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ÿêà ÿâëÿº ñîáîþ ñóìó êâàäðàò³â óñ³õ íåâ’ÿçîê. Öèì ôàêòîì<br />
³ îáóìîâëåíà íàçâà ìåòîäó.<br />
Ïðèíöèï ìåòîäó íàéìåíøèõ êâàäðàò³â ïîëÿãຠâ òîìó,<br />
ùî òðåáà òàê ï³ä³áðàòè ïàðàìåòðè k ³ b, ùîá ñóìà êâàäðàò³â<br />
íåâ’ÿçîê áóëà ì³í³ìàëüíîþ. Î÷åâèäíî ïðè öüîìó, ùî ñóìà<br />
àáñîëþòíèõ âåëè÷èí íåâ’ÿçîê áóäå òåæ ì³í³ìàëüíîþ. Öå<br />
ïðèâîäèòü äî òîãî, ùî øóêàíà ïðÿìà íàéêðàùèì ÷èíîì ó<br />
ñóêóïíîñò³ íàáëèæàºòüñÿ äî òî÷îê M i (x i , y i ).<br />
Ó çâ’ÿçêó ç âèùåñêàçàíèì áóäåìî äîñë³äæóâàòè ôóíêö³þ<br />
S(k, b), ÿêà âèçíà÷åíà ôîðìóëîþ (10.8.5) íà åêñòðåìóì. ßê<br />
ìè çíàºìî, äëÿ öüîãî íåîáõ³äíî ñïî÷àòêó çíàéòè ñòàö³îíàðí³<br />
òî÷êè, ÿê³ çíàõîäÿòüñÿ ³ç ñèñòåìè<br />
( k b)<br />
( k b)<br />
⎧ ⎪S′<br />
k<br />
, = 0<br />
⎨<br />
⎪⎩ Sb<br />
′ , = 0<br />
àáî<br />
n<br />
⎧<br />
∑ ( kx + − ) =<br />
⎪<br />
i<br />
b yi xi<br />
0<br />
i=<br />
1<br />
⎨<br />
n<br />
⎪ ∑ ( kxi<br />
+ b − y<br />
i)<br />
= 0 .<br />
⎩i=<br />
1<br />
ϳñëÿ ïðîñòèõ àëãåáðà¿÷íèõ ïåðåòâîðåíü îñòàííÿ ñèñòåìà<br />
ïðèéìຠâèãëÿä:<br />
2<br />
( ) ( )<br />
n<br />
( ∑ i)<br />
n n n<br />
⎧<br />
∑xi k + ∑xi b = ∑xy<br />
i i<br />
⎪ i= 1 i= 1 i=<br />
1<br />
⎨<br />
n<br />
⎪ x k+ n⋅ b = ∑y .<br />
i<br />
⎪⎩ i= 1 i=<br />
1<br />
(10.8.6)<br />
² çðàçó æ âèíèêຠïèòàííÿ: ³ñíóº ÷è í³ ðîçâ’ÿçîê ö³º¿<br />
ñèñòåìè? ßê â³äîìî (äèâ. ï. 3.3.2), â³äïîâ³äü íà öå çàïèòàííÿ<br />
äຠçíà÷åííÿ âèçíà÷íèêà ñèñòåìè (10.8.6). Âèçíà÷íèê<br />
ñèñòåìè (10.8.6) ìຠâèãëÿä:<br />
2 2<br />
( 1 2) ( 1 2) ( 1 2)<br />
2 2<br />
2 x + x − x + x = x − x > 0, ÿêùî x 1 ≠ x 2 .<br />
Îòæå, çã³äíî ç ïðàâèëîì Êðàìåðà ñèñòåìà ìຠºäèíèé<br />
ðîçâ’ÿçîê. Öå îçíà÷àº, ùî ñòàö³îíàðíà òî÷êà ò³ëüêè îäíà:<br />
M 0 (k 0 , b 0 ). Ïðè öüîìó ìè ùå íå çíàºìî, ÷è áóäå â ö³é òî÷ö³<br />
åêñòðåìóì, ÷è í³, à ÿêùî áóäå, òî ÿêèé â³í. Ïåâíà ð³÷, ùî<br />
ìàêñèìóì íàì íå ï³äõîäèòü. Ùîá âçíàòè âñå öå, çðîáèìî<br />
ùå îäèí êðîê: çíàéäåìî äðóã³ ÷àñòèíí³ ïîõ³äí³ â òî÷ö³<br />
M 0 (k 0 , b 0 ):<br />
n<br />
n<br />
2<br />
( ) ∑<br />
( ) ∑ ( )<br />
S′′ k , b = 2 x = A, S′′ k , b = 2 x = B, S′′<br />
k , b = 2n = C.<br />
kk 0 0 i<br />
kb 0 0 i<br />
bb 0 0<br />
i= 1 i=<br />
1<br />
Îñê³ëüêè ∆ 1 = AB – C 2 =4∆ > 0 (ïåðåâ³ðòå!), òî ðàçîì ç<br />
óìîâîþ S′′ ( k0, b<br />
0)<br />
> 0 (äèâ. òåîðåìó 10.6.2) ïðîáëåìà âèð³øåíà<br />
ïîçèòèâíî, à ñàìå: â òî÷ö³ M 0 (k 0 , b 0 )<br />
kk<br />
ôóíêö³ÿ<br />
( , )<br />
n<br />
∑ 2<br />
i<br />
i=<br />
1<br />
S k b = ε ìຠì³í³ìóì. Öå îçíà÷àº, ùî ïðÿìà y = k 0 x + b 0<br />
íàéìåíøå âñüîãî â³äõèëÿºòüñÿ â³ä òî÷îê M i (x i , y i ) ³ º íàä³ÿ,<br />
ùî îòðèìàíà åìï³ðè÷íà ôîðìóëà º âäàëîþ.<br />
10.8.2. Ìåòîä íàéìåíøèõ êâàäðàò³â ìîæëèâî¿ ñòåïåíåâî¿<br />
çàëåæíîñò³ x òà y<br />
ϳñëÿ âäàëîãî çàâåðøåííÿ ïðîáëåìè ï. 10.8.1. âñå æ<br />
òàêè çíîâó âèíèêຠïèòàííÿ: à ùî ðîáèòè, êîëè òî÷êè<br />
M i (x i , y i ) íå ãðóïóþòüñÿ íàâêîëî ïðÿìî¿, íàïðèêëàä ÿê öå<br />
ïîêàçàíî íà ðèñ. 10.23.<br />
n n<br />
2<br />
∑ xi ∑ xi n n<br />
i= 1 i=<br />
1 2<br />
∆= = ⋅∑<br />
−<br />
n<br />
i<br />
i= 1 i=<br />
1<br />
∑ xi<br />
n<br />
i=<br />
1<br />
( ∑ i)<br />
n x x<br />
2<br />
. (10.8.7)<br />
Ìîæíà ïîêàçàòè, ùî ÿêùî x i ≠ x n , i ≠ n, òî âèçíà÷íèê,<br />
ÿêèé îá÷èñëþºòüñÿ çà ôîðìóëîþ (10.8.7), â³äì³ííèé â³ä<br />
íóëÿ ³, á³ëüø òîãî, éîãî çíà÷åííÿ äîäàòíå.<br />
Îáìåæèìîñÿ äîâåäåííÿì öüîãî ôàêòó ïðè n =2:<br />
Ðèñ. 10.23<br />
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