(J<0);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1470#1470|; $key = q/d_{0};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|113#113|; $key = q/GIVENS;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|816#816|; $key = q/{displaymath}sigma=left[frac{1}{n-1}sum_{i=1}^n(x_i-bar{x})^2right]^{1slash2}.{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|222#222|; $key = q/n-1;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1728#1728|; $key = q/ODEy'=y;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1443#1443|; $key = q/p;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|603#603|; $key = q/f(x)=0.5x_1^2+2.5x_2^2;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1328#1328|; $key = q/Heun;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1537#1537|; $key = q/{displaymath}y=left[array{{c}1.7890.6321.336array{right]{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|742#742|; $key = q/Gauss.;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|498#498|; $key = q/{displaymath}f(t,x)=x_1+x_2t+x_3t^2+cdots+x_nt^{n-1}{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|719#719|; $key = q/FM;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|352#352|; $key = q/x^=g(x^);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1105#1105|; $key = q/{displaymath}p(t)=0.086+0.4t+1.4t^{2}.{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|727#727|; $key = q/m $key = q/y_{k}=y(t_{k});MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1460#1460|; $key = q/hat{x}-x=-0.019171.;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|83#83|; $key = q/P^{-1}=P^{T};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|400#400|; $key = q/n=1;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1366#1366|; $key = q/k $key = q/http:slashslashwww-or.stanford.edu;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|294#294|; $key = q/Mathematica;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|356#356|; $key = q/1slash2;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1436#1436|; $key = q/{displaymath}x_i=(b_i-sum_{j=1}^{i-1}a_{ij}x_j)slasha_{ii},qquadi=2,ldots,n.{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|420#420|; $key = q/x_1=-1;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|427#427|; $key = q/{displaymath}u(0,x)=f(x),u(t,0)=0,u(t,1)=0.{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1561#1561|; $key = q/f(x^*)=0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1132#1132|; $key = q/{displaymath}A=left[matrix{0.641&0.242cr0.321&0.121cr0.962&0.363}right]{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|873#873|; $key = q/e_{j};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|577#577|; $key = q/Ritz;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|10#10|; $key = q/f_1=f_2;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1249#1249|; $key = q/k;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|432#432|; $key = q/r^2-r-1=0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1150#1150|; $key = q/m=alpha+(b-alpha)slash2;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1069#1069|; $key = q/mathbf{x=Atextbackslashb};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|890#890|; $key = q/x_0=1;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1142#1142|; $key = q/u_{k-1};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1447#1447|; $key = q/A^{-1}B;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|553#553|; $key = q/hat{x},;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|35#35|; $key = q/(1+epsilon)+epsilon=1;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|178#178|; $key = q/0slash0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|158#158|; $key = q/Deltax=hat{x}-x,;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|677#677|; $key = q/{displaymath}g_1=nablaf(x_1)=left[array{{c}3.333-3.333array{right].{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1347#1347|; $key = q/x+y=1.93039times10^2,;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|169#169|; $key = q/{displaymath}left[matrix{4&-1&-1&0cr-1&4&0&-1cr-1&0&4&-1cr0&-1&-1&4}right]left[m,2}+u_{2,3}}right]=left[matrix{0cr0cr1cr1}right]{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1652#1652|; $key = q/beta=16;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|96#96|; $key = q/M_{k}=I-me_{k}^{T};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|439#439|; $key = q/{displaymath}L_1=M_1^{-1}=left[array{{rcc}1&0&02&1&0-1&0&1array{right],L_2=M_2^{array{{crc}1&0&00&1&00&-frac{1}{2}&1array{right]{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|452#452|; $key = q/{displaymath}y(t_{k+1})=y(t_{k})+f(t_{k},y(t_{k}))h_{k}+O(h_{k}^{2}).{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1457#1457|; $key = q/{displaymath}L=L_1L_2=left[array{{rcc}1&0&02&1&0-1&0&1array{right]left[array{{cct]=left[array{{rcc}1&0&02&1&0-1&1&1array{right],{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|478#478|; $key = q/f;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|51#51|; $key = q/P_k,;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|487#487|; $key = q/r=1.324;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1263#1263|; $key = q/x_k^TAx_kslashx_k^Tx_k;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|958#958|; $key = q/ΜΑ=Μ_{n-1}cdotsM_1A=M_{n-1}cdotsΜ_1b=Mb;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|462#462|; $key = q/{displaymath}Q^{T}A=left[matrix{2.236&0&1.118cr0&1.581&0cr0&0&0.935cr0&0&0cr0&0&eft[matrix{1.789cr0.632cr1.336cr0.338cr0}right].{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|839#839|; $key = q/{displaymath}4u_{1,1}-u_{0,1}-u_{2,1}-u_{1,0}-u_{1,2}=0,4u_{2,1}-u_{1,1}-u_{3,1}3}=0,4u_{2,2}-u_{1,2}-u_{3,2}-u_{2,1}-u_{2,3}=0.{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1651#1651|; $key = q/ODEs;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1407#1407|; $key = q/p+1;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1452#1452|; $key = q/f(x_0)=0.5,f(x_1)=0.081,f(x_2)=0.216;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1267#1267|; $key = q/Daubechies;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1737#1737|; $key = q/{displaymath}A^TA=left[array{{cc}1+epsilon^2&11&1+epsilon^2array{right]{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|745#745|; $key = q/x_2=x_1;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1251#1251|; $key = q/f(t,x);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|712#712|; $key = q/G(x);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1196#1196|; $key = q/{displaymath}bar{x}=frac{1}{n}sum_{i=1}^nx_i,{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|221#221|; $key = q/s_k^TH_f(x_k)s_k<0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1310#1310|; $key = q/x_{i};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1580#1580|; $key = q/{displaymath}x_{k}=(ax_{k+1}+b)(modM){displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1754#1754|; $key = q/{displaymath}left[array{{lll}5.0&0.0&2.50.0&2.5&0.02.5&0.0&2.125array{right]=lef2.236&0&1.1180&1.581&00&0&0.935array{right]=LL^T{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|741#741|; 1;(x)slash(f(x))^2.{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|1131#1131|;
