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: $BH>1"E*$J : $B>oHyJ,J}Dx<0$N?tCM2rK!(B : $B7W;;Nc(B

$B7x$$J}Dx<0(B

$B7x$$J}Dx<0(B(stiff equation)$B$H$O!"4JC1$K8@$&$H(B $B!V2r$N4K$d$+$JJQ2=$H5^7c$JJQ2=$,:.:_$9$kJ}Dx<0!W(B[7] $B$N;v$G$"$k!#(B NR3$B$G$O(B $B!V%9%1!<%k$,Bg$-$/0[$J$kFHN)JQ?t$,J#?t8DB8:_$7!"=>B0JQ?t$,$=$l$i$NFHN)JQ?t$K$h$C$FJQ2=$9$k>l9g!W$K(BStiffness$B$,LdBj$K$J$k$H$7$F$$$k!#(B

$BNc$($P(B

$\displaystyle \frac{du}{dx}$ $\displaystyle =$ $\displaystyle 998 u + 1998 v$  
$\displaystyle \frac{dv}{dx}$ $\displaystyle =$ $\displaystyle -999 u - 1999 v$  

$B$H$$$&(BODE$B$r(B $ u(0)=1, v(0)=0$ $B$H$$$&=i4|>r7o$G2r$$$?;~$N2r$O(B
$\displaystyle u$ $\displaystyle =$ $\displaystyle 2e^{-x}-e^{-1000x}$  
$\displaystyle v$ $\displaystyle =$ $\displaystyle -e^{-x}+e^{-1000x}$  

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$\displaystyle y(x) = \sum_i c_i e^{-\lambda_i x} +$   $BFC2r(B (43)

$B$N7A$GI=$5$l$k>l9g!"85$N(BODE$B$,0BDj$G$"$k$?$a$K$O(B$ \lambda_i$ $B$N $ x \to \infty$ $B$GH/;6$7$J$$$h$&$J>r7o(B)$B!#(B $B$3$N(B$ \lambda_i$ $B$NBg$-$5$KBg$-$J$P$i$D$-$,$"$k$H$-!V7x$$!W$H9M$($i$l$k!#(B

$B:#$3$N(BODE$B$r!"(B $B$$$^$^$G@bL@$7$F$-$?J}K!$G?tCM@QJ,$9$k:]$K$O!"(B $B%"%k%4%j%:%`$N0BDj>r7o$H$7$F!"(B $B%9%F%C%WI}(B$ h$ $B$KBP$7$F!J>/$J$/$H$b!K(B $ h < 1/1000$ $B$,MW5a$5$l$k!#(B $B$3$3$GG:$^$7$$$N$O!"(B $ e^{-1000x}$ $B$H$$$&9`$,(B $ x=0$ $B$N6aK5$rD6$($l$P(B$ u, v$ $B$NCM$r5a$a$k>e$Gr7o$+$i(B $BHs>o$K>.$5$J%9%F%C%WI}$r;H$$B3$1$J$/$F$O$J$i$J$$(B $B!J=>$C$F7W;;Ii2Y$,Bg$-$/$J$C$F$7$^$&!KE@$G$"$k!#(B

$B$3$N$h$&$JG:$^$7$$LdBj$r2sHr$9$k$K$O!"0BDj@-NN0h$N9-$$%"%k%4%j%:%`$r;HMQ$9$l$PNI$$!#(B $B!VF~Lg(B $B$=$N#3!W(B[8] $B$G4{$K=R$Y$?$h$&$K!"1"E*$JJ}K!$OM[E*$JJ}K!$KHf$Y$F0BDjNN0h$,9-$$!#(B $B=>$C$F!"(B$B7x$$J}Dx<0$r2r$/>l9g$O1"E*$JJ}K!$rMQ$$$k(B$B!#(B

$B$3$l$^$G=R$Y$?%k%s%2!&%/%C%?K!!J%I%k%^%s%I!&%W%j%s%9K!!K!"%V%j%k%7%e!&%9%H%"K!$O$$$:$l$bM[E*$J%"%k%4%j%:%`$G$"$k!#(B $B$=$l$>$l$KBP1~$9$k1"E*$JJ}K!$H$7$F(B NR3$B$G$O!"(B $B%m!<%<%s%V%m%C%/K!(B(Rosenbrock method)$B$H(B $BH>1"E*Jd30K!(B(semi-implicit extrapolation method)$B$rDs6!$7$F$$$k(B3$B!#(B $B%9%F%C%Q%k!<%A%sL>$O!"$=$l$>$l(BStepperRoss (ROsenbrock Stiffly Stable)$B$*$h$S(B StepperSie (Semi-Implicit Extrapolation) $B$G$"$k!#(B $B$3$l$i$N%k!<%A%s$N>\:Y$K$D$$$F$O(Bweb note 23[9], 24[10] $B$r;2>H$N$3$H!#(B





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