复合辛普森公式

1、实验5复合辛普森公式李涛 201226100108 计自 1201、实验目的48亍4用复合辛普森公式计算积分0、1 cos xdx，使误差不超过10-（注意所给积分特点,做出相应的处理后再计算）、实验步骤1算法原理复合辛普森原理:将区间a,b划分为等分，在每个子区间Xk,Xk 1上采用辛普森公式，若记xk 1 2 xk1丄h,则得2bf(x)dx an 1f (x)dxk 0Snhn6k1f(Xk)04 f(xk 12)f(Xk) Rn(f ).1f(Xk)04f (Xk 12)f(Xk)-f(a)1f(Xk 12)01f(Xk) f (b),1称为复合辛普森求积公式，其余项为Rn(f ) I

2、Sn蓝（Vk0(k ), k (Xk, Xk 1 )于是当 f(x) c4a,b时，与复合梯形公式相似有Rn(f) 1 Sn需”()，(ab)易知误差阶为h4，收敛性是显然的，实际上，只要f(x) Ca,b则可得到收敛性，即blim Sn f (x)dxna此外，由于Sn中求积公系数均为正数，故知辛普森公式计算稳定。2算法步骤复合辛普森:首先将区间a,b划分为n等分，在每个子区间1Xki2 Xk -h,则得2Xk,Xki上采用辛普森公式，若记Snh n 16kof(xk) 4f(xk12)f(xk)n 1n 1-f(a) 4 f(Xk12)2 f(xj f(b)6k 0k 1算法过程：这里将辛

3、普森公式写为Sn()函数，然后在 Solve()函数里依次计算 S1,S2,S4,S6当相邻的精度小于eps时退出循环，贝U S2n保存结果。三程序代码#in elude #in elude #defi ne eps 1e-6 using n amespaee std;double f(double x)retur n sqrt(1+cos(x)*cos(x);/被积函数double Sn( double a,double b,double n)double h=(a+b)/(2* n);double sum=0;for(i nt k=1;keps)coutn = i S1 = S1 n S2

4、 = S2 n n： i+;S1n=S2 n;S2n=S n(a,b,2*i);return S2n;int mai n()coutSolve(0,48)Z =bb.lbSlA1=2St = 5G.1&31S2 =56.202Sn = 3Gt = &G.5028S2 =ES.M34ri4St - 5B.B624S2 -5G.2041=5St =.24 iSZ =58.n = 6SI = SS.BL22S2 =58.1339n = ?St = GS.13?9S2 =57.9518ri - 8St - 57.951SS2 -59.0773(1=9SL 土 5y .MV73S2 -58.S825n

5、= JJdSI = 5S.5825S2=58.Sit?n = 1121 =.5169S2=C8.4921ii 一 12SI - 58.421S2-SB .4865i1 = 1J畀=58 .铀砧82=5K.47yin = 14SI = 51.4791S2=51.4742n 一 J_Si = E.474252=58.4G0?ii 一 JLhSI - SA.4607S2-5B.4797n = 171 = bt.479782=5B.47bbn = 1HSI = St.4755S2=S8.474n - 1 ?1 = 8.474&2 =58.4733ii 一 2QSi - 5S.4733S2-5B.472

6、7n = Z1il = bS.4727S2=5B.472S口 = J?I?SI = 51.4725=58.472&n = 2 31 = 5S.47262=58.4611ii 一 24Si - S8.46丄4S2-58.470Sn = 25SI = 58.47%S2=58.4708KR.47HSpeturned 8 execution tine 二 0.117 sPress any key to cont inueVxn=12!1=SS.4V2182=b.4H6bn=13S1=51M652=51.4791n-14S1=E.49912=S-4742ii一1531-5ft.4742$2-5.4607

7、n=lbS1=5K.4bW782=SB.4797n=17S1=51-4797SS=M -4755n1&31=5I.475Eas=51_474n=19S丄=58.474S2 =:58.4733n=2&S1=58.4733S2=58.4727n=211=58.4729S2=S8.472En2231-GS.472GS2-58.472&n=23S1=58.4726S2=58.4614n=24S1=51.4614S2=58.470Bn=25S1=58 .47%S2=A8.470Rn一26S1一58 .4708G2一8.4709n=27齿1=b8 .?WHSZ=bti . 47HBn=281=5S.470

8、8S2=58.4708n=29S1=&S .4708G2=58.470Bn一3BS1一58 .4?08S2一8.470Bn=31ii=b8.47tS2=bH.47n=32S1=.4705S2=58.4706execut ion tine = U.Wb4 sProcess returned U微软挤音简捷半=58.470551.4705 5ft.4?0S E.40G 5S.47S5 bS.47Mb 疆.47館CS.406SS.405 bM .4?S5b55.470558.470658.470558.47055S.4785 5B.470S55.470558.4705 bK.47(*=58.4705 =58.4705=58.47A?；=-5A.4705=b.47Hb =58.4705=S8.4705-58.4705=5S.470b=58.4705=58.470S-58.4705=58.4705=58.4705=S8.47PS=58.4705-58.4705=5.47Ubrruccss r*ct iirnGil H 0x0?e xc匚ul,丄tin txnie - 3.1J4 sFre ss any key