Khoá luận tốt nghiệp một số phương pháp biến đổi ngược tuyến tính

❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ❍⑨ ◆❐■ ✷
❑❍❖❆ ❚❖⑩◆
✖✖✖✖✕♦✵♦✖✖✖✖✖

◆●❯❨➍◆ ❚❍➚ ❍❖⑨■ ❚❍❯

▼❐❚ ❙➮ P❍×❒◆● P❍⑩P ❇■➌◆ ✣✃■ ◆●×Ñ❈
❚❯❨➌◆ ❚➑◆❍

❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P ✣❸■ ❍➴❈
❈❤✉②➯♥ ♥❣➔♥❤✿ ❚♦→♥ ù♥❣ ❞ö♥❣

❍⑨ ◆❐■ ✲ ✷✵✶✾


❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ❍⑨ ◆❐■ ✷
❑❍❖❆ ❚❖⑩◆
✖✖✖✖✕♦✵♦✖✖✖✖✖

◆●❯❨➍◆ ❚❍➚ ❍❖⑨■ ❚❍❯

▼❐❚ ❙➮ P❍×❒◆● P❍⑩P ❇■➌◆ ✣✃■ ◆●×Ñ❈
❚❯❨➌◆ ❚➑◆❍

❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P ✣❸■ ❍➴❈
❈❤✉②➯♥ ♥❣➔♥❤✿ ❚♦→♥ ù♥❣ ❞ö♥❣
◆❣÷í✐ ❤÷î♥❣ ❞➝♥ ❦❤♦❛ ❤å❝
❚❤❙✳ ❚❘❺◆ ❚❯❻◆ ❱■◆❍

❍⑨ ◆❐■ ✲ ✷✵✶✾

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

▼ët sè ♣❤÷ì♥❣ ♣❤→♣ ❜✐➳♥ ✤ê✐ ♥❣÷ñ❝ t✉②➳♥ t➼♥❤

▲❮■ ❈❷▼ ❒◆
❚r÷î❝ ❦❤✐ tr➻♥❤ ❜➔② ❦❤â❛ ❧✉➟♥ ❝õ❛ ♠➻♥❤✱ tæ✐ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥
❝❤➙♥ t❤➔♥❤ tî✐ ❝→❝ t❤➛② ❣✐→♦ ✈➔ ❝æ ❣✐→♦ tr♦♥❣ ❑❤♦❛ ❚♦→♥ ✕ ❚r÷í♥❣ ✣↕✐
❤å❝ ❙÷ ♣❤↕♠ ❍➔ ◆ë✐ ✷✱ ✤➣ t➟♥ t➻♥❤ ❣✐ó♣ ✤ï ❝❤➾ ❜↔♦ tr♦♥❣ s✉èt t❤í✐ ❣✐❛♥
tæ✐ t❤❡♦ ❤å❝ t↕✐ ❦❤♦❛ ✈➔ tr♦♥❣ t❤í✐ ❣✐❛♥ ❧➔♠ ❦❤â❛ ❧✉➟♥✳
✣➦❝ ❜✐➺t tæ✐ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ tî✐ ❚❤❙✳❚r➛♥ ❚✉➜♥ ❱✐♥❤✕
❚r÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ❍➔ ◆ë✐ ✷✱ ♥❣÷í✐ trü❝ t✐➳♣ ❤÷î♥❣ ❞➝♥ tæ✐✱ ❧✉æ♥
t➟♥ t➙♠ ❝❤➾ ❜↔♦ ✈➔ ✤à♥❤ ❤÷î♥❣ ❝❤♦ tæ✐ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❧➔♠ ❦❤â❛
❧✉➟♥ ✤➸ tæ✐ ❝â ✤÷ñ❝ ❦➳t q✉↔ ♥❤÷ ♥❣➔② ❤æ♠ ♥❛②✳
▼➦❝ ❞ò ✤➣ ❝â r➜t ♥❤✐➲✉ ❝è ❣➢♥❣✱ s♦♥❣ t❤í✐ ❣✐❛♥ ✈➔ ❦✐♥❤ ♥❣❤✐➺♠ ❜↔♥
t❤➙♥ ❝á♥ ♥❤✐➲✉ ❤↕♥ ❝❤➳ ♥➯♥ ❦❤â❛ ❧✉➟♥ ❦❤æ♥❣ t❤➸ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉
sât r➜t ♠♦♥❣ ✤÷ñ❝ sü ✤â♥❣ ❣â♣ þ ❦✐➳♥ ❝õ❛ ❝→❝ t❤➛② ❝æ ❣✐→♦✱ ❝→❝ ❜↕♥ s✐♥❤
✈✐➯♥ ✈➔ ❜↕♥ ✤å❝✳
❚æ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✦
❍➔ ◆ë✐✱ t❤→♥❣ ✺ ♥➠♠ ✷✵✶✾
❚→❝ ❣✐↔

◆❣✉②➵♥ ❚❤à ❍♦➔✐ ❚❤✉

◆❣✉②➵♥ ❚❤à ❍♦➔✐ ❚❤✉

✶

❑✹✶❇ ❚♦→♥ ✣❍❙P ❍➔ ◆ë✐ ✷

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

▼ët sè ♣❤÷ì♥❣ ♣❤→♣ ❜✐➳♥ ✤ê✐ ♥❣÷ñ❝ t✉②➳♥ t➼♥❤

▲❮■ ❈❆▼ ✣❖❆◆
❑❤â❛ ❧✉➟♥ ♥➔② ❧➔ ❦➳t q✉↔ ♥❣❤✐➯♥ ❝ù✉ ❝õ❛ ❜↔♥ t❤➙♥ tæ✐ ❞÷î✐ sü ❤÷î♥❣
❞➝♥ t➟♥ t➻♥❤ ❝õ❛ t❤➛② ❣✐→♦ ❚❤❙✳❚r➛♥ ❚✉➜♥ ❱✐♥❤✳
❚r♦♥❣ ❦❤✐ ♥❣❤✐➯♥ ❝ù✉ ❤♦➔♥ t❤➔♥❤ ✤➲ t➔✐ ♥❣❤✐➯♥ ❝ù✉ ♥➔② tæ✐ ✤➣ t❤❛♠
❦❤↔♦ ♠ët sè t➔✐ ❧✐➺✉ ✤÷ñ❝ ❧✐➺t ❦➯ tr♦♥❣ ♣❤➛♥ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✳
❚æ✐ ①✐♥ ❦❤➥♥❣ ✤à♥❤ ❦➳t q✉↔ ❝õ❛ ✤➲ t➔✐ ✧▼ët sè ♣❤÷ì♥❣ ♣❤→♣ ❜✐➳♥ ✤ê✐
♥❣÷ñ❝ t✉②➳♥ t➼♥❤✧ ❧➔ ❦➳t q✉↔ ❝õ❛ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉✱ ❤å❝ t➟♣ ✈➔ ♥é ❧ü❝ ❝õ❛
❜↔♥ t❤➙♥✱ ❦❤æ♥❣ ❝â sü trò♥❣ ❧➦♣ ✈î✐ ❦➳t q✉↔ ❝õ❛ ❝→❝ ✤➲ t➔✐ ❦❤→❝✳ ◆➳✉
s❛✐ tæ✐ ①✐♥ ❝❤à✉ ❤♦➔♥ t♦➔♥ tr→❝❤ ♥❤✐➺♠✳
❍➔ ◆ë✐✱ t❤→♥❣ ✺ ♥➠♠ ✷✵✶✾
❚→❝ ❣✐↔

◆❣✉②➵♥ ❚❤à ❍♦➔✐ ❚❤✉

◆❣✉②➵♥ ❚❤à ❍♦➔✐ ❚❤✉

✷

❑✹✶❇ ❚♦→♥ ✣❍❙P ❍➔ ◆ë✐ ✷


▼ö❝ ❧ö❝
▲í✐ ❝↔♠ ì♥

✶

▲í✐ ❝❛♠ ✤♦❛♥

✷

▼Ð ✣❺❯
✶✳ ▲➼ ❞♦ ❝❤å♥ ✤➲ t➔✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✷✳ ▼ö❝ t✐➯✉ ✈➔ ♥❤✐➺♠ ✈ö ♥❣❤✐➯♥ ❝ù✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✸✳ P❤÷ì♥❣ ♣❤→♣ ♥❣❤✐➯♥ ❝ù✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✹✳ ❈➜✉ tró❝ ❦❤â❛ ❧✉➟♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✻

✶ ❚ê♥❣ q✉❛♥

✽

✶✳✶

✶✳✷

✻
✻
✻
✼

●✐î✐ t❤✐➺✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✽

✶✳✶✳✶

❇✐➳♥ ✤ê✐ ♥❣÷ñ❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✽

✶✳✶✳✷

❇✐➳♥ ✤ê✐ ♥❣÷ñ❝ t✉②➳♥ t➼♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✾

▲à❝❤ sû ♥❣❤✐➯♥ ❝ù✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✵

✷ ❇✐➳♥ ✤ê✐ ♥❣÷ñ❝ t✉②➳♥ t➼♥❤ ✈î✐ ♣❤÷ì♥❣ ♣❤→♣ ❜➻♥❤ ♣❤÷ì♥❣
tè✐ t❤✐➸✉
✶✷
✷✳✶

❳➨t ❜➔✐ t♦→♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✷

✷✳✷

❙❛✐ sè tr✉♥❣ ❜➻♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✸

✷✳✸

P❤÷ì♥❣ ♣❤→♣ ❜➻♥❤ ♣❤÷ì♥❣ tè✐ t❤✐➸✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✸

✷✳✸✳✶

❚➻♠ ❤➔♠ ①➜♣ ①➾ ❝â ❞↕♥❣ ✤❛ t❤ù❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✺

✷✳✸✳✷

❳➜♣ ①➾ ❤➔♠ ✤❛ t❤ù❝ ❧÷ñ♥❣ ❣✐→❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✽

✸


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

▼ët sè ♣❤÷ì♥❣ ♣❤→♣ ❜✐➳♥ ✤ê✐ ♥❣÷ñ❝ t✉②➳♥ t➼♥❤

✸ ❇✐➳♥ ✤ê✐ ♥❣÷ñ❝ t✉②➳♥ t➼♥❤ ✈î✐ ♣❤÷ì♥❣ ♣❤→♣ ❝➛✉ ♣❤÷ì♥❣ ✷✹
✸✳✶

P❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❋r❡❞❤♦❧♠ ❧♦↕✐ ♠ët ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✳✷

P❤➨♣ ❝➛✉ ♣❤÷ì♥❣✱ ♣❤➨♣ ✤÷❛ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ✈➲ ❤➺

✷✹

♣❤÷ì♥❣ tr➻♥❤ t✉②➳♥ t➼♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✺

✸✳✸

P❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✼

✸✳✹

⑩♣ ❞ö♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✽

❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦

✸✹

❑➳t ❧✉➟♥

✸✺

◆❣✉②➵♥ ❚❤à ❍♦➔✐ ❚❤✉

✹

❑✹✶❇ ❚♦→♥ ✣❍❙P ❍➔ ◆ë✐ ✷


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

▼ët sè ♣❤÷ì♥❣ ♣❤→♣ ❜✐➳♥ ✤ê✐ ♥❣÷ñ❝ t✉②➳♥ t➼♥❤

❉❛♥❤ ♠ö❝ ❜↔♥❣ ❜✐➸✉
❚➯♥ ❜↔♥❣
❇↔♥❣ ✷✳✶
❇↔♥❣ ✸✳✶✳ ●✐→ trà ❝õ❛

❚r❛♥❣
✶✼

1

xe−y x f (x)d(x)

✷✾

0

✤÷ñ❝ t➼♥❤ ❜➡♥❣ ♣❤➨♣ ❝➛✉
♣❤÷ì♥❣ ✈➔ ❝❤➼♥❤ ①→❝✳

❇↔♥❣ ✸✳✷✳ ❑➳t q✉↔ ❝õ❛ ♣❤➨♣ ❜✐➳♥ ✤ê✐

✸✷

♥❣÷ñ❝ trü❝ t✐➳♣ f = A−1 g
❇↔♥❣ ✸✳✸✳ ❙♦ s→♥❤ gi ❜❛♥ ✤➛✉ ✈î✐ ❣✐→ trà

✸✸

✤÷ñ❝ ✤÷❛ r❛ ❜ð✐ ♥❣❤✐➺♠ f

◆❣✉②➵♥ ❚❤à ❍♦➔✐ ❚❤✉

✺

❑✹✶❇ ❚♦→♥ ✣❍❙P ❍➔ ◆ë✐ ✷


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

▼ët sè ♣❤÷ì♥❣ ♣❤→♣ ❜✐➳♥ ✤ê✐ ♥❣÷ñ❝ t✉②➳♥ t➼♥❤

▼Ð ✣❺❯
✶✳ ▲þ ❞♦ ❝❤å♥ ✤➲ t➔✐✿
❈→❝ ♣❤÷ì♥❣ ♣❤→♣ ❝õ❛ ❜➔✐ t♦→♥ ❜✐➳♥ ✤ê✐ ♥❣÷ñ❝ t✉②➳♥ t➼♥❤ ✤÷ñ❝ ù♥❣
❞ö♥❣ tr♦♥❣ ♥❤✐➲✉ ♥❣➔♥❤ ❦❤♦❛ ❤å❝ ✈➔ ❦ÿ t❤✉➟t ♥❤÷ ✈✐➵♥ t❤→♠✱ r❛❞❛r✱
q✉❛♥❣ ❤å❝✱ ② ❤å❝✱✳✳✳ ❈→❝ ♣❤÷ì♥❣ ♣❤→♣ ❜✐➳♥ ❞ê✐ ♥❣÷ñ❝ ❞ò♥❣ ✤➸ ❣✐↔✐ q✉②➳t
❝→❝ ❜➔✐ t♦→♥ ❦❤✐ ✤➣ ❜✐➳t ❦➳t q✉↔ ✈➔ ❝➛♥ ♣❤↔✐ ①→❝ ✤à♥❤ ❝→❝ ♥❤➙♥ tè t↕♦ r❛
❝→❝ ❦➳t q✉↔ ✤â ✲ ❣å✐ ❧➔ ❤➺ sè ♥❤➙♥ q✉↔ ✭❝❛✉s❛❧ ❢❛❝t♦r✮✳
❚✉② ♥❤✐➯♥ s✐♥❤ ✈✐➯♥ ❙÷ ♣❤↕♠ ❚♦→♥ ❤å❝ ♥â✐ ❝❤✉♥❣ ❝❤÷❛ ❝â ♥❤✐➲✉ ✤✐➲✉

❦✐➺♥ ✤➸ t➻♠ ❤✐➸✉ ✈➲ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ❜✐➳♥ ✤ê✐ ♥❣÷ñ❝ t✉②➳♥ t➼♥❤✳ ❱➻ ✈➟②
tæ✐ ❝❤å♥ ✤➲ t➔✐ ✏ ▼ët sè ♣❤÷ì♥❣ ♣❤→♣ ❜✐➳♥ ✤ê✐ ♥❣÷ñ❝ t✉②➳♥ t➼♥❤ ✑ ❧➔♠
❦❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ♥❤➡♠ ✤÷❛ r❛ ♠ët sè ❧➼ t❤✉②➳t ❝ì ❜↔♥ ✈➲ ❜✐➳♥ ✤ê✐
♥❣÷ñ❝ ❝ô♥❣ ♥❤÷ ❜✐➳♥ ✤ê✐ ♥❣÷ñ❝ t✉②➳♥ t➼♥❤✱ ♠ët sè ♣❤÷ì♥❣ ♣❤→♣ ❝õ❛
❜✐➳♥ ✤ê✐ ♥❣÷ñ❝ t✉②➳♥ t➼♥❤✳

✷✳ ▼ö❝ t✐➯✉ ✈➔ ♥❤✐➺♠ ✈ö ♥❣❤✐➯♥ ❝ù✉
✷✳✶ ▼ö❝ t✐➯✉
❚➻♠ ❤✐➸✉ ✈➲ ❜✐➳♥ ✤ê✐ ♥❣÷ñ❝✱ ❜✐➳♥ ✤ê✐ ♥❣÷ñ❝ t✉②➳♥ t➼♥❤ ✈➔ tø ✤â ✤÷❛
r❛ ♠ët sè ♣❤÷ì♥❣ ♣❤→♣ ❜✐➳♥ ✤ê✐ ♥❣÷ñ❝ t✉②➳♥ t➼♥❤✳

✷✳✷ ◆❤✐➺♠ ✈ö

✲ ◆❣❤✐➯♥ ❝ù✉ ❝ì sð ❧➼ ❧✉➟♥ ✈➲ ♣❤÷ì♥❣ ♣❤→♣ ❜✐➳♥ ✤ê✐ ♥❣÷ñ❝✳
✲ ◆❣❤✐➯♥ ❝ù✉ ❝ì sð ❧➼ ❧✉➟♥ ✈➲ ♣❤÷ì♥❣ ♣❤→♣ ❜✐➳♥ ✤ê✐ ♥❣÷ñ❝ t✉②➳♥ t➼♥❤✳

✸✳ P❤÷ì♥❣ ♣❤→♣ ♥❣❤✐➯♥ ❝ù✉✿
✲ P❤➙♥ t➼❝❤✱ tê♥❣ ❤ñ♣ ❝→❝ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✈➲ ❝❤õ ✤➲ ❜✐➳♥ ✤ê✐ ♥❣÷ñ❝
t✉②➳♥ t➼♥❤ ✈➔ ù♥❣ ❞ö♥❣✳

◆❣✉②➵♥ ❚❤à ❍♦➔✐ ❚❤✉

✻

❑✹✶❇ ❚♦→♥ ✣❍❙P ❍➔ ◆ë✐ ✷


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

▼ët sè ♣❤÷ì♥❣ ♣❤→♣ ❜✐➳♥ ✤ê✐ ♥❣÷ñ❝ t✉②➳♥ t➼♥❤

✲ ❍ä✐ þ ❦✐➳♥ ❝❤✉②➯♥ ❣✐❛✳

✹✳ ❈➜✉ tró❝ ❦❤â❛ ❧✉➟♥
◆❣♦➔✐ ♣❤➛♥ ♠ð ✤➛✉✱ ❦➳t ❧✉➟♥ ✈➔ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✱ ❦❤â❛ ❧✉➟♥ ❣ç♠ ✸
❝❤÷ì♥❣✿
❈❤÷ì♥❣ ✶✳ ❚ê♥❣ q✉❛♥
❈❤÷ì♥❣ ✷✳ ❇✐➳♥ ✤ê✐ ♥❣÷ñ❝ t✉②➳♥ t➼♥❤ ✈î✐ ♣❤÷ì♥❣ ♣❤→♣ ❜➻♥❤ ♣❤÷ì♥❣
tè✐ t❤✐➸✉
❈❤÷ì♥❣ ✸✳ ❇✐➳♥ ✤ê✐ ♥❣÷ñ❝ t✉②➳♥ t➼♥❤ ✈î✐ ♣❤÷ì♥❣ ♣❤→♣ ❝➛✉ ♣❤÷ì♥❣

◆❣✉②➵♥ ❚❤à ❍♦➔✐ ❚❤✉

✼

❑✹✶❇ ❚♦→♥ ✣❍❙P ❍➔ ◆ë✐ ✷


ữỡ
ờ q
ợ t
ờ ữủ
t ờ ữủ
t ờ ữủ t rs r
tr ồ tt q tr t t r số q
s trs ỹ t t q t t tứ ú ử t t
tr ử t ợ tr t ỗ t
ữủ t t trồ ừ r t tứ t q trữớ trồ
ỹ[3]
t ữủ ồ t ờ ữủ õ t ợ t q

s õ t t r số q õ ữủ ợ t t
t t ợ số q s õ t t q
t ờ ữủ ởt tr số t q trồ t tr
ồ tt ú t t số t ổ t
trỹ t t t ữủ ú õ ự ử rở r tr q ồ
rr ồ ỵ tt tr tổ ỷ ỵ t ồ
t t tr s t ỵ ữỡ ồ t
ồ t ỷ ỵ ổ ỳ tỹ ồ r
ỹ



❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

▼ët sè ♣❤÷ì♥❣ ♣❤→♣ ❜✐➳♥ ✤ê✐ ♥❣÷ñ❝ t✉②➳♥ t➼♥❤

✶✳✶✳✶✳✷ ▼ö❝ t✐➯✉
▼ö❝ t✐➯✉ ❝õ❛ ♠ët ❜➔✐ t♦→♥ ❜✐➳♥ ✤ê✐ ♥❣÷ñ❝ ❧➔ t➻♠ r❛ ❝→❝ t❤❛♠ sè ♠æ
❤➻♥❤ tèt ♥❤➜t s❛♦ ❝❤♦ ♠ ✭①➜♣ ①➾ ♥❤ä ♥❤➜t✮ t❤ä❛ ♠➣♥✿

d = G (m)
ð ✤â ● ❧➔ ♠ët ❤➔♠ ♠æ t↔ ♠è✐ q✉❛♥ ❤➺ rã r➔♥❣ ❣✐ú❛ ❝→❝ ❞ú ❧✐➺✉ ❞ ✤÷ñ❝
q✉❛♥ s→t✱ ✈➔ ❝→❝ t❤❛♠ sè ♠æ ❤➻♥❤ ♠✳ ● ✤↕✐ ❞✐➺♥ ❝❤♦ ❝→❝ ❤➔♠ ❧✐➯♥ ❤➺
❣✐ú❛ ❝→❝ t❤❛♠ sè ♠æ ❤➻♥❤ ✈î✐ ❞ú ❧✐➺✉ ✤÷ñ❝ q✉❛♥ s→t ✳[3]

✶✳✶✳✷ ❇✐➳♥ ✤ê✐ ♥❣÷ñ❝ t✉②➳♥ t➼♥❤
❚r♦♥❣ tr÷í♥❣ ❤ñ♣ ❝â ♠ët ❜➔✐ t♦→♥ ❜✐➳♥ ✤ê✐ ♥❣÷ñ❝ t✉②➳♥ t➼♥❤ ♠æ t↔
♠ët ❤➺ t✉②➳♥ t➼♥❤✱ ❞ ✭❞ú ❧✐➺✉✮ ✈➔ ♠ ✭t❤❛♠ sè ♠æ ❤➻♥❤ tèt ♥❤➜t✮ ❧➔ ❝→❝
✈❡❝tì ✈➔ ❜➔✐ t♦→♥ ❝â t❤➸ ✤÷ñ❝ ✈✐➳t ❧➔✿

d = G (m)
ð ✤â ● ❧➔ ♠ët ♠❛ tr➟♥✳[3]
❱➼
 ❞ö✿ ●✐↔✐ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ s❛✉✿

3x + y − z + t = 0




 2x + 3y − t = 0


x + 5y − 3z = 7




 3y + 2z + t = 2
❇➔✐ t♦→♥
tr➯♥ ❧➔ ♠ët
t➼♥❤ ✈î✐✿
 
 ❜➔✐ t♦→♥ ❜✐➳♥✤ê✐ ♥❣÷ñ❝
 t✉②➳♥

0
3 1 −1 1
x
 



 
0
 2 3 0 −1 
y 
 


 
d= G=
m= 
7
 1 5 −3 0 
z 
 


 
2
0 3 2 1
t
❑❤✐ ✤â✿

d = G (m)

◆❣✉②➵♥ ❚❤à ❍♦➔✐ ❚❤✉

✾

❑✹✶❇ ❚♦→♥ ✣❍❙P ❍➔ ◆ë✐ ✷


õ tốt

ởt số ữỡ ờ ữủ t t

ừ t ờ ữủ t t
t ờ ữủ t t ử tở tt
ừ ởt t tr r õ ỳ ữủ t t
tr t ừ t số ỹ tr ữỡ tr
ữỡ tr t t ỳ t số ợ ỳ q st ữủ
ỳ ỳ t t ữủ s õ s s ợ ỳ tỹ t
ữủ tr q tr ờ ữủ ữủ t
tr ởt q tr t ữủ sỹ t ủ ỳ ỹ
q st r ữợ t số ữủ
t sỹ ũ ủ ỳ ỳ ữủ q st t t tr
ữủ t tử t ữủ t t ứ ìợ t
t ữủ tr ữợ ố ũ ữủ
t ờ ữủ

sỷ ự
ởt tr ỳ ử t ờ ữủ
ữủ r t r t ổ t t
ừ tr r ừ t tỷ tr

[i]

ữủ ồ t ừ s ữủ t ồ t
ờ ữủ s õ ụ ữủ ự t
ỵ ữ r t q trồ tr t ỵ ú ỵ
ự tr ỳ t ỵ õ rt ợ
t tr ỵ tt t ỵ ố
t t ở tr ừ r t tổ q ỳ ữủ
t rữợ ữỡ ữủ t tr ừ
ỹ tr ổ tr ừ s ừ rt
[ii] [iii]

r tố t trú t ồ ừ

ỗ ố ừ sỹ t tr ừ ữỡ





P ở


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

▼ët sè ♣❤÷ì♥❣ ♣❤→♣ ❜✐➳♥ ✤ê✐ ♥❣÷ñ❝ t✉②➳♥ t➼♥❤

♣❤→♣ ❣✐↔✐ ✤♦→♥ ❞ú ❧✐➺✉ ✤à❛ ✈➟t ❧þ✳ ✣➸ ❣✐↔✐ q✉②➳t ❝→❝ ✈➜♥ ✤➲ ♥❣❤à❝❤ ✤↔♦
♥❤➜t ✤à♥❤ ❧✐➯♥ q✉❛♥ ✤➳♥ ✈✐➺❝ ①→❝ ✤à♥❤ ❝→❝ ♥❣✉ç♥ ✤÷ñ❝ ♠æ t↔ ❜ð✐ ♠ët
♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ t✉②➳♥ t➼♥❤ ❝õ❛ ❧♦↕✐ ✤➛✉ t✐➯♥✱ ❝→❝ t→❝ ❣✐↔ ♥➔② ✤➣
✤➲ ①✉➜t ♠ët ♣❤÷ì♥❣ ♣❤→♣ sè ♠❛♥❣ t➯♥ ❝õ❛ ❤å✱ ✈➔ ✤➣ ✤÷❛ r❛ ♥❤✐➲✉ ❜➔✐
❜→♦ ✈➔ s→❝❤✳[4]

◆❣➔② ♥❛②✱ ❝→❝ ✈➜♥ ✤➲ ♥❣❤à❝❤ ✤↔♦ t✐➳♣ tö❝ ✤÷ñ❝ ♥❣❤✐➯♥ ❝ù✉ tr♦♥❣ ❝→❝
❧➽♥❤ ✈ü❝ ♥❣♦➔✐ t♦→♥ ❤å❝ ✈➔ ✈➟t ❧þ✱ ♥❤÷ ❤â❛ ❤å❝✱ ❦✐♥❤ t➳✱ ❦❤♦❛ ❤å❝ ♠→②
t➼♥❤✱ ✳✳✳
[i]

☎
❲❡②❧✱ ❍❡r♠❛♥♥✳✧❯❜❡r
❞✐❡ ❛s②♠♣t♦t✐s❝❤❡ ❱❡rt❡✐❧✉♥❣ ❞❡r ❊✐❣❡♥✇❡rt❡✧✳

◆❛❝❤r✐❝❤t❡♥ ❞❡r ❑☎
♦♥✐❣❧✐❝❤❡♥ ●❡s❡❧❧s❝❤❛❢t ❞❡r ❲✐ss❡♥s❝❤❛❢t❡♥ ③✉ ●☎♦tt✐♥❣❡♥✳
[ii]

●✳❊✳ ❇❛❝❦✉s ❛♥❞ ❏✳❋✳ ●✐❧❜❡rt✳ ✧◆✉♠❡r✐❝❛❧ ❛♣♣❧✐❝❛t✐♦♥s ♦❢ ❛ ❢♦r✲

♠❛❧✐s♠ ❢♦r ❣❡♦♣❤②s✐❝❛❧ ✐♥✈❡rs❡ ♣r♦❜❧❡♠s✧✳ ●❡♦♣❤②s✳ ❏✳ ❘♦②✳ ❆str✳ ❙♦❝✳✱
✶✾✻✼✳
[iii]

●✳❊✳ ❇❛❝❦✉s ❛♥❞ ❏✳❋✳ ●✐❧❜❡rt✳ ✧❚❤❡ r❡s♦❧✈✐♥❣ ♣♦✇❡r ♦❢ ❣r♦ss ❡❛rt❤

❞❛t❛✧✳ ●❡♦♣❤②s✳ ❏✳ ❘♦②✳ ❆str✳ ❙♦❝✳✱ ✶✾✻✽✳

◆❣✉②➵♥ ❚❤à ❍♦➔✐ ❚❤✉

✶✶

❑✹✶❇ ❚♦→♥ ✣❍❙P ❍➔ ◆ë✐ ✷

ữỡ
ờ ữủ t t ợ
ữỡ ữỡ tố t
t t
Pữỡ ữỡ tố t tữớ ữủ sỷ ử
t s
t
sỷ t tr yi ừ y = f (x) t tữỡ ự

x = xi ợ i = 1, 2, ..., n Pm (x) ợ f (x) tr õ
m

Pm (x) =



aj j (x)
j=0

ợ j (x) ỳ t aj ỳ t số qt
t ồ Pm (x) s q tr t t ỡ õ
ỗ tớ ữ s số i õ t t t t
ữủ ỳ yi ữủ ỵ tr q tr t t r
t t ồ ừ Pm (x) tũ
tở ỵ tỹ t ừ f (x)[2]




❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

▼ët sè ♣❤÷ì♥❣ ♣❤→♣ ❜✐➳♥ ✤ê✐ ♥❣÷ñ❝ t✉②➳♥ t➼♥❤

✷✳✷ ❙❛✐ sè tr✉♥❣ ❜➻♥❤
◆❤ú♥❣ ❤➔♠ tr♦♥❣ t❤ü❝ ♥❣❤✐➺♠ t❤✉ ✤÷ñ❝ t❤÷í♥❣ ♠➢❝ ♣❤↔✐ ♥❤ú♥❣ s❛✐
sè ❝â t➼♥❤ ❝❤➜t ♥❣➝✉ ♥❤✐➯♥✳ ◆❤ú♥❣ s❛✐ sè ♥❣➝✉ ♥❤✐➯♥ ♥➔② ①✉➜t ❤✐➺♥ ❞♦
sü t→❝ ✤ë♥❣ ❝õ❛ ♥❤ú♥❣ ②➳✉ tè ♥❣➝✉ ♥❤✐➯♥ ✈➔ ❦➳t q✉↔ t❤ü❝ ♥❣❤✐➺♠ ✤➸ t❤✉
✤÷ñ❝ ❝→❝ ❣✐→ trà ❝õ❛ ❤➔♠✳
❈❤➼♥❤ ✈➻ ❧þ ❞♦ tr➯♥✱ ✤➸ ✤→♥❤ ❣✐→ sü s❛✐ ❦❤→❝ ❣✐ú❛ ❤❛✐ ❤➔♠ tr♦♥❣ t❤ü❝
♥❣❤✐➺♠ t❛ ❝➛♥ ✤÷❛ r❛ ❦❤→✐ ♥✐➺♠ ✈➲ s❛✐ sè ✭❤♦➦❝ ✤ë ❧➺❝❤✮✳ ◆➳✉ ❤❛✐ ❤➔♠
t❤ü❝ ❝❤➜t ❦❤→ ❣➛♥ ♥❤❛✉ t❤➻ s❛✐ sè ❝❤ó♥❣ t❛ ✤÷❛ r❛ ♣❤↔✐ ❦❤→ ❜➨ tr➯♥ ♠✐➲♥
✤❛♥❣ ①➨t✳ ❑❤→✐ ♥✐➺♠ ✈➲ s❛✐ sè ♥â✐ tr➯♥ ❦❤æ♥❣ ❝❤ó þ tî✐ ♥❤ú♥❣ ❦➳t q✉↔
❝â t➼♥❤ ❝❤➜t ❝→ ❜✐➺t ♠➔ ①➨t tr➯♥ ♠ët ♠✐➲♥ ♥➯♥ ✤÷ñ❝ ❣å✐ ❧➔ s❛✐ sè tr✉♥❣
❜➻♥❤✳
❚❛ ❣å✐ σn ❧➔ s❛✐ sè ✭ ❤♦➦❝ ✤ë ❧➺❝❤✮ tr✉♥❣ ❜➻♥❤ ❝õ❛ ❤❛✐ ❤➔♠ f (x) ✈➔

Pm (x) tr➯♥ t➟♣ x = (x1 , x2, ..., xn ) ♥➳✉
σn =

1
n

2

n

[f (xi ) − Pm (xi )]
i=1

✣➸ ❣✐↔✐ ✤÷ñ❝ ❜➔✐ t♦→♥ ✶ t❛ s➩ sû ❞ö♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❜➻♥❤ ♣❤÷ì♥❣ tè✐

t❤✐➸✉ ✤➸ q✉② ♥â ✈➲ ❜➔✐ t♦→♥ ❜✐➳♥ ✤ê✐ ♥❣÷ñ❝ t✉②➳♥ t➼♥❤ ✈➔ ❣✐↔✐ ✤➸ t➻♠ r❛
❝→❝ t❤❛♠ sè ❝➛♥ t➻♠✳[2]

✷✳✸ P❤÷ì♥❣ ♣❤→♣ ❜➻♥❤ ♣❤÷ì♥❣ tè✐ t❤✐➸✉
●✐↔ sû ❝â ♠➝✉ q✉❛♥ s→t (xi , yi ) ❝õ❛ ❤➔♠ y = f (x)✳ ❚❤❡♦ ❜➔✐ t♦→♥ ✶ t❛
❝â

y = Pm (x) + ε
❚r♦♥❣ ✤â Pm (x) = a0 ϕ0 (x) + a1 ϕ1 (x) + ... + am ϕm x ❧➔ ❤➔♠ ①➜♣ ①➾ ❝õ❛
❤➔♠ y = f (x) ✈➔ ❝→❝ ❤➔♠ ϕ0 (x) , ϕ1 (x) , ..., ϕm (x) ❧➔ ✭♥✰✶✮ ❤➔♠ ✤ë❝ ❧➟♣
t✉②➳♥ t➼♥❤ ♠➔ t❛ ❝â t❤➸ ❝❤å♥ tò② þ ✈➔ ❝→❝ ❤➺ sè aj ❧➔ ❝→❝ t❤❛♠ sè ❝❤÷❛
❜✐➳t ♠➔ t❛ ♣❤↔✐ ①→❝ ✤à♥❤ ❞ü❛ ✈➔♦ ❤➺ ❤➔♠ ✤➣ ❝❤å♥ ✈➔ ❝→❝ ✤✐➸♠ q✉❛♥ s→t✱

ε ❧➔ s❛✐ sè ❣✐ú❛ ❣✐→ trà ✤♦ ✤÷ñ❝ ✈➔ ❣✐→ trà t➼♥❤ ✤÷ñ❝✳

◆❣✉②➵♥ ❚❤à ❍♦➔✐ ❚❤✉

✶✸

❑✹✶❇ ❚♦→♥ ✣❍❙P ❍➔ ◆ë✐ ✷


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

▼ët sè ♣❤÷ì♥❣ ♣❤→♣ ❜✐➳♥ ✤ê✐ ♥❣÷ñ❝ t✉②➳♥ t➼♥❤

❱✐➳t ❞÷î✐ ❞↕♥❣ ♠❛ tr➟♥ ✿

y = Xa + ε 





ϕ (x ) ϕ1 (x0 )
y
 0 0
 0 
 ϕ (x ) ϕ (x )
y 
1
1
 0 1
 1 
X=
❚r♦♥❣ ✤â y = 

 ...
 ... 
...



ϕ0 (xn ) ϕ1 (xn )
yn




ε

a
 0 
 0 
ε 
a 
 1 
 1 
ε=
a=


 ... 
 ... 




εn
am
X ❧➔ ♠❛ tr➟♥ ✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤✳[5]

... ϕm (x0 )




... ϕm (x1 ) 




...
...

... ϕm (xn )

❙❛✐ sè ❣✐ú❛ ❣✐→ trà ✤♦ ✤÷ñ❝ ✈➔ ❣✐→ trà t➼♥❤ ✤÷ñ❝ t❤❡♦ ✭✷✳✶✮ ❧➔✿

εi = yi − Pm (xi )
s❛✐ sè ♥➔② ❝â t❤➸ ➙♠ ❤❛② ❞÷ì♥❣ tò② t❤✉ë❝ ✈➔♦ ❣✐→ trà ❝õ❛ yi ✳
❚❛ ①➨t ❜➻♥❤ ♣❤÷ì♥❣ ❝õ❛ s❛✐ sè t↕✐ ✤✐➸♠ ✐✿

ε2i = [yi − Pm (xi )]2
❱î✐ ♥ ✤✐➸♠ tê♥❣ ❜➻♥❤ ♣❤÷ì♥❣ s❛✐ sè s➩ ❧➔ ✿
n

S=

n

ε2i =

i=1

(yi − Pm (x))2

i=1

❱✐➳t ❞÷î✐ ❞↕♥❣✿

n

yi −

S=

2

m

i=0

Xij aj

= y − Xa

2

j=0

❘ã r➔♥❣ ❙ ❧➔ ❤➔♠ ❝õ❛ ❝→❝ ❣✐→ trà ❝➛♥ t➻♠ aj ✈➔ ❝❤ó♥❣ t❛ s➩ ❝❤å♥ ❝→❝ aj
s❛♦ ❝❤♦ ❙ ✤↕t ❣✐→ trà ♠✐♥✱ ♥❣❤➽❛ ❧➔

∂S
∂aj

=0✳

j = (1, ..., m)

T

❑❤✐ ✤â t❛ t❤✉ ✤÷ñ❝ ✿ X T X a = X y

⇒ a = XT X


[5]

✭✷✳✷✮
−1

XT y



a
 0 
 a 
 1 
❱î✐ a = 

 ... 


am
❧➔ t❤❛♠ sè tèt ♥❤➜t s❛♦ ❝❤♦ s❛✐ sè tr✉♥❣ ❜➻♥❤ ❣✐ú❛ Pm (x) ✈➔ f (x) ❧➔ ♥❤ä

◆❣✉②➵♥ ❚❤à ❍♦➔✐ ❚❤✉

✶✹

❑✹✶❇ ❚♦→♥ ✣❍❙P ❍➔ ◆ë✐ ✷


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

▼ët sè ♣❤÷ì♥❣ ♣❤→♣ ❜✐➳♥ ✤ê✐ ♥❣÷ñ❝ t✉②➳♥ t➼♥❤

♥❤➜t✳
❚❛ ①➨t ♠ët sè tr÷í♥❣ ❤ñ♣ ❝ö t❤➸ ❝õ❛ ❜➔✐ t♦→♥✳

✷✳✸✳✶ ❚➻♠ ❤➔♠ ①➜♣ ①➾ ❝â ❞↕♥❣ ✤❛ t❤ù❝
❚r♦♥❣ tr÷í♥❣ ❤ñ♣ tê♥❣ q✉→t t❛ ❝❤å♥ ❤➺ ❤➔♠ ①➜♣ ①➾ ❧➔ ♠ët ✤❛ t❤ù❝✱
♥❣❤➽❛ ❧➔✿

Pm (x) = a0 + a1 x + ... + am xm
❚❛ ❝â

1 x0

 1 x
1

X=
 ... ...

1 xn

...

xm
0
xm
1





...


... .. 

m
... xn





a
 0 
a 
 1 
a=

 ... 



am

❱➟② ❤➔♠ S = y − a0 − a1 x − a2 x2 − ... − am x

m 2

n

yi −

=
i=0

= y − Xa

2

m

Xij aj
j=0

2

❘ã r➔♥❣ ❙ ❧➔ ❤➔♠ ❝õ❛ ❝→❝ ❣✐→ trà ❝➛♥ t➻♠ aj ✈➔ ❝❤ó♥❣ t❛ s➩ ❝❤å♥ ❝→❝ aj
s❛♦ ❝❤♦ ❙ ✤↕t ❣✐→ trà ♠✐♥✱ ♥❣❤➽❛ ❧➔
⑩♣
  t❛ ❝â ✿
❞ö♥❣ ❝æ♥❣ t❤ù❝ ✭✷✳✷✮
1 x0

1 1 ... 1


 x x ... x   1 x
1
1
n 
 0


 ... ... ... ...   ... ...


m
m
m
1 xn
x x1 ... xn
 0
 

1 1 ... 1
y

  0 
 x x ... x   y 
1
n   1 
 0
=

.

 ... ... ... ...   ... 

 

m
m
m
x0 x1 ... xn
yn

∂S
∂aj

...

=0

xm
0
xm
1

 

j = (1, ..., m)


a

  0 
  a 
...
  1 
 

  ... 
... ... 
 

m
... xn
am

❚ø ✤â t❛ s✉② r❛✿

◆❣✉②➵♥ ❚❤à ❍♦➔✐ ❚❤✉

✶✺

❑✹✶❇ ❚♦→♥ ✣❍❙P ❍➔ ◆ë✐ ✷


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣
n













▼ët sè ♣❤÷ì♥❣ ♣❤→♣ ❜✐➳♥ ✤ê✐ ♥❣÷ñ❝ t✉②➳♥ t➼♥❤

n

xi
i=1
n

n

xi
i=1

xm
i

i=1

i=1

✣➦t 






G=





x2i

...
i=1

...

xm+1
...
i

xm+1
i

...
n

x2m
i

i=1

n

n
n

xi
i=1

x2i

...

xm
i

i=1

n
i=1

❑❤✐ ✤â ✿
Ga = H

xm
i

...
i=1
n

...

i=1

...
n

n

xi
i=1
n



i=1

...

xm+1
i

...
n

xm+1
...
i

a0





n



y
  i=1 i

  n

 

xy
  a1  
 =  i=1 i i

 

  ...

 

  ...   n



xm
i yi
i=1
an

i=1
n

...
n



xm
i

...

i=1

...
n

n

x2m
i


























n

y
 i=1 i
 n

x i yi


i=1
H=

 ...

 n
xm
i yi












i=1

i=1

⇒ a = G−1 H
●✐↔✐ ♥â t❛ ♥❤➟♥ ✤÷ñ❝ ❝→❝ ❣✐→ trà aj
✭❈❤ó þ✿ ❝→❝ ❣✐→ trà xi ✈➔ yi ❧➔ ❝→❝ ❣✐→ trà ✤➲ ❜➔✐ ✤➣ ❝❤♦ tr÷î❝✮✳

❙❛✐ sè tr✉♥❣ ❜➻♥❤

❙❛✐ sè tr✉♥❣ ❜➻♥❤ ❝õ❛ ✤❛ t❤ù❝ ①➜♣ ①➾ ❝â ❞↕♥❣ ✿

σn =

1
n

2

n

[yi − Pm (xi )]
i=1

❚r♦♥❣ ❜➔✐ t♦→♥ ❝ö t❤➸ ✤➸ t➻♠ ❝→❝ ❤➺ sè ❝õ❛ ❝→❝ ♠❛ tr➟♥ ● ✈➔ ❍ t❛
❧➔♠ t❤❡♦ ❧÷ñ❝ ✤ç tr♦♥❣ ❜↔♥❣ s❛✉✳ ▼❛ tr➟♥ ● ❝→❝ ❤➺ sè ❝õ❛ ❞á♥❣ ✤➛✉ t✐➯♥
❝❤♦ ❜ð✐ ❝→❝ tê♥❣ æ ❧➛♥ ❧÷ñt tø ❝ët ✶ ✤➳♥ ❝ët ♠✱ ❝õ❛ ❞á♥❣ t❤ù ✷ ❝❤♦ ð ❝→❝
tê♥❣ ❧➛♥ ❧÷ñt tø ❝ët ✷ ✤➳♥ ❝ët ✭♠✰✶✮✱✳ ✳ ✳ ✱ ❝á♥ ♠❛ tr➟♥ ● ❝→❝ ❤➺ sè ❝❤♦
❜ð✐ ❝→❝ tê♥❣ ð ❧➛♥ ❧÷ñt tø ❝ët ✭✷♠✰✷✮ ✤➳♥ ❝ët ❝✉è✐ ❝ò♥❣ ✭✸♠✰✷✮

◆❣✉②➵♥ ❚❤à ❍♦➔✐ ❚❤✉

✶✻

❑✹✶❇ ❚♦→♥ ✣❍❙P ❍➔ ◆ë✐ ✷


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

▼ët sè ♣❤÷ì♥❣ ♣❤→♣ ❜✐➳♥ ✤ê✐ ♥❣÷ñ❝ t✉②➳♥ t➼♥❤

❱➼ ❞ö✿ ❙û ❞ö♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❜➻♥❤ ♣❤÷ì♥❣ tè✐ t❤✐➸✉ ✤➸ t➻♠ ✤❛ t❤ù❝
❜➟❝ ✷ P2 (x) = a0 + a1 x + a2 x2 ①➜♣ ①➾ ✈î✐ ❤➔♠ ❝❤♦ ❜ð✐ ❜↔♥❣ s❛✉✿
① ✵✳✼✽ ✶✱✺✻ ✷✱✸✹ ✸✱✶✷ ✸✱✽✶
②

✷✳✺

✶✳✷

✶✳✶✷ ✷✳✷✺ ✹✳✷✽

ð ✤➙② ♠❂✷✱ ♥❂✺✳ ❚ø ❜↔♥❣ ✶ t❛ t❤✉ ✤÷ñ❝ ❜↔♥❣ s❛✉ ✤➸ t➼♥❤ t❤❛♠ sè ❝õ❛
♣❤÷ì♥❣ tr➻♥❤ ❝❤✉➞♥
✭q✉→ tr➻♥❤ t➼♥❤ t♦→♥ t❤ü❝ ❤✐➺♥ ✈î✐ ✸ ❝❤ú sè s❛✉ ❞➜✉ ♣❤➞②✮✳

x0

x1

x2

x3

x4

y

xy

x2 y

✶

✵✳✼✽

✵✳✻✵✽

✵✳✹✼✺

✵✳✸✼✵

✷✳✺✵

✶✳✾✺

✶✳✺✷

✶

✶✳✺✻

✷✳✹✸✹

✸✳✼✾✻

✺✳✾✷✷

✶✳✷✵

✶✳✽✼✷

✷✱✾✷✶

✶

✷✳✸✹

✺✳✹✼✻

✶✷✳✽✸✶

✷✾✳✾✽✷

✶✳✶✷

✷✳✻✷✶

✻✳✶✸✸

✶

✸✳✶✷

✾✳✼✸✹

✸✵✳✸✼✶

✾✹✳✼✺✾

✷✳✷✺

✼✳✵✷✵

✷✶✳✾✵✷

✶

✸✳✽✶

✶✹✳✺✶✻

✺✺✳✸✵✻

✷✶✵✳✼✶✼

✹✳✷✽

✶✻✳✸✵✼ ✻✷✳✶✷✽

✺

✶✶✳✻✶ ✸✷✳✼✻✽ ✶✵✷✳✼✻✶ ✸✹✶✳✼✺✵ ✶✶✳✸✺ ✷✾✳✼✼✵ ✾✹✳✻✵✹

❚ø ✤â t❛ s✉② r❛ ❝→❝ ❤➺ sè a0 , a1 , a2 ❝õ❛ ✤❛ t❤ù❝ ①➜♣ ①➾ P2 (x) t❤ä❛
♠➣♥✿










11, 35
5
11, 61 32, 768
a0


 

 11, 61 32, 768 102, 761   a1  =  29, 770 


 

a2
94, 604
32, 768 102, 761 341, 75

◆❣✉②➵♥ ❚❤à ❍♦➔✐ ❚❤✉

✶✼

❑✹✶❇ ❚♦→♥ ✣❍❙P ❍➔ ◆ë✐ ✷

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣




▼ët sè ♣❤÷ì♥❣ ♣❤→♣ ❜✐➳♥ ✤ê✐ ♥❣÷ñ❝ t✉②➳♥ t➼♥❤
−1 





a0
5
11, 61 32, 768
11, 35

 
 






⇒
=
.
 a1   11, 61 32, 768 102, 761   29, 770 
a2

32, 768 102, 761 341, 75
94, 604
●✐↔✐ ❜➔✐ t♦→♥ tr➯♥ t❛ ✤÷ñ❝✿
a0 = 5, 045, a1 = −4, 043, a2 = 1, 009
❉♦ ✤â ✤❛ t❤ù❝ ①➜♣ ①➾ ❝➛♥ t➻♠ ❝â ❞↕♥❣✿

P2 (x) = 5, 045 − 4, 043x + 1, 009x2
✣➸ s♦ s→♥❤ ❝→❝ yi ✈î✐ P2 (xi ) ✈➔ ❝❤✉➞♥ ❜à t➼♥❤ s❛✐ sè tr✉♥❣ ❜➻♥❤ σ5 t❛
t❤ü❝ ❤✐➺♥ t➼♥❤ t♦→♥ tr➯♥ ❜↔♥❣ s❛✉✿
①
② P2 (x) y − P2 (x) [y − P2 (x)]2
✵✱✼✽

✷✱✺

✷✱✺✵✺

✲✵✱✵✵✺

✵✱✵✵✵✵✷✺

✶✱✺✻

✶✱✷

✶✱✶✾✹

✵✱✵✵✻

✵✱✵✵✵✵✸✻

✷✱✸✹ ✶✱✶✷

✶✱✶✵

✵✱✵✶✵

✵✱✵✵✵✶✵✵

✸✱✶✷ ✷✱✷✺ ✷✱✷✺✷

✲✵✱✵✵✷

✵✱✵✵✵✵✹✵

✸✱✽✶ ✹✱✷✽ ✹✱✷✽✽

✲✵✱✵✵✽

✵✱✵✵✵✵✻✹

❙✉② r❛

5

[P2 (x) − y]2 = 0, 00029

i

✈➔ s❛✐ sè tr✉♥❣ ❜➻♥❤ σ5 =

1
5 .0, 00029

= 0.07

✷✳✸✳✷ ❳➜♣ ①➾ ❤➔♠ ✤❛ t❤ù❝ ❧÷ñ♥❣ ❣✐→❝
✷✳✸✳✷✳✶ ❳➙② ❞ü♥❣ ❤➔♠ ✤❛ t❤ù❝ ❧÷ñ♥❣ ❣✐→❝
❚r♦♥❣ t❤ü❝ t➳ ❦❤✐ t➼♥❤ t♦→♥ t❛ ❣➦♣ ♥❤ú♥❣ ❤➔♠ f (x) ❝â t➼♥❤ ❝❤➜t t✉➛♥
❤♦➔♥✳ ❚❛ t➻♠ ❝→❝❤ ①➜♣ ①➾ ♠ët ❤➔♠ ✤➸ ♣❤↔♥ →♥❤ ✤÷ñ❝ ✤➦❝ ✤✐➸♠ r✐➯♥❣
❝õ❛ ♥â✳ ❑❤✐ ✤â tø ✤❛ t❤ù❝✿
m

Pm (x) =

✭✷✳✸✮

aj ϕj (x)
j=1

▲➜② ❤➺ ❤➔♠ ❧÷ñ♥❣ ❣✐→❝ ❧➔♠ ❝ì sð✳ t❛ ❣✐↔ t❤✐➳t r➡♥❣ ❝→❝ ❤➔♠ f (x) ①➨t
tr➯♥ ✤♦↕♥ 0 ≤ x ≤ 2π ✳ ❚r➯♥ ✤♦↕♥ ❝â ✤ë ❞➔✐ 2π t❤➻ ❤➺ ❤➔♠ ❧÷ñ♥❣ ❣✐→❝

{Pj (x)} = {1, cos x, sinx, cos2x, sin2x, ..., cosmx, sinmx} ❧➔ t✉➛♥ ❤♦➔♥
✈➔ ✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤✳ ❑❤❛✐ tr✐➸♥ ❤➔♠ f (x) t❤❡♦ ❝ì sð ✭✷✳✸✮ ❣å✐ ❧➔ ❦❤❛✐

◆❣✉②➵♥ ❚❤à ❍♦➔✐ ❚❤✉

✶✽

❑✹✶❇ ❚♦→♥ ✣❍❙P ❍➔ ◆ë✐ ✷


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

▼ët sè ♣❤÷ì♥❣ ♣❤→♣ ❜✐➳♥ ✤ê✐ ♥❣÷ñ❝ t✉②➳♥ t➼♥❤

tr✐➸♥ ❧÷ñ♥❣ ❣✐→❝ ❤❛② ❦❤❛✐ tr✐➸♥ ❋♦✉r✐❡r✳ ❚ù❝ ❧➔ ❤➔♠ ①➜♣ ①➾ ❧➔ ♠ët ✤❛
t❤ù❝ ❧÷ñ♥❣ ❣✐→❝ ❝â ❞↕♥❣✿
k

Tk (x) = a0 +

(ar cos rx + br sinrx)
r=1

❚r♦♥❣ ✤â a0 , ar , br ❧➔ ♥❤ú♥❣ t❤❛♠ sè ✈➔ k ∈ N

[2]

✷✳✸✳✷✳✷ ❚➻♠ ❝→❝ t❤❛♠ sè a0, ar , br ✈î✐ r = 1, 2, ...k
❇➙② ❣✐í t❛ ①➨t tr÷í♥❣ ❤ñ♣ ❝→❝ ✤✐➸♠ xt ✱ t = 1, 2, ..., n ♥➡♠ tr➯♥ ❦❤♦↔♥❣
tø (0, 2π) ✈➔ ❝→❝❤ ✤➲✉ ♥❤❛✉ ❧➔✿ 0 < x1 < x2 < ... < xn = 2π
tr♦♥❣ ✤â xt = th t = 1, 2, ..., n; h =

2π
n

.

❇ê ✤➲ ✶✿[2]
❱î✐ n ≥ 2k + 1 ✈➔ xt ❧➔ ♥❤ú♥❣ ✤✐➸♠ ❝õ❛ t➟♣ ❤ñ♣ ❳✱ t❛ ❝â ❝→❝ ✤➥♥❣
t❤ù❝ s❛✉✿

n

n

cos rxt =
t=1
n

r = 1, k

✭✷✳✹✮

r = 1, k , s = 1, k

✭✷✳✺✮

sin rxt = 0
t=1

cos rxt . sin sxt = 0
t=1
n

n

cos rxt . cos sxt =

t=1

✭✷✳✻✮

sin rxt .sinsxt = 0
t=1

r = 1, k , s = 1, k , r = s
n

cos2 rxt =

t=1

n

sin2 rxt =

t=1

n
2

✭✷✳✼✮

r = 1, k

❈❤ù♥❣ ♠✐♥❤
❚❛ ❝â ❝æ♥❣ t❤ù❝ ❊✉❧❡r s❛✉✿

eix = cos x + i sin x

✭x ❧➔ sè t❤ü❝✱ i ❧➔ ✤ì♥ ✈à ↔♦✱ i2 = −1✮

✭✷✳✽✮

❇➡♥❣ ❝→❝ ✤ç♥❣ ♥❤➜t t❤ù❝ ♣❤➛♥ t❤ü❝ ✈➔ ♣❤➛♥ ↔♦ t❛ t❤➜②✳

eix = 1 ⇔ x = 2pπ (p ∈ Z)
❚ø ✤â t❛ ♥❤➟♥ t❤➜② q = −2k, ..., −2, −1, 1, 2, ..., 2k (2k + 1 ≤ n) t❤➻
q

eiqh = ei n 2π = 1 ✈➔ →♣ ❞ö♥❣ ❝æ♥❣ t❤ù❝ t➼♥❤ tê♥❣ ♥ tø ♠ët ❝❤✉é✐ sè
♥❤➙♥ ✭❝æ♥❣ ❜ë✐ ❧➔ eiqh = 1 ✮ t❛ ❝â
n

e
t=1

iqxi

n

=

e

iqth

=

t=1

◆❣✉②➵♥ ❚❤à ❍♦➔✐ ❚❤✉

eiq(n+1)h −eiqh
eiqh −1

=

(ei2qπ −1).eiqh
eiqh −1

✶✾

✭✷✳✾✮

❑✹✶❇ ❚♦→♥ ✣❍❙P ❍➔ ◆ë✐ ✷


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

▼ët sè ♣❤÷ì♥❣ ♣❤→♣ ❜✐➳♥ ✤ê✐ ♥❣÷ñ❝ t✉②➳♥ t➼♥❤

❱➻ q = −2k, ..., −2, −1, 1, 2, ..., 2k (2k + 1 ≤ n) ♥➯♥ t❤❡♦ ♥❤➟♥ ①➨t rót r❛
tø ❝æ♥❣ t❤ù❝ ❊✉❧❡r✿ ei2qπ = 1

n

❚ø ✤â ❞ü❛ tr➯♥ ✭✷✳✾✮ t❛ ❝â

✭✷✳✶✵✮

eiqxt = 0

t=1

tr♦♥❣ ✤â q = −2k, ..., −2, −1, 1, 2, ..., 2k
❚ø ✭✷✳✽✮ ✈➔ ✭✷✳✶✵✮ t❛ ❝â
n

n

cos qxt + i

n

sin qxt =

t=1

t=1

n

(cos qxt + i sin qxt ) =
t=1

✭✷✳✶✶✮

eiqxt = 0

t=1

✣ç♥❣ ♥❤➜t ❝→❝ ♣❤➛♥ t❤ü❝ ✈➔ ♣❤➛♥ ↔♦ tø ✈➳ ✤➛✉ t✐➯♥ ✈➔ ✈➳ s❛✉ ❝ò♥❣ ❝õ❛
✭✷✳✶✶✮ t❛ ✤÷ñ❝
n

n

cos qxt =
t=1

sin qxt = 0

✭✷✳✶✷✮

(q = −2k, ..., −2, −1, 1, 2, ..., 2k)

t=1

❚r♦♥❣ ✭✷✳✶✷✮ ❧➜② q = r = 1, k t❛ t❤✉ ✤÷ñ❝ ✭✷✳✹✮ ✈➔ ❧➜② q❂r✰s
✭ r = 1, k, s = 1, k ✮
t❛ ❝â✿

n

cos(r + s)xt = 0

✭✷✳✶✸✮

sin(r + s)xt = 0

(2.14)

t=1
n
t=1

❚r♦♥❣ ✭✷✳✶✷✮ ❧➜② q❂r✲s
❚❛ ❝â

r = 1, k, s = 1, k,r = s

n

cos(r − s)xt = 0

✭✷✳✶✺✮

sin(r − s)xt = 0

(2.16)

t=1
n
t=1

❑❤✐ r❂s t❤➻ sin (r − s) xt = 0 t = 1, n ♥➯♥ ✭✷✳✶✻✮ ❞ò♥❣ ❝❤♦ ❝↔ ❤❛✐ tr÷í♥❣
❤ñ♣ r❂s

❍❛②

n

sin(r − s)xt = 0

r = 1, k, s = 1, k

✭✷✳✶✼✮

t=1

❚❤❡♦ ❜✐➳♥ ✤ê✐ ❧÷ñ♥❣ ❣✐→❝ t❛ ❧↕✐ ❝â✿
n

cos rxt cos sxt =
t=1
n

sin rxt sin sxt =
t=1
n

cos rxt sin sxt =
t=1

1
2
1
2

1
2

n

cos (r + s) xt +
t=1
n

cos (r − s) xt −
t=1
n

cos (r + s) xt −
t=1

1
2
1
2
1
2

n

cos (r − s) xt

✭✷✳✶✽✮

cos (r + s) xt

✭✷✳✶✾✮

cos (r − s) xt

✭✷✳✷✵✮

t=1
n
t=1
n
t=1

❉ü❛ tr➯♥ ✭✷✳✶✸✮✱✭✷✳✶✺✮ tø ✭✷✳✶✽✮✱✭✷✳✶✾✮ t❛ t❤✉ ✤÷ñ❝ ✭✷✳✻✮
✈➔ ❞ü❛ tr➯♥ ✭✷✳✶✹✮✱ ✭✷✳✶✻✮ tø ✭✷✳✶✼✮ ✭✷✳✷✵✮ t❛ t❤✉ ✤÷ñ❝ ✭✷✳✺✮
t❛ ①➨t ❜✐➳♥ ✤ê✐ ❧÷ñ♥❣ ❣✐→❝

◆❣✉②➵♥ ❚❤à ❍♦➔✐ ❚❤✉

✷✵

❑✹✶❇ ❚♦→♥ ✣❍❙P ❍➔ ◆ë✐ ✷


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣
n

2

cos rxt =

t=1
n

cos2 rxt =

t=1

1
2
1
2

▼ët sè ♣❤÷ì♥❣ ♣❤→♣ ❜✐➳♥ ✤ê✐ ♥❣÷ñ❝ t✉②➳♥ t➼♥❤

n

[1 + cos 2rxt ] =
t=1
n

[1 − cos 2rxt ] =
t=1

❚ø ✭✷✳✶✷✮ t❛ ❧↕✐ ❝â

n

n
2

+

n
2

−

cos 2rxt

✭✷✳✷✶✮

cos 2rxt

✭✷✳✷✷✮

t=1
n

n

t=1

✭✷✳✷✸✮

cos 2rxt = 0
t=1

❚ø ✭✷✳✷✶✮✱ ✭✷✳✷✷✮✱ ✭✷✳✷✸✮ t❛ s✉② r❛ ✭✷✳✼✮
❱➟② ❜ê ✤➲ ❤♦➔♥ t♦➔♥ ✤÷ñ❝ ❝❤ù♥❣ ♠✐♥❤
◗✉❛② ❧↕✐ ❜➔✐ t♦→♥✿

❚ø✿

k

Tk (x) = α0 +

(ar cos rx + βr sinrx)
r=1

❚r♦♥❣ ✤â α0
, αr , βr ❧➔ ♥❤ú♥❣ t❤❛♠ sè ✈➔ k ∈ N
1 cos x0 ... cos kx0 sin x0 ...

 1 cos x ... cos kx sin x ...
1
1
1

❚❛ ❝â X = 
 ...
...
...
...
... ...

1 cos xn ... cos kxn sin xn ...
❱➟② ❤➔♠

y − a0 −


sin kx1 


... 

sin kxn

2

k

S=

sin kx0



(ar cos rx + br sinrx)
r=1

n

2

k

yt −

=

t=0

Xt,(j+l) aj bl

= y − Xa

2

l = k + 1, 2k

j=0

❑❤✐ ✤â t❛ ❝❤♦ ❙ ✤↕t ❣✐→ trà ♠✐♥ tù❝ ❧➔

∂S
∂a

=0

⑩♣ ❞ö♥❣ ❦➳t q✉↔ ❜ê ✤➲ ✶ ✈➔ ❝æ♥❣ t❤ù❝ ✭✷✳✸✮
❚❛ rót r❛ ❝→❝ ❤➺ sè ❝õ❛ ✤❛ t❤ù❝ ❧➔

a0 =
ar =
br =

1
n
2
n

2
n

n

yt
t=1
n

yt cos rxt

r = 1, k

yt sin rxt

r = 1, k

t=1
n
t=1

✷✳✸✳✷✳✸ ❙❛✐ sè tr✉♥❣ ❜➻♥❤ ❝õ❛ ✤❛ t❤ù❝ ❧÷ñ♥❣ ❣✐→❝
❝â ❞↕♥❣✿

◆❣✉②➵♥ ❚❤à ❍♦➔✐ ❚❤✉

✷✶

❑✹✶❇ ❚♦→♥ ✣❍❙P ❍➔ ◆ë✐ ✷

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣
σn =

1
n

▼ët sè ♣❤÷ì♥❣ ♣❤→♣ ❜✐➳♥ ✤ê✐ ♥❣÷ñ❝ t✉②➳♥ t➼♥❤
2

n

[yt − Tk (xt )]
t=1

❱➼ ❞ö
❚➻♠ ✤❛ t❤ù❝ ❧÷ñ♥❣ ❣✐→❝ ❝➜♣ ✷✿ T2 (x) ①➜♣ ①➾ ❤➔♠ ❝❤♦ tr➯♥ ❝→❝ ❝ët ✭✷✮✱✭✸✮
❝õ❛ ❜↔♥❣ s❛✉✿

◆❣✉②➵♥ ❚❤à ❍♦➔✐ ❚❤✉

✷✷

❑✹✶❇ ❚♦→♥ ✣❍❙P ❍➔ ◆ë✐ ✷


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

▼ët sè ♣❤÷ì♥❣ ♣❤→♣ ❜✐➳♥ ✤ê✐ ♥❣÷ñ❝ t✉②➳♥ t➼♥❤

❚➼♥❤ a0 , a1 , a2 , b1 , b2 ❞ü❛ tr➯♥ ❝æ♥❣ t❤ù❝ t➼♥❤ t❛ ✈ø❛ t➻♠ ✤÷ñ❝ t❛ ❝â✿

a0 =
a1 =
a2 =
b1 =
b2 =

1
12
1
6
1
6
1
6
1
6

12

yt =
t=1
12

0,088
12

= 0, 007

yt cos xt =
t=1
12

5,136
6

yt cos 2xt =
t=1
12

sin xt =
t=1
12

−0,124
6

18,022
6

yt sin 2xt =
t=1

= 0, 860
= −0, 021

= 3, 004

2,859

6

= 0, 432

❱➟② t❛ t❤✉ ✤÷ñ❝ ❤➔♠ ①➜♣ ①➾ ❧➔

T2 (x) = 0, 007 + 0, 860 cos x + 3, 004 sin x − 0, 021 cos 2x + 0, 432 sin 2x.

◆❣✉②➵♥ ❚❤à ❍♦➔✐ ❚❤✉

✷✸

❑✹✶❇ ❚♦→♥ ✣❍❙P ❍➔ ◆ë✐ ✷