4.13-bo'lim : Nyuton usuli
1. Agar \({x_{\,0}} = 5\) bo'lsa, \(f\left( x \right) = {x^3} - 7{x^2} + 8x - 3\) uchun \({x_{\,2}}\) ni aniqlash uchun Nyuton usulidan foydalaning.
Yechim
Haqiqatan ham, bu muammoni hal qilish uchun unchalik ko'p narsa yo'q. Biz bilamizki, Nyuton usulining asosiy formulasi:
\[{x_{n + 1}} = {x_n} - \frac{{f\left( {{x_n}} \right)}}{{f'\left( {{x_n}} \right)}}\]shuning uchun biz qilishimiz kerak bo'lgan narsa buni ikki marta bosib o'tishdir.
Mana funktsiyaning hosilasi, chunki bizga bu kerak bo'ladi.
\[f'\left( x \right) = 3{x^2} - 14x + 8\]Endi biz yuqoridagi formulani ikki marta bajarishimiz kerak.
2-qadam
\({x_{\,1}}\) formulasi orqali birinchi iteratsiya:
\[{x_{\,1}} = {x_{\,0}} - \frac{{f\left( {{x_{\,0}}} \right)}}{{f'\left( {{x_{\,0}}} \right)}} = 5 - \frac{{f\left( 5 \right)}}{{f'\left( 5 \right)}} = 5 - \frac{{ - 13}}{{13}} = 6\]3-qadam
\({x_{\,2}}\) formulasi orqali ikkinchi iteratsiya:
\[{x_{\,2}} = {x_{\,1}} - \frac{{f\left( {{x_{\,1}}} \right)}}{{f'\left( {{x_{\,1}}} \right)}} = 6 - \frac{{f\left( 6 \right)}}{{f'\left( 6 \right)}} = 6 - \frac{9}{{32}} = 5.71875\] Shunday qilib, bu muammoning javobi \(\require{bbox} \bbox[2pt,border:1px solid black]{{{x_{\,2}} = 5.71875}}\).Muammo bayonotida so'ralmagan bo'lsa-da, haqiqiy ildiz 5,68577952608963. Shuni ham yodda tutingki, bu olish uchun ba'zi bir hisoblash yordami talab qilinadi va bu umuman qo'lda ololmaydigan narsa emas.