Êóðñîâàÿ ðàáîòà
ïî èíôîðìàòèêå íà òåìó:
«×èñëåííûå ìåòîäû ðåøåíèÿ
ñèñòåì ëèíåéíûõ óðàâíåíèé»
Âûïîëíèë:
ñòóäåíò 06–ÈÑÒ, Ôàäååâà Ò.Â.
Ïðîâåðèë:
Ëîâûãèíà Ì.Á.
ã. Ïàâëîâî
2008
Ñîäåðæàíèå.
I Òåîðåòè÷åñêàÿ ÷àñòü
1. Ââåäåíèå....................................................................3
2. ×èñëåííûå ìåòîäû ..................................................6
1) Ìàòðè÷íûé ìåòîä........................................6
2) Ìåòîä Êðàìåðà.............................................9
3) Ìåòîä Ãàóññà …………...............................12
4) Èòåðàöèè äëÿ ëèíåéíûõ ñèñòåì….…..…..17
a) Èòåðàöèÿ ßêîáè..………………...…..18
b) Èòåðàöèÿ Ãàóññà – Çåéäåëÿ..……...…20
II Ïðàêòè÷åñêàÿ ÷àñòü
1) Ìàòðè÷íûé ìåòîä........................................22
2) Ìåòîä Êðàìåðà.............................................24
3) Ìåòîä Ãàóññà……........................................26
4) Ëèñòèíã ïðîãðàììû.……………………….28
III Ïîëüçà ââåäåíèÿ ðàñ÷¸òîâ.……………………………….65
IV Ëèòåðàòóðà……….................................................................66
I. Òåîðåòè÷åñêàÿ ÷àñòü.
Ââåäåíèå.
Ëèíåéíàÿ àëãåáðà – ÷àñòü àëãåáðû, èçó÷àþùàÿ âåêòîðíûå (ëèíåéíûå) ïðîñòðàíñòâà è èõ ïîäïðîñòðàíñòâà, ëèíåéíûå îòîáðàæåíèÿ (îïåðàòîðû), ëèíåéíûå, áèëèíåéíûå, è êâàäðàòè÷íûå ôóíêöèè íà âåêòîðíûõ ïðîñòðàíñòâàõ.
Ëèíåéíàÿ àëãåáðà, ÷èñëåííûå ìåòîäû – ðàçäåë âû÷èñëèòåëüíîé ìàòåìàòèêè, ïîñâÿùåííûé ìàòåìàòè÷åñêîìó îïèñàíèþ è èññëåäîâàíèþ ïðîöåññîâ ÷èñëåííîãî ðåøåíèÿ çàäà÷ ëèíåéíîé àëãåáðû.
Ñðåäè çàäà÷ ëèíåéíîé àëãåáðû íàèáîëüøåå çíà÷åíèå èìåþò äâå: ðåøåíèå ñèñòåìû ëèíåéíûõ àëãåáðàè÷åñêèõ óðàâíåíèé îïðåäåëåíèå ñîáñòâåííûõ çíà÷åíèé è ñîáñòâåííûõ âåêòîðîâ ìàòðèöû. Äðóãèå ÷àñòî âñòðå÷àþùèåñÿ çàäà÷è: îáðàùåíèå ìàòðèöû, âû÷èñëåíèå îïðåäåëèòåëÿ è ò.ä.
Ëþáîé ÷èñëåííûé ìåòîä ëèíåéíîé àëãåáðû ìîæíî ðàññìàòðèâàòü êàê íåêîòîðóþ ïîñëåäîâàòåëüíîñòü âûïîëíåíèÿ àðèôìåòè÷åñêèõ îïåðàöèé íàä ýëåìåíòàìè âõîäíûõ äàííûõ. Åñëè ïðè ëþáûõ âõîäíûõ äàííûõ ÷èñëåííûé ìåòîä ïîçâîëÿåò íàéòè ðåøåíèå çàäà÷è çà êîíå÷íîå ÷èñëî àðèôìåòè÷åñêèõ îïåðàöèé, òî òàêîé ìåòîä íàçûâàåòñÿ ïðÿìûì.  ïðîòèâîïîëîæíîì ñëó÷àå ÷èñëåííûé ìåòîä íàçûâàåòñÿ èòåðàöèîííûì. Ïðÿìûå ìåòîäû - ýòî òàêèå, êàê ìåòîä Ãàóññà, ìåòîä îêàéìëåíèÿ, ìåòîä ïîïîëíåíèÿ, ìåòîä ñîïðÿæ¸ííûõ ãðàäèåíòîâ è äð. Èòåðàöèîííûå ìåòîäû – ýòî ìåòîä ïðîñòîé èòåðàöèè, ìåòîä âðàùåíèé, ìåòîä ïåðåìåííûõ íàïðàâëåíèé, ìåòîä ðåëàêñàöèè è äð. Çäåñü áóäóò ðàññìàòðèâàòüñÿ ìàòðè÷íûé ìåòîä, ìåòîä Ãàóññà è ìåòîä Êðàìåðà.
 äàííîé ðàáîòå áóäóò ðàññìîòðåíû ÷èñëåííûå ìåòîäû â ýëåêòðîííûõ òàáëèöàõ Excel è ïðîãðàììå MathCAD, Microsoft Visual Basic.
MathCAD.
Ïðîãðàììà MathCAD ïî ñâîåìó íàçíà÷åíèþ ïîçâîëÿåò ìîäåëèðîâàòü â ýëåêòðîííîì äîêóìåíòå íàó÷íî–òåõíè÷åñêèå, à òàêæå ýêîíîìè÷åñêèå ðàñ÷¸òû â ôîðìå, äîñòàòî÷íî áëèçêîé ê îáùåïðèíÿòûì ðó÷íûì ðàñ÷¸òàì. Ýòî óïðîùàåò ñîñòàâëåíèå ïðîãðàììû ðàñ÷¸òà, àâòîìàòèçèðóåò ïåðåðàñ÷¸ò è ïîñòðîåíèå ãðàôè÷åñêèõ èëëþñòðàöèé ïîäîáíî ýëåêòðîííûì òàáëèöàì Excel, äîêóìåíòèðîâàíèå ðåçóëüòàòîâ êàê â òåêñòîâîì ðåäàêòîðå Word.
Ïðîãðàììà Mathcad èçâåñòíà çà ë¸ãêîñòü, ñ êîòîðîé ìàòåìàòè÷åñêèå óðàâíåíèÿ, òåêñò, è ãðàôèêà ìîãóò áûòü îáúåäèíåíû â îäíîì äîêóìåíòå. Êðîìå òîãî, âû÷èñëèòåëüíûå ñïîñîáíîñòè Mathcad ðàñïðîñòðàíÿþòñÿ îò ñëîæåíèÿ ñòîëáöà ÷èñåë ê ðåøåíèþ èíòåãðàëîâ è ïðîèçâîäíûõ, ðåøåíèå ñèñòåì óðàâíåíèé è áîëüøå.
Äîñòîèíñòâîì MathCAD ÿâëÿåòñÿ òàêæå íàëè÷èå â åãî ñîñòàâå ýëåêòðîííûõ êíèã. Îäíà èç íèõ – ó÷åáíèê ïî ñàìîé ïðîãðàììå, äðóãèå – ñïðàâî÷íèê ïî ðàçëè÷íûì ðàçäåëàì ìàòåìàòèêè, ôèçèêè, ðàäèîýëåêòðîíèêè è äð.
Microsoft Office Excel.
Åñëè æå ãîâîðèòü î ïðîãðàììå Excel, êîòîðàÿ ÿâëÿåòñÿ îäíîé èç íàèáîëåå èçâåñòíûõ â îáðàáîòêå ýëåêòðîííûõ òàáëèö, òî áåç ïðåóâåëè÷åíèÿ ìîæíî óòâåðæäàòü, ÷òî åå âîçìîæíîñòè ïðàêòè÷åñêè íåèñ÷åðïàåìû.
Îáðàáîòêà òåêñòà, óïðàâëåíèå áàçàìè äàííûõ - ïðîãðàììà íàñòîëüêî ìîùíà, ÷òî âî ìíîãèõ ñëó÷àÿõ ïðåâîñõîäèò ñïåöèàëèçèðîâàííûå ïðîãðàììû - ðåäàêòîðû èëè ïðîãðàììû áàç äàííûõ. Òàêîå ìíîãîîáðàçèå ôóíêöèé ìîæåò ïîíà÷àëó çàïóòàòü, íåæåëè çàñòàâèòü ïðèìåíÿòü èõ íà ïðàêòèêå. Íî ïî ìåðå ïðèîáðåòåíèÿ îïûòà íà÷èíàåøü ïî äîñòîèíñòâó öåíèòü òî, ÷òî ãðàíèö âîçìîæíîñòåé Excel òÿæåëî äîñòè÷ü.
Çà âñþ èñòîðèþ òàáëè÷íûõ ðàñ÷åòîâ ñ ïðèìåíåíèåì ïåðñîíàëüíûõ êîìïüþòåðîâ òðåáîâàíèÿ ïîëüçîâàòåëåé ê ïîäîáíûì ïðîãðàììàì ñóùåñòâåííî èçìåíèëèñü.  íà÷àëå îñíîâíîé àêöåíò â òàêîé ïðîãðàììå, êàê, íàïðèìåð, Visi Calc, ñòàâèëñÿ íà ñ÷åòíûå ôóíêöèè. Ñåãîäíÿ, ïîëîæåíèå äðóãîå. Íàðÿäó ñ èíæåíåðíûìè è áóõãàëòåðñêèìè ðàñ÷åòàìè îðãàíèçàöèÿ è ãðàôè÷åñêîå èçîáðàæåíèå äàííûõ ïðèîáðåòàþò âñå âîçðàñòàþùåå çíà÷åíèå. Êðîìå òîãî, ìíîãîîáðàçèå ôóíêöèé, ïðåäëàãàåìîå òàêîé ðàñ÷åòíîé è ãðàôè÷åñêîé ïðîãðàììîé, íå äîëæíî îñëîæíÿòü ðàáîòó ïîëüçîâàòåëÿ. Ïðîãðàììû äëÿ Windows ñîçäàþò äëÿ ýòîãî èäåàëüíûå ïðåäïîñûëêè.
 ïîñëåäíåå âðåìÿ ìíîãèå êàê ðàç ïåðåøëè íà èñïîëüçîâàíèå Windows â êà÷åñòâå ñâîåé ïîëüçîâàòåëüñêîé ñðåäû. Êàê ñëåäñòâèå, ìíîãèå ôèðìû, ñîçäàþùèå ïðîãðàììíîå îáåñïå÷åíèå, íà÷àëè ïðåäëàãàòü áîëüøîå êîëè÷åñòâî ïðîãðàìì äëÿ Windows.
Visual Basic.
Microsoft Visual Basic – ýòî ìîùíàÿ ñèñòåìà ïðîãðàììèðîâàíèÿ, ïîçâîëÿþùàÿ áûñòðî è ýôôåêòèâíî ñîçäàâàòü ïðèëîæåíèÿ äëÿ Microsoft Windows.  îòëè÷èå îò Excel è MathCAD ýòî íàèáîëåå óäîáíàÿ ïðîãðàììà äëÿ ðåøåíèÿ ñèñòåì ëèíåéíûõ óðàâíåíèé. Ïðîñòîé ïîëüçîâàòåëüñêèé èíòåðôåéñ, ïîçâîëÿþùèé ëåãêî ïåðåêëþ÷àòüñÿ ñ ïðîåêòà ôîðìû íà ñàì êîä ïðîãðàììû.
Óäîáíîå îêíî äëÿ êîäà ñàìîé ïðîãðàììû:
×èñëåííûå ìåòîäû.
Ðàçðåøèìîñòü ñèñòåìû ëèíåéíûõ óðàâíåíèé.
Êîãäà ìû ãîâîðèì î ãëàâíîé ìàòðèöå ñèñòåìû ëèíåéíûõ óðàâíåíèé, òî âñåãäà èìååì â âèäó êâàäðàòíóþ ìàòðèöó n×n, ò. å. ìàòðèöó ñ îäèíàêîâûì êîëè÷åñòâîì ñòðîê è ñòîëáöîâ. Ýòî âàæíî.
Åñëè, íàïðèìåð, êîëè÷åñòâî ñòðîê (êîëè÷åñòâî óðàâíåíèé â ñèñòåìå) áóäåò ìåíüøå, ÷åì êîëè÷åñòâî ñòîëáöîâ (ôàêòè÷åñêè, êîëè÷åñòâà íåèçâåñòíûõ), òî ñèñòåìà áóäåò íåîïðåäåëåííîé, ò. å. ìû íå ñìîæåì îäíîçíà÷íî îïðåäåëèòü âñå íåèçâåñòíûå (ðåøèòü ñèñòåìó).
Íî ýòî íå åäèíñòâåííîå îãðàíè÷åíèå. Èç âåêòîðíîé àëãåáðû èçâåñòíî, ÷òî ñèñòåìà ëèíåéíûõ óðàâíåíèé èìååò ðåøåíèå (îäíîçíà÷íîå) òîãäà è òîëüêî òîãäà, êîãäà åå ãëàâíûé îïðåäåëèòåëü íå ðàâåí íóëþ: Δ ≠ 0.
Ðàññìîòðèì ñëó÷àé, êîãäà îïðåäåëèòåëü ñèñòåìû ðàâåí íóëþ. Çäåñü âîçìîæíû äâà âàðèàíòà:
1. Δ = 0 è êàæäûé èç äîïîëíèòåëüíûõ îïðåäåëèòåëåé Δxi = 0. Ýòî èìååò ìåñòî òîëüêî òîãäà, êîãäà êîýôôèöèåíòû ïðè íåèçâåñòíûõ xi ïðîïîðöèîíàëüíû, ò. å. êàæäîå óðàâíåíèå ñèñòåìû ïîëó÷àåòñÿ èç ïåðâîãî óðàâíåíèÿ óìíîæåíèåì îáåèõ åãî ÷àñòåé íà ÷èñëî k. Ïðè ýòîì ñèñòåìà èìååò áåñ÷èñëåííîå ìíîæåñòâî ðåøåíèé.
2. Δ = 0 è õîòÿ áû îäèí äîïîëíèòåëüíûé îïðåäåëèòåëü Δxi ≠ 0. Ýòî èìååò ìåñòî òîëüêî òîãäà, êîãäà êîýôôèöèåíòû ïðè âñåõ íåèçâåñòíûõ xi, ïðîïîðöèîíàëüíû. Ïðè ýòîì ïîëó÷àåòñÿ ñèñòåìà èç ïðîòèâîðå÷èâûõ óðàâíåíèé, êîòîðàÿ íå èìååò ðåøåíèé.
Ìàòðè÷íûé ìåòîä ðåøåíèÿ ñèñòåì ëèíåéíûõ óðàâíåíèé.
Ïóñòü äàíà ñèñòåìà ëèíåéíûõ óðàâíåíèé:
Ðàññìîòðèì ìàòðèöó, ñîñòàâëåííóþ èç êîýôôèöèåíòîâ ïðè íåèçâåñòíûõ:
Ñâîáîäíûå ÷ëåíû è íåèçâåñòíûå ìîæíî çàïèñàòü â âèäå ìàòðèöû ñòîëáöîâ:
Òîãäà, èñïîëüçóÿ ïðàâèëî óìíîæåíèå ìàòðèö, ýòó ñèñòåìó óðàâíåíèé ìîæíî çàïèñàòü òàê:
èëè
A·x = b. (1)
Ðàâåíñòâî (1) íàçûâàåòñÿ ìàòðè÷íûì óðàâíåíèåì èëè ñèñòåìîé óðàâíåíèé â ìàòðè÷íîì âèäå.
Ìàòðèöà À êîýôôèöèåíòîâ ïðè íåèçâåñòíûõ íàçûâàåòñÿ ãëàâíîé ìàòðèöåé ñèñòåìû.
Èíîãäà ðàññìàòðèâàþò òàêæå ðàñøèðåííóþ ìàòðèöó ñèñòåìû, ò. å. ãëàâíóþ ìàòðèöó ñèñòåìû, äîïîëíåííóþ ñòîëáöîì ñâîáîäíûõ ÷ëåíîâ, êîòîðóþ çàïèñûâàþò â ñëåäóþùåì âèäå:
Ëþáóþ ëèíåéíóþ ñèñòåìó óðàâíåíèé ìîæíî çàïèñàòü â ìàòðè÷íîì âèäå. Íàïðèìåð, ïóñòü äàíà ñèñòåìà:
Ýòà ñèñòåìà èç äâóõ óðàâíåíèé ñ òðåìÿ íåèçâåñòíûìè – x, y,.  âûñøåé ìàòåìàòèêå ìîæíî ðàññìàòðèâàòü ñèñòåìû èç î÷åíü áîëüøîãî ÷èñëà óðàâíåíèé ñ áîëüøèì êîëè÷åñòâîì íåèçâåñòíûõ è ïîýòîìó íåèçâåñòíûå ïðèíÿòî îáîçíà÷àòü òîëüêî áóêâîé õ, íî ñ èíäåêñàìè:
Çàïèøåì ýòó ñèñòåìó â ìàòðè÷íîì âèäå:
Çäåñü ãëàâíàÿ ìàòðèöà ñèñòåìû:
Ðàñøèðåííàÿ ìàòðèöà áóäåò èìåòü âèä:
Ðåøåíèÿ ìàòðè÷íûõ óðàâíåíèé.
Ìàòðè÷íûå óðàâíåíèÿ ðåøàþòñÿ ïðè ïîìîùè îáðàòíûõ ìàòðèö. Óðàâíåíèå ðåøàåòñÿ ñëåäóþùèì îáðàçîì. Ïóñòü ìàòðèöà À – íåâûðîæäåííàÿ (D ≠ 0), òîãäà ñóùåñòâóåò îáðàòíàÿ ìàòðèöà À-1. Óìíîæèâ íà íåå îáå ÷àñòè ìàòðè÷íîãî óðàâíåíèÿ, èìååì À-1(ÀÕ) = À-1Â. Èñïîëüçóÿ ñî÷åòàòåëüíûé çàêîí óìíîæåíèÿ, ïåðåïèøåì ýòî ðàâåíñòâî â âèäå
(À-1À) Õ = À-1Â.
Ïîñêîëüêó À-1 À = Å è ÅÕ = Õ, íàõîäèì:
Õ = À-1Â.
Òàêèì îáðàçîì, ÷òîáû ðåøèòü ìàòðè÷íîå óðàâíåíèå, íóæíî:
1. Íàéòè îáðàòíóþ ìàòðèöó À-1.
2. Íàéòè ïðîèçâåäåíèå îáðàòíîé ìàòðèöû À-1 íà ìàòðèöó ñòîëáåö ñâîáîäíûõ ÷ëåíîâ Â, ò. å À-1Â.
Ïîëüçóÿñü îïðåäåëåíèåì ðàâíûõ ìàòðèö, çàïèñàòü îòâåò.
Ïðè ýòîì ñîáñòâåííî íàõîæäåíèå îáðàòíîé ìàòðèöû – ïðîöåññ äîñòàòî÷íî òðóäîåìêèé è åãî ïðîãðàììèðîâàíèå âðÿä ëè ìîæíî íàçâàòü ýëåìåíòàðíîé çàäà÷åé. Ïîýòîìó íà ïðàêòèêå ÷àùå ïðèìåíÿþò ÷èñëåííûå ìåòîäû ðåøåíèÿ ñèñòåì ëèíåéíûõ óðàâíåíèé.
Ê ÷èñëåííûì ìåòîäàì ðåøåíèÿ ñèñòåì ëèíåéíûõ óðàâíåíèé îòíîñÿò òàêèå êàê: ìåòîä Ãàóññà, ìåòîä Êðàìåðà, èòåðàòèâíûå ìåòîäû.  ìåòîäå Ãàóññà, íàïðèìåð, ðàáîòàþò íàä ðàñøèðåííîé ìàòðèöåé ñèñòåìû. À â ìåòîäå Êðàìåðà – ñ îïðåäåëèòåëÿìè ñèñòåìû, îáðàçîâàííûìè ïî ñïåöèàëüíîìó ïðàâèëó.
Ìåòîä Êðàìåðà.
Ïðè ðåøåíèè ñèñòåì ëèíåéíûõ óðàâíåíèé ïî ìåòîäó Êðàìåðà ïîñëåäîâàòåëüíî âûïîëíÿåòñÿ ñëåäóþùèé àëãîðèòì:
1. Çàïèñûâàþò ñèñòåìó â ìàòðè÷íîì âèäå (åñëè ýòî åùå íå ñäåëàíî).
2. Âû÷èñëÿþò ãëàâíûé îïðåäåëèòåëü ñèñòåìû:
3. Âû÷èñëÿþò âñå äîïîëíèòåëüíûå îïðåäåëèòåëè ñèñòåìû:
4. Åñëè ãëàâíûé îïðåäåëèòåëü ñèñòåìû íå ðàâåí íóëþ, òî âûïîëíÿþò ïóíêò 5. Èíà÷å ðàññìàòðèâàþò âîïðîñ î ðàçðåøèìîñòè äàííîé ñèñòåìû (èìååò áåñ÷èñëåííîå ìíîæåñòâî ðåøåíèé èëè íå èìååò ðåøåíèé). Íàõîäÿò çíà÷åíèÿ âñåõ íåèçâåñòíûõ ïî ôîðìóëàì Êðàìåðà äëÿ ðåøåíèÿ ñèñòåìû n ëèíåéíûõ óðàâíåíèé ñ n íåèçâåñòíûìè, êîòîðûå èìåþò âèä:
Ïðèìåð 1
Ðåøèòü ïî ìåòîäó Êðàìåðà ñèñòåìó èç òðåõ óðàâíåíèé ñ òðåìÿ íåèçâåñòíûìè:
Ðåøåíèå
Çàïèøåì ãëàâíûé è ïîáî÷íûå îïðåäåëèòåëè ñèñòåìû:
Âû÷èñëèì ýòè îïðåäåëèòåëè:
Δ = 3*4*(-4)+7*(-3)*5+(-2)*(-8)*5-5*4*5-3*(-3)*(-8)-7*(-2)*(-4) = 48-105+80-100-72-56 = 128-333 = -205.
Δ1 = -112+(-45)+(-192)-(-240)-24-168 = -112-45-192+240-24-168 = 240-541 = -301.
Δ2 = -36-420-280-75+196-288 = 196-1099 = -903.
Δ3 = -144-147-30-140+27-168 = -629+27 = -602.
Ãëàâíûé îïðåäåëèòåëü ñèñòåìû íå ðàâåí íóëþ. Íàõîäèì íåèçâåñòíûå ïî ôîðìóëàì Êðàìåðà.
Ïîäñòàâèì íàéäåííûå çíà÷åíèÿ îïðåäåëèòåëåé â ôîðìóëû Êðàìåðà:
x1 = Δ1/Δ = -301/(-205) = 1,468292682927 ≈ 1,47;
x2 = Δ2/Δ = -903/(-205) = 4,40487804878 ≈ 4,4;
x3 = Δ3/Δ = -602/(-205) = 2,936585365854 ≈ 2,93.
Âûâîä.
Ïðè ðåøåíèè ñèñòåì ëèíåéíûõ óðàâíåíèé ïî ìåòîäó Êðàìåðà èñïîëüçóþòñÿ ôîðìóëû, â êîòîðûõ ó÷àñòâóþò êàê ãëàâíûé, òàê è äîïîëíèòåëüíûå îïðåäåëèòåëè ñèñòåìû:
Íàïîìíèì, ÷òî ãëàâíûì îïðåäåëèòåëåì ñèñòåìû íàçûâàåòñÿ îïðåäåëèòåëü ãëàâíîé ìàòðèöû ñèñòåìû, ñîñòàâëåííîé èç êîýôôèöèåíòîâ ïðè íåèçâåñòíûõ:
Åñëè â ãëàâíîì îïðåäåëèòåëå ñèñòåìû çàìåíèòü ïîî÷åðåäíî ñòîëáöû êîýôôèöèåíòîâ ïðè x1, x2,...xn íà ñòîëáåö ñâîáîäíûõ ÷ëåíîâ, òî ïîëó÷èì n äîïîëíèòåëüíûõ îïðåäåëèòåëåé (äëÿ êàæäîãî èç n íåèçâåñòíûõ):
Ïðè ýòîì âàæåí âîïðîñ î ðàçðåøèìîñòè äàííîé ñèñòåìû, êîòîðûé ðåøàåòñÿ ñðàâíåíèåì ãëàâíîãî è äîïîëíèòåëüíûõ îïðåäåëèòåëåé ñèñòåìû ñ íóëåì:
Ìåòîä Ãàóññà – ïðÿìîé è îáðàòíûé õîä.
Ðàññìîòðèì ìåòîä Ãàóññà. Íàïðèìåð, ïóñòü äàíà ðàñøèðåííàÿ ìàòðèöà íåêîòîðîé ñèñòåìû m ëèíåéíûõ óðàâíåíèé c n íåèçâåñòíûìè:
Áóäåì ñ÷èòàòü, ÷òî a11 ≠ 0 (åñëè ýòî íå òàê, òî äîñòàòî÷íî ïåðåñòàâèòü ïåðâóþ è íåêîòîðóþ äðóãóþ ñòðîêó ðàñøèðåííîé ìàòðèöû ìåñòàìè). Ïðîâåäåì ñëåäóþùèå ýëåìåíòàðíûå ïðåîáðàçîâàíèÿ:
C2-(a21/a11)*C1,
...
Cm-(am1/a11)*C1,
ò.å. Ci-(ai1/a11)*C1, i = 2, 3, ..., m.
Ò. å. îò êàæäîé ñòðîêè ðàñøèðåííîé ìàòðèöû (êðîìå ïåðâîé) îòíèìàåì ïåðâóþ ñòðîêó, óìíîæåííóþ íà ÷àñòíîå îò äåëåíèÿ ïåðâîãî ýëåìåíòà ýòîé ñòðîêè íà äèàãîíàëüíûé ýëåìåíò à11.
 ðåçóëüòàòå ïîëó÷èì ìàòðèöó:
Ò. å. ïåðâàÿ ñòðîêà îñòàëàñü áåç èçìåíåíèé, à â ñòîëáöå ïîä à11 íà âñåõ ìåñòàõ îêàçàëèñü íóëè. Îáðàòèì âíèìàíèå, ÷òî ïðåîáðàçîâàíèÿ êîñíóëèñü âñåõ ýëåìåíòîâ ñòðîê, íà÷èíàÿ ñî âòîðîé, âñåé ðàñøèðåííîé ìàòðèöû ñèñòåìû.
Òåïåðü íàøà çàäà÷à ñîñòîèò â òîì, ÷òîáû ïîëó÷èòü íóëè ïîäî âñåìè äèàãîíàëüíûìè ýëåìåíòàìè ìàòðèöû À – aij, ãäå I = j.
Ïîâòîðèì íàøè ýëåìåíòàðíûå ïðåîáðàçîâàíèÿ, íî óæå äëÿ ýëåìåíòà α22.
C1-(a12/α22)*C2,
...
Cm-(αm2/α22)*C2,
ò.å. Ci-(αi2/α22)*C2, i = 3, ..., m.
Ò. å. îò êàæäîé ñòðîêè ðàñøèðåííîé ìàòðèöû (òåïåðü êðîìå ïåðâîé è âòîðîé) îòíèìàåì âòîðóþ ñòðîêó, óìíîæåííóþ íà ÷àñòíîå îò äåëåíèÿ ïåðâîãî ýëåìåíòà ýòîé (òåêóùåé) ñòðîêè íà äèàãîíàëüíûé ýëåìåíò α22.
Òàêèå ïðåîáðàçîâàíèÿ ïðîäîëæàþòñÿ äî òåõ ïîð, ïîêà ìàòðèöà íå ïðèâåäåòñÿ ê âåðõíåå - òðåóãîëüíîìó âèäó. Ò. å. ïîä ãëàâíîé äèàãîíàëüþ íå îêàæóòñÿ âñå íóëè:
Âñïîìíèâ, ÷òî êàæäàÿ ñòðîêà ïðåäñòàâëÿåò ñîáîé îäíî èç óðàâíåíèé ëèíåéíîé ñèñòåìû óðàâíåíèé, ëåãêî çàìåòèòü, ÷òî ïîñëåäíåå m-îå óðàâíåíèå ïðèíèìàåò âèä:
γmn*xn = δm.
Îòñþäà ëåãêî ìîæíî íàéòè çíà÷åíèå ïåðâîãî êîðíÿ – xn = δm/γmn.
Ïîäñòàâèâ ýòî çíà÷åíèå â ïðåäûäóùåå m-1-å óðàâíåíèå, ëåãêî ïîëó÷èì çíà÷åíèå xn-1-îãî êîðíÿ.
Òàêèì îáðàçîì, ïîäíèìàÿñü äî ñàìîãî âåðõà îáðàòíûì õîäîì ìåòîäà Ãàóññà, ìû ïîñëåäîâàòåëüíî íàéäåì âñå êîðíè ñèñòåìû óðàâíåíèé.
Ïðèìåð 1
Ðàññìîòðèì ñèñòåìó óðàâíåíèé:
Ãëàâíûé îïðåäåëèòåëü äàííîé ñèñòåìû:
Δ = [1*(-4)*(-2)+2*2*1+(-1)*(-1)*(-1)]-[1*(-4)*(-1)+2*(-1)*(-2)+2*(-1)*1] = [8+4-1]-[4+4-2] = 11-6 =5,
ò. å. Δ ≠ 0.
Ò. å. ñèñòåìà îïðåäåëåíà è ðàçðåøèìà. Ðåøèì åå ïî ìåòîäó Ãàóññà.
Ïðîâåäåì ïðÿìîé õîä ìåòîäà Ãàóññà, âûïèñàâ ïðåäâàðèòåëüíî ðàñøèðåííóþ ìàòðèöó ñèñòåìû:
Ïîëó÷èì íóëè ïîä ãëàâíîé äèàãîíàëüþ â ïåðâîì ñòîëáöå ðàñøèðåííîé ìàòðèöû. Äëÿ ïîëó÷åíèÿ íóëÿ â ýëåìåíòå a21 (ò. å. ïîä äèàãîíàëüþ âî âòîðîé ñòðîêå ìàòðèöû) âòîðóþ ñòðîêó ìàòðèöû ïðåîáðàçóåì ïî ôîðìóëå C2-(a21/a11)*C1 = C2-(2/1)*C1 = C2-2*C1:
Àíàëîãè÷íî ïîñòóïàåì è ñ ýëåìåíòîì à31 (ò. å. ïîä äèàãîíàëüþ â òðåòüåé ñòðîêå ìàòðèöû). Òðåòüþ ñòðîêó ìàòðèöû ïðåîáðàçóåì ïî ôîðìóëå C3-(a31/a11)*C1 = C3-(-1/1)*C1 = C3+C1:
Òàêèì îáðàçîì, ìû ïîëó÷èëè íóëè ïîä ãëàâíîé äèàãîíàëüþ â ïåðâîì ñòîëáöå ðàñøèðåííîé ìàòðèöû. Îñòàëîñü ïîëó÷èòü íóëü ïîä ãëàâíîé äèàãîíàëüþ âî âòîðîì ñòîëáöå ìàòðèöû, ò. å. íà ìåñòå ýëåìåíòà à32. Äëÿ ýòîãî òðåòüþ ñòðîêó ìàòðèöû ïðåîáðàçóåì ïî ôîðìóëå C3-(a32/a22)*C2 = C3-(1/-2)*C2 = C3+1/2C2:
Òàêèì îáðàçîì, ïðîâåäÿ ïðÿìîé õîä ìåòîäà Ãàóññà, ìû ïîëó÷èëè ðàñøèðåííóþ ìàòðèöó ñèñòåìû, ïðèâåäåííóþ ê âåðõíå-òðåóãîëüíîìó âèäó:
Ýòà ìàòðèöà ýêâèâàëåíòíà ñèñòåìå:
Îáðàòíûì õîäîì ìåòîäà Ãàóññà íàéäåì êîðíè ñèñòåìû. Èç ïîñëåäíåãî óðàâíåíèÿ íàéäåì êîðåíü õ3:
-5/2x3 = 3/2,
x3 = (3/2):(-5/2) = 3/2*(-2/5) = -3/5.
Êîðåíü x3 = -3/5 íàéäåí. Ïîäñòàâèì åãî â âåðõíåå (âòîðîå) óðàâíåíèå ñèñòåìû (-2x2-3x3 = 1):
-2x2-3(-3/5) = 1,
-2x2+9/5 = 1,
-2x2 = 1-9/5,
-2x2 = -4/5,
x2 = (-4/5):(-2) = (-4/5)*(-1/2) = 2/5.
Êîðåíü x2 = 2/5 íàéäåí. Ïîäñòàâèì åãî è êîðåíü õ3 â âåðõíåå (ïåðâîå) óðàâíåíèå ñèñòåìû (x1-x2+x3 = 0):
x1-2/5+(-3/5) = 0,
x1-5/5 = 0,
x1 = 5/5 = 1.
Ïðîâåðêà:
ò. å.
ò. å.
è ò. ä.
Âûâîä.
Èòàê, ìåòîä Ãàóññà (èëè, èíà÷å, ìåòîä ïîñëåäîâàòåëüíîãî èñêëþ÷åíèÿ íåèçâåñòíûõ) ñîñòîèò â ñëåäóþùåì:
1. Ïóòåì ýëåìåíòàðíûõ ïðåîáðàçîâàíèé ñèñòåìó óðàâíåíèé ïðèâîäÿò ê ýêâèâàëåíòíîé åé ñèñòåìå ñ âåðõíå-òðåóãîëüíîé ìàòðèöåé. Ýòè äåéñòâèÿ íàçûâàþò ïðÿìûì õîäîì.
2. Èç ïîëó÷åííîé òðåóãîëüíîé ñèñòåìû ïåðåìåííûå íàõîäÿò ñ ïîìîùüþ ïîñëåäîâàòåëüíûõ ïîäñòàíîâîê (îáðàòíûé õîä).
3. Ïðè ýòîì âñå ïðåîáðàçîâàíèÿ ïðîâîäÿòñÿ íàä òàê íàçûâàåìîé ðàñøèðåííîé ìàòðèöåé ñèñòåìû, êîòîðóþ è ïðèâîäÿò ê âåðõíåå - òðåóãîëüíîìó âèäó â ïðÿìîì õîäå ìåòîäà.
Èòåðàöèÿ äëÿ ëèíåéíûõ ñèñòåì.
Ñïîñîá èòåðàöèé äàåò âîçìîæíîñòü ïîëó÷èòü ïîñëåäîâàòåëüíîñòü ïðèáëèæåííûõ çíà÷åíèé, ñõîäÿùèõñÿ ê òî÷íîìó ðåøåíèþ ñèñòåìû, ïîäîáíî òîìó, êàê ýòî äåëàåòñÿ äëÿ îäíîãî óðàâíåíèÿ.
Äëÿ îïðåäåëåííîñòè îãðàíè÷èìñÿ ñèñòåìîé èç ÷åòûðåõ óðàâíåíèé ñ ÷åòûðüìÿ íåèçâåñòíûìè (ñèñòåìà ÷åòâåðòîãî ïîðÿäêà), êîòîðóþ çàïèøåì â âèäå:
Ðàçðåøèì ïåðâîå óðàâíåíèå ñèñòåìû îòíîñèòåëüíî õ1:
õ1 = (-a12/a11)õ2-a13/a11õ3-a14/a11õ4-a15/a11.
Çàòåì ðàçðåøèì âòîðîå óðàâíåíèå îòíîñèòåëüíî õ2 è ò. ä. Òîãäà ñèñòåìó ìîæíî ïåðåïèñàòü â âèäå:
ãäå α = -aik/aii, i = 1, 2, 3, 4; k = 1, 2, 3, 4, 5.
Ñèñòåìà ÿâëÿåòñÿ ÷àñòíûì ñëó÷àåì çàïèñè âèäà:
Ïðè ýòîì ëèíåéíàÿ ôóíêöèÿ L1 ôàêòè÷åñêè íå çàâèñèò îò õ1.
Çàäàäèì êàêèå-ëèáî íà÷àëüíûå çíà÷åíèÿ íåèçâåñòíûõ (íóëåâûå ïðèáëèæåíèÿ):
õ1(0), õ2(0), õ3(0), õ4(0).
Ïîäñòàâëÿÿ ýòè çíà÷åíèÿ â ïðàâûå ÷àñòè ñèñòåìû (*), ïîëó÷èì ïåðâûå ïðèáëèæåíèÿ:
Ïîëó÷åííûå ïåðâûå ïðèáëèæåíèÿ ìîãóò áûòü òàê æå èñïîëüçîâàíû äëÿ ïîëó÷åíèÿ âòîðûõ, òðåòüèõ è ò. ä. ïðèáëèæåíèé. Ò. å. ìîæíî çàïèñàòü:
Óñëîâèÿ ñõîäèìîñòè èòåðàöèîííîãî ïðîöåññà.
Óñòàíîâèì óñëîâèÿ, âûïîëíåíèå êîòîðûõ îáåñïå÷èò ñõîäèìîñòü ïîëó÷àþùèõñÿ ïðèáëèæåíèé ê èñòèííîìó (òî÷íîìó) ðåøåíèþ ñèñòåìû õ1, õ2, õ3, õ4.
Íå âäàâàÿñü â ïîäðîáíîñòè, ñêàæåì, ÷òî äëÿ òîãî ÷òîáû èòåðàöèîííûé ïðîöåññ ñõîäèëñÿ ê òî÷íîìó ðåøåíèþ, äîñòàòî÷íî, ÷òîáû âñå êîýôôèöèåíòû ñèñòåìû áûëè ìàëû ïî ñðàâíåíèþ ñ äèàãîíàëüíûìè.
Ýòî óñëîâèå ìîæíî ñôîðìóëèðîâàòü è áîëåå òî÷íî:
Äëÿ ñõîäèìîñòè ïðîöåññà èòåðàöèé äîñòàòî÷íî, ÷òîáû â êàæäîì ñòîëáöå ñóììà îòíîøåíèé êîýôôèöèåíòîâ ñèñòåìû ê äèàãîíàëüíûì ýëåìåíòàì, âçÿòûì èç òîé æå ñòðîêè, áûëà ñòðîãî ìåíüøå åäèíèöû:
Èòåðàöèÿ ßêîáè.
Ðàññìîòðèì ñèñòåìó ëèíåéíûõ óðàâíåíèé:
Óðàâíåíèÿ ìîæíî çàïèñàòü â âèäå:
Ýòî ïîçâîëÿåò ïðåäëîæèòü ñëåäóþùèé èòåðàöèîííûé ïðîöåññ:
èëè (äðóãîé âèä çàïèñè)
Ïîêàæåì, ÷òî åñëè íà÷àòü ñ òî÷êè P0 = (õ1(0), õ2(0), õ3(0), õ4(0)) = (1, 2, 2), òî èòåðàöèÿ (3) ñõîäèòñÿ ê ðåøåíèþ (2, 4, 3). Ïîäñòàâèì õ1 = 1, õ2 = 2, õ2 = 2 â ïðàâóþ ÷àñòü êàæäîãî óðàâíåíèÿ èç (3), ÷òîáû ïîëó÷èòü íîâûå çíà÷åíèÿ:
Íîâàÿ òî÷êà P1 = (õ1(1), õ2(1), õ3(1), õ4(1)) = (1.75, 3.375, 3), áëèæå, ÷åì P0.
Èòåðàöèÿ, èñïîëüçóþùàÿ (3), ãåíåðèðóåò ïîñëåäîâàòåëüíîñòü òî÷åê {Pk}, êîòîðàÿ ñõîäèòñÿ ê ðåøåíèþ (2, 4, 3):
k | õ1(k) | õ2(k) | õ3(k) |
0 | 1.0 | 2.0 | 2.0 |
1 | 1.75 | 3.375 | 3.0 |
2 | 1.84375 | 3.875 | 3.025 |
3 | 1.9625 | 3.925 | 2.9625 |
4 | 1.990625 | 3.9765625 | 3.0 |
5 | 1.99414063 | 3.9953125 | 3.0009375 |
… | … | … | … |
15 | 1.99999993 | 3.99999985 | 3.0009375 |
… | … | … | … |
19 | 2.0 | 4.0 | 3.0 |
Ýòîò ïðîöåññ íàçûâàåòñÿ èòåðàöèåé ßêîáè è ìîæåò èñïîëüçîâàòüñÿ äëÿ ðåøåíèÿ îïðåäåëåííûõ òèïîâ ëèíåéíûõ ñèñòåì.
Èòåðàöèÿ Ãàóññà-Çåéäåëÿ.
Ïðîöåññ èòåðàöèè ßêîáè èíîãäà ìîæíî ìîäèôèöèðîâàòü äëÿ óñêîðåíèÿ ñõîäèìîñòè.
Îòìåòèì, ÷òî èòåðàòèâíûé ïðîöåññ ßêîáè ïðîèçâîäèò òðè ïîñëåäîâàòåëüíîñòè – {õ1(k)}, {õ2(k)}, {õ3(k)}, {õ4(k)}. Êàæåòñÿ ðàçóìíûì, ÷òî õ1(k+1) ìîæåò áûòü èñïîëüçîâàíî âìåñòî õ2(k). Àíàëîãè÷íî õ1(k+1) è õ2(k+1) ìîæíî èñïîëüçîâàòü â âû÷èñëåíèè õ3(k+1). Íàïðèìåð, äëÿ óðàâíåíèé èç ñèñòåìû (1) ýòî äàñò ñëåäóþùèé âèä èòåðàöèîííîãî ïðîöåññà Ãàóññà-Çåéäåëÿ, èñïîëüçóþùèé (3*):
Òàêîé èòåðàöèîííûé ïðîöåññ äàñò ðåçóëüòàòû:
k | õ1(k) | õ2(k) | õ3(k) |
0 | 1.0 | 2.0 | 2.0 |
1 | 1.75 | 3.75 | 2.95 |
2 | 1.95 | 3.96875 | 2.98625 |
3 | 1.995625 | 3.99609375 | 2.99903125 |
… | … | … | … |
8 | 1.99999983 | 3.99999988 | 2.99999996 |
9 | 1.99999998 | 3.99999999 | 3.0 |
10 | 2.0 | 4.0 | 3.0 |
Ò. å. ê òî÷íîìó ðåøåíèþ ìû ïðèøëè óæå íà 10-îì øàãå èòåðàöèè, à íå íà 19, êàê â èòåðàöèè ßêîáè.
Âûâîä.
1. Ñïîñîá èòåðàöèé äàåò âîçìîæíîñòü ïîëó÷èòü ïîñëåäîâàòåëüíîñòü ïðèáëèæåííûõ çíà÷åíèé, ñõîäÿùèõñÿ ê òî÷íîìó ðåøåíèþ ñèñòåìû. Äëÿ ýòîãî ñèñòåìà ïðèâîäèòñÿ ê âèäó (äëÿ ñëó÷àÿ ñèñòåìû èç ÷åòûðåõ óðàâíåíèé):
Ýòè ôîðìóëû êàê ðàç è çàäàþò ñîáñòâåííî èòåðàöèîííûé ïðîöåññ.
2. Ïðè ýòîì ÷òîáû èòåðàöèîííûé ïðîöåññ ñõîäèëñÿ ê òî÷íîìó ðåøåíèþ, äîñòàòî÷íî, ÷òîáû âñå êîýôôèöèåíòû ñèñòåìû áûëè ìàëû ïî ñðàâíåíèþ ñ äèàãîíàëüíûìè.
Ýòî óñëîâèå ìîæíî ñôîðìóëèðîâàòü è áîëåå òî÷íî:
Äëÿ ñõîäèìîñòè ïðîöåññà èòåðàöèé äîñòàòî÷íî, ÷òîáû â êàæäîì ñòîëáöå ñóììà îòíîøåíèé êîýôôèöèåíòîâ ñèñòåìû ê äèàãîíàëüíûì ýëåìåíòàì, âçÿòûì èç òîé æå ñòðîêè, áûëà ñòðîãî ìåíüøå åäèíèöû:
3. Ñëåäóåò òàê æå ñêàçàòü, ÷òî èòåðàöèîííûé ïðîöåññ ìîæåò ïðîâîäèòüñÿ êàê â âèäå èòåðàöèè ßêîáè, òàê è â âèäå èòåðàöèè Ãàóññà-Çåéäåëÿ.  ïîñëåäíåì ñëó÷àå ñõîäèìîñòü èòåðàöèîííîãî ïðîöåññà ìîæåò ñóùåñòâåííî óëó÷øèòüñÿ.
II. Ïðàêòè÷åñêàÿ ÷àñòü.
1) Ìåòîä îáðàòíîé ìàòðèöû.
Ìåòîä îáðàòíîé ìàòðèöû | ||||||
| ||||||
x1 | x2 | x3 | x4 | |||
12 | -4 | 0 | 6 | 2 | ||
A= | -4 | 21 | 5 | 3 | B= | 4 |
-3 | 2 | -22 | 1 | -2 | ||
-2 | -3 | 5 | 23 | 4 | ||
0,083 | 0,013 | -0,002 | -0,023 | |||
A-1= | 0,016 | 0,048 | 0,009 | -0,011 | ||
-0,009 | 0,003 | -0,044 | 0,004 | |||
0,011 | 0,007 | 0,010 | 0,039 | |||
x= | 0,129 | |||||
0,165 | ||||||
0,097 | ||||||
0,186 |
2) Ìåòîä Êðàìåðà.
Ìåòîä Êðàìåðà | |||||||
| |||||||
x1 | x2 | x3 | x4 | ||||
12 | -4 | 0 | 6 | 2 | |||
A= | -4 | 21 | 5 | 3 | B= | 4 | |
-3 | 2 | -22 | 1 | -2 | |||
-2 | -3 | 5 | 23 | 4 | |||
'A'= | -134088 | ||||||
2 | -4 | 0 | 6 | ||||
A1= | 4 | 21 | 5 | 3 | |||
-2 | 2 | -22 | 1 | ||||
4 | -3 | 5 | 23 | ||||
'A1'= | -17296 | x1= | 0,129 | ||||
12 | 2 | 0 | 6 | ||||
A2= | -4 | 4 | 5 | 3 | |||
-3 | -2 | -22 | 1 | ||||
-2 | 4 | 5 | 23 | ||||
'A2'= | -22188 | x2= | 0,165 | ||||
12 | -4 | 2 | 6 | ||||
A3= | -4 | 21 | 4 | 3 | |||
-3 | 2 | -2 | 1 | ||||
-2 | -3 | 4 | 23 | ||||
'A3'= | -12980 | x3= | 0,097 | ||||
12 | -4 | 0 | 2 | ||||
A4= | -4 | 21 | 5 | 4 | |||
-3 | 2 | -22 | -2 | ||||
-2 | -3 | 5 | 4 | ||||
'A4'= | -24896 | x4= | 0,186 | ||||
x= | 0,129 | ||||||
0,165 | |||||||
0,097 | |||||||
0,186 |
3) Ìåòîä Ãàóññà.
Ìåòîä Ãàóññà | |||||||
| |||||||
x1 | x2 | x3 | x4 | ||||
12 | -4 | 0 | 6 | 2 | |||
A= | -4 | 21 | 5 | 3 | B= | 4 | |
-3 | 2 | -22 | 1 | -2 | |||
-2 | -3 | 5 | 23 | 4 | |||
'A'= | -134088 | ||||||
1,000 | -0,333 | 0,000 | 0,500 | 0,167 | |||
-4,000 | 21,000 | 5,000 | 3,000 | 4,000 | |||
-3,000 | 2,000 | -22,000 | 1,000 | -2,000 | |||
-2,000 | -3,000 | 5,000 | 23,000 | 4,000 | |||
1,000 | -0,333 | 0,000 | 0,500 | 0,167 | |||
0,000 | 25,333 | 5,000 | 5,000 | 4,667 | |||
0,000 | 1,000 | -22,000 | 2,500 | -1,500 | |||
0,000 | -3,667 | 5,000 | 24,000 | 4,333 | |||
1,000 | -0,333 | 0,000 | 0,500 | 0,167 | |||
0,000 | 1,000 | 0,197 | 0,197 | 0,184 | |||
0,000 | 0,000 | -22,197 | 2,303 | -1,684 | |||
0,000 | 0,000 | 5,724 | 24,724 | 5,009 | |||
1,000 | -0,333 | 0,000 | 0,500 | 0,167 | |||
0,000 | 1,000 | 0,197 | 0,197 | 0,184 | |||
0,000 | 0,000 | 1,000 | -0,104 | 0,076 | |||
0,000 | 0,000 | 0,000 | 25,317 | 4,574 | |||
x= | 0,120 | ||||||
0,130 | |||||||
0,095 | |||||||
0,181 |
4) Ëèñòèíã ïðîãðàììû (Ìåòîä Êðàìåðà, Ìåòîä Ãàóññà, Ìåòîä îáðàòíîé ìàòðèöû).
Begin VB.Form frmAriel
BorderStyle = 1 'Åäèíñòâåííûé Ôèêñèðîâàííûé
Caption = "Ðåøåíèå ñèñòåì ëèíåéíûõ óðàâíåíèé"
ClientHeight = 6315
ClientLeft = 4365
ClientTop = 2430
ClientWidth = 7815
BeginProperty Font
Name = "MS Sans Serif"
Size = 12
Charset = 204
Weight = 700
Underline = 0 'False
Italic = -1 'True
Strikethrough = 0 'False
EndProperty
LinkTopic = "Ôîðìà1"
MaxButton = 0 'False
MinButton = 0 'False
ScaleHeight = 6315
ScaleWidth = 7815
Begin VB.TextBox txtMOMZ
Alignment = 2 'Âûðàâíèâàíèå ïî Öåíòðó
BeginProperty Font
Name = "Times New Roman"
Size = 15.75
Charset = 204
Weight = 400
Underline = 0 'False
Italic = 0 'False
Strikethrough = 0 'False
EndProperty
Height = 375
Left = 3960
TabIndex = 45
Top = 5520
Width = 975
End
Begin VB.TextBox txtMOMY
Alignment = 2 'Âûðàâíèâàíèå ïî Öåíòðó
BeginProperty Font
Name = "Times New Roman"
Size = 15.75
Charset = 204
Weight = 400
Underline = 0 'False
Italic = 0 'False
Strikethrough = 0 'False
EndProperty
Height = 375
Left = 2640
TabIndex = 44
Top = 5520
Width = 975
End
Begin VB.TextBox txtMOMX
Alignment = 2 'Âûðàâíèâàíèå ïî Öåíòðó
BeginProperty DataFormat
Type = 1
Format = "0,0000"
HaveTrueFalseNull= 0
FirstDayOfWeek = 0
FirstWeekOfYear = 0
LCID = 1049
SubFormatType = 1
EndProperty
BeginProperty Font
Name = "Times New Roman"
Size = 15.75
Charset = 204
Weight = 400
Underline = 0 'False
Italic = 0 'False
Strikethrough = 0 'False
EndProperty
Height = 375
Left = 1320
TabIndex = 43
Top = 5520
Width = 975
End
Begin VB.TextBox txtMGZ
Alignment = 2 'Âûðàâíèâàíèå ïî Öåíòðó
BeginProperty Font
Name = "Times New Roman"
Size = 15.75
Charset = 204
Weight = 400
Underline = 0 'False
Italic = 0 'False
Strikethrough = 0 'False
EndProperty
Height = 375
Left = 3960
TabIndex = 42
Top = 4800
Width = 975
End
Begin VB.TextBox txtMGY
Alignment = 2 'Âûðàâíèâàíèå ïî Öåíòðó
BeginProperty Font
Name = "Times New Roman"
Size = 15.75
Charset = 204
Weight = 400
Underline = 0 'False
Italic = 0 'False
Strikethrough = 0 'False
EndProperty
Height = 375
Left = 2640
TabIndex = 41
Top = 4800
Width = 975
End
Begin VB.TextBox txtMGX
Alignment = 2 'Âûðàâíèâàíèå ïî Öåíòðó
BeginProperty DataFormat
Type = 1
Format = "0,0000"
HaveTrueFalseNull= 0
FirstDayOfWeek = 0
FirstWeekOfYear = 0
LCID = 1049
SubFormatType = 1
EndProperty
BeginProperty Font
Name = "Times New Roman"
Size = 15.75
Charset = 204
Weight = 400
Underline = 0 'False
Italic = 0 'False
Strikethrough = 0 'False
EndProperty
Height = 375
Left = 1320
TabIndex = 40
Top = 4800
Width = 975
End
Begin VB.TextBox txtMKZ
Alignment = 2 'Âûðàâíèâàíèå ïî Öåíòðó
BeginProperty Font
Name = "Times New Roman"
Size = 15.75
Charset = 204
Weight = 400
Underline = 0 'False
Italic = 0 'False
Strikethrough = 0 'False
EndProperty
Height = 375
Left = 3960
TabIndex = 39
Top = 4080
Width = 975
End
Begin VB.TextBox txtMKY
Alignment = 2 'Âûðàâíèâàíèå ïî Öåíòðó
BeginProperty Font
Name = "Times New Roman"
Size = 15.75
Charset = 204
Weight = 400
Underline = 0 'False
Italic = 0 'False
Strikethrough = 0 'False
EndProperty
Height = 375
Left = 2640
TabIndex = 38
Top = 4080
Width = 975
End
Begin VB.TextBox txtMKX
Alignment = 2 'Âûðàâíèâàíèå ïî Öåíòðó
BeginProperty DataFormat
Type = 1
Format = "0,0000"
HaveTrueFalseNull= 0
FirstDayOfWeek = 0
FirstWeekOfYear = 0
LCID = 1049
SubFormatType = 1
EndProperty
DataMember = "0,0000"
BeginProperty Font
Name = "Times New Roman"
Size = 15.75
Charset = 204
Weight = 400
Underline = 0 'False
Italic = 0 'False
Strikethrough = 0 'False
EndProperty
Height = 375
Left = 1320
Locked = -1 'True
TabIndex = 37
Top = 4080
Width = 975
End
Begin VB.TextBox txtA33
Alignment = 2 'Âûðàâíèâàíèå ïî Öåíòðó
BeginProperty DataFormat
Type = 1
Format = "0,000"
HaveTrueFalseNull= 0
FirstDayOfWeek = 0
FirstWeekOfYear = 0
LCID = 1049
SubFormatType = 1
EndProperty
BeginProperty Font
Name = "Times New Roman"
Size = 14.25
Charset = 204
Weight = 700
Underline = 0 'False
Italic = 0 'False
Strikethrough = 0 'False
EndProperty
Height = 375
Left = 4440
TabIndex = 30
Top = 2640
Width = 1095
End
Begin VB.TextBox txtA23
Alignment = 2 'Âûðàâíèâàíèå ïî Öåíòðó
BeginProperty DataFormat
Type = 1
Format = "0,000"
HaveTrueFalseNull= 0
FirstDayOfWeek = 0
FirstWeekOfYear = 0
LCID = 1049
SubFormatType = 1
EndProperty
BeginProperty Font
Name = "Times New Roman"
Size = 14.25
Charset = 204
Weight = 700
Underline = 0 'False
Italic = 0 'False
Strikethrough = 0 'False
EndProperty
Height = 375
Left = 4440
TabIndex = 29
Top = 2040
Width = 1095
End
Begin VB.TextBox txtA13
Alignment = 2 'Âûðàâíèâàíèå ïî Öåíòðó
BeginProperty DataFormat
Type = 1
Format = "0,000"
HaveTrueFalseNull= 0
FirstDayOfWeek = 0
FirstWeekOfYear = 0
LCID = 1049
SubFormatType = 1
EndProperty
BeginProperty Font
Name = "Times New Roman"
Size = 14.25
Charset = 204
Weight = 700
Underline = 0 'False
Italic = 0 'False
Strikethrough = 0 'False
EndProperty
Height = 375
Left = 4440
TabIndex = 28
Top = 1440
Width = 1095
End
Begin VB.TextBox txtA32
Alignment = 2 'Âûðàâíèâàíèå ïî Öåíòðó
BeginProperty DataFormat
Type = 1
Format = "0,000"
HaveTrueFalseNull= 0
FirstDayOfWeek = 0
FirstWeekOfYear = 0
LCID = 1049
SubFormatType = 1
EndProperty
BeginProperty Font
Name = "Times New Roman"
Size = 14.25
Charset = 204
Weight = 700
Underline = 0 'False
Italic = 0 'False
Strikethrough = 0 'False
EndProperty
Height = 375
Left = 2520
TabIndex = 24
Top = 2640
Width = 1095
End
Begin VB.TextBox txtA22
Alignment = 2 'Âûðàâíèâàíèå ïî Öåíòðó
BeginProperty DataFormat
Type = 1
Format = "0,000"
HaveTrueFalseNull= 0
FirstDayOfWeek = 0
FirstWeekOfYear = 0
LCID = 1049
SubFormatType = 1
EndProperty
BeginProperty Font
Name = "Times New Roman"
Size = 14.25
Charset = 204
Weight = 700
Underline = 0 'False
Italic = 0 'False
Strikethrough = 0 'False
EndProperty
Height = 375
Left = 2520
TabIndex = 23
Top = 2040
Width = 1095
End
Begin VB.TextBox txtA12
Alignment = 2 'Âûðàâíèâàíèå ïî Öåíòðó
BeginProperty DataFormat
Type = 1
Format = "0,000"
HaveTrueFalseNull= 0
FirstDayOfWeek = 0
FirstWeekOfYear = 0
LCID = 1049
SubFormatType = 1
EndProperty
BeginProperty Font
Name = "Times New Roman"
Size = 14.25
Charset = 204
Weight = 700
Underline = 0 'False
Italic = 0 'False
Strikethrough = 0 'False
EndProperty
Height = 375
Left = 2520
TabIndex = 22
Top = 1440
Width = 1095
End
Begin VB.CommandButton cmdExit
Caption = "Âûõîä"
BeginProperty Font
Name = "Times New Roman"
Size = 14.25
Charset = 204
Weight = 700
Underline = 0 'False
Italic = -1 'True
Strikethrough = 0 'False
EndProperty
Height = 495
Left = 5400
TabIndex = 18
Top = 5520
Width = 2055
End
Begin VB.CommandButton cmdCount
Caption = "Âû÷èñëèòü"
BeginProperty Font
Name = "Times New Roman"
Size = 14.25
Charset = 204
Weight = 700
Underline = 0 'False
Italic = -1 'True
Strikethrough = 0 'False
EndProperty
Height = 495
Left = 5400
TabIndex = 17
Top = 4680
Width = 2055
End
Begin VB.CommandButton cmdClean
Caption = "Î÷èñòèòü"
BeginProperty Font
Name = "Times New Roman"
Size = 14.25
Charset = 204
Weight = 700
Underline = 0 'False
Italic = -1 'True
Strikethrough = 0 'False
EndProperty
Height = 495
Left = 5400
TabIndex = 16
Top = 3840
Width = 2055
End
Begin VB.TextBox txtB3
Alignment = 2 'Âûðàâíèâàíèå ïî Öåíòðó
BeginProperty DataFormat
Type = 1
Format = "0,000"
HaveTrueFalseNull= 0
FirstDayOfWeek = 0
FirstWeekOfYear = 0
LCID = 1049
SubFormatType = 1
EndProperty
BeginProperty Font
Name = "Times New Roman"
Size = 14.25
Charset = 204
Weight = 700
Underline = 0 'False
Italic = 0 'False
Strikethrough = 0 'False
EndProperty
Height = 375
Left = 6480
TabIndex = 12
Top = 2640
Width = 1095
End
Begin VB.TextBox txtB2
Alignment = 2 'Âûðàâíèâàíèå ïî Öåíòðó
BeginProperty DataFormat
Type = 1
Format = "0,000"
HaveTrueFalseNull= 0
FirstDayOfWeek = 0
FirstWeekOfYear = 0
LCID = 1049
SubFormatType = 1
EndProperty
BeginProperty Font
Name = "Times New Roman"
Size = 14.25
Charset = 204
Weight = 700
Underline = 0 'False
Italic = 0 'False
Strikethrough = 0 'False
EndProperty
Height = 375
Left = 6480
TabIndex = 11
Top = 2040
Width = 1095
End
Begin VB.TextBox txtB1
Alignment = 2 'Âûðàâíèâàíèå ïî Öåíòðó
BeginProperty DataFormat
Type = 1
Format = "0,000"
HaveTrueFalseNull= 0
FirstDayOfWeek = 0
FirstWeekOfYear = 0
LCID = 1049
SubFormatType = 1
EndProperty
BeginProperty Font
Name = "Times New Roman"
Size = 14.25
Charset = 204
Weight = 700
Underline = 0 'False
Italic = 0 'False
Strikethrough = 0 'False
EndProperty
Height = 375
Left = 6480
TabIndex = 10
Top = 1440
Width = 1095
End
Begin VB.TextBox txtA31
Alignment = 2 'Âûðàâíèâàíèå ïî Öåíòðó
BeginProperty DataFormat
Type = 1
Format = "0,000"
HaveTrueFalseNull= 0
FirstDayOfWeek = 0
FirstWeekOfYear = 0
LCID = 1049
SubFormatType = 1
EndProperty
BeginProperty Font
Name = "Times New Roman"
Size = 14.25
Charset = 204
Weight = 700
Underline = 0 'False
Italic = 0 'False
Strikethrough = 0 'False
EndProperty
Height = 375
Left = 600
TabIndex = 9
Top = 2640
Width = 1095
End
Begin VB.TextBox txtA21
Alignment = 2 'Âûðàâíèâàíèå ïî Öåíòðó
BeginProperty DataFormat
Type = 1
Format = "0,000"
HaveTrueFalseNull= 0
FirstDayOfWeek = 0
FirstWeekOfYear = 0
LCID = 1049
SubFormatType = 1
EndProperty
BeginProperty Font
Name = "Times New Roman"
Size = 14.25
Charset = 204
Weight = 700
Underline = 0 'False
Italic = 0 'False
Strikethrough = 0 'False
EndProperty
Height = 375
Left = 600
TabIndex = 8
Top = 2040
Width = 1095
End
Begin VB.TextBox txtA11
Alignment = 2 'Âûðàâíèâàíèå ïî Öåíòðó
BeginProperty DataFormat
Type = 1
Format = "0,0000"
HaveTrueFalseNull= 0
FirstDayOfWeek = 0
FirstWeekOfYear = 0
LCID = 1049
SubFormatType = 1
EndProperty
BeginProperty Font
Name = "Times New Roman"
Size = 14.25
Charset = 204
Weight = 700
Underline = 0 'False
Italic = 0 'False
Strikethrough = 0 'False
EndProperty
Height = 375
Left = 600
TabIndex = 7
Top = 1440
Width = 1095
End
Begin VB.Line Ñòðîêà3
X1 = 5760
X2 = 5760
Y1 = 1080
Y2 = 3360
End
Begin VB.Label Ìåòêà29
Caption = "Z"
BeginProperty Font
Name = "Times New Roman"
Size = 12
Charset = 204
Weight = 700
Underline = 0 'False
Italic = 0 'False
Strikethrough = 0 'False
EndProperty
Height = 255
Left = 4920
TabIndex = 36
Top = 1080
Width = 255
End
Begin VB.Label Ìåòêà28
Caption = "Y"
BeginProperty Font
Name = "Times New Roman"
Size = 12
Charset = 204
Weight = 700
Underline = 0 'False
Italic = 0 'False
Strikethrough = 0 'False
EndProperty
Height = 255
Left = 3000
TabIndex = 35
Top = 1080
Width = 255
End
Begin VB.Label Ìåòêà27
Caption = "X"
BeginProperty Font
Name = "Times New Roman"
Size = 12
Charset = 204
Weight = 700
Underline = 0 'False
Italic = 0 'False
Strikethrough = 0 'False
EndProperty
Height = 375
Left = 1080
TabIndex = 34
Top = 1080
Width = 255
End
Begin VB.Label Ìåòêà23
Caption = "Z"
BeginProperty Font
Name = "Times New Roman"
Size = 12
Charset = 204
Weight = 700
Underline = 0 'False
Italic = 0 'False
Strikethrough = 0 'False
EndProperty
Height = 255
Left = 4440
TabIndex = 33
Top = 3480
Width = 255
End
Begin VB.Label Ìåòêà22
Caption = "Y"
BeginProperty Font
Name = "Times New Roman"
Size = 12
Charset = 204
Weight = 700
Underline = 0 'False
Italic = 0 'False
Strikethrough = 0 'False
EndProperty
Height = 255
Left = 3120
TabIndex = 32
Top = 3480
Width = 255
End
Begin VB.Label Ìåòêà21
Caption = "X"
BeginProperty Font
Name = "Times New Roman"
Size = 12
Charset = 204
Weight = 700
Underline = 0 'False
Italic = 0 'False
Strikethrough = 0 'False
EndProperty
Height = 255
Left = 1800
TabIndex = 31
Top = 3480
Width = 255
End
Begin VB.Label Ìåòêà20
Caption = "À33"
BeginProperty Font
Name = "Times New Roman"
Size = 12
Charset = 204
Weight = 700
Underline = 0 'False
Italic = 0 'False
Strikethrough = 0 'False
EndProperty
Height = 375
Left = 3840
TabIndex = 27
Top = 2640
Width = 375
End
Begin VB.Label Ìåòêà19
Caption = "À23"
BeginProperty Font
Name = "Times New Roman"
Size = 12
Charset = 204
Weight = 700
Underline = 0 'False
Italic = 0 'False
Strikethrough = 0 'False
EndProperty
Height = 375
Left = 3840
TabIndex = 26
Top = 2040
Width = 375
End
Begin VB.Label Ìåòêà18
Caption = "À13"
BeginProperty Font
Name = "Times New Roman"
Size = 12
Charset = 204
Weight = 700
Underline = 0 'False
Italic = 0 'False
Strikethrough = 0 'False
EndProperty
Height = 375
Left = 3840
TabIndex = 25
Top = 1440
Width = 375
End
Begin VB.Label Ìåòêà17
Caption = "À32"
BeginProperty Font
Name = "Times New Roman"
Size = 12
Charset = 204
Weight = 700
Underline = 0 'False
Italic = 0 'False
Strikethrough = 0 'False
EndProperty
Height = 375
Left = 1920
TabIndex = 21
Top = 2640
Width = 375
End
Begin VB.Label Ìåòêà16
Caption = "À22"
BeginProperty Font
Name = "Times New Roman"
Size = 12
Charset = 204
Weight = 700
Underline = 0 'False
Italic = 0 'False
Strikethrough = 0 'False
EndProperty
Height = 375
Left = 1920
TabIndex = 20
Top = 2040
Width = 375
End
Begin VB.Label Ìåòêà15
Caption = "À12"
BeginProperty Font
Name = "Times New Roman"
Size = 12
Charset = 204
Weight = 700
Underline = 0 'False
Italic = 0 'False
Strikethrough = 0 'False
EndProperty
Height = 375
Left = 1920
TabIndex = 19
Top = 1440
Width = 375
End
Begin VB.Line Ñòðîêà2
X1 = 0
X2 = 7800
Y1 = 3360
Y2 = 3360
End
Begin VB.Label Ìåòêà12
Alignment = 2 'Âûðàâíèâàíèå ïî Öåíòðó
Caption = "Ìåòîä îáðàòíîé ìàòðèöû"
BeginProperty Font
Name = "Times New Roman"
Size = 11.25
Charset = 204
Weight = 700
Underline = 0 'False
Italic = 0 'False
Strikethrough = 0 'False
EndProperty
Height = 855
Left = 0
TabIndex = 15
Top = 5280
Width = 1095
End
Begin VB.Label Ìåòêà11
Alignment = 2 'Âûðàâíèâàíèå ïî Öåíòðó
Caption = "Ìåòîä Ãàóññà"
BeginProperty Font
Name = "Times New Roman"
Size = 11.25
Charset = 204
Weight = 700
Underline = 0 'False
Italic = 0 'False
Strikethrough = 0 'False
EndProperty
Height = 615
Left = 0
TabIndex = 14
Top = 4560
Width = 1095
End
Begin VB.Label Ìåòêà10
Alignment = 2 'Âûðàâíèâàíèå ïî Öåíòðó
Caption = "Ìåòîä Êðàìåðà"
BeginProperty Font
Name = "Times New Roman"
Size = 11.25
Charset = 204
Weight = 700
Underline = 0 'False
Italic = 0 'False
Strikethrough = 0 'False
EndProperty
Height = 615
Left = 0
TabIndex = 13
Top = 3840
Width = 1095
End
Begin VB.Label Ìåòêà7
Caption = "Â3"
BeginProperty Font
Name = "Times New Roman"
Size = 12
Charset = 204
Weight = 700
Underline = 0 'False
Italic = 0 'False
Strikethrough = 0 'False
EndProperty
Height = 375
Left = 6000
TabIndex = 6
Top = 2640
Width = 255
End
Begin VB.Label Ìåòêà6
Caption = "Â2"
BeginProperty Font
Name = "Times New Roman"
Size = 12
Charset = 204
Weight = 700
Underline = 0 'False
Italic = 0 'False
Strikethrough = 0 'False
EndProperty
Height = 375
Left = 6000
TabIndex = 5
Top = 2040
Width = 255
End
Begin VB.Label Ìåòêà5
Caption = "Â1"
BeginProperty Font
Name = "Times New Roman"
Size = 12
Charset = 204
Weight = 700
Underline = 0 'False
Italic = 0 'False
Strikethrough = 0 'False
EndProperty
Height = 375
Left = 6000
TabIndex = 4
Top = 1440
Width = 255
End
Begin VB.Label Ìåòêà4
Caption = "À31"
BeginProperty Font
Name = "Times New Roman"
Size = 12
Charset = 204
Weight = 700
Underline = 0 'False
Italic = 0 'False
Strikethrough = 0 'False
EndProperty
Height = 375
Left = 0
TabIndex = 3
Top = 2640
Width = 375
End
Begin VB.Label Ìåòêà3
Caption = "À21"
BeginProperty Font
Name = "Times New Roman"
Size = 12
Charset = 204
Weight = 700
Underline = 0 'False
Italic = 0 'False
Strikethrough = 0 'False
EndProperty
Height = 375
Left = 0
TabIndex = 2
Top = 2040
Width = 375
End
Begin VB.Label Ìåòêà2
Caption = "À11"
BeginProperty Font
Name = "Times New Roman"
Size = 12
Charset = 204
Weight = 700
Underline = 0 'False
Italic = 0 'False
Strikethrough = 0 'False
EndProperty
Height = 375
Left = 0
TabIndex = 1
Top = 1440
Width = 375
End
Begin VB.Label Ìåòêà1
Caption = $"frmSlay.frx":0000
BeginProperty Font
Name = "Times New Roman"
Size = 12
Charset = 204
Weight = 700
Underline = 0 'False
Italic = 0 'False
Strikethrough = 0 'False
EndProperty
Height = 855
Left = 120
TabIndex = 0
Top = 120
Width = 7575
End
End
Attribute VB_Name = "frmAriel"
Attribute VB_GlobalNameSpace = False
Attribute VB_Creatable = False
Attribute VB_PredeclaredId = True
Attribute VB_Exposed = False
Private Sub cmdClean_Click()
Dim ñ As Byte
ñ = MsgBox("Õîòèòå î÷èñòèòü?", 33, "Î÷èñòêà")
If ñ = 1 Then
txtA11.Text = ""
txtA12.Text = ""
txtA13.Text = ""
txtA21.Text = ""
txtA22.Text = ""
txtA23.Text = ""
txtA31.Text = ""
txtA32.Text = ""
txtA33.Text = ""
txtB1.Text = ""
txtB2.Text = ""
txtB3.Text = ""
txtMKX.Text = ""
txtMGX.Text = ""
txtMOMX.Text = ""
txtMKY.Text = ""
txtMGY.Text = ""
txtMOMY.Text = ""
txtMKZ.Text = ""
txtMGZ.Text = ""
txtMOMZ.Text = ""
End If
End Sub
Private Sub cmdCount_Click()
If (Val(txtA11) * Val(txtA22) * Val(txtA33) + Val(txtA12) * Val(txtA23) * Val(txtA31) + Val(txtA21) * Val(txtA32) * Val(txtA13) - Val(txtA13) * Val(txtA22) * Val(txtA31) - Val(txtA12) * Val(txtA21) * Val(txtA33) - Val(txtA23) * Val(txtA32) * Val(txtA11)) = 0 Then
Dim g As Byte
g = MsgBox("Ðåøèòü äàííóþ ñèñòåìó ìåòîäîì Êðàìåðà è ìåòîäîì îáðàòíîé ìàòðèöû íåâîçìîæíî", 32, "Îïðåäåëèòåëü ñèñòåìû ðàâåí 0")
Else
ds = (Val(txtA11) * Val(txtA22) * Val(txtA33) + Val(txtA12) * Val(txtA23) * Val(txtA31) + Val(txtA21) * Val(txtA32) * Val(txtA13) - Val(txtA13) * Val(txtA22) * Val(txtA31) - Val(txtA12) * Val(txtA21) * Val(txtA33) - Val(txtA23) * Val(txtA32) * Val(txtA11))
dx = (Val(txtB1) * Val(txtA22) * Val(txtA33) + Val(txtA12) * Val(txtA23) * Val(txtB3) + Val(txtB2) * Val(txtA32) * Val(txtA13) - Val(txtA13) * Val(txtA22) * Val(txtB3) - Val(txtA12) * Val(txtB2) * Val(txtA33) - Val(txtA23) * Val(txtA32) * Val(txtB1))
dy = (Val(txtA11) * Val(txtB2) * Val(txtA33) + Val(txtB1) * Val(txtA23) * Val(txtA31) + Val(txtA21) * Val(txtB3) * Val(txtA13) - Val(txtA13) * Val(txtB2) * Val(txtA31) - Val(txtB1) * Val(txtA21) * Val(txtA33) - Val(txtA23) * Val(txtB3) * Val(txtA11))
dz = (Val(txtA11) * Val(txtA22) * Val(txtB3) + Val(txtA12) * Val(txtB2) * Val(txtA31) + Val(txtA21) * Val(txtA32) * Val(txtB1) - Val(txtB1) * Val(txtA22) * Val(txtA31) - Val(txtA12) * Val(txtA21) * Val(txtB3) - Val(txtB2) * Val(txtA32) * Val(txtA11))
txtMKX.Text = (dx / ds)
txtMKY.Text = (dy / ds)
txtMKZ.Text = (dz / ds)
txtMOMX.Text = (((Val(txtA22) * Val(txtA33) - Val(txtA23) * Val(txtA32)) / ds) * Val(txtB1) + ((Val(txtA13) * Val(txtA32) - Val(txtA12) * Val(txtA33)) / ds) * Val(txtB2) + ((Val(txtA12) * Val(txtA23) - Val(txtA13) * Val(txtA22)) / ds) * Val(txtB3))
txtMOMY.Text = (((Val(txtA23) * Val(txtA31) - Val(txtA21) * Val(txtA33)) / ds) * Val(txtB1) + ((Val(txtA11) * Val(txtA33) - Val(txtA13) * Val(txtA31)) / ds) * Val(txtB2) + ((Val(txtA13) * Val(txtA21) - Val(txtA11) * Val(txtA23)) / ds) * Val(txtB3))
txtMOMZ.Text = (((Val(txtA21) * Val(txtA32) - Val(txtA22) * Val(txtA31)) / ds) * Val(txtB1) + ((Val(txtA12) * Val(txtA31) - Val(txtA11) * Val(txtA32)) / ds) * Val(txtB2) + ((Val(txtA11) * Val(txtA22) - Val(txtA12) * Val(txtA21)) / ds) * Val(txtB3))
End If
If Val(txtA11) = 0 Then
Dim gg As Byte
gg = MsgBox("Ðåøèòü äàííóþ ñèñòåìó ìåòîäîì Ãàóññà íåâîçìîæíî. Ïîìåíÿéòå óðàâíåíèÿ ìåñòàìè, è ïîïðîáóéòå ñíîâà", 48, "Êîýôôèöèåíò ïðè õ â ïåðâîì óðàâíåíèè ðàâåí 0")
Else
s1 = ((Val(txtB3) * Val(txtA11)) - (Val(txtB1) * Val(txtA31)))
s2 = ((Val(txtA22) * Val(txtA11)) - (Val(txtA12) * Val(txtA21)))
s3 = ((Val(txtB2) * Val(txtA11)) - (Val(txtB1) * Val(txtA21)))
s4 = ((Val(txtA32) * Val(txtA11)) - (Val(txtA12) * Val(txtA31)))
s5 = ((Val(txtA33) * Val(txtA11)) - (Val(txtA12) * Val(txtA31)))
s6 = s2
s7 = ((Val(txtA23) * Val(txtA11)) - (Val(txtA13) * Val(txtA21)))
s8 = s4
s12 = s1 * s2
s34 = s3 * s4
s56 = s5 * s6
s78 = s7 * s8
sh = s12 - s34
sz = s56 - s78
If sz = 0 Then
Dim ñ As Byte
ñ = MsgBox("Äåëèòü íà íîëü íåëüçÿ.", 48, "Îøèáêà")
Else
txtMGZ.Text = (sh / sz)
End If
sy1 = s3
sy2 = s2
sy3 = ((Val(txtA23) * Val(txtA11)) - (Val(txtA13) * Val(txtA21)))
sy4 = s2
sy12 = (sy1) / (sy2)
sy34 = (sy3) / (sy4)
txtMGY.Text = ((sy12) - ((sy34) * (sh / sz)))
sx1 = ((Val(txtB1)) / (Val(txtA11)))
sx2 = ((Val(txtA13)) / (Val(txtA11)))
sx3 = ((Val(txtA12)) / (Val(txtA11)))
txtMGX.Text = ((sx1) - ((sx2) * (sh / sz)) - ((sx3) * ((sy12) - ((sy34) * (sh / sz)))))
End If
End Sub
Private Sub cmdExit_Click()
Dim a As Byte
a = MsgBox("Õîòèòå âûéòè?", 33, "Âûõîä")
If a = 1 Then
End
End If
End Sub
Ïîëüçà ââåäåíèÿ ðàñ÷åòîâ.
Îãðîìíîå êîëè÷åñòâî ÷èñëåííûõ ìåòîäîâ ñòàâèò àêòóàëüíîé çàäà÷åé íå ñòîëüêî ñîçäàíèå íîâûõ, ñêîëüêî èññëåäîâàíèå è êëàññèôèêàöèþ ñòàðûõ, âûÿâëåíèå ëó÷øèõ. Àíàëèç âëèÿíèÿ îøèáîê ïîêàçàë, ÷òî ìåæäó ëó÷øèìè ìåòîäàìè íåò ïðèíöèïèàëüíîé ðàçíèöû ñ òî÷êè çðåíèÿ óñòîé÷èâîñòè ê îøèáêàì îêðóãëåíèÿ. Ñîçäàíèå ìîùíûõ êîìïüþòåðîâ ñóùåñòâåííî îñëàáèëî çíà÷åíèå ðàçëè÷èÿ ìåæäó ìåòîäàìè (â òàêèõ õàðàêòåðèñòèêàõ, êàê îáú¸ì òðåáóåìîé ïàìÿòè, êîëè÷åñòâî àðèôìåòè÷åñêèõ îïåðàöèé).  ýòèõ óñëîâèÿ íàèáîëåå ïðåäïî÷òèòåëüíûìè ñòàíîâÿòñÿ òå ìåòîäû, êîòîðûå íå î÷åíü îòëè÷àþòñÿ îò ëó÷øèõ ïî ñêîðîñòè è óäîáñòâó ðåàëèçàöèè íà êîìïüþòåðàõ, ïîçâîëÿþò ðåøàòü øèðîêèé êëàññ çàäà÷ êàê õîðîøî, òàê è ïëîõî îáóñëîâëåííûõ è äàâàòü ïðè ýòîì îöåíêó òî÷íîñòè âû÷èñëèòåëüíîãî ðåøåíèÿ.
 MathCAD è Excel ÷èñëåííûå ìåòîäû ïðåäñòàâëÿþò ñîáîé òå æå ñàìûå îáùåïðèíÿòûå ðó÷íûå ðàñ÷¸òû, íî âûïîëíÿåìûå íå ÷åëîâåêîì, à êîìïüþòåðîì, ÷òî ïîíèæàåò âîçìîæíîñòü îøèáêè äî íóëÿ. Ïðîãðàììà íà Visual Basic íàìíîãî óïðîùàåò çàäà÷ó. Ñ ïîìîùüþ åäèíîæäû ñîçäàííîé ïðîãðàììû ìîæíî ðåøàòü ñèñòåìû ëèíåéíûõ óðàâíåíèé, ââîäÿ ìèíèìóì çíà÷åíèé. Òàêæå ýòà ïðîãðàììà ìîæåò áûòü èñïîëüçîâàíà íå òîëüêî âàìè, íî è ïðîñòûìè ïîëüçîâàòåëÿìè.
III. Ëèòåðàòóðà
1) Ì. Äîäæ, Ê. Êèíàòà, Ê. Ñòèíñîí "Ýôôåêòèâíàÿ ðàáîòà â Microsoft Excel 97", èçäàòåëüñòâî "Ïèòåð"; Ñàíêò-Ïåòåðáóðã, 1998ã.
2) Å.Ê. Îâ÷àðåíêî, Î.Ï. Èëüèíà, Å.Â. Áàëûáåðäèí "Ôèíàíñîâî - ýêîíîìè÷åñêèå ðàñ÷åòû â Excel", Ìîñêâà, 1999 ã.
3) Éîðã Øèá, Excel 7,0: Ñîòíè ïîëåçíûõ ðåöåïòîâ, Äþññåëüäîðô-Êèåâ-Ìîñêâà- Ñàíêò-Ïåòåðáóðã, 1997 ã.
4) Ñèìîíîâè÷ Ñ.Â. è äð. Èíôîðìàòèêà Áàçîâûé êóðñ: Ó÷åá, äëÿ òåõíè÷åñêèõ âóçîâ. ÑÏÁ: Èçä. «Ïèòåð», 2004.–640ñ
5) Êàëèòêèí Í.Í. è äð. ×èñëåííûå ìåòîäû. Ì.: Íàóêà, 1982
6) Òóð÷àê Ë.È. Îñíîâû ÷èñëåííûõ ìåòîäîâ. Ì.: Íàóêà, 1987
7) Äüÿêîíîâ Â.Ï. Ñèñòåìà MathCAD. Ì.: Ðàäèî è ñâÿçü, 1993