❇❐ ●■⑩❖ ❉Ö❈ ❱⑨ ✣⑨❖ ❚❸❖
✣❸■ ❍➴❈ ✣⑨ ◆➂◆●
❑❍❖❆ ❚❖⑩◆
✯✯✯✯✯✯✯✯✯
❍❯Ý◆❍ ❚❍➚ ▼ß ❍❸◆❍
❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P
Ù◆● ❉Ư◆● P❍❺◆ ▼➋▼ ▼❆❚❍❊▼❆❚■❈❆
❈❍❖ Pì PP
ữợ ❚❘❯◆●
✣➔ ◆➤♥❣✱ ✵✺✴✷✵✶✺
✷
▼ö❝ ❧ö❝
▼Ð ✣❺❯
✶ ▼❐❚ ❙➮ ❑■➌◆ ❚❍Ù❈ ❈❍❯❽◆ ❇➚
✶✳✶
✹
✼
❑❍⑩■ ◆■➏▼ ❱➋ ❙➮ ●❺◆ ✣Ó◆● ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✼
✶✳✶✳✶
❙❛✐ sè t✉②➺t ✤è✐✱ s❛✐ sè t÷ì♥❣ ✤è✐
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✼
✶✳✶✳✷
❙❛✐ sè t❤✉ ❣å♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✼
✶✳✷
❙❆■ ❙➮ ❚➑◆❍ ❚❖⑩◆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✽
✶✳✸
◆●❍■➏▼ ❱⑨ ❑❍❖❷◆● P❍❹◆ ▲■ ◆●❍■➏▼
✳ ✳ ✳ ✳ ✳ ✳
✾
✶✳✸✳✶
◆❣❤✐➺♠ t❤ü❝ ❝õ❛ ữỡ tr ởt
ị ❤➻♥❤ ❤å❝ ❝õ❛ ♥❣❤✐➺♠
✳ ✳ ✳ ✳ ✳ ✳ ✳
ỹ tỗ t tỹ ừ ữỡ tr➻♥❤ ♠ët ➞♥
✶✵
✶✳✸✳✹
❑❤♦↔♥❣ ♣❤➙♥ ❧✐ ♥❣❤✐➺♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✷
✷ P❍×❒◆● P❍⑩P ❉❹❨ ❈❯◆● ❚➐▼ ◆●❍■➏▼ ●❺◆
✣Ĩ◆● ❈Õ❆ P❍×❒◆● ❚❘➐◆❍
✶✺
✷✳✶
✷✳✷
✷✳✸
●■❰■ ❚❍■➏❯ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✺
✷✳✶✳✶
✣➦t ✈➜♥ ✤➲
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✺
✷✳✶✳✷
❈→❝❤ ❣✐↔✐ q✉②➳t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✻
P❍×❒◆● P❍⑩P ❉❹❨ ❈❯◆●
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✽
✷✳✷✳✶
▼ỉ t↔ ♣❤÷ì♥❣ ♣❤→♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✽
✷✳✷✳✷
❙ü ❤ë✐ tư ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ✈➔ ✤→♥❤ ❣✐→ s❛✐ sè
✷✸
✳ ✳
▼❐❚ ❙➮ ❇⑨■ ❚❖⑩◆ ❚➐▼ ◆●❍■➏▼ ●❺◆ ✣Ĩ◆● ❱❰■
P❍×❒◆● P❍⑩P ❉❹❨ ❈❯◆● ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✷✼
✸ Ù◆● ❉Ư◆● P❍❺◆ ▼➋▼ ▼❆❚❍❊▼❆❚■❈❆ ❈❍❖ P❍×❒◆●
P❍⑩P ❉❹❨ ❈❯◆●
✷✾
✸✳✶
▼❐❚ ❱⑨■ ◆➆❚ ❱➋ P❍❺◆ ▼➋▼ ▼❆❚❍❊▼❆❚■❈❆ ✳ ✳
✸✳✶✳✶
●✐ỵ✐ t❤✐➺✉
❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✷✾
✷✾
❙❱❚❍✿ ❍❯Ý◆❍ ❚❍➚ ▼ß ❍❸◆❍
✸
✸✳✷
✸✳✶✳✷
●✐❛♦ ❞✐➺♥ t÷ì♥❣ t→❝ ❝õ❛ ▼❛t❤❡♠❛t✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✸✵
✸✳✶✳✸
❈→❝ t➼♥❤ ♥➠♥❣ ❝õ❛ ▼❛t❤❡♠❛t✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✸✶
✸✳✶✳✹
▼ët sè ❤➔♠ t❤æ♥❣ ❞ö♥❣
✸✷
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
Ù◆● ❉Ư◆● P❍❺◆ ▼➋▼ ▼❆❚❍❊▼❆❚■❈❆ ❈❍❖ P❍×❒◆●
P❍⑩P ❉❹❨ ❈❯◆●
❑➌❚ ▲❯❾◆
❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦
❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✸✸
✹✵
✹✶
❙❱❚❍✿ ❍❯Ý◆❍ ❚❍➚ ▼ß ❍❸◆❍
é
ỵ ỹ ồ t
t t❤ü❝ t➳ ✭tr♦♥❣ ❦❤♦❛ ❤å❝ ❦ÿ t❤✉➟t✱ tr♦♥❣ t❤✐➯♥ ✈➠♥✱ ✤♦ ✤↕❝
r✉ë♥❣ ✤➜t✱ ✳✳✳✮ ❞➝♥ ✤➳♥ ✈✐➺❝ ❝➛♥ ♣❤↔✐ ❣✐↔✐ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ♣❤✐ t✉②➳♥✱ t✉②
♥❤✐➯♥ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ♥➔② t❤÷í♥❣ ♣❤ù❝ t↕♣✱ ❞♦ ✤â ♥â✐ ❝❤✉♥❣ ❦❤â ❝â t❤➸
❣✐↔✐ ✤÷đ❝ ✭✤÷❛ ✤÷đ❝ ✈➲ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ❝ì ❜↔♥✮ ❜➡♥❣ ❝→❝ ❜✐➳♥ ✤ê✐ ✤↕✐
sè t❤➟♠ ❝❤➼ tr♦♥❣ ♠ët sè tr÷í♥❣ ❤đ♣ ❝ơ♥❣ ❦❤ỉ♥❣ t❤➸ t➻♠ ✤÷đ❝ ♥❣❤✐➺♠
t÷í♥❣ ♠✐♥❤✳ ❍ì♥ ♥ú❛✱ ổ tự tữớ ự t ỗ
❝❤♦ ❞ị ❝â ❝ỉ♥❣ t❤ù❝ ♥❣❤✐➺♠✱ ✈✐➺❝ ❦❤↔♦ s→t ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛
♥â ❝ô♥❣ ❣➦♣ ♣❤↔✐ r➜t ♥❤✐➲✉ ❦❤â ❦❤➠♥✳ ❱➻ ✈➟②✱ ♥❣❛② tø t❤í✐ ❆r❝❤✐♠❡❞❡s✱
❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ❣➛♥ ✤ó♥❣ ✤➣ ✤÷đ❝ ①➙② ❞ü♥❣✳ ◆❤✐➲✉ ♣❤÷ì♥❣ ♣❤→♣ ✤➣
trð t❤➔♥❤ ❦✐♥❤ ✤✐➸♥ ✈➔ ✤÷đ❝ sû ❞ư♥❣ rë♥❣ r➣✐ tr♦♥❣ t❤ü❝ t➳✳ ❇ð✐ ✈➟②✱ ✈✐➺❝
♥❣❤✐➯♥ ❝ù✉ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ❣➛♥ ✤ó♥❣ ✤➸ t➻♠ ♥❣❤✐➺♠ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤
♥➔② trð ♥➯♥ ❝➜♣ t❤✐➳t ✈➔ tü ♥❤✐➯♥✳
❈ị♥❣ ✈ỵ✐ sü ♣❤→t tr✐➸♥ ❝õ❛ t✐♥ ❤å❝✱ ❝→❝ ữỡ ú
õ ỵ tỹ t➳ ❤ì♥✳ ✣➸ ❣✐↔✐ ♠ët ♣❤÷ì♥❣ tr➻♥❤ ❜➡♥❣ t❛② tr➯♥
❣✐➜②✱ ❝â ❦❤✐ ♣❤↔✐ ♠➜t ❤➔♥❣ ♥❣➔② ✈ỵ✐ ♥❤ú♥❣ s❛✐ sât ❞➵ ①↔② r❛✱ t❤➻ ✈ỵ✐ sü
❤é trđ ❝õ❛ ❝→❝ ♣❤➛♥ ♠➲♠ ❝❤✉②➯♥ ❞ư♥❣ ❝❤ó♥❣ t❛ ❝❤➾ ❝➛♥ ✈➔✐ ♣❤ót t❤➟♠
❝❤➼ ✈➔✐ ❣✐➙②✳ ▼➦t ❦❤→❝✱ ♥❤✐➲✉ ✈➜♥ ✤➲ ❧➼ t❤✉②➳t ✭sü ❤ë✐ tö✱ tè❝ ✤ë ❤ë✐ tö✱
✤ë ❝❤➼♥❤ ①→❝✱ ✤ë ♣❤ù❝ t↕♣ t➼♥❤ t♦→♥✱ ✳✳✳✮ s➩ ✤÷đ❝ ♥❤➻♥ t❤➜② rã ❤ì♥ ❦❤✐
sû ❞ö♥❣ ❝→❝ ♣❤➛♥ ♠➲♠ ♥➔②✳ ❱➻ ✈➟②✱ ✈✐➺❝ sû ❞ư♥❣ t❤➔♥❤ t❤↕♦ ❝→❝ ❝ỉ♥❣
❝ư t➼♥❤ t♦→♥ ❧➔ ❝➛♥ t❤✐➳t ❝❤♦ ❝ỉ♥❣ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉✱ ♥❤➜t ❧➔ ✤è✐ ✈ỵ✐ ❤å❝
s✐♥❤ ✈➔ s✐♥❤ ✈✐➯♥✳
❱ỵ✐ ♠♦♥❣ ♠✉è♥ ❧➔ t➻♠ ❤✐➸✉ ✈➔ ♥❣❤✐➯♥ ❝ù✉ ✈➲ ♣❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ❣➛♥
✤ó♥❣ ♥❤➡♠ ✤→♣ ù♥❣ ♥❣✉②➺♥ ồ ự ồ ừ t
ỗ tớ ữủ sỹ ủ ỵ ở ừ ữợ ❞➝♥ ✕ ❚❙✳ ▲➯
❍↔✐ ❚r✉♥❣ ♥➯♥ tæ✐ ❧ü❛ ❝❤å♥ ✤➲ t➔✐✿ ✓Ù♥❣ ❞ư♥❣ ♣❤➛♥ ♠➲♠ ▼❛t❤❡♠❛t✐❝❛
❝❤♦ ♣❤÷ì♥❣ ♣❤→♣ ❞➙② ❝✉♥❣ ❣✐↔✐ ❣➛♥ ✤ó♥❣ ♣❤÷ì♥❣ tr➻♥❤✔ ❝❤♦ ❧✉➟♥ ✈➠♥
❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P
❙❱❚❍✿ ❍❯Ý◆❍ ❚❍➚ ▼ß ❍❸◆❍
✺
tèt ♥❣❤✐➺♣ ❝õ❛ ♠➻♥❤✳
✷✳ ▼ö❝ ✤➼❝❤ ♥❣❤✐➯♥ ❝ù✉
▼ö❝ ✤➼❝❤ ❝õ❛ ✤➲ t➔✐ ❧➔ sû ❞ư♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❞➙② ❝✉♥❣ ✤➸ ①❡♠ ①➨t
✈➔ t➻♠ ♥❣❤✐➺♠ ❣➛♥ ✤ó♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤✱ tø õ s s s số ợ
ừ ữỡ tr õ ỗ tớ ự ự ử
tt ✤➸ ✈✐➳t ❝❤÷ì♥❣ tr➻♥❤ t➻♠ ♥❣❤✐➺♠ ❣➛♥ ✤ó♥❣ t❤❡♦
♣❤÷ì♥❣ ♣❤→♣ ❞➙② ❝✉♥❣ ✈➔ ♠ỉ t↔ ♥❣❤✐➺♠ ❝❤➼♥❤ ①→❝ ❝õ❛ ♣❤÷ì♥❣ tr
ỗ t tổ q õ ✤÷đ❝ ❧➟♣ tr➻♥❤✳
✸✳ ✣è✐ t÷đ♥❣ ✈➔ ♣❤↕♠ ✈✐ ♥❣❤✐➯♥ ❝ù✉
✣è✐ t÷đ♥❣ ♥❣❤✐➯♥ ❝ù✉✿ ◆❣❤✐➯♥ ❝ù✉ ♣❤÷ì♥❣ ♣❤→♣ ❞➙② ❝✉♥❣ ✤➸ t➻♠
♥❣❤✐➺♠ ❣➛♥ ✤ó♥❣ ❝❤♦ ♣❤÷ì♥❣ tr➻♥❤ ✈➔ ❧➟♣ tr➻♥❤ ♣❤÷ì♥❣ ♣❤→♣ ❞➙② ❝✉♥❣
tr♦♥❣ ▼❛t❤❡♠❛t✐❝❛✳
P❤↕♠ ✈✐ ♥❣❤✐➯♥ ❝ù✉✿ ◆❣❤✐➯♥ ❝ù✉ ♣❤÷ì♥❣ ♣❤→♣ ❞➙② ❝✉♥❣ ❝❤♦ ❝→❝
♣❤÷ì♥❣ tr➻♥❤ ♣❤✐ t✉②➳♥✳
P❤÷ì♥❣ ♣❤→♣ ♥❣❤✐➯♥ ❝ù✉✿ ❚➻♠ ✤å❝ t➔✐ ❧✐➺✉ ✈➲ ♣❤÷ì♥❣ ♣❤→♣ ❞➙② ❝✉♥❣
✈➔ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ t➼♥❤ ①➜♣ ①➾ ❦❤→❝❀ ♣❤➙♥ t➼❝❤ t➔✐ ❧✐➺✉❀ ❤➺ t❤è♥❣ ❤â❛❀
❦❤→✐ q✉→t ❤â❛ t➔✐ ❧✐➺✉ ✈➔ ❦✐➸♠ ❝❤ù♥❣✳
✹✳ Þ ♥❣❤➽❛ ❦❤♦❛ ❤å❝ ✈➔ t❤ü❝ t✐➵♥ ❝õ❛ ✤➲ t
t õ ỵ t ỵ tt õ t❤➸ sû ❞ư♥❣ ♥❤÷ ❧➔ t➔✐ ❧✐➺✉
t❤❛♠ ❦❤↔♦ ❞➔♥❤ ❝❤♦ s✐♥❤ ✈✐➯♥ ✈➔ ❝→❝ ✤è✐ t÷đ♥❣ ❝â ♠è✐ q✉❛♥ t➙♠ ✤➳♥
♣❤÷ì♥❣ ♣❤→♣ ❞➙② ❝✉♥❣ ✈➔ ♣❤➛♥ ♠➲♠ ▼❛t❤❡♠❛t✐❝❛✳
✺✳ ❈➜✉ tró❝
t
ỗ ✸ ❝❤÷ì♥❣
❈❤÷ì♥❣ ✶✿ ▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à✳
❈❤÷ì♥❣ ✷✿ P❤÷ì♥❣ ♣❤→♣ ❞➙② ❝✉♥❣ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ t➻♠ ♥❣❤✐➺♠ ❣➛♥
✤ó♥❣✳
❈❤÷ì♥❣ ✸✿ Ù♥❣ ❞ư♥❣ ♣❤➛♥ ♠➲♠ ▼❛t❤❡♠❛t✐❝❛ ❝❤♦ ♣❤÷ì♥❣ ♣❤→♣ ❞➙②
❝✉♥❣✳
❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P
❙❱❚❍✿ ❍❯Ý◆❍ ❚❍➚ ▼ß ❍❸◆❍
✻
▲❮■ ❈❷▼ ❒◆
❊♠ ①✐♥ ❜➔② tä sü ❜✐➳t ì♥ ❝❤➙♥ t❤➔♥❤ ✤➳♥ ❇❛♥ ●✐→♠ ❍✐➺✉ tr÷í♥❣ ✣↕✐
❤å❝ ❙÷ P❤↕♠ ✲ ✣↕✐ ❍å❝ ✣➔ ◆➤♥❣✱ ❇❛♥ ❝❤õ ♥❤✐➺♠ ❦❤♦❛ ❚♦→♥✱ ✤➣ t↕♦ ❝ì
❤ë✐ ❝❤♦ ❝❤ó♥❣ ❡♠ ✤÷đ❝ ❧➔♠ ❧✉➟♥ ✈➠♥ tèt ♥❣❤✐➺♣✳ ❈❤ó♥❣ ❡♠ ①✐♥ ❣û✐ ❧í✐
❝↔♠ ì♥✱ ❧í✐ tr✐ ➙♥ s➙✉ s➢❝ ✤➳♥ t➜t ❝↔ ❝→❝ t❤➛② ❝æ ❣✐→♦ tr♦♥❣ tr÷í♥❣✱ ✤➦❝
❜✐➺t ❧➔ ❝→❝ t❤➛② ❝ỉ ❣✐→♦ tr♦♥❣ ❦❤♦❛ ❚♦→♥ ✤➣ t➟♥ t➻♥❤ ❝❤➾ ❞↕②✱ tr✉②➲♥ ✤↕t
❝❤♦ ❝❤ó♥❣ ❡♠ ♥❤ú♥❣ tự ờ qỵ tr sốt tớ ❣✐❛♥
✈ø❛ q✉❛✳ ❳✐♥ ❝↔♠ ì♥ sü ❣✐ó♣ ✤ï✱ ❝❤✐❛ s➫ ❝õ❛ t➜t ❝↔ ❝→❝ ❜↕♥ tr♦♥❣ ❧ỵ♣
tr♦♥❣ t❤í✐ ❣✐❛♥ ❝❤ó♥❣ ❡♠ ❧➔♠ ♥❣❤✐➯♥ ❝ù✉✳
❈✉è✐ ❝ò♥❣✱ ❡♠ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ỡ t r ữớ
trỹ t ữợ ❞➝♥ ❝❤ó♥❣ ❡♠✱ ❧✉ỉ♥ q✉❛♥ t➙♠✱ ✤ë♥❣ ✈✐➯♥ ❝❤➾ ❞➝♥ t➟♥
t➻♥❤ ✤➸ ❝❤ó♥❣ ❡♠ ❤♦➔♥ t❤➔♥❤ tèt ❧✉➟♥ ✈➠♥ tèt ♥❣❤✐➺♣ ♥➔②✳
❚✉② ✤➣ ❝â ♥❤✐➲✉ ❝è ❣➢♥❣ s♦♥❣ ❧✉➟♥ ✈➠♥ tèt ♥❣❤✐➺♣ ✈➝♥ ❦❤æ♥❣ tr→♥❤
❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉ sât ✈➲ ♥ë✐ ❞✉♥❣ ❧➝♥ ❤➻♥❤ t❤ù❝ tr➻♥❤ ❜➔②✱ ❝❤ó♥❣ ❡♠ r➜t
♠♦♥❣ ♥❤➟♥ ữủ sỹ õ õ ừ qỵ t ổ ✤å❝✳
❊♠ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✦
❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P
❙❱❚❍✿ ❍❯Ý◆❍ ❚❍➚ ▼ß ❍❸◆❍
✼
❈❤÷ì♥❣ ✶
▼❐❚ ❙➮ ❑■➌◆ ❚❍Ù❈ ❈❍❯❽◆ ❇➚
✶✳✶ ❑❍⑩■ ◆■➏▼ ❱➋ ❙➮ ●❺◆ ✣Ĩ◆●
✶✳✶✳✶ ❙❛✐ sè t✉②➺t ✤è✐✱ s❛✐ sè t÷ì♥❣ ố
r t t t tữớ ợ ❣✐→ trà ❣➛♥ ✤ó♥❣ ❝õ❛
a ❧➔ sè ❣➛♥ ✤ó♥❣ ❝õ❛ a∗ ♥➳✉ a ❦❤ỉ♥❣ s❛✐ ❦❤→❝ a∗
∗
♥❤✐➲✉✳ ✣↕✐ ❧÷đ♥❣ ∆ := |a − a | ❣å✐ ❧➔ s❛✐ sè t❤➟t sü ❝õ❛ a✳ ❉♦ ❦❤ỉ♥❣ ❜✐➳t
a∗ ♥➯♥ t❛ ❝ơ♥❣ ❦❤ỉ♥❣ ❜✐➳t ∆✳ ❚✉② ♥❤✐➯♥✱ t❛ ❝â t❤➸ t➻♠ ✤÷đ❝ ∆a ≥ 0 ✱
❣å✐ ❧➔ s❛✐ sè t✉②➺t ✤è✐ ❝õ❛ a✱ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥✿
❝→❝ ✤↕✐ ❧÷đ♥❣✳ ❚❛ ♥â✐
|a − a∗ | ≤ ∆a
❤❛②
a − ∆a ≤ a∗ ≤ a + ∆a✳
✣÷ì♥❣ ♥❤✐➯♥
❝➔♥❣ ♥❤ä ❝➔♥❣ tèt✳ ❙❛✐ sè t÷ì♥❣ ✤è✐ ❝õ❛
δa :=
a
∆a
✭✶✳✶✮
t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ✭✶✳✶✮
❧➔
∆a
|a|
✶✳✶✳✷ ❙❛✐ sè t❤✉ ❣å♥
▼ët sè t❤➟♣ ♣❤➙♥
a
❝â ❞↕♥❣ tê♥❣ q✉→t ♥❤÷ s❛✉✿
a = ±(βp 10p + βp−1 10p−1 + ... + βp−s 10p−s )
0 ≤ βi ≤ 9(i = p − 1, p − s); βp > 0 ❧➔ ♥❤ú♥❣ sè ♥❣✉②➯♥✳ ◆➳✉
p − s ≥ 0 t❤➻ a ❧➔ sè ♥❣✉②➯♥❀ p − s = m(m > 0) t a õ ỗ
m ❝❤ú sè✳ ◆➳✉ s = +∞✱ a ❧➔ sè t❤➟♣ ♣❤➙♥ ✈æ ❤↕♥✳ ❚❤✉ ❣å♥ ♠ët sè a ❧➔
✈ùt ❜ä ♠ët sè ❝→❝ ❝❤ú sè ❜➯♥ ♣❤↔✐ a ✤➸ ✤÷đ❝ ♠ët sè a ♥❣➢♥ ❣å♥ ❤ì♥ ✈➔
❣➛♥ ✤ó♥❣ ♥❤➜t ✈ỵ✐ a✳
❚r♦♥❣ ✤â
❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P
❙❱❚❍✿ ❍❯Ý◆❍ ❚❍➚ ▼ß ❍❸◆❍
✽
◗✉✐ t➢❝ t❤✉ ❣å♥✿ ●✐↔ sû
a = (βp 10p + ... + βj 10j + ... + βp−s 10p−s )
✈➔ t❛ ❣✐ú ❧↕✐ ✤➳♥ sè ❤↕♥❣ t❤ù ❥✳ ●å✐ ♣❤➛♥ ✈ùt ❜ä ❧➔
µ✱
t❛ ✤➦t
a = βp 10p + ... + βj+1 10j+1 + βj 10j )
tr♦♥❣ ✤â✿
βj + 1 ♥➳✉
βj
j :=
à = 0.5 ì 10j
t
j = j
0.5 ì 10j < à < 10j ,
0 < à < 0.5 ì 10j ,
βj
βj = βj+1
❧➔ ❝❤➤♥ ✈➔
✭✶✳✷✮
♥➳✉
βj
❧➫ ✈➻
t➼♥❤ t♦→♥ ✈ỵ✐ sè ❝❤➤♥ t❤✉➟♥ t✐➺♥ ❤ì♥✳
✶✳✷ ❙❆■ ❙➮ ❚➑◆❍ ❚❖⑩◆
❚r♦♥❣ t➼♥❤ t♦→♥ t❛ t❤÷í♥❣ ❣➦♣ ✹ ❧♦↕✐ s❛✐ sè s❛✉✿
❛✮ ❙❛✐ sè tt ổ õ ỵ tữ õ ❜➔✐ t♦→♥ t❤ü❝ t➳✳
❙❛✐ sè ♥➔② ❦❤ỉ♥❣ ❧♦↕✐ trø ✤÷đ❝✳
❜✮ ❙❛✐ sè ♣❤÷ì♥❣ ♣❤→♣ ✲ ❈→❝ ❜➔✐ t♦→♥ t❤÷í♥❣ ❣➦♣ r➜t ♣❤ù❝ t↕♣✱ ❦❤ỉ♥❣
t❤➸ ❣✐↔✐ ✤ó♥❣ ✤÷đ❝ ♠➔ ♣❤↔✐ sû ❞ư♥❣ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ❣➛♥ ✤ó♥❣✳ ❙❛✐ sè
♥➔② s➩ ✤÷đ❝ ♥❣❤✐➯♥ ❝ù✉ ❝❤♦ tø♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❝ư t❤➸✳
❝✮ ❙❛✐ sè ❝→❝ sè ❧✐➺✉ ✲ ❈→❝ sè ❧✐➺✉ t❤÷í♥❣ t❤✉ ✤÷đ❝ ❜➡♥❣ t❤ü❝ ♥❣❤✐➺♠
❞♦ ✤â ❝â s❛✐ sè✳
❞✮ ❙❛✐ sè t➼♥❤ t♦→♥ ✲ ❈→❝ sè ✈è♥ ✤➣ ❝â s❛✐ sè✱ ❝á♥ t❤➯♠ s❛✐ sè t❤✉ ❣å♥
♥➯♥ ❦❤✐ t➼♥❤ t♦→♥ s➩ ①✉➜t ❤✐➺♥ s❛✐ sè t➼♥❤ t♦→♥✳
●✐↔ sû ♣❤↔✐ t➻♠ ✤↕✐ ❧÷đ♥❣ ② t❤❡♦ ❝æ♥❣ t❤ù❝✿
y = f (x1 , x2 , ..., xn )
●å✐
x∗i , y ∗ (i = 1, n)
✈➔
❝→❝ ✤è✐ sè ✈➔ ❤➔♠ sè✳
xi , y(i = 1, n) ❧➔ ❝→❝ ❣✐→ trà
◆➳✉ f ❦❤↔ ✈✐ ❧✐➯♥ tư❝ t❤➻
✤ó♥❣ ✈➔ ❣➛♥ ✤ó♥❣ ❝õ❛
n
∗
|y − y | = |f (x1 , ..., xn ) −
f (x∗1 , ..., x∗n )|
|fi ||xi − x∗i |.
=
i=1
df
fi ❧➔ ✤↕♦ ❤➔♠ dx
t➼♥❤ t↕✐
i
∆xi ❦❤→ ❜➨ t❛ ❝â t❤➸ ❝♦✐
tr♦♥❣ ✤â
tư❝ ✈➔
❝→❝ t❤í✐ ✤✐➸♠ tr✉♥❣ ❣✐❛♥✳ ❉♦
df
dxi ❧✐➯♥
n
|fi (x1 , ..., xn )|∆xi .
∆y =
i=1
❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P
❙❱❚❍✿ ❍❯Ý◆❍ ❚❍➚ ▼ß ❍❸◆❍
✾
❉♦ ✤â
∆y
δy =
=
|y|
n
|
i=1
d
ln f |∆xi .
dxi
✶✳✸ ◆●❍■➏▼ ❱⑨ ❑❍❖❷◆● P❍❹◆ ▲■ ◆●❍■➏▼
✶✳✸✳✶ ◆❣❤✐➺♠ t❤ü❝ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ♠ët ➞♥
❳➨t ♣❤÷ì♥❣ tr➻♥❤ ♠ët ➞♥✿
f (x) = 0
tr♦♥❣ ✤â
f
✭✶✳✸✮
❧➔ ♠ët ❤➔♠ sè trữợ ừ ố số
x
tỹ ừ ữỡ tr ❧➔ sè t❤ü❝
❧➔ ❦❤✐ t❤❛②
α
✈➔♦
x
α
t❤ä❛ ♠➣♥ ✭✶✳✸✮ tù❝
ð ✈➳ tr→✐ t ữủ
f () = 0.
ị ồ ừ
ỗ t ừ số
y = f (x)
tr ♠ët ❤➺ tå❛ ✤ë ✈✉æ♥❣ ❣â❝ ❖①② ✭❍➻♥❤ ✶✳✶✮✳ ●✐↔ sỷ ỗ t t trử
t ởt
x =
M
t ✤✐➸♠
M
♥➔② ❝â t✉♥❣ ✤ë
y=0
✈➔ ❤♦➔♥❤ ✤ë
❚❤❛② ❝❤ó♥❣ ✈➔♦ ✭✶✳✺✮ t❛ ✤÷đ❝✿
0 = f (α)
❱➟② ❤♦➔♥❤ ✤ë
α
❝õ❛ ❣✐❛♦ ✤✐➸♠
M
✭✶✳✻✮
❝❤➼♥❤ ❧➔ ♠ët ♥❣❤✐➺♠ ❝õ❛ ✭✶✳✸✮✳
❍➻♥❤ ✶✳✶✿ Þ ♥❣❤➽❛ ❤➻♥❤ ❤å❝ ❝õ❛ ♥❣❤✐➺♠
❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P
❙❱❚❍✿ ❍❯Ý◆❍ ❚❍➚ ▼ß ❍❸◆❍
rữợ ỗ t t ụ õ t t ữỡ tr
ữỡ tr tữỡ ữỡ
g(x) = h(x)
ỗ ỗ t ừ số
y = g(x), y = h(x)
sỷ ỗ t t t↕✐ ✤✐➸♠ ▼ ❝â ❤♦➔♥❤ ✤ë
✭✶✳✽✮
x=α
g(α) = h(α)
t❤➻ t❛ ❝â✿
✭✶✳✾✮
❍➻♥❤ ỗ t số g(x), h(x)
ở
ừ ừ ỗ t ởt
ừ tự ừ
ỹ tỗ t tỹ ừ ữỡ tr ởt
rữợ t t ❣➛♥ ✤ó♥❣ ♥❣❤✐➺♠ t❤ü❝ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✸✮
t❛ ♣❤↔✐ tü ọ tỹ õ tỗ t ổ tr ớ t
õ t ũ ữỡ ỗ t ð ♠ư❝ ✶✳✸✳✷ tr➯♥✳ ❚❛ ❝ơ♥❣ ❝â t❤➸ ❞ị♥❣
✤à♥❤ ❧➼ s
ỵ
õ số tỹ
a b(a < b) s❛♦ ❝❤♦ f (a) ✈➔ f (b) tr→✐
❞➜✉ tù❝
f (a).f (b) < 0
ỗ tớ
f (x)
tử tr
[a, b]
t ð tr♦♥❣ ❦❤♦↔♥❣
(a, b)
❝â ➼t ♥❤➜t
♠ët ♥❣❤✐➺♠ t❤ü❝ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✸✮✳
❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P
❙❱❚❍✿ ❍❯Ý◆❍ ❚❍➚ ▼ß ❍❸◆❍
✶✶
❈❤ù♥❣ ♠✐♥❤✿
❈❤✐❛ ✤♦↕♥
[a, b]
•
◆➳✉
f ( a+b
2 ).f (a) > 0
t❤➻ ✤➦t
a1 =
•
◆➳✉
f ( a+b
2 ).f (a) < 0
t❤➻ ✤➦t
a1 = a; b1 =
❈❤✐❛ ✤♦↕♥
[a1 , b1 ]
a+b
2 ✳
t❤➔♥❤ ❤❛✐ ♣❤➛♥ ❜➡♥❣ ♥❤❛✉ ❜ð✐ ✤✐➸♠
a+b
2 ; b1
= b✳
a+b
2 ✳
t❤➔♥❤ ❤❛✐ ♣❤➛♥ ❜➡♥❣ ♥❤❛✉ ❜ð✐ ✤✐➸♠
•
◆➳✉
1
f ( a1 +b
2 ).f (a1 ) > 0
t❤➻ ✤➦t
a2 =
a1 +b1
2 ; b2
= b1 ✳
•
◆➳✉
1
f ( a1 +b
2 ).f (a1 ) < 0
t❤➻ ✤➦t
a2 = a1 ; b2 =
a1 +b1
2 ✳
a1 +b1
2 ✳
...
{an }, {bn } ♠➔ {an }
✈ỵ✐ f (a)✱ f (bn ) ❝ị♥❣
❈ù t✐➳♣ tư❝ q✉→ tr➻♥❤ tr➯♥ t❛ ①→❝ ✤à♥❤ ✤÷đ❝ ❤❛✐ ❞➣②
❧➔ ❞➣② t➠♥❣✱
{bn }
❧➔ ❞➣② ❣✐↔♠ ✈➔
f (b)✳
⇒ f (an ).f (bn ) < 0, ∀n ∈ N ✳
❉➣② {an } t➠♥❣✱ ❜à ❝❤➦♥ tr➯♥
f (an )
❝ị♥❣ ❞➜✉
❞➜✉ ✈ỵ✐
❉➣②
{bn }
❜ð✐
b ⇒ lim an
tỗ t
ữợ
a lim bn
tỗ t↕✐✳
n→∞
n→∞
α = lim an ✱ β = lim bn ✱
n→∞
n→∞
⇒ lim (bn − an ) = β − α✳
✣➦t
n→∞
bn − an = b−a
2n ; ∀n ∈ N ✳
♥➯♥ lim (bn − an ) = 0 ⇒ β − α = 0 ⇒ β = α✳
n→∞
⇒ lim bn = lim an = α = β ✳
n→∞
n→∞
⇒ lim f (bn ) = f (α) = lim f (an )✳
▼➔
n→∞
n→∞
⇒ 0 ≥ f (α) ≥ 0 ⇒ f (α) = 0
✈➔
α ∈ (a, b)✳
❱➟② ✤à♥❤ ❧➼ ✶✳✶ ✤➣ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤
✣✐➲✉ ✤â ❝â t ữủ ồ tr ỗ t ỗ t❤à ❝õ❛ ❤➔♠
sè
y = f (x)
t↕✐
a≤x≤b
❧➔ ♠ët ✤÷í♥❣ ❧✐➲♥ ♥è✐
ữợ tr trö❝ ❤♦➔♥❤✱ ♥➯♥ ♣❤↔✐ ❝➢t trö❝ ❤♦➔♥❤ t↕✐ ➼t ♥❤➜t ♠ët
✤✐➸♠ ð tr♦♥❣ ❦❤♦↔♥❣ tø ❛ ✤➳♥ ❜✳ ❱➟② ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✸✮ ❝â ➼t ♥❤➜t ♠ët
♥❣❤✐➺♠ ð tr♦♥❣ ❦❤♦↔♥❣
(a, b)✳
❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P
❙❱❚❍✿ ❍❯Ý◆❍ ❚❍➚ ▼ß ❍❸◆❍
ỗ t ừ số y(x) t a ≤ x ≤ b
✶✳✸✳✹ ❑❤♦↔♥❣ ♣❤➙♥ ❧✐ ♥❣❤✐➺♠
✣à♥❤ ♥❣❤➽❛ ✶✳✶✳
❑❤♦↔♥❣
(a, b)
♥➔♦ ✤â ❣å✐ ❧➔ ❦❤♦↔♥❣ ♣❤➙♥ ❧✐ ♥❣❤✐➺♠
❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✸✮ ♥➳✉ ♥â ❝❤ù❛ ♠ët ✈➔ ❝❤➾ ♠ët ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣
tr➻♥❤ ✤â✳
✣➸ t➻♠ ❦❤♦↔♥❣ ♣❤➙♥ ❧✐ ♥❣❤✐➺♠ t❛ ❝â ✤à♥❤
ỵ
(a, b) ởt
tớ f (a) f (b) tr
ỡ ỗ
tr õ số
f (x)
tö❝
❞➜✉✱ tù❝ ❧➔ ❝â ✭✶✳✶✵✮ t❤➻
(a, b)
❧➔ ♠ët ❦❤♦↔♥❣ ♣❤➙♥ ❧✐ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✸✮✳
❈❤ù♥❣ ♠✐♥❤✿
❚ø ❣✐↔ t❤✐➳t✱ ✈➻
f (x)
❧✐➯♥ tư❝ ✈➔ ✤ì♥ ✤✐➺✉ ♥➯♥ tr➯♥
(a, b)✱ f (x)
t➠♥❣
❤♦➦❝ ❣✐↔♠✳
❍ì♥ ♥ú❛✱ tø ✤✐➲✉ ❦✐➺♥
❤➔♠ sè
f (x)
f (a).f (b) < 0
ự tọ út ừ ỗ t
♣❤➼❛ ❝õ❛ trư❝ ❤♦➔♥❤✳
❑➳t ❤đ♣ ✈ỵ✐ t➼♥❤ ✤ì♥ ✤✐➺✉ ❝õ❛ ❤➔♠ sè t❛ s✉② r❛
❧✐ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤
(a, b)
❧➔ ❦❤♦↔♥❣ ♣❤➙♥
f (x) = 0✳
❱➟② ✤à♥❤ ❧➼ ✶✳✷ ✤➣ ✤÷đ❝ ự
õ t ồ ỗ t ỗ t ừ
y = f (x) t trö❝ ❤♦➔♥❤ t↕✐ ♠ët ✈➔ ❝❤➾ ♠ët ✤✐➸♠
(a, b) ❝❤ù❛ ♠ët ✈➔ ❝❤➾ ♠ët ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤
sè
❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P
ð tr♦♥❣
(a, b)✳
❱➟②
✭✶✳✸✮✳
❙❱❚❍✿ ❍❯Ý◆❍ ❚❍➚ ▼ß ❍❸◆❍
✶✸
❍➻♥❤ ✶✳✹✿ ❑❤♦↔♥❣ ♣❤➙♥ ❧✐ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ f (x) = 0
f (x) ❝â ✤↕♦ ❤➔♠ t❤➻ ✤✐➲✉ ❦✐➺♥ ✤ì♥ ✤✐➺✉ ❝â t❤➸
❦❤ỉ♥❣ ✤ê✐ ❞➜✉ ❝õ❛ ✤↕♦ ❤➔♠ ✈➻ ✤↕♦ ❤➔♠ ❦❤ỉ♥❣ ✤ê✐
◆➳✉
❦✐➺♥
t❤❛② ❜➡♥❣ ✤✐➲✉
❞➜✉ t❤➻ ❤➔♠ sè
✤ì♥ ✤✐➺✉✳ õ
ỵ
f (x)
(a, b) ởt tr ✤â ❤➔♠ f (x) ❧✐➯♥ tö❝✱ ✤↕♦
✤ê✐ ❞➜✉ ✈➔ f (a)✱ f (b) tr→✐ ❞➜✉ t❤➻ (a, b) ❧➔ ♠ët ❦❤♦↔♥❣
◆➳✉
❦❤ỉ♥❣
♣❤➙♥ ❧✐ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✸✮✳
▼✉è♥ t➻♠ ❝→❝ ❦❤♦↔♥❣ ♣❤➙♥ ❧✐ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✸✮ t❤÷í♥❣
♥❣÷í✐ t❛ ♥❣❤✐➯♥ ❝ù✉ sü ❜✐➳♥ t❤✐➯♥ ❝õ❛ ❤➔♠ sè
y = f (x) rỗ ử
ử
ữỡ tr
f (x) = x3 − x − 1.
✭✶✳✶✶✮
❍➣② ❝❤ù♥❣ tä ♣❤÷ì♥❣ tr➻♥❤ õ tỹ t
rữợ t t❛ ①➨t sü ❜✐➳♥ t❤✐➯♥ ❝õ❛ ❤➔♠ sè
tö❝ t↕✐ ♠å✐
x✱
f (x)
õ
ỗ tớ
1
f (x) = 3x2 1 = 0 ⇐⇒ x = ± √ .
3
❚❛ ❝â ❜↔♥❣ ❜✐➳♥ t❤✐➯♥ ♥❤÷ tr♦♥❣ ❍➻♥❤ ✶✳✺✿
tr♦♥❣ ✤â✿
1
1
1
2
f (M ) = f (− √ ) = − √ + √ − 1 = √ − 1 < 0
3
3 3 3 3
3 3
❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P
❙❱❚❍✿ ❍❯Ý◆❍ ❚❍➚ ▼ß ❍❸◆❍
✶✹
❍➻♥❤ ✶✳✺✿ ❇↔♥❣ ❜✐➳♥ t❤✐➯♥ ❝õ❛ ❤➔♠ sè f (x) = x3 x 1
ỗ t t trử ❤♦➔♥❤ t↕✐ ♠ët ✤✐➸♠ ❞✉② ♥❤➜t ✭❍➻♥❤ ✶✳✻✮✱ ❞♦ ✤â
♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✶✶✮ ❝â ♠ët ♥❣❤✐➺♠ t❤ü❝ ❞✉② ♥❤➜t✱ ❦➼ ❤✐➺✉ õ
ỗ t số f (x) = x3 − x − 1 tr♦♥❣ ❦❤♦↔♥❣ [1, 2]
f (1) = 13 − 1 − 1 = −1 < 0❀ f (2) = 23 − 2 − 1 = 5 > 0✳
◆❤÷ ✈➟②✱ f (1).f (2) < 0✳
❱➟② ❦❤♦↔♥❣ (1, 2) ❝❤ù❛ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✶✶✮✳
❚❛ t➼♥❤ t❤➯♠✿
❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P
❙❱❚❍✿ ❍❯Ý◆❍ ❚❍➚ ▼ß ❍❸◆❍
ữỡ
Pì PP
ể ế
Pì
●■❰■ ❚❍■➏❯
✷✳✶✳✶ ✣➦t ✈➜♥ ✤➲
❈❤ó♥❣ t❛ t❤÷í♥❣ t➻♠ ❤✐➸✉ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ t➻♠ ♥❣❤✐➺♠ trü❝ t✐➳♣✳ ◆➳✉
♠å✐ t➼♥❤ t♦→♥ ❝õ❛ t❛ ❧➔ ❝❤➼♥❤ ①→❝ t❤➻ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ✤â ❝❤♦ ❦➳t q✉↔
❤♦➔♥ t♦➔♥ ❝❤➼♥❤ ①→❝✳ ❚✉② ♥❤✐➯♥ tr♦♥❣ t❤ü❝ t➳ ❦❤✐ t➼♥❤ t♦→♥ ❝❤ó♥❣ t❛
t❤÷í♥❣ ①✉②➯♥ ♣❤↔✐ ❧➔♠ trá♥ ❝→❝ sè✱ ♥❣❤➽❛ ❧➔ t❛ ❝❤➾ t➼♥❤ t♦→♥ tr➯♥ ❝→❝
sè ❣➛♥ ú tổ õ ữ rt ợ t q ố ũ
ố ợ ữỡ tr số t✉②➳♥ t➼♥❤ ✏♥❤↕② ❝↔♠✑ ✈ỵ✐ s❛✐ sè✳ ❱➻ ✈➟②✱
❝❤ó♥❣ t❛ ♥➯♥ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ❣➛♥ ✤ó♥❣ ✤➸ t➻♠ ♥❣❤✐➺♠ ❝õ❛
♠ët ♣❤÷ì♥❣ tr➻♥❤✳
❈❤♦ ♣❤÷ì♥❣ tr➻♥❤✿
f (x) = 0
tr♦♥❣ ✤â
f
✭✷✳✶✮
❧➔ ♠ët ❤➔♠ ✤↕✐ sè ❤♦➦❝ s✐➯✉ ✈✐➺t✳ ❚➻♠ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣
tr➻♥❤ ✭✷✳✶✮ ❧➔ ♠ët ❜➔✐ t♦→♥ t❤÷í♥❣ ❣➦♣ tr♦♥❣ ❦ÿ t❤✉➟t✳ ◆➳✉ ♣❤÷ì♥❣ tr➻♥❤
✭✶✳✸✮ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ✤↕✐ sè ❜➟❝ ♥ ❝â ❞↕♥❣✿
a0 xn + a1 xn−1 + ... + an−1 x + an = 0(a0 = 0),
t❤➻ ✈ỵ✐
n = 1, n = 2✱
✭✷✳✷✮
t❛ ❝â ❝æ♥❣ t❤ù❝ t➼♥❤ ♥❣❤✐➺♠ ♠ët ❝→❝❤ ✤ì♥ ❣✐↔♥✳
◆❣÷í✐ t❛ ❝ơ♥❣ t➻♠ r❛ ♥❤ú♥❣ ❝ỉ♥❣ t❤ù❝ t➼♥❤ ♥❣❤✐➺♠ ❝õ❛ ✭✷✳✷✮ ❦❤✐
❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P
n=3
❙❱❚❍✿ ❍❯Ý◆❍ ❚❍➚ ▼ß ❍❸◆❍
✶✻
✈➔
n = 4✱ ♥❤÷♥❣ ✈✐➺❝ sû ❞ư♥❣ r➜t ♣❤ù❝ t↕♣✳ ỏ ợ ỳ ữỡ tr
số tứ tr ❧➯♥ ❤♦➦❝ ♣❤÷ì♥❣ tr➻♥❤ s✐➯✉ ✈✐➺t t❤➻ ❦❤ỉ♥❣ ❝â ❝ỉ♥❣
t❤ù❝ t➼♥❤ ♥❣❤✐➺♠✳ ❱➻ ✈➟②✱ ✈✐➺❝ t➻♠ ♥❤ú♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❣➛♥ ✤ó♥❣ ✤➸ ❣✐↔✐
♣❤÷ì♥❣ tr➻♥❤ ✤↕✐ sè ✈➔ s✐➯✉ ✈✐➺t ❝ơ♥❣ ♥❤÷ ✈✐➺t ✤→♥❤ ❣✐→ ♠ù❝ ✤ë ❝❤➼♥❤
①→❝ ❝õ❛ ♥❣❤✐➺♠ ❣➛♥ ✤ó♥❣ t➻♠ ✤÷đ❝ ❝â ♠ët ✈❛✐ trá q✉❛♥ trå♥❣✳
✷✳✶✳✷ ❈→❝❤ ❣✐↔✐ q✉②➳t
❚❤ỉ♥❣ t❤÷í♥❣✱ q✉→ tr➻♥❤ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✶✮ ❜❛♦ ỗ ữợ s
ã
ữợ sỡ ở é t t ởt ừ ự
ừ
ã
f (x)
ữợ ❣✐↔✐ ❦✐➺♥ t♦➔♥✿ ❚➻♠ ♥❣❤✐➺♠ ✈ỵ✐ ✤ë ❝❤➼♥❤ ①→❝ ❝➛♥ t❤✐➳t✳
✣➸ ❣✐↔✐ sì ❜ë ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✶✮ t❛ ❝â t❤➸ sû ❞ư♥❣ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ✤ì♥
❣✐↔♥ ♥❤÷ ♣❤÷ì♥❣ ♣❤→♣ ❝❤✐❛ ổ ữỡ ỗ t
Pữỡ ổ
sû ❤➔♠ sè f (x) ❧✐➯♥ tö❝ tr➯♥ ✤♦↕♥ ❬❛✱ ❜❪ ✈➔ f (a).f (b) < 0.
●å✐ ∆0 := [a, b]✱ t❛ ❝❤✐❛ ✤æ✐ ∆0 ✈➔ ❝❤å♥ ∆1 := [a1 , b1 ] ❧➔ ♠ët tr♦♥❣ ❤❛✐
♥û❛ ❝õ❛ ∆0 s❛♦ ❝❤♦ f (a1 ).f (b1 ) ≤ 0.
◆â✐ ❝❤✉♥❣ ð ữợ tự n t õ
n = [an , bn ] ⊂ ∆n−1 ⊂ ... ⊂ ∆0 .
bn − an =
(b−a)
2n
→ 0✭❦❤✐ n → ∞✮✳ ❉➵ t❤➜② ❞➣② an ✤ì♥ ✤✐➺✉
t➠♥❣✱ ❜à ❝❤➦♥ tr➯♥ ❜ð✐ b ❝á♥ ❞➣② bn ✤ì♥ ✤✐➺✉ ữợ a
ỡ ỳ bn an → 0 s✉② r❛ an , bn → α(n → ∞).
2
❱➻ f (an ).f (bn ) ≤ 0 ♥➯♥ ❝❤♦ n → ∞✱ t❛ ❝â [f (α)] ≤ 0✱ s✉② r❛ f (α) = 0✳
◆❣♦➔✐ r❛
◆❣♦➔✐ r❛✱ t❛ ❝â ÷ỵ❝ ❧÷đ♥❣ s❛✐ sè s❛✉✿
0 ≤ α − an ≤ b n an =
ba
.
2n
ì ừ ữỡ ✤ỉ✐ ❧➔ t❤✉➟t t♦→♥ r➜t ✤ì♥ ❣✐↔♥✱ ❞♦ ✤â ❞➵
❧➟♣ tr➻♥❤ tr➯♥ ♠→② t➼♥❤✳ ▼➦t ❦❤→❝✱ ✈➻ ♣❤÷ì♥❣ ♣❤→♣ ❝❤✐❛ ✤ỉ✐ sû ❞ư♥❣ r➜t
f ♥➯♥ tè❝ ✤ë ❤ë✐ tư ❦❤→
Pữỡ ỗ t
ỗ t số y = f (x) tr➯♥ ❣✐➜② ❦➫ æ ✈✉æ♥❣✳
➼t t❤æ♥❣ t✐♥
ở ừ
ừ ỗ t õ tr ✈ỵ✐ trư❝ ❤♦➔♥❤ ❝❤➼♥❤ ❧➔ ♥❣❤✐➺♠ ❝➛♥ t➻♠✳ ◆❤✐➲✉
❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P
❙❱❚❍✿ ❍❯Ý◆❍ ❚❍➚ ▼ß ❍❸◆❍
✶✼
f (x) = 0 ✈➲ ❞↕♥❣ t÷ì♥❣ ✤÷ì♥❣ ϕ(x) = (x)
ở ừ ừ ỗ t y = ϕ(x) ✈➔
❦❤✐ t❛ ❜✐➯♥ ✤ê✐ ♣❤÷ì♥❣ tr➻♥❤
◆❣❤✐➺♠ ❝➛♥ t➻♠ ❧➔ ❤♦➔♥❤
y = ψ(x)✳
❱➼ ❞ö ✷✳✶✿
❚➻♠ ❦❤♦↔♥❣ ♣❤➙♥ ❧✐ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ s❛✉✿
√
3x − 2 8 x − 5 = 0.
✭✷✳✸✮
√
y = f (x) = 3x − 2 8 x − 5✳ ❚❛ ❝â t❤➸ ❞➵ ❞➔♥❣ t➼♥❤ ✤÷đ❝ f (0) = −5
√
8
✈➔ f (5) = 3.5 − 2 5 − 5
7, 5543109 > 0✳ ❉♦ ✤â✱ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✸✮ ❝â
➼t ♥❤➜t ✶ ♥❣❤✐➺♠ tr♦♥❣ ❦❤♦↔♥❣ (0, 5)✳ ✣➸ ①❡♠ ồ ừ
t
ỗ t t õ t ớ tt ỗ t tr
(0, 5).
ỗ t t t ỗ t t trử ❤♦➔♥❤ t↕✐ ♠ët ✤✐➸♠ tr♦♥❣
❦❤♦↔♥❣
t↕✐ ❝→❝
(2, 3)✳ ❚✉② ♥❤✐➯♥✱ ✤➸ ❝❤➼♥❤ ①→❝ ❤ì♥ t❛ ❝➛♥ t➼♥❤ ❣✐→ trà ❝õ❛ ❤➔♠ sè
✤✐➸♠ x = 2 ✈➔ x = 3✳ ❚❛ ❝â✿ f (2) = −1, 181 < 0, f (3) = 1, 706 >
0✳
❱➟② ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✸✮ ❝â ♠ët ♥❣❤✐➺♠ ❞✉② ♥❤➜t tr♦♥❣ ❦❤♦↔♥❣
(2, 3)
❙❛✉ ❦❤✐ ✤➣ t→❝❤ ✤÷đ❝ ♥❣❤✐➺♠ t❤➻ ❝æ♥❣ ✈✐➺❝ t✐➳♣ t❤❡♦ ❧➔ ❝❤➼♥❤ ①→❝ ❤â❛
♥❣❤✐➺♠ ✤➳♥ ✤ë tt tỹ ữợ t ❝â t❤➸ sû
❞ư♥❣ ♠ët tr♦♥❣ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ s❛✉✿ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣✱ ♣❤÷ì♥❣ ♣❤→♣
❞➙② ❝✉♥❣✱ ♣❤÷ì♥❣ ♣❤→♣ t✐➳♣ t✉②➳♥✱ ✳✳✳ ◆❤÷♥❣ ❞♦ t❤í✐ ❣✐❛♥ ❝â ❤↕♥ ♥➯♥
tr♦♥❣ ♣❤↕♠ ✈✐ ❜➔✐ ❧✉➟♥ ✈➠♥ ♥➔②✱ tỉ✐ s➩ tr➻♥❤ ❜➔② ♣❤÷ì♥❣ ♣❤→♣ ❞➙② ❝✉♥❣
❣✐↔✐ ❣➛♥ ✤ó♥❣ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤✳
❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P
❙❱❚❍✿ ❍❯Ý◆❍ ❚❍➚ ▼ß ❍❸◆❍
Pì PP
ổ t ữỡ
sỷ r➡♥❣ t❛ ✤➣ t➻♠ ✤÷đ❝ ♠ët ❦❤♦↔♥❣ ❧✐ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✶✮
❧➔
(a, b)✳
❚❛ ❧✉ỉ♥ ❣✐↔ t❤✐➳t ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ t❤ã❛ ♠➣♥✿
❞✉② ♥❤➜t tr➯♥
(a, b)
❦❤æ♥❣ ✤ê✐ ❞➜✉ tr➯♥
(a, b)
❛✳ P❤÷ì♥❣ tr➻♥❤ ✭✷✳✶✮ ❝â ♥❣❤✐➺♠
❜✳
f ∈ C 2 [a, b]
f (x), f (x)
✈➔
α
❱➲ ♥❣✉②➯♥ t➢❝ ♣❤÷ì♥❣ ♣❤→♣ ❞➙② ❝✉♥❣ ❝ơ♥❣ ❣✐è♥❣ ♥❤÷ ♣❤÷ì♥❣ ♣❤→♣
❝❤✐❛ ✤ỉ✐✱ ♥❣❤➽❛ ❧➔ ❞ü❛ ✈➔♦ ❤❛✐ ✤✐➸♠
t✐➳♣ ❝→❝ ✤✐➸♠
xn
a0 = a, b0 = b
❜❛♥ ✤➛✉✱ t❛ s➩ ❝❤å♥
♥➡♠ tr♦♥❣ ❦❤♦↔♥❣ ♥➡♠ tr♦♥❣ ❦❤♦↔♥❣
(an , bn )
s❛♦ ❝❤♦
❦❤♦↔♥❣ ❝❤å♥ ❧✉ỉ♥ ❧✉ỉ♥ ❝❤ù❛ ♥❣❤✐➺♠ ✤ó♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤✳ Ð ♣❤÷ì♥❣
♣❤→♣ ❝❤✐❛ ✤ỉ✐✱
xn
✤÷đ❝ ❝❤å♥ ❧➔ ✤✐➸♠ ♥➡♠ ❣✐ú❛ ❝õ❛ ❦❤♦↔♥❣
(an , bn )
✈➔
(an+1 , bn+1 ) t✐➳♣ t❤❡♦ s➩ ❧➔ ❦❤♦↔♥❣ ❝❤ù❛ ♥❣❤✐➺♠ tr♦♥❣ ✷ ❦❤♦↔♥❣
❝♦♥ (an , xn ) ❤♦➦❝ (xn , bn )✳ ◆❤÷ ✈➟②✱ ❦❤♦↔♥❣ (an , bn ) s➩ ♥❤ä ❞➛♥ tỵ✐ 0✱ ❝❤♦
❦❤♦↔♥❣
✤➳♥ ❧ó❝ t❛ ❝â t❤➳ ①❡♠ t➜t ❝↔ ❝→❝ ✤✐➸♠ ♥➡♠ tr♦♥❣ ❦❤♦↔♥❣ ❧➔ ①➜♣ ①➾ ❝õ❛
♥❣❤✐➺♠✳
❈á♥ ð ♣❤÷ì♥❣ ♣❤→♣ ❞➙② ❝✉♥❣ ❣✐→ trà
xn
✤÷đ❝ ❝❤å♥ t✐➳♣ t❤❡♦ ❧↕✐ ❧➔ ❣✐❛♦
✤✐➸♠ ❝õ❛ ❞➙② ố ỗ t t ợ
trử
ữ ỵ tr ữỡ ♣❤→♣ ❝❤✐❛ ✤ỉ✐ ✤ë ❞➔✐ ❦❤♦↔♥❣ ❝♦♥
❞➛♥ tỵ✐
0✱
(an , bn ) t✐➳♥
♥❤÷♥❣ tr♦♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❞➙② ❝✉♥❣ ✤✐➲✉ ♥➔② ❦❤ỉ♥❣ ✤ó♥❣✳ ❈â
t❤➸ ♠ët tr♦♥❣ ❤❛✐ ❣✐→ trà
a
♥➔② s➩ ❧✉ỉ♥ ✤â♥❣ ✈❛✐ trá ❧➔
❧✉æ♥ ✤â♥❣ ✈❛✐ trá
x0 , x1 , ..., xn , ...
bn
t❤➻
an
❤♦➦❝
an
b
❤♦➦❝
✤÷đ❝ ❣✐ú ❧↕✐ s➩ ❣✐ú ♥❣✉②➯♥✳ ●✐→ trà
bn ✳
❱➼ ❞ö ♥➳✉
s➩ t❤❛② ✤ê✐ ✈➔ ❝❤➼♥❤ ❧➔
b ❣✐ú ♥❣✉②➯♥ ✈➔ ❧✉æ♥
xn−1 , n = 1, 2, ... ❉➣②
❧➔ ❞➣② ✤ì♥ ✤✐➺✉ t➠♥❣ ❤♦➦❝ ❣✐↔♠ ✈➔ ❤ë✐ tư ✤➳♥ ♥❣❤✐➺♠
✤ó♥❣✳ ❑❤ỉ♥❣ ♥❤÷ ð ♣❤÷ì♥❣ ♣❤→♣ ❝❤✐❛ ✤ỉ✐✱ ✤✐➲✉ ❦✐➺♥ ❞ø♥❣ ð ✤➙② ❦❤æ♥❣
(an , bn ) ♠➔ ❧➔ ✤ë ❞➔✐ ❦❤♦↔♥❣ (xn , xn−1 )✳ ❚❛ s➩ ❞ø♥❣
xn ❧➔ ♥❣❤✐➺♠ ①➜♣ ①➾ ♥➳✉ |xn − xn−1 | ≤ ε ❤♦➦❝ ✭✈➔✮
❝á♥ ❧➔ ✤ë ❞➔✐ ❦❤♦↔♥❣
t❤✉➟t t♦→♥ ✈➔ ①❡♠
|f (xn )| ≤ δ ✳
❚❛ ❜✐➳t r➡♥❣ ♣❤÷ì♥❣ tr➻♥❤ ✤÷í♥❣ t❤➥♥❣ ✤✐ q✉❛ ❤❛✐ ✤✐➸♠
A(a, f (a)), B(b, f (b))
❝â ❞↕♥❣✿
y − f (a)
x−a
=
.
f (b) − f (a)
b−a
❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P
✭✷✳✹✮
❙❱❚❍✿ ❍❯Ý◆❍ ❚❍➚ ▼ß ❍❸◆❍
✶✾
❉➙② ❝✉♥❣
AB
❝➢t trö❝ ❤♦➔♥❤ t↕✐ ✤✐➸♠ ❝â tå❛ ✤ë
(c, 0)✱
❞♦ ✤â t❛ ❝â✿
c−a
−f (a)
=
f (b) − f (a) b − a
ứ õ s r
c=a
rữợ t t t
>0
f (a)(b a)
.
f (b) − f (a)
a0 = a, b0 = b
✭✷✳✺✮
✈➔ ❝❤♦ trữợ ởt tr
> 0
ừ ọ ũ ✤✐➲✉ ❦✐➺♥ ①➜♣ ①➾ ♥❣❤✐➺♠ ✈➔ ❞ø♥❣ q✉→ tr➻♥❤
t➼♥❤ t♦→♥✳ ụ trữợ ởt số
k
số ữợ tố
k
ữợ tt t ữ t tú t t tổ số ữợ q ợ
ữ ữủ ❦➳t q✉↔ ✈➔ ❦➳t t❤ó❝✳
❙❛✉ ✤â t❛ t❤ü❝ ❤✐➺♥ ❝→❝ ữợ s
ã
ữợ
t
x0 =
a0 f (b0 ) b0 f (a0 )
f (b0 ) − f (a0 )
✳
❱➻
f (a0 )f (b0 ) < 0✱
❞♦ ✤â ♠ët tr♦♥❣ ❤❛✐ tr÷í♥❣ ❤đ♣ s❛✉ ✤➙② s➩ ①↔②
r❛✿
❛✳
|f (x0 )| ≤ δ ✳
❜✳
f (x0 ) = 0✳
◆➳✉
❚❛ ❝â
x0
f (a)f (x0 ) < 0
❧➔ ♥❣❤✐➺♠ ①➜♣ ①➾ ✈➔ ❦➳t t❤ó❝✳
t❤➻ ♥❣❤✐➺♠ s➩ ð tr♦♥❣ ❦❤♦↔♥❣
(a, x0 )
❞♦ ✤â
t❛ ✤➦t✿
a1 = a0 , b1 = x0 .
◆➳✉
f (x0 )f (b) < 0
t❤➻ ♥❣❤✐➺♠ s➩ ð tr♦♥❣ ❦❤♦↔♥❣
(x0 , b)
❞♦ ✤â t❛
✤➦t✿
a1 = x0 , b1 = b0 .
s ữợ
ã
ữợ
t
x1 =
P
a1 f (b1 ) − b1 f (a1 )
.
f (b1 ) − f (a1 )
❙❱❚❍✿ ❍❯Ý◆❍ ❚❍➚ ▼ß ❍❸◆❍
✷✵
❱➻
f (a1 )f (b1 ) < 0✱
❞♦ ✤â ♠ët tr♦♥❣ ❤❛✐ tr÷í♥❣ ❤đ♣ s❛✉ ✤➙② s➩ ①↔②
r❛✿
❛✳
|x1 − x0 | ≤ ε
✈➔
|f (x1 )| ≤ δ ✳
❚❛ ❝â
x1
❧➔ ♥❣❤✐➺♠ ①➜♣ ①➾ ✈➔ ❦➳t
t❤ó❝✳
❜✳
f (x1 ) = 0✳
◆➳✉
f (a1 )f (x1 ) < 0
t❤➻ ♥❣❤✐➺♠ s➩ ð tr♦♥❣ ❦❤♦↔♥❣
(a1 , x1 )
❞♦ ✤â
(x1 , b1 )
❞♦ ✤â
t❛ ✤➦t✿
a2 = a1 , b2 = x1 .
◆➳✉
f (x1 )f (b1 ) < 0
t❤➻ ♥❣❤✐➺♠ s➩ ð tr♦♥❣ ❦❤♦↔♥❣
t❛ ✤➦t✿
a2 = x1 , b2 = b1 .
s ữợ
...
ã
ữợ
t
xn =
an f (bn ) − bn f (an )
.
f (bn ) − f (an )
f (an )f (bn ) < 0✱
❞♦ ✤â ♠ët tr♦♥❣ ❤❛✐ tr÷í♥❣ ❤đ♣ s❛✉ ✤➙② s➩ ①↔②
|xn − xn−1 | ≤ ε
✈➔
r❛✿
❛✳
|f (xn )| ≤ δ ✳
❚❛ ❝â
xn
❧➔ ♥❣❤✐➺♠ ①➜♣ ①➾ ✈➔ ❦➳t
t❤ó❝✳
❜✳
f (xn ) = 0✳
◆➳✉
f (an )f (xn ) < 0 t❤➻ ♥❣❤✐➺♠ s➩ ð tr♦♥❣ ❦❤♦↔♥❣ (an , xn ) ❞♦ ✤â
t❛ ✤➦t✿
an+1 = an , bn+1 = xn .
◆➳✉
f (xn )f (bn ) < 0
t❤➻ ♥❣❤✐➺♠ s➩ ð tr♦♥❣ ❦❤♦↔♥❣
(xn , bn )
❞♦ ✤â
t❛ ✤➦t✿
an+1 = xn , bn+1 = bn .
◆➳✉
n>k
✭tr♦♥❣ ✤â ❦ ❧➔ số ữợ tố t tổ số
ữợ q✉→ ❧ỵ♥ ✈➔ ❦➳t t❤ó❝✳
❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P
❙❱❚❍✿ ❍❯Ý◆❍ ❚❍➚ ▼ß ❍❸◆❍
✷✶
❱➼ ❞ư ✷✳✷✿
❚❛ ①➨t ♣❤÷ì♥❣ tr➻♥❤✿
f (x) = sin x − x2 cos x = 0.
x=0
f (a) = −0, 6988; f (b) = 2, 5739✱
P❤÷ì♥❣ tr➻♥❤ ♥➔② ❝â ♥❣❤✐➺♠ ✤ó♥❣ ❧➔
❚❛ t❤➜② ♥➳✉
a = −0, 5; b = 2
t❤➻
tù❝
❧➔ tr→✐ ❞➜✉✳ ❱➟② t❛ ❝â t❤➸ →♣ ❞ư♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❞➙② ❝✉♥❣✳
✣➦t
ε = 10−3
✈➔ ♥❤í sü ❤é trđ tø ♣❤➛♥ ♠➲♠ tt ú t
t t t s ữợ t ữủ
0, 00044
ỵ tữ t ♣❤÷ì♥❣ ♣❤→♣ ❞➙② ❝✉♥❣ r➜t ✤ì♥ ❣✐↔♥ ✈➔ ❞➵ ❤✐➸✉✳ ❚✉②
♥❤✐➯♥ ✤➸ ❦❤↔♦ s→t ♠ët ❝→❝❤ ❝❤➦t ❝❤➩ t❤➻ t❛ r ởt số
ỗ ó tr♦♥❣ ❦❤♦↔♥❣
[a, b]✳
Ð ✤➙②✱ ❝❤ó♥❣ t❛ s➩ ❦❤ỉ♥❣ ✤✐
q✉→ ✈➔♦ ❝→❝ ❝❤✐ t✐➳t ♥➔②✳
❚❛ ✈✐➳t ❧↕✐ ❝æ♥❣ t❤ù❝ ✭✷✳✺✮
c=a−
❱➻ ❦❤✐ t➼♥❤
c
t❤➻
a
✈➔
c=b−
f (a)(a − b)
f (a)(b − a)
=a−
.
f (b) − f (a)
f (a) − f (b)
b
✤➲✉ ❜➻♥❤ ✤➥♥❣ ♥➯♥
f (b)(a − b)
f (b)(b − a)
=b−
.
f (b) − f (a)
f (a) − f (b)
❇➙② ❣✐í ✤➸ ✤ì♥ ❣✐↔♥ t❛ ❝❤➾ ①➨t tr÷í♥❣ ủ
f (x)
ỗ
(f (x) < 0)
(f (x) > 0) tr ✤♦↕♥ [a, b]✳ ❚r♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔②✱ ♠ët tr♦♥❣
❤❛✐ ✤✐➸♠ a ❤♦➦❝ b s➩ ✤÷đ❝ ❝è ✤à♥❤✳ ◆➳✉ t❛ ❣å✐ ❣✐→ trà ❝è ✤à♥❤ ♥➔② ❧➔ d ✈➔
❣✐→ trà ❝á♥ ❧↕✐ ❧➔ x0 ✭tù❝ ❧➔ ♥➳✉ d = a t❤➻ x0 = b✱ ♥➳✉ d = b t❤➻ x0 = a✮
t❤➻ ❝→❝ ❣✐→ trà xn ✤÷đ❝ t➼♥❤ t❤❡♦ ❝ỉ♥❣ t❤ù❝✿
❤♦➦❝ ❧ã♠
xn = xn−1 −
d
✈➔
x0
f (xn−1 )(xn−1 − d)
, n = 0, 1, 2, ...
f (xn−1 ) − f (d)
✭✷✳✻✮
✤÷đ❝ ❝❤å♥ ❝ư t❤➸ tr♦♥❣ ❝→❝ tr÷í♥❣ ❤đ♣ s❛✉✿
f (x) ❧➔ ❤➔♠ ỗ tr [a, b] tự f (x) 0 t ồ d ũ
ợ f (x) r trữớ ủ ❛✮ t❤➻ d = a✱ x0 = b ✈➔ tr♦♥❣ tr÷í♥❣
❤đ♣ ❜✮ t❤➻ d = b✱ x0 = a✳
❛✳ ◆➳✉
❜✳ ◆➳✉
f (x)
❧➔ ❤➔♠ ❧ã♠ tr➯♥
f (x)✳ ❚r♦♥❣ tr÷í♥❣
❜✮ t❤➻ d = b x0 = a
ũ ợ
trữớ ủ
[a, b]
❚➮❚ ◆●❍■➏P
f (x) ≥ 0✱ t❛ ❝ô♥❣ ❝❤å♥ d
❛✮ t❤➻ d = a✱ x0 = b ✈➔ tr♦♥❣
tù❝ ❧➔
❤ñ♣
❙❱❚❍✿ ❍❯Ý◆❍ ❚❍➚ ▼ß ❍❸◆❍
t ổ ồ
d
ũ ợ
f (x)
r ữỡ ụ õ t tỹ t ữợ
s
t
x0 = a, B = b khi f (a).f ” < 0,
x0 = b, B = a khi f (a).f > 0.
t
xk+1
t ổ tự q
xk+1 = xk
ú ỵ ❧➔ ❦❤✐
f
(B − xk ).f (xk )
.
f (B) − f (xk )
✭✷✳✽✮
❦❤ỉ♥❣ ✤ê✐ ❞➜✉✱ t❤➻ ❞➣② ♥❣❤✐➺♠ ❣➛♥ ✤ó♥❣ t❤❡♦ ữỡ
ỡ õ ợ s số
s ữợ
f (xk ).f (xk + s.ε)✱ tr♦♥❣ ✤â ♥➳✉ ❞➣② ♥❣❤✐➺♠ ❣➛♥ ✤ó♥❣
t➠♥❣ ✭tù❝ ❧➔ ❦❤✐ x0 = a✮ t❤➻ ❧➜② s = 1❀ ❣✐↔♠ ✭x0 = b✮ t❤➻ ❧➜② s = −1✳ ◆➳✉
f (xk ).f (xk + s.ε) < 0 t❤➻ ❝â ✤→♣ sè
t❤ù
k
✭✷✳✼✮
t❛ ①➨t ❞➜✉
x = xk ± ε.
❈❤➼♥❤ ①→❝ ❤ì♥✿
x = (xk +
❱➼ ❞ư ✷✳✸✿
ε
sε
)± .
2
2
▲↕✐ ①➨t ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✶✶✮
x3 − x − 1 = 0.
(1, 2)
(1, 2) ❝â f (x) = 6x > 0, f (1) = −1✳ ❉♦ ✤â ✈ỵ✐ ♣❤÷ì♥❣
❝❤å♥ x0 = a = 1; B = b = 2; s = 1✳ ❚✐➳♣ t❤❡♦ ❧➔ t➼♥❤
❱ỵ✐ ❦❤♦↔♥❣ ♣❤➙♥ ❧✐ ♥❣❤✐➺♠ ✤➣ ❜✐➳t ❧➔
❚r➯♥ ❦❤♦↔♥❣
♣❤→♣ ❞➙② ❝✉♥❣
t♦→♥ t❤❡♦ ❝æ♥❣ t❤ù❝ ❧➦♣ ✭✷✳✽✮✳
❙û ❞ư♥❣ ♣❤➛♥ ♠➲♠ ▼❛t❤❡♠❛t✐❝❛ ✺✳✷ ✈ỵ✐ s❛✐ số
= 104 t
s ữợ t❛ ♥❤➟♥ ✤÷đ❝ ❝→❝ ❣✐→ trà ♥❣❤✐➺♠ ①➜♣ ①➾ ♥❤÷ tr♦♥❣ ❜↔♥❣
✭✷✳✶✮
❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P
❙❱❚❍✿ ❍❯Ý◆❍ ❚❍➚ ▼ß ❍❸◆❍
ố ữợ
tr ừ ữỡ tr ✭✷✳✽✮
✶✱✶✻✻✻✼
✶✱✷✺✸✶✶
✶✱✷✾✸✹✹
✶✱✸✶✶✷✽
✶✱✸✶✽✾✾
✶✱✸✷✷✷✽
✶✱✸✷✸✻✽
✶✱✸✷✹✷✽
✶✱✸✷✹✺✸
✶✱✸✷✹✻✹
❇↔♥❣ ✷✳✶✿ ❈→❝ ❣✐→ trà ♥❣❤✐➺♠ ①➜♣ ①➾ ✈ỵ✐ ε = 10−4
✷✳✷✳✷ ❙ü ❤ë✐ tư ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ✈➔ ✤→♥❤ ❣✐→ s❛✐ sè
❛✮ ❚➼♥❤ ❤ë✐ tö✿
❉➣② ①➜♣ ①➾ ❧✐➯♥ t✐➳♣ ❧➔ ♠ët ❞➣② t➠♥❣✱ ❜à ❝❤➦♥ tr➯♥✿
a = x0 < x1 < ... < xn < xn+1 < α < b,
ữợ
b = x0 > x1 > ... > xn > xn+1 > α > a✱
❉♦ ✤â ❤ë✐ tư ✤➳♥ ❣✐→ trà α✳
❍ì♥ ♥ú❛✱ ❝❤✉②➸♥ q✉❛ ❣✐ỵ✐ ❤↕♥ tr♦♥❣ ❝ỉ♥❣ t❤ù❝✿
xn = xn−1 −
f (xn−1 )(xn−1 − d)
.
f (xn−1 ) − f (d)
t❛ ✤÷đ❝✿
α=α−
f (α)(a − d)
.
f (α) − f (d)
f (α) = 0 ❤❛② α ❝❤➼♥❤ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✶✮
(a, b)✳
❚ø ✤➙② s✉② r❛
tr♦♥❣ ❦❤♦↔♥❣
❜✮ ✣→♥❤ ❣✐→ s❛✐ sè
f (x)
∀x ∈ (a, b).
●✐↔ sû
✈ỵ✐
❦❤ỉ♥❣ ✤ê✐ ❞➜✉ tr➯♥
(a, b)
✈➔
0 < m ≤ |f (x)| ≤ M < ∞
❚❛ ❝â ❝→❝ ❝æ♥❣ t❤ù❝ ✤→♥❤ ❣✐→ s❛✐ sè s❛✉ ✤➙②✿
❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P
❙❱❚❍✿ ❍❯Ý◆❍ ❚❍➚ ▼ß ❍❸◆❍
✷✹
|xn − α| ≤
❈❤ù♥❣ ♠✐♥❤✿
|f (xn )|
M −m
; |xn − α| ≤
|xn − xn−1 |.
m
m
⑩♣ ❞ö♥❣ ✣à♥❤ ❧➼ ❣✐→ trà tr✉♥❣ ❜➻♥❤ ▲❛❣r❛♥❣❡ ✭❈æ♥❣ t❤ù❝ sè ❣✐❛ ❤ú✉
❤↕♥✮✱ t❛ ❝â✿
f (xn ) − f (α) = f (c)(xn − α)(xn − α)
❱➻ f (α) = 0 ✈➔ 0 < m ≤ |f (x)| ♥➯♥
✈ỵ✐
c ∈ (xn , α) ⊂ (a, b).
|f (xn ) − f (α)| = |f (c)(xn − α)| ≥ m|xn − α|.
❙✉② r❛✿
|xn − α| ≤
|f (xn )|
.
m
◆❤÷ ✈➟②✱ ✤➸ ✤→♥❤ ❣✐→ ✤ë ❝❤➼♥❤ ①→❝ ❝õ❛ ♥❣❤✐➯♠ ♥❤➟♥ ✤÷đ❝ ❜➡♥❣ ♣❤÷ì♥❣
♣❤→♣ ❞➙② ❝✉♥❣✱ t❛ ❝â t❤➸ sû ❞ư♥❣ ❝ỉ♥❣ t❤ù❝✿
|xn − α| ≤
|f (xn )| max |f (x)|, x ∈ [a, b]
≤
.
m
m
◆❣♦➔✐ r❛✱ ♥➳✉ ❜✐➳t ✷ ❣✐→ trà ❣➛♥ ✤ó♥❣ ❧✐➯♥ t✐➳♣✱ t❛ ❝â t❤➸ ✤→♥❤ ❣✐→ s❛✐
sè ♥❤÷ s❛✉✿
❚ø tr➯♥ ✭❝❤ù♥❣ ♠✐♥❤ sü ❤ë✐ tư ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ❞➙② ❝✉♥❣✮ t❛ ❝â✿
xn = xn−1 −
f (xn−1 )(xn−1 − d)
, n = 0, 1, 2, ...
f (xn−1 ) − f (d)
❙✉② r❛✿
−f (xn−1 ) =
❱➻
α
f (xn−1 ) − f (d)
(xn − xn−1 ).
xn−1 − d
❧➔ ♥❣❤✐➺♠ ✤ó♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤
f (α) − f (xn−1 ) =
f (x) = 0
♥➯♥ t❛ ❝â t❤➸ ✈✐➳t✿
f (xn−1 ) − f (d)
(xn − xn−1 ).
xn−1 − d
⑩♣ ❞ư♥❣ ✤à♥❤ ❧➼ ❣✐ỵ✐ ❤↕♥ tr✉♥❣ ❜➻♥❤ ▲❛❣r❛♥❣❡✱ t❛ ❝â✿
f (c1 )(α − xn−1 ) = f (α) − f (xn−1 )
✈➔
❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P
❙❱❚❍✿ ❍❯Ý◆❍ ❚❍➚ ▼ß ❍❸◆❍
✷✺
f (c2 )(xn−1 − d) = f (xn−1 ) − f (d)
tr♦♥❣ ✤â✱
c1
♥➡♠ ❣✐ú❛
α
✈➔
xn−1 ✱ c2
♥➡♠ ❣✐ú❛
xn−1
✈➔
d✳
❙✉② r❛
f (c1 )(α − xn−1 ) = f (α) − f (xn−1 ) =
=
f (xn−1 ) − f (d)
(xn − xn−1 )
xn−1 − d
f (c2 )(xn−1 − d)
(xn − xn−1 ) = f (c2 )(xn − xn−1 ).
xn−1 − d
❱➟②
f (c1 )(α − xn + xn − xn−1 ) = f (c2 )(xn − xn−1 ),
❤❛②
f (c1 )(α − xn−1 ) = [f (c2 ) − f (c1 )](xn − xn−1 ),
✈➔
|α − xn | =
|f (c2 ) − f (c1 )|
|xn − xn−1 |.
|f (c1 )|
❚❤❡♦ ❣✐↔ t❤✐➳t t❛ ❝â✿
|f (c2 ) − f (c1 )| ≤ |M − m|,
❞♦ ✤â✿
|xn − α| ≤
M −m
|xn − xn−1 |.
m
◆❤÷ ✈➟②✱ t❛ ❝â ✷ ❝æ♥❣ t❤ù❝ ✤→♥❤ ❣✐→ s❛✐ sè✿
|xn − α| ≤
|f (xn )|
M −m
; |xn − α| ≤
|xn − xn−1 |.
m
m
❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P
❙❱❚❍✿ ❍❯Ý◆❍ ❚❍➚ ▼ß ❍❸◆❍
