2.2.2 DK $BK!(B -- Durand $B$N2r

$BBe?tJ}Dx<0(B

$$p(x)\equiv x^n+a_{n-1}x^{n-1}+\cdots+a_{1}x+a_0=0$$

$B$N:,$r(B $\xi_1$, $\cdots$, $\xi_n$ $B$H$9$k$H(B

$$p(x)=(x-\xi_1)\cdots(x-\xi_n),$$

$$p'(\xi_i)=\prod_{j\ne i}(\xi_{i}-\xi_{j}) =(\xi_i-\xi_1)\cdots(\xi_i-\xi_{i-1}) (\xi_i-\xi_{i+1})\cdots(\xi_i-\xi_{n}).$$ (2.3)

Newton $BK!$G$O8=:_$N6a;wCM(B $x^{(k)}$ $B$+$i $B$r5a$a$k$K$O(B

$$x^{(k+1)}=x^{(k)}-\frac{p(x^{(k})}{p'(x^{(k)})}$$ (2.4)

$B$H$9$k!#$3$3$GF34X?t$N7W;;$r$7$J$$$G:Q$^$;$k$?$a$K!"$3$NJ,Jl$r(B (2.3) $B$GBeMQ$9$k$3$H$r9M$($k!#(B $B$D$^$j(B $x^{(k)}$ $B$,(B $\xi_i$ $B$N6a;wCM$@$H9M$($F(B $p'(x^{(k)})$ $B$r(B $p' (\xi_i)$ $B$GCV$-49$($k$o$1$G$"$k!#3F:,(B $\xi_i$ $B$KBP$9$k(B $B6a;wCM(B $x_i^{(k)}$ $B$,(B $i=1$, $\cdots$, $n$ $B$K$D$$$FA4It$=$m$C$F$$$k$H$7(B $B$F!"(B(2.4) $B$NBe$o$j$K(B

$$x_i^{(k+1)}=x_i^{(k)}-\frac{p(x_i^{(k)})} {\dsp\prod_{j\ne i}\left(x_i^{(k)}-x_j^{(k)}\right)}.$$ (2.5)

$B$rMQ$$$k!#(B

Durand $B$O(B (2.5) $B$N<}B+@-$r<($7!":G=*E*$K(B $2$ $B>h<}B+(B $B$K$J$k$3$H$r>ZL@$7$?!#(B $\qedsymbol$

$\xi_1$, $\cdots$, $\xi_n$ $B$,$9$Y$FAj0[$J$j!"(B$x_i^{(k)}$ $B$,(B $\xi_i$ $B$K==J,6a$$$H$7$F!"(B $\eps_i^{(k)}:= x_i^{(k)}-\xi_i$ $B$H$*$1$P!"(B

$$\eps_i^{(k+1)} = \eps_i^{(k)} \left[ 1-\frac{\prod_{j\ne i}(x_i^{(k)}-\xi_j)} {\prod_{j\ne i}(x_i^{(k)}-x_j^{(k)})} \right]$$

$$= \eps_{i}^{(k)} \left[ 1-\prod_{j\ne i} \left( 1+\frac{\eps_j^{(k)}}{x_i^{(k)}-x_{j}^{(k)}} \right) \right]$$

$2$ $B $\eps_j^{(k)}\cdot\eps_{\ell}^{(k)}$ $B$rL5;k$9$k$H(B

$$\eps_{i}^{(k)}\kinji -\eps_{i}^{(k)}\sum_{j\ne i}\frac{\eps_j^{(k)}} {x_i^{(k)}-x_j^{(k)}}.$$

$$\vert x_i^{(k)}-x_j^{(k)}\vert\ge \frac{1}{C}$$   $$\mbox{($C$\ $B$O(B $i$, $j$, $k$\ $B$K$h$i$J$$(B)}$$

$B$H$9$l$P(B

$$\left\vert \eps_i^{(k+1)}\vert\le C\vert\eps_{i}^{(k)}\cdot\sum_{j\ne i}\vert\eps_j^{(k)} \right\vert$$

$B$3$l$O(B $2$ $B>h<}B+$r0UL#$9$k$,!"$5$i$K>\$7$/$_$k$H(B

Proof. ($B!"%9!"%O!"%R%(%O%F%"!"%L!"%^!"%O!"!"!#!W!(B[13] $B$N(B 20 $B$d!"(B $B;3K\!&KL@n(B [17] $B$r;2>H!#(B) $\qedsymbol$ ARRAY(0x12b7160) $\qedsymbol$

ARRAY(0x12cbae0)