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楼主 |
发表于 2020-7-25 07:58:05
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我们设\(x_k=\frac{a_k}{a_{k+1}},k=1..n\),则有极值条件:
\(x_{k+2}^2=x_{k+1}^2+x_{k+1}-\frac{x_{k}^2}{x_{k+1}}\)
对于n=7,我们得到:
\(x_1^2=x_7^2+x_7-\frac{x_6^2}{x_7}\)
\(x_2^2=x_1^2+x_1-\frac{x_7^2}{x_1}\)
\(x_3^2=x_2^2+x_2-\frac{x_1^2}{x_2}\)
\(x_4^2=x_3^2+x_3-\frac{x_2^2}{x_3}\)
\(x_5^2=x_4^2+x_4-\frac{x_3^2}{x_4}\)
\(x_6^2=x_5^2+x_5-\frac{x_4^2}{x_5}\)
\(x_7^2=x_6^2+x_6-\frac{x_5^2}{x_6}\)
通过艰辛的计算,我们消元得到(超出想象的结果,\(x_1,x_2,x_3,x_4,x_5,x_6,x_7\)为下列关于y方程的根,详细表达式见附件):
\(y^{1575}+237y^{1574}+27960y^{1573}+2189167y^{1572}+127967055y^{1571}+
5956542111y^{1570}+229966468039y^{1569}+7573740288747y^{1568}+217195916251734y^
{1567}+5509224934586649y^{1566}+125136090704063355y^{1565}+2570697121556978829y^
{1564}+48156540002089434273y^{1563}+828275961122268316399y^{1562}+
13156417844524062956717y^{1561}+193959627276535346931317y^{1560}+
2665471049415845840780533y^{1559}+34274013692711101754535718y^{1558}+
413737859400601836305780338y^{1557}+4702496720097649823655021586y^{1556}+
50455171343554283441379325067y^{1555}+512229855678807802192283264868y^{1554}+
4930683484539424975189646038889y^{1553}+45085742320062585790190905818592y^{1552}
+392269284750521293203307247108736y^{1551}+3252297692356540986585666729214139y
^{1550}+25729825589613508859427216137618490y^{1549}+
194464748622873082362576842998104622y^{1548}+
1405597526808002206155756147100493505y^{1547}+
9725246507446609876489454720237816927y^{1546}+\dots+-71793810348771483375391108000228145
y^{30}-13100909023550720373698929424205502y^{29}-\
2296874872643888083204839838615397y^{28}-386643414798137816051162164327259y^{27}
-62440236022275419435274445188562y^{26}-9664237519210817455694811960715y^{25}-\
1431896440799115559001206901440y^{24}-202819370853057431834671945578y^{23}-\
27421542096528982240583950780y^{22}-3532672638900873004315266906y^{21}-\
432808771922657726356622019y^{20}-50317756233030382350028192y^{19}-\
5537563785067957159898811y^{18}-575310020991268693223646y^{17}-\
56252156607650435518263y^{16}-5158569399568432942730y^{15}-441947025365891770693
y^{14}-35214265373062071162y^{13}-2596193691265762792y^{12}-176045275408316310y^
{11}-10902389172376620y^{10}-611481429273170y^9-30746301739012y^8-1368654303590
y^7-53085298369y^6-1757100738y^5-48242847y^4-1054508y^3-17199y^2-186y-1=0\)
解得:
{-2.5313754361572083274, -2.2897435314085089836, -2.1834962340662334071, -2.1253313509433058988, -1.7875749231432755320, -1.7423863064272187694,
-1.6770641724445321307, -1.6243773891112491293, -1.5415473045208114337, -1.3881929471272967637, -1.3328098201143265808, -1.3146415925398947908,
-1.3077838701985277477, -1.3059632128632167189, -1.2151831983216832552, -1.2100157148300979496, -1.2064913022022663453, -1.2034626937323212323,
-1.1498770384577404686, -1.1377798045432585345, -1.0312842256127762898, -1.0308402673719049890, -1.0221997159959176423, -0.92331967132561932280,
-0.90883720222180845989, -0.78607477684269818152, -0.78412461246543316888, -0.78397910048394251469, -0.77526882129506576722, -0.76910744508814982268,
-0.69212265623708541717, -0.66663655058268905479, -0.64591036580872210630, -0.37970446355950123768, -0.36499631361226621680, -0.29905968201527480038,
-0.22359585813136146356, -0.18249015231659363020, -0.17387273230980682958, -0.10301647889050310476, 0.79863335015334249027, 0.81831197137120640244,
0.84713240968364325923, 0.90020230214287103567, 0.90667579762331956742, 0.91222229382793599056, 0.93953457502418666744, 0.94450390184391910245,
0.97923337970653848310, 0.98724364494215151574, 0.99197152733571484758, 1.0151757673093954822, 1.0201960406524940163, 1.0407750733403786538,
1.0497215743716858584, 1.0644601871440451014, 1.0889696744406762472, 1.0889944527757833722, 1.0995090477768181806, 1.1099664528720066446,
1.1188511257163850983, 1.1327891624940687097, 1.1413799553375473493, 1.1442564456368120404, 1.2049578377713863665, 1.2200423816708315112,
1.2298948000619613306, 1.2311306286792130033, 1.2833612147705397827, 1.3423373691686473528, 1.3870442957496284705, 1.3954604220135717331,
1.5511049214726247284, 1.6117716462358246318, 1.6202221715093102880, 1.7227089659927969040, 1.9699437824343871243}
可以对比wayne给出的结果:(说明结果是准确的)
\(x_1 =\frac{a_1}{a_2}=-1.206491302, x_2 = \frac{a_2}{a_3}=1.109966453, x_3 =\frac{a_3}{a_4}=1.015175767,
x_4 =\frac{a_4}{a_5}=0.9122222938, x_5 =\frac{a_5}{a_6}=-0.7839791009, x_6 =\frac{a_6}{a_7}=0.9445039018, x_7 =\frac{a_7}{a_1}=1.088969674\)
N=7结果.pdf
(266.94 KB, 下载次数: 3)
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