1.2.1 $B4pAC;v9`(B

$X$ $B$r(B Banach $B6u4V!"(B$U$ $B$r(B $X$ $B$N3+=89g!"(B $f\colon U\to X$ $B$r(B Frechet $BHyJ,2DG=$J

$$f(x)=0$$ (1.3)

$B$r9M$($k!#E,Ev$J(B $x_0\in U$ $B$r

$$x_{k+1}=x_k - \left(f'(x_k)\right)^{-1}f(x_k)$$ (1.4)

$B$GNs(B $\{x_k\}_{k\in\N}$ $B$r@8@.$7$?$H$-!"$3$l$,6K8B(B $x_\infty$ $B$r(B $B;}$D$3$H$,$"$k!#$3$N$H$-(B $x_\infty$ $B$O(B (1.3) $B$N(B $B2r$H$J$k!#(B

$B0J2<(B $\left(f'(x_k)\right)^{-1}$ $B$N$3$H$r(B $f'(x_k)^{-1}$ $B$H=q$/!#(B

$B$=$3$G==J,Bg$-$JHV9f(B $k$ $B$r $B$r(B (1.4) $B$N6a;w2r$H$7$F:NMQ$9$kJ}K!$,(B $B9M$($i$l$k$,!"$=$l$r(B Newton $BK!(B$B$H8F$V!#7A<0E*$K$O(B

$$F(x):= x-f'(x)^{-1} f(x)$$

$B$GDj$a$i$l$k(B $F$ $B$NITF0E@$rH?I|K!$G5a$a$F$$$k$3$H$K$J$k(B^1.1$B!#(B

$B@~7A<}B+$N>l9g!"(B

$$\Vert x^{(k)}-a\Vert\le L^{k}\Vert x^{(0)}-a\Vert$$

$B$,@.$jN)$D!#$D$^$jEyHf?tNsE*(B ($B$"$k$$$O;X?t4X?tE*(B) $B$K<}B+$9$k$o$1$G$"$k!#(B

Proof. ($B35N,(B) $f$ $B$O(B $C^2$-$B5i$G!"(B$f(a)=0$ $B$H2>Dj$9$k$H!"==J,(B $a$ $B$K(B $B6a$$(B $x_k$ $B$KBP$7$F(B

$$x_{k+1}-a = \left[x_k-f'(x_k)^{-1}f(x_k)\right]-a$$

$$= x_k-a-f'(x_k)^{-1}f(x_k)$$

$$= x_k-a-f'(x_k)^{-1}\left(f(x_k)-f(a)\right)$$

$$= x_k-a-f'(x_k)^{-1} \left[f'(x_k)(x_k-a)-O\left((a-x_k)^2\right)\right]$$

$$= O\left((a-x_k)^2\right).$$

$B>\$7$/$ONc$($P;3K\(B [19] $B$r8+$h!#(B $\qedsymbol$ ARRAY(0x12231f4) $\qedsymbol$

$B=E:,$N>l9g$O<}B+$NB.$5$,@~7A<}B+$K$J$j(B ($B$D$^$j!V<}B+$,CY$$!W(B)$B!"(B $B
>$K$h$k$b$N$@$,!(BSchwartz \cite{Schwartz} $B$b8+$F$*$3$&!#(B \qed \end{jremark}">