1、實驗一非線性方程的數值解法(一)信息與計算科學金融崔振威2010020340311、 實驗目的：熟悉二分法和簡單迭代法的算法實現。2、 實驗內容：教材P402.1.53、 實驗要求1 根據實驗內容編寫二分法和簡單迭代法的算法實現2 簡單比較分析兩種算法的誤差3 試構造不同的迭代格式，分析比較其收斂性(一)、二分法程序：functionef=bisect(fx,xa,xb,n,delta)%fx是由方程轉化的關于x的函數，有fx=0。%xa解區間上限%xb解區間下限%n最多循環步數，防止死循環。%delta為允許誤差x=xa;fa=eval(fx);x=xb;fb=eval(fx);disp(&

2、#39;nxaxbxcfc');fori=1:nxc=(xa+xb)/2;x=xc;fc=eval(fx);X=i,xa,xb,xc,fc;disp(X),iffc*fa<0xb=xc;elsexa=xc;endif(xb-xa)<delta,break,endend(二)、簡單迭代法程序：functionx0,k=iterate(f,x0,eps,N)ifnargin<4N=500;endifnargin<3ep=1e-12;endx=x0;x0=x+2*eps;k=0;whileabs(x-x0)>eps&k<Nx0=x;x=feval(

3、f,x0);k=k+1;endx0=x;ifk=Nend解：a、g(x)=x5-3x3-2x2+2二分法求方程：(1)、在matlab的命令窗口中輸入命令：>>fplotCxA5-3*xA3-2*xA2+2',-3,3);grid得下圖：由上圖可得知：方程在卜3,3區間有根。(2)、二分法輸出結果>> f='xA5-3*xA3-2*xA2+2'f =xA5-3*xA3-2*xA2+2>> bisect(f,-3,3,20,10A(-12)2.0000-3.00000-1.50000.03133.0000-3.0000-1.5000-2

4、.2500-31.61824.0000-2.2500-1.5000-1.8750-8.43015.0000-1.8750-1.5000-1.6875-2.96326.0000-1.6875-1.5000-1.5938-1.21817.0000-1.5938-1.5000-1.5469-0.53828.0000-1.5469-1.5000-1.5234-0.24059.0000-1.5234-1.5000-1.5117-0.101510.0000-1.5117-1.5000-1.5059-0.034311.0000-1.5059-1.5000-1.5029-0.001412.0000-1.502

5、9-1.5000-1.50150.015013.0000-1.5029-1.5015-1.50220.006814.0000-1.5029-1.5022-1.50260.002715.0000-1.5029-1.5026-1.50270.000716.0000-1.5029-1.5027-1.5028-0.000317.0000-1.5028-1.5027-1.50280.000218.0000-1.5028-1.5028-1.5028-0.000119.0000-1.5028-1.5028-1.50280.000120.0000-1.5028-1.5028-1.5028-0.00002、迭代

6、法求方程：迭代法輸出結果：>>f=inline('xA5-3*xA3-2*xA2+2');>>x0,k=iterate(fun1,2)x0=2k=1>>x0,k=iterate(fun1,1.5)x0=NaNk=6>>x0,k=iterate(fun1,2.5)x0=NaNk=5(3)、誤差分析：由二分法和迭代法輸出結果可知，通過定點迭代法得出方程的解誤差比二分法大，而利用二分法求出的結果中，可以清楚看出方程等于零時的解，其誤差比迭代法小。b、g(x)=cos(sin(x)二分法求方程：(1)、在matlab的命令窗口中輸入命令:

7、>>fplot('cos(sin(x)',-4,4D;grid得下圖：由上圖可得知：方程在卜4,4區間無根。(2)、二分法輸出結果>>f='cos(sin(x)'f=cos(sin(x)>>bisect(f,-4,4,20,10A(-12)2.000004.00002.00000.61433.00002.00004.00003.00000.99014.00003.00004.00003.50000.93915.00003.50004.00003.75000.84116.00003.75004.00003.87500.78427

8、.00003.87504.00003.93750.75548.00003.93754.00003.96880.74129.00003.96884.00003.98440.734110.00003.98444.00003.99220.730511.00003.99224.00003.99610.728812.00003.99614.00003.99800.727913.00003.99804.00003.99900.727514.00003.99904.00003.99950.727315.00003.99954.00003.99980.727116.00003.99984.00003.9999

9、0.727117.00003.99994.00003.99990.727118.00003.99994.00004.00000.727019.00004.00004.00004.00000.727020.00004.00004.00004.00000.72702、迭代法求方程：迭代法輸出結果：>>f=inline('cos(sin(x)');>>x0,k=iterate(f,0.5)x0=0.7682k=15>>x0,k=iterate(f,1)x0=0.7682k=15>>x0,k=iterate(f,1.5)x0=0.7682

10、k=16>>x0,k=iterate(f,2)x0=0.7682k=15>>x0,k=iterate(f,2.5)x0=0.7682k=14(3)、由于該方程無解，所以無法比較誤差。c、g(x)=x2-sin(x+0.15)二分法求方程：1)、在matlab的命令窗口中輸入命令：>>fplot('xA2-sin(x+0.15)',-10,10);grid得下圖:)1zuwo806040200onIIIIJI/I/III>II-1Q-(3-64-2024e810由上圖可得知：方程在卜3,3區間有根。(2)、二分法輸出結果>>f

11、='xA2-sin(x+0.15)'f=xA2-sin(x+0.15)>>bisect(f,-3,3,30,10A(-12)1.0000-3.00003.00000-0.14942.0000-3.00000-1.50003.22573.0000-1.50000-0.75001.12714.0000-0.75000-0.37500.36375.0000-0.37500-0.18750.07266.0000-0.18750-0.0938-0.04747.0000-0.1875-0.0938-0.14060.01048.0000-0.1406-0.0938-0.1172-

12、0.01919.0000-0.1406-0.1172-0.1289-0.004510.0000-0.1406-0.1289-0.13480.002911.0000-0.1348-0.1289-0.1318-0.000812.0000-0.1348-0.1318-0.13330.001113.0000-0.1333-0.1318-0.13260.000114.0000-0.1326-0.1318-0.1322-0.000315.0000-0.1326-0.1322-0.1324-0.000116.0000-0.1326-0.1324-0.13250.000017.0000-0.1325-0.13

13、24-0.1324-0.000018.0000-0.1325-0.1324-0.1325-0.000019.0000-0.1325-0.1325-0.13250.000020.0000-0.1325-0.1325-0.13250.000021.0000-0.1325-0.1325-0.13250.000022.0000-0.1325-0.1325-0.13250.000023.0000-0.1325-0.1325-0.1325-0.000024.0000-0.1325-0.1325-0.13250.000025.0000-0.1325-0.1325-0.1325-0.000026.0000-0

14、.1325-0.1325-0.13250.000027.0000-0.1325-0.1325-0.13250.000028.0000-0.1325-0.1325-0.13250.000029.0000-0.1325-0.1325-0.13250.000030.0000-0.1325-0.1325-0.1325-0.00002、迭代法求方程：迭代法輸出結果：>>f=inline('xA2-sin(x+0.15)');>>x0,k=iterate(f,1.96)x0=NaNk=12>>x0,k=iterate(f,0,2)x0=-0.1494k=

15、1>>x0,k=iterate(f,0.2)x0=0.3234k=500>>x0,k=iterate(f,0.3)x0=0.3234k=500>>x0,k=iterate(f,0.001)x0=0.3234k=500(3)、誤差分析：由二分法和迭代法輸出結果可知，利用二分法求出的結果中，可以清楚看出方程等于零時的解，其誤差比迭代法小。d、g(x)=xx-cos(x)二分法求方程：(1)、在matlab的命令窗口中輸入命令：>>fplot('xA(x-cos(x)',-1,1);grid得下圖：12108G420X1014一一L_L

16、1ri-O.a-0.64-0.200.20.4D.60.8.1由上圖可得知：方程在-1,1區間有根。(2)、二分法輸出結果>>f='xA(x-COS(x)'f=xA(x-COS(x)>>bisect(f,-0.1,0.1,20,10A(-12)1.0000-0.10000.1000Inf2.0000-0.10000-0.0500-22.8740+3.5309i3.0000-0.05000-0.0250-43.6821+3.3947i4.0000-0.02500-0.0125-84.4110+3.2958i1.0e+002*0.0500-0.00010-0

17、.0001-1.6511+0.0323i1.0e+002*0.0600-0.00010-0.0000-3.2580+0.0319i1.0e+002*0.0700-0.00000-0.0000-6.4648+0.0317i1.0e+003*0.0080-0.00000-0.0000-1.2872+0.0032i1.0e+003*0.0090-0.00000-0.0000-2.5679+0.0032i1.0e+003*0.0100-0.00000-0.0000-5.1285+0.0031i1.0e+004*0.0011-0.00000-0.0000-1.0249+0.0003i1.0e+004*0

18、.0012-0.00000-0.0000-2.0490+0.0003i1.0e+004*0.0013-0.00000-0.0000-4.0971+0.0003i1.0e+004*0.0014-0.00000-0.0000-8.1931+0.0003i1.0e+005*0.0001-0.00000-0.0000-1.6385+0.0000i1.0e+005*0.0002-0.00000-0.0000-3.2769+0.0000i1.0e+005*0.0002-0.0000-6.5537+0.0000i1.0e+006*0.0000-0.0000-1.3107+0.0000i1.0e+006*0.0000-0.0000-2.6215+0.0000i1.0e+006*0.0000-0.0000-5.2429+0.0000i2、迭代法求方程：迭代法輸出結果：>&