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/* {\hrulefill} *
{\bf Newton-Raphson ˡ}
$f(x)=0$ ˤϡ
$\displaystyle\qquad
x_0=a,\quad x_{n+1}=x_n-{f(x_n)\over f'(x_n)}
\qquad\cdots\cdots\ (\star)$
Ŭʽ $a$ ФƲơ $\{x_n\}$ 롣
$a$ ޤͿСοϾҤ
Ĥβ $\alpha$ ˡ«뤳ȤΤƤ, {\it i.e.},
$\displaystyle\qquad
\exists\ \alpha=\lim_{n\to\infty}x_n
\quad such\ that\quad f(\alpha)=0\ .$
$(\star)$ ΰ̣Ȥμ«ͻҤϡοޤȼ
Ǥ롣
{\special{epsfile=newton.eps hscale=.7 vscale=.7 hoffset=25}
\vskip 12cm}
$\displaystyle\qquad
y-f(x_n)=f'(x_n)(x-x_n)
\qquad\cdots\cdots\ \hbox{}\ (x_n,f(x_n))\ \hbox{̤}$
$\displaystyle\qquad
x_n-{f(x_n)\over f'(x_n)}
\qquad\cdots\cdots\ \hbox{嵭}\ x\ \hbox{Ȥθ}
\ x\ \hbox{ɸ}$
$\{x_n\}$ μ«ξӼ«Υɾϡ
ۤɴñʤȤǤϤʤ$\epsilon$-$\delta$ ˡȲϳؤ˴ؤ
μɬפȤ롣̣ΤˤϡʸҲ𤹤롣
ʲC newton.c ϡ
$\qquad\displaystyle x^2-5=0$
βΣĤ $\sqrt{5}$ Ǥ뤳ȤܤơNewton-Raphson ˡ $\sqrt{5}$
ץǤ롣{\tt A, F(X), DF(X)}
Ѥơƿͤǿͼ¸ԤäƤߤ褦
* {\hrulefill} */
/* {\hrulefill\ newton.c\ \hrulefill} */
#include <stdio.h>
#define A 4. /* {\hfill} */
#define F(X) ((X)*(X)-5.) /* Ϳ줿ؿ {\hfill} */
#define DF(X) (2.*(X)) /* Ƴؿ {\hfill} */
main()
{
double x, y; /* ٤Ƿ {\hfill} */
x = A; /* $\displaystyle x_0=a$ {\hfill} */
printf("%.16f\n", x); /* $x_0$ ɽ {\hfill} */
y = x - F(x) / DF(x); /* $\displaystyle
x_1=x_0 - {f(x_0)\over f'(x_0)}$
{\hfill} */
printf("%.16f\n", y); /* $x_1$ ɽ {\hfill} */
while (x != y) /* {\ iteration äƤ
ͤѤʤʤä齪λ
\hfill} */
{
x = y;
y = x - F(x) / DF(x); /* $\displaystyle x_{n+1}=x_n
- {f(x_n)\over f'(x_n)}$ {\hfill} */
printf("%.16f\n", y); /* $x_{n+1}$ ɽ {\hfill} */
}
}
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