Численные методы. Ионкин (2009), страница 7

Ôîðìóëà23Ñèìïñîíà áóäåò òî÷íà è äëÿ êóáè÷åñêèõ ìíîãî÷ëåíîâ (f (x) = a0 + a1 x + a2 x + a3 x ). ×òîáûRxi 3ïîêàçàòü ýòî, íàéäåìx dx:Åñëèxi−1Zxih11x3 dx = (x4i − x4i−1 ) = (x2i − x2i−1 )(x2i + x2i−1 ) = (xi + xi−1 )(x2i + x2i−1 )444xi−1Òåïåðü çàïèøåì ôîðìóëó Ñèìïñîíà äëÿRxix3 dxè ïðåîáðàçóåì åå:xi−1 hh 3xi−1 + 4x3i− 1 + x3i =266(xi + xi−1 )333xi + xi−1 + 4=2x2 + 2xi xi−1 + x2i−1h(xi + xi−1 )(x2i − xi xi−1 + x2i−1 + i)62 23xi + 3x2i−1hh= (xi + xi−1 )= (xi + xi−1 )(x2i + x2i−1 )624=Òàêèì îáðàçîì, ìû ïîêàçàëè, ÷òî ôîðìóëà Ñèìïñîíà òî÷íà è äëÿ ìíîãî÷ëåíîâ òðåòüåé ñòåïåíè.Ïðèáëèçèì ïîäûíòåãðàëüíóþ ôóíêöèþf (x)ïîëèíîìîì ÝðìèòàH3 (x):H3 (xi−1 ) = fi−1H3 (xi− 1 ) = fi− 122H3 (xi ) = fi0H30 (xi− 1 ) = fi−122f (x) = H3 (x) + ψH3 (x)ZxiZxiZxif (x)dx =H3 (x)dx +ψH3 (x)dx =xi−1xi−1xi−1h= (H3 (xi−1 ) + 4H3 (xi− 1 ) + H3 (xi )) +26ZxiψH3 (x)dx =xi−1=h(fi−1 + 4fi− 1 + fi ) +26ZxiψH3 (x)dxxi−1Íàéäåì ïîãðåøíîñòü íà i-îì ÷àñòè÷íîì îòðåçêå:Zxiψi =xi−1hf (x)dx − (fi−1 + 4fi− 1 + fi ) =26ZxiψH3 (x)dxxi−1Èñïîëüçîâàíèå ïîëèíîìà Ýðìèòà òðåòüåé ñòåïåíè äëÿ ïîëó÷åíèÿ òî÷íîé îöåíêèïîãðåøíîñòè êâàäðàòóðíîé ôîðìóëû Ñèìïñîíà49Ïîãðåøíîñòü äëÿ ïîëèíîìà Ýðìèòà èìååò âèä:ψH3 (x) =ÏóñòüM4 =|f (4) (ξ)|,supf (4) (ξ)(x − xi−1 )(x − xi− 1 )2 (x − xi )24!òîãäà ñïðàâåäëèâà îöåíêà:ξ∈[xi−1 ,xi ]M4|ψH3 (x)| ≤4!Zxi(x − xi−1 )(x − xi− 1 )2 (xi − x)dx = O(h5 )2xi−1Çàäà÷à.

Äîêàçàòü, ÷òîRxi(x − xi−1 )(x − xi− 1 )2 (xi − x)dx =2xi−1h5.120Äîêàçàòåëüñòâî. Ïðîâåäåì çàìåíó â ïîäûíòåãðàëüíîì âûðàæåíèè:0≤t≤1x = xi−1 + th,Òîãäà:dx = hdtx − xi−1 = thxi − x = h(1 − t)1(x − xi− 1 )2 = h2 (t − )222Òàêèì îáðàçîì, òðåáóåìûé èíòåãðàë ëåãêî âû÷èñëèòü:Zxi(x − xi−1 )(x − xi− 1 )2 (xi − x)dx =2xi−15Z1=h2Z1 1th55 2453t (1 − t) t −dt = hdt =2t − t − t +24412000Òåïåðü ìû ìîæåì îöåíèòü ïîãðåøíîñòü íà âñåì îòðåçêåΨ=nX[a, b]:ψi (h)i=1|Ψ| ≤Ó÷òåì, ÷òîM4 h5 n4! 120hn = b − a:M4 h4 (b − a)|Ψ| ≤=4!120 4h M4 (b − a)2180Çàìå÷àíèå. Åñëè ïîäûíòåãðàëüíóþ ôóíêöèþ çàìåíèòü ñîîòâåòñòâóþùèì ïîëèíîìîì Ëàãðàíæà, òî ïîãðåøíîñòü êâàäðàòóðíîé ôîðìóëû Ñèìïñîíà óâåëè÷èòñÿ.Íàèëó÷øåå ñðåäíåêâàäðàòè÷íîå ïðèáëèæåíèå ôóíêöèèŸ650Íàèëó÷øåå ñðåäíåêâàäðàòè÷íîå ïðèáëèæåíèå ôóíêöèèÎïðåäåëåíèå.

Ôóíêöèÿ f(x) íàçûâàåòñÿ èíòåãðèðóåìîé ñ êâàäðàòîì íà îòðåçêå [a, b], åñëèRbf 2 (x)dx < ∞.aÐàññìîòðèì ëèíåéíîå ïðîñòðàíñòâî[a, b].H = L2 ôóíêöèé, èíòåãðèðóåìûõ ñ êâàäðàòîì íà îòðåçêå∈ L2 ):Ââåäåì ñêàëÿðíîå ïðîèçâåäåíèå ôóíêöèé f(x) è g(x) (f, gZb(f, g) =f (x)g(x)dxaÒåïåðü îïðåäåëèì íîðìó:Zb1||f || = (f, f ) = ( f 2 (x)dx) 212aÐàññìîòðèì ñîâîêóïíîñòü ôóíêöèé:φ0 (x), φ1 (x), . .

. , φn (x) ËÍÇ è èíòåãðèðóåìûå ñ êâàäðàòîì, φi∈ L2(1)Ðàññìîòðèì îáîáùåííûé ìíîãî÷ëåí:φ(x) =nXCk φk (x),Ck ÷èñëà(2)k=0Ñðåäè âñåõ îáîáùåííûõ ìíîãî÷ëåíîâ íàì íåîáõîäèìî íàéòè îáîáùåííûé ìíîãî÷ëåíφ(x), òàêîé÷òî:||f (x) − φ(x)|| = min ||f (x) − φ(x)|| =φ∈L2Zbf (x) −= min φ∈L2!2Ck φk (x) 21dxk=0aÎáîáùåííûé ìíîãî÷ëåínXφ(x) íàçûâàåòñÿ íàèëó÷øèì ñðåäíåêâàäðàòè÷íûì ïðèáëèæåíèåì ôóíê-öèè f(x).

Ïîêàæåì, ÷òî îíî ñóùåñòâóåò è åäèíñòâåííî.Óòâåðæäåíèå. Íàèëó÷øåå ñðåäíåêâàäðàòè÷íîå ïðèáëèæåíèå ñóùåñòâóåò è åäèíñòâåííî.Äîêàçàòåëüñòâî. Ðàññìîòðèì ñíà÷àëà ñëó÷àén = 0:φ0 (x) ∈ L2 , φ(x) = C0 φ0 (x)Ââåäåì ôóíêöèþF (C0 ):2ZbF (C0 ) = ||f − φ(x)|| =a(f (x) − φ(x))2 dx =Íàèëó÷øåå ñðåäíåêâàäðàòè÷íîå ïðèáëèæåíèå ôóíêöèèZb=f 2 (x)dx − 2C0aZb51f (x)φ0 (x)dx + C02Zbaφ20 (x)dx =aC02 (φ0 , φ0 )= (f, f ) − 2C0 (f, φ0 ) +Íåîáõîäèìîå óñëîâèå ìèíèìóìà äëÿ ôóíêöèè F:dF=0dC0C0Ìèíèìóì ýòîé ôóíêöèè ïî ïåðåìåííîéíàõîäèòñÿ â âåðøèíå ïàðàáîëû:(f, φ0 )(φ0 , φ0 )C0 =Òàêèì îáðàçîì:φ(x) = C0 φ0 (x)Ðàññìîòðèì ïðèìåð.

Ïóñòüφ0 (x) = 1,òîãäà:Rbf (x)dxaC0 =Rbdx1=b−aZbf (x)dxaa1φ(x) =b−aZbf (x)dxaÏîëó÷èëè ñðåäíåå çíà÷åíèå ôóíêöèèf (x)íà[a, b].Ðàññìîòðèì òåïåðü îáùèé ñëó÷àé. Ïóñòü çàäàíà ñèñòåìà ôóíêöèé (1). Ââåäåì ôóíêöèþF (C0 , C1 , . . . , Cn ):2ZbF (C0 , C1 , . . . , Cn ) = ||f − φ(x)|| =f (x) −2f (x)dx − 2=nXZbCkk=0a= (f, f ) − 2f (x)φ0 (x)dx +nXCkk=0anXCk (f, φk ) +k=0!2Ck φk (x)dx =k=0aZbnXnXCkZbCll=0nXk=0nXφ20 (x)dx =aCl (φk , φl )l=0Çàïèøåì íåîáõîäèìîå óñëîâèå ìèíèìóìà äëÿ ôóíêöèè F:∂F (C0 , C1 , .

. . , Cn )= 0,∂Ckk = 0, 1, . . . , nÒîãäà ïîëó÷èì ñèñòåìó óðàâíåíèé äëÿ íàõîæäåíèÿ êîýôôèöèåíòîânXl=0Cl (φk , φl ) = (f, φk ),k = 0, 1, . . . , nCi(3)Íàèëó÷øåå ñðåäíåêâàäðàòè÷íîå ïðèáëèæåíèå ôóíêöèè52Ìàòðèöåé ýòîé ñèñòåìû ÿâëÿåòñÿ ìàòðèöà Ãðàìà:(φ0 , φ0 ) (φ0 , φ1 ) · · · (φ0 , φn ) (φ1 , φ0 ) (φ1 , φ1 ) · · · (φ1 , φn ) G =  ........ ....(φ0 , φn ) (φ1 , φn ) · · · (φn , φn )Òàê êàê ñèñòåìà ôóíêöèé (1) ëèíåéíî íåçàâèñèìà, òî îïðåäåëèòåëü ÃðàìàdetG 6= 0,à çíà-÷èò íàèëó÷øåå ñðåäíåêâàäðàòè÷íîå ïðèáëèæåíèå ñóùåñòâóåò è åäèíñòâåííî (ìîæíî îäíîçíà÷íîíàéòè êîýôôèöèåíòûCi ).Çàìå÷àíèå. Åñëè ñèñòåìà ôóíêöèé(f, φk )(1) - îðòîíîðìèðîâàííàÿ, òî åñòü(φk , φl ) = δkl ,òîCk =- êîýôôèöèåíòû Ôóðüå.Çàìå÷àíèå.

Ïóñòü çàäàíà ñèñòåìà ôóíêöèé:1, x, x2 . . . , xnÏóñòüρ(x) > 0- âåñîâàÿ ôóíêöèÿ.Zβρ(x)φk (x)φl (x)dx = 0αÂûáèðàÿÏóñòüρ(x), α, β{φi }n0 -ìîæíî ïîëó÷èòü îðòîãîíàëüíûå ìíîãî÷ëåíû.îðòîíîðìèðîâàííàÿ ñèñòåìà. Òîãäà íàèìåíüøåå îòêëîíåíèå:Zb0≤f (x) −aZnX!2Ck φk (x)dx =k=0bf 2 (x)dx − 2anXCk (φk , f ) +k=0(f, f ) −nXCk2 =k=0nXCk2 ≥ 0k=0Òàêèì îáðàçîì, ìû ïîëó÷èëè íåðàâåíñâòî Áåññåëÿ:nXCk2 ≤ ||f ||2k=0Åñëè ñèñòåìà{φk }n0 îðòîíîðìèðîâàííûé áàçèñ, èÁåññåëÿ ñòàíåò ðàâåíñòâîì Ïàðñåâàëÿ:2||f || =∞Xk=0Ck2(φk , φl ) = δkl ,òî ïîëó÷åííîå íåðàâåíñòâîÃëàâà III×èñëåííîå ðåøåíèå íåëèíåéíûõóðàâíåíèé è ñèñòåì íåëèíåéíûõóðàâíåíèéŸ1ÂâåäåíèåÏóñòü çàäàíà ôóíêöèÿf (x), x ∈ R, ïðè÷åì[a, b]ôóíêöèÿ f íåïðåûâíà.Áóäåì ðåøàòü óðàâíåíèå íà îòðåçêåf (x) = 0, x ∈ [a, b]Ïðîöåññ ðåøåíèÿ ðàçáèâàþò íà 2 ýòàïà:1.

Ëîêàëèçóåì êîðíè (ïðè ýòîì êîðíè ìîãóò áûòü êîìïëåêñíûå)2. Ñòðîèì èòåðàöèîííûé ìåòîä íàõîæäåíèÿ êîðíÿÎïðåäåëåíèå. a-îêðåñòíîñòüþ êîðíÿ x∗ íàçûâàåòñÿ ìíîæåñòâî òî÷åêUa (x∗ ) = {x : |x − x∗ | ≤ a}Ðàññìîòðèì ñïîñîáû ëîêàëèçàöèè êîðíÿ:1. Ðàçîáüåì îòðåçîê[a, b]ìíîæåñòâîì òî÷åê{xi }N1a ≤ x0 < x1 < . . . < xN ≤ bÒîãäà ìîæíî óòâåðæäàòü, ÷òî åñëèf (xi−1 )f (xi ) < 0),[xi−1 ..xi ] åñòü ïî êðàéÅñëè æå f (xi−1 )f (xi ) > 0),òî íà îòðåçêåíåé ìåðå îäèí êîðåíü (òàêæå èõ ìîæåò áûòü íå÷åòíîå ÷èñëî).òî ñêàçàòü íè÷åãî íåëüçÿ, òàê êàê íà ýòîì îòðåçêå ëèáî ÷åòíîå ÷èñëî êîðíåé, ëèáî êîðíåéíåò âîîáùå.2. Ìåòîä áèñåêöèè (äåëåíèÿ ïîïîëàì) Ïóñòüf (x) ∈ C[a, b]; f (a) < 0, f (b) > 0Âîçüìåìx0 =a+b.253Ìåòîä ïðîñòîé èòåðàöèèÅñëèÅñëèÅñëè54f (x0 ) > 0, òî êîðåíü óðàâíåíèÿ x∗ ∈ (a, x0 )f (x0 ) < 0 , òî êîðåíü óðàâíåíèÿ x∗ ∈ (x0 , b)f (x0 ) = 0, òî ìû íàøëè êîðåíü óðàâíåíèÿ.a+x0, âî âòîðîì2äóðó ëîêàëèçàöèè êîðíÿ, è òàê äàëåå.Âî ïåðâîì ñëó÷àå âîçüìåìx1 =x1 =x0 +b, è àíàëîãè÷íî ïîâîòðèì ïðîöå2 ñëó÷àå, åñëè äàíà ñèñòåìà óðàâíåíèéf1 (x1 , .

. . , xm ) = 0,...fm (x1 , . . . , xm ) = 0,åå ìîæíî ïðåäñòàâèòü â âèäåŸ2f~(~x) = 0,ãäåf~ = (f1 , f2 , . . . , fm )T , ~x = (x1 , x2 , . . . , xm )TÌåòîä ïðîñòîé èòåðàöèèÈòàê, ìû ðåøàåì óðàâíåíèåf (x) = 0x∗- êîðåíü óðàâíåíèÿ, ëîêàëèçîâàííûé íà(1)Ua (x∗ )Çàìåíèì óðàâíåíèå íà ýêâèâàëåíòíîåãäå ôóíêöèÿr(x)x = S(x)(2)S(x) = x + r(x)f (x)(3)íå ìåíÿåò çíàê íàÏîñòðîèì ïîñëåäîâàòåëüíîñòü{xn }Ua (x∗ )ñëåäóþùèì îáðàçîì:x0 ∈ Ua (x∗ )xn+1 = S(xn ), n = 0, 1, . . .(4)Îïðåäåëåíèå. Ôóíêöèÿ S(x) Ëèïøèö-íåïðåðûâíà(óäîâëåòâîðÿåò óñëîâèþ Ëèïøèöà) ñ êîíñòàíòîé q >0, åñëè|S(x1 ) − S(x2 )| ≤ q|x1 − x2 |, ∀x1 , x2 ∈ (a, b)Óòâåðæäåíèå. Åñëè S(x) óäîâëåòâîðÿåò óñëîâèþ Ëèïøèöà ñ 0 < q < 1 íà Ua (x∗ ) è |x − x0 | <a,òî ìåòîä ïðîñòîé èòåðàöèè (4) ðåøåíèÿ óðàâíåíèÿ (1) ñõîäèòñÿ, ïðè÷åì ñî ñêîðîñòüþãåîìåòðè÷åñêîé ïðîãðåññèè ñî çíàìåíàòåëåì q.Äîêàçàòåëüñòâî.

Ïî ïîñòðîåíèþ|x0 − x∗ | < a,çíà÷èò|xn+1 − x∗ | = |S(xn ) − S(x∗ )| ≤ q|xn − x∗ | ⇒|xn − x∗ | ≤ q n alimn→∞ qn = 0,òàê êàê0 < q < 1.Ñëåäîâàòåëüíî, ìåòîä ñõîäèòñÿ, ïðè÷åì ñî ñêîðîñòüþ ãåîìåòðè÷åñêîé ïðîãðåññè ñî çíàìåíàòåëåì q.Ìåòîä Íüþòîíà è ìåòîä ñåêóùèõ55Çàìå÷àíèå.

Åñëè S(x) äèôôåðåíöèðóåìà íà Ua (x∗ ), òî q = supx∈Ua (x∗ ) |S 0 (x)|Çàìå÷àíèå. Ïóñòü f(x) äèôôåðåíöèðóåìà, f 0 (x) > 0 íà Ua (x∗ ) è ∃M1 = supx∈Ua (x∗ ) |f 0 (x)|Òîãäà çàïèøåì ìåòîä ïðîñòîé èòåðàöèè â âèäå:xn+1 − xn+ f (xn ) = 0,τxn+1 = S(xn ),τ >0S(x) = x − τ f (x)∃S 0 (x) = 1 − τ f 0 (x) íà Ua (x∗ ). Äëÿq = supx∈Ua (x∗ ) |1 − τ f 0 (x)| < 1, ò.å. ÷òîáû 0 < τ < M21Ñëåäîâàòåëüíî,ñõîäèìîñòè ìåòîäà íåîáõîäèìî, ÷òîáûÌåòîä Ýéòêåíà (óñêîðåíèå ñõîäèìîñòè)Ìåòîä Ýéòêåíà íå ÿâëÿåòñÿ òåîðåòè÷åñêè îáîñíîâàííûì, íî ïðè ïðèáëèæåííûõ çíà÷åíèÿõïàðàìåòðîâ ïîçâîëÿÿåò óâåëè÷èòü ñêîðîñòü ñõîäèìîñòè.nÏóñòü xn − x∗ ' Aq , ãäå A è q - íåêîòîðûå êîíñòàíòû. Òîãäà:xn−1 − x∗ = Aq n−1xn − x∗ = Aq nxn+1 − x∗ = Aq n+1ñëåäîâàòåëüíî,(xn+1 − xn )2 = A2 q 2n (q − 1)2(xn+1 − 2xn + xn−1 ) = Aq n−1 (q − 1)2Îòêóäà ïîëó÷àåì:(xn+1 − xn )2= Aq n+1 = xn+1 − x∗xn+1 − 2xn + xn−1Ñòàëî áûòü:x∗ ' xn+1 −(xn+1 − xn )2xn+1 − 2xn + xn−1Èç-çà íåòî÷íîñòè â êà÷åñòâå ñëåäóþùåé èòåðàöèè ìû äîëæíû âçÿòü çíà÷åíèå, áëèçêîå êŸ3x∗Ìåòîä Íüþòîíà è ìåòîä ñåêóùèõÌû ðåøàåì óðàâíåíèåf (x) = 0Ïóñòü êîðåíü ëîêàëèçîâàí íàÐàçëîæèìf (x∗ )Ua (x∗ ), f (x) ∈ C 1 (Ua (x∗ )),(1)ïðè ýòîìf 0 (x) 6= 0íàïî Òåéëîðó:0 = f (x∗ ) = f (x) + f 0 (x)(x∗ − x) + o(x∗ − x) ≈ f (x) + f 0 (x)(x∗ − x)Ïîëîæèì â ýòîé ôîðìóëåx = xn , x∗ = xn+1 ,òîãäà ïîëó÷èì:xn+1 = xn −f (xn )f 0 (xn )Ua (x∗ ).Ìåòîä Íüþòîíà è ìåòîä ñåêóùèõÂçÿâx0 ∈ Ua (x∗ ),56ïîëó÷àåì ìåòîä Íüþòîíà:xn+1 = xn −f (xn ), n = 0, 1, 2, .

. .f 0 (xn )Íà êàæäîé èòåðàöèè ñ÷èòàòü ïðîèçâîäíóþ çàòðàòíî, â òî æå âðåìÿ íà íåáîëüøîì èíòåðâàëåîíà, êàê ïðàâèëî, ìåíÿåòñÿ íå ñèëüíî. Ñëåäîâàòåëüíî, ìîæíî èñïîëüçîâàòü ïðîèçâîäíóþ, îäèíðàç âû÷èñëåííóþ íà ïåðâîé èòåðàöèè. Ïîëó÷àåì ìîäèôèöèðîâàííûé ìåòîä Íüþòîíà:xn+1 = xn −f (xn ), n = 0, 1, 2, . . . ; x0 ∈ Ua (x∗ )f 0 (x0 )Ìîäèôèöèðîâàííûé ìåòîä Íüþòîíà ñõîäèòñÿ ìåäëåííåå îáû÷íîãî ìåòîäà Íüþòîíà, íî áûñòðåå ìåòîäà ïðîñòîé èòåðàöèè.Ìåòîä Íüþòîíà äëÿ ñèñòåìû óðàâíåíèéÐàññìîòðèì ñèñòåìó:(f1 (x1 , x2 ) = 0,f2 (x1 , x2 ) = 0,Ïóñòü(x∗1 , x∗2 )- åå ðåøåíèå. Ðàçëîæèìf1èf2(2)â îêðåñòíîñòè êîðíÿ:0 = f1 (x∗1 , x∗2 ) = f1 (x1 , x2 ) +∂f1 (x1 , x2 ) ∗∂f1 (x1 , x2 ) ∗(x1 − x1 ) +(x2 − x2 ) + .

. .∂x1∂x20 = f2 (x∗1 , x∗2 ) = f2 (x1 , x2 ) +∂f2 (x1 , x2 ) ∗∂f2 (x1 , x2 ) ∗(x1 − x1 ) +(x2 − x2 ) + . . .∂x1∂x2ÇàìåíÿÿxiÎáîçíà÷èìíàxnièx∗iíàxn+1,iïîëó÷èì:f1 (xn1 , xn2 ) +∂f1 (xn1 , xn2 ) n+1∂f1 (xn1 , xn2 ) n+1(x1 − xn1 ) +(x2 − xn2 ) = 0∂x1∂x2f2 (xn1 , xn2 ) +∂f2 (xn1 , xn2 ) n+1∂f2 (xn1 , xn2 ) n+1(x1 − xn1 ) +(x2 − xn2 ) = 0∂x1∂x2xn = (xn1 , xn2 )T , f n = (f1n , f2n )T ,"I(xn ) =à òàêæån∂f1 (xn1 ,x2 )∂x1n∂f2 (xn1 ,x2 )∂x1n∂f1 (xn1 ,x2 )∂x2n∂f2 (xn1 ,x2 )∂x2#(3)Òîãäà óðàâíåíèå ìîæíî çàïèñàòü â âèäå:f (xn ) + I(xn )(xn+1 − xn ) = 0Åñëè∀n ∃I −1 (xn ),(4)òîxn+1 = xn − I −1 (xn )f (xn ), n = 0, 1, 2, .

. . ;x0 çàäàíî(5)Ìåòîä Íüþòîíà è ìåòîä ñåêóùèõ57Çàìå÷àíèå. Ñ÷èòàòü I −1 (xn ) íå î÷åíü óäîáíî, ïîýòîìó îáû÷íî ââîäÿò ïîãðåøíîñòüv n+1 = xn+1 − xnè ðåøàþò íà êàæäîé èòåðàöèè óðàâíåíèå:I(xn )v n+1 = −f (xn )Çàìå÷àíèå.  ñëó÷àå ñèñòåìû ìîæíî ïðèìåíèòü ìîäèôèöèðîâàííûé ìåòîä Íüþòîíà:xn+1 = xn − I −1 (x0 )f (xn )Íî â ýòîì ñëó÷àå ñêîðîñòü ñõîäèìîñòè áóäåò çíà÷èòåëüíî ìåíüøå.Åñëè äàíà ñèñòåìà èç m óðàâíåíèé:f1 (x1 , . .