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âàòè ïðîöåñ çàëåæíîñò³ çì³ííèõ x òà y. Ïðè öüîìó áóäóòü<br />
çåêîíîìëåí³ êîøòè, îñê³ëüêè â³äïàäຠïîòðåáà â äîäàòêîâèõ<br />
åêñïåðèìåíòàõ.<br />
Òåïåð âèíèêຠïèòàííÿ: ÿê îòðèìàòè çà åêñïåðèìåíòàëüíèìè<br />
äàíèìè âäàëó ôîðìóëó?<br />
³äïîâ³äü òàêà: òðåáà äîáðå îð³ºíòóâàòèñÿ â íàáëèæåíèõ<br />
ìàòåìàòè÷íèõ ìåòîäàõ, â³ä ñàìèõ ïðîñòèõ äî ñó÷àñíèõ.<br />
Ïðî äåÿê³ ç íèõ ìè êîðîòêî ðîçïîâ³ìî.<br />
Ñàìèé ïðîñòèé ïîâ’ÿçàíèé ç ë³í³éíîþ ³íòåðïîëÿö³ºþ.<br />
Ñóòü éîãî ïîëÿãຠó íàñòóïíîìó: îñê³ëüêè äàí³ òàáëèö³ ìîæíà<br />
³íòåðïðåòóâàòè ÿê êîîðäèíàòè n òî÷îê: M 1 (x 1 , y 1 ), K,<br />
M n (x n , y n ), òî ìè ¿õ çîáðàçèìî íà ïðÿìîêóòí³é êîîðäèíàòí³é<br />
ïëîùèí³. Ïîò³ì ö³ òî÷êè ç’ºäíàºìî ïðÿìèìè (ðèñ. 10.19).<br />
Ðèñ. 10.19<br />
Ïðè öüîìó, ïåâíà ð³÷, ïðèïóñêàºòüñÿ, ùî çàëåæí³ñòü ì³æ<br />
x òà y íà ñåãìåíòàõ [x i–1 , x i ] º ë³í³éíà.<br />
Òåïåð åìï³ðè÷íó ôîðìóëó ìîæíà ïîáóäóâàòè. Êîðèñòóþ-<br />
÷èñü ìåòîäàìè àíàë³òè÷íî¿ ãåîìåò𳿠(ï. 4.3.1), ¿¿ çîáðàæóþòü<br />
ó âèãëÿä³:<br />
⎧ y2 − y1<br />
⎪y1 + ( x−x1) , x∈( x1,<br />
x2)<br />
⎪<br />
x2 − x1<br />
⎪LLLLLLLLLLLLLLLL<br />
⎪<br />
⎪ yi<br />
− yi−<br />
1<br />
f( x)<br />
= ⎨yi−1 + ( x−xi− 1) , x∈( xi−<br />
1,<br />
xi)<br />
⎪ xi<br />
− xi−<br />
1<br />
⎪LLLLLLLLLLLLLLLL . (10.8.1)<br />
⎪<br />
⎪ yn<br />
− yn−<br />
1<br />
⎪<br />
yn− 1<br />
+ ( x−xn− 1) , x∈( xn−<br />
1,<br />
xn)<br />
⎩ xn<br />
− xn−<br />
1<br />
Åìï³ðè÷íó ôîðìóëó ïîáóäîâàíî. Âäàëà âîíà ÷è í³, öå<br />
âæå äðóãå ïèòàííÿ, íà ÿêå, ãàäàþ, ÷èòà÷ çìîæå âæå â³äïîâ³ñòè.<br />
Á³ëüø ñêëàäíà ïîáóäîâà åìï³ðè÷íèõ ôîðìóë ïîâ’ÿçàíà ç<br />
òàê çâàíèì ìåòîäîì ñïëàéíà.<br />
Ñóòü éîãî ïîëÿãຠâ òîìó, ùî íà êîæíîìó ñåãìåíò³<br />
[x i–1 , x i ], äå x i — äàí³ åêñïåðèìåíòó, òî÷êè M i–1 òà M i ç’ºäíóþòüñÿ<br />
íå ïðÿìèìè, à êðèâèìè, ÿê³ çàäàþòüñÿ äëÿ çðó÷íîñò³<br />
â³äîìèìè ôóíêö³ÿìè (ÿê ïðàâèëî, âîíè áåðóòüñÿ ³ç íàáîðó<br />
îñíîâíèõ åëåìåíòàðíèõ ôóíêö³é).<br />
Ñïëàéí-ìåòîä ïðèïóñêàº, ùî ïîáóäîâàíà åìï³ðè÷íî ôóíêö³ÿ<br />
y = f(x) äîñòàòíüî “ãëàäêà” (äîñòàòíüî ðàç äèôåðåíö³éîâíà).<br />
Åìï³ðè÷íà ôóíêö³ÿ, ÿêà ïîáóäîâàíà çà ôîðìóëîþ<br />
(10.8.1), íå º òàêîþ, îñê³ëüêè â òî÷êàõ x i , i = 1 , n âîíà íåäèôåðåíö³éîâíà.<br />
Öå º ãîëîâíèì íåäîë³êîì ôîðìóëè (10.8.1).<br />
Ñïëàéí-ìåòîä, ãðóáî êàæó÷è, áàçóºòüñÿ íà “ñêëåþâàíí³”<br />
÷àñòèí ð³çíèõ ãðàô³ê³â åëåìåíòàðíèõ ôóíêö³é. ² òîìó ôàíòàç³ÿ,<br />
äîñâ³ä ³ âèíàõ³äëèâ³ñòü äîñë³äíèêà äîïîìàãàþòü éîìó<br />
“ñêëå¿òè” ãðàô³êè åëåìåíòàðíèõ ôóíêö³é òàê, ùîá ïîáóäîâà<br />
åìï³ðè÷íî¿ ôóíêö³¿ áóëà âäàëîþ.<br />
Íàâåäåìî ìîæëèâ³ âàð³àíòè (ðèñ. 10.20).<br />
Ðèñ. 10.20<br />
ßê ïîêàçóº ³ñòîð³ÿ, åìï³ðè÷í³ ôîðìóëè áóâàþòü íå ò³ëüêè<br />
âäàë³, à ïðîñòî ãåí³àëüí³. Íàïðèêëàä, äëÿ âñòàíîâëåííÿ çàëåæíîñò³<br />
ì³æ ñèëîþ ñòðóìó, îïîðó ³ íàïðóãè Îì ïðîâ³â<br />
äåê³ëüêà åêñïåðèìåíò³â. ³í çàô³êñóâàâ íàïðóãó ³ äàí³ åêñïåðèìåíò³â<br />
çîáðàçèâ ó âèãëÿä³ òî÷îê íà ïëîùèí³<br />
(ðèñ. 10.21)<br />
394 395
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