$B7x$$J}Dx<0(B

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

$BNc$($P(B

$$\frac{du}{dx} = 998 u + 1998 v$$

$$\frac{dv}{dx} = -999 u - 1999 v$$

$B$H$$$&(BODE$B$r(B $u(0)=1, v(0)=0$ $B$H$$$&=i4|>r7o$G2r$$$?;~$N2r$O(B

$$u = 2e^{-x}-e^{-1000x}$$

$$v = -e^{-x}+e^{-1000x}$$

$B$H$J$k$,!"[email protected],(B($e^{-x}$ )$B$H5^[email protected],(B( $e^{-1000x}$ )$B$,:.:_$7$F$$$k!#(B $B=>$C$F85$N(BODE$B$O7x$$J}Dx<0$G$"[email protected]$($k!#(B

$B!J"(!K0lHL$K(BODE$B$N2r$,(B

$$y(x) = \sum_i c_i e^{-\lambda_i x} +$$ (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$$$^$^[email protected]@$7$F$-$?J}[email protected],$9$k:]$K$O!"(B $B%"%k%4%j%:%`$N0BDj>r7o$H$7$F!"(B $B%9%F%C%WI}(B$h$ $B$KBP$7$F!J>/$J$/$H$b!K(B $h < 1/1000$ $B$,MW5a$5$l$k!#(B $B$3$3$GG:$^$7$$$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$7$^[email protected]$G$"$k!#(B

$B$3$N$h$&$JG:$^$7$$LdBj$r2sHr$9$k$K$O!"[email protected]$N9-$$%"%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