利用Romberg算法计算下面定积分的近似值:
dx,,Ax,,,7,75,dt,,,1.
证明
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题:求初值问题 ,其中 A,,8,8,5,,,T,,,0,50,,,,x0,3,,2,1,,,,
x(t)3,,,,1,,,,x(t),x(t)解: C,-22 ,,,,
,,,,1x(t)3,,,,
,,77,5
det(,E,A),8,,85,(,,5)(,-5)(,,15)A的最小多项式为
05,
At2设 e,a(t)E,a(t)A,a(t)A,T(At),,,(A),5,,5,,15012,
2,t由与在A上相等得: ,,T,t,a(t),a(t),,a(t),,,f,t,e012
5t,e,a(t),5a(t),25a(t)012,-5tatatate,(),5(),25() ,012
-15t,atatate,(),15(),225()012,
解得,
1,5t,5t,15ta(t),(3e,6e,e)0,8,1,5t,5ta(t),(e,e),1 10,15t,5t,15t,a(t),(e,2e,e)2,200,
At2 ,,fAt,e,a(t)E,a(t)A,a(t)A,T(At)012故,
,,,,,,1-7-75105800,,,,,,,()1,()-8-8-5,()1201450atatat 012,,,,,,,,,,,,10-50404025,,,,,,aaaaa,,a,7,105,7,805012121,,aaaaa,,8,120a,8,145,5120121,,,,aaaa40,5,40a,2521202,, tttttttt5,5,155,5,155,5,,2e44e3ee4e5e5e,,,,,,e
,,1tttttttt,15,555,5,5,556e4e2e3ee6e5e5e,,,,,,,,10t5t,5,t15t5,t5,15t5t,5t,,2e4e2e3ee2e5e5e,,,,,,,,
At所以 ,,xt,eC
5515551555t,t,tt,t,tt,t,,2e,4,4e,3e,e,4e5e,5e3e,,1,,,,5515551555t,t,tt,t,tt,t,,2e,4e,6e3e,e,6e,5e,5e-2,,,,105515551555t,t,tt,t,tt,t,,,,2e,4e,2e,3e,e,2e5e,5e1,,,,
5-5-15ttt,,17e,9e,4e
1,,5-5-15ttt,-17e,11e,6e,,105-5-15ttt,,17e-9e,2e,,
解得:
ttt,15-5-15tx(),(17e,9e,4e)1,10,1,ttt5-5-15tx(),(-17e,11e,6e),2 10,1t5-5t-15t,x(t),(17e-9e,2e)3,10,
2. 利用Romberg算法计算下面定积分的近似值:
12,x-6edx,,10 ,其中误差界为 ,0
2b,a1,xa,0,b,1,h,, 解:, f(x),enn
1T,,,f(0),f(1),0.6839397 ; 12
n,11hT,T,f(x) ; ,2nn1k,22k,02
111T,T,f(),0.7313703 ; 21222
1113,, ; T,T,f(),f(),0.742984142,,2444,,
111357,, ; T,T,f(),f(),f(),f(),0.745865684,,288888,,
41S,T,T,0.7471805 ; 12133
41S,T,T,0.7468554 ; 24233
41 ; S,T,T,0.746826148433
161 ; C,S,S,0.74683371211515
161 ; C,S,S,0.74682412421515
641 ; R,C,C,0.74682401216363
TSCRk kk,1k,2k,322220 0.6839397 1 0.7313703 0.7471805 2 0.7429841 0.7468554 0.7468337 3 0.7458656 0.7468261 0.7468241 0.7468240
,7,所以,近似值为0.7468240。 R,C,10,,12