北航数值分析上机作业 二 (报告+所有程序大全)

1、数值分析上机作业2qr分解求特征值题目：双步位移qr分解法求a的全部特征值，对每一个实特征值求特征向量。已知1. 算法设计方案总体算法与老师的讲义和课本一致，在这里对自己的设计和改进做说明。算法的关键在于对于被运算的矩阵a的左下角理论为0的元素不再计算。1) a拟上三角化对于，记，则，这样去掉了系数，便于计算。设对a到的拟上三角化进行到到第k步，后半段不为0的部分是，则，只需要计算矩阵的右上角，右下角，和左下角即可，计算过程中按照行向量左乘矩阵以及列向量右乘矩阵的运算方法进行，左下角矩阵只有最右列有非零元素，可以简化运算。2) qr分解对于，记，则，这样去掉了系数，便于计算。设对a到的qr分解

2、进行到到第k步，。后半段不为0的部分是，则，这样只需要计算矩阵的右上角和右下角矩阵。计算矩阵的右上角和右下角矩阵时，计算过程中按照行向量左乘矩阵以及列向量右乘矩阵的运算方法进行。3) 求特向量的方法对进行选主元的高斯消去法，化成上三角阵，进行带入求解，对于第k行，如果出现，并且的情况，直接令为1。特向量完成求解后再进行归一化。2. 拟上三角的a1) 12位有效数字的形式-8.82751675883e-001-9.93313649183e-002-1.10334928599e+000-7.60044358564e-0011.54910107991e-001-1.94659186287e+000-

3、8.78243638293e-002-9.25588938718e-0016.03259944053e-0011.51886095647e-001-2.34787836242e+0002.37237010494e+0001.81929082221e+0003.23780410155e-0012.20579844032e-0012.10269266255e+0001.81613808610e-0011.27883908999e+000-6.38057812440e-001-4-0010.00000000000e+0001.72827459997e+000-1.17146

4、764279e+000-1.24383926270e+000-6.39975834174e-001-2.00283307904e+0002.92494720612e-001-6.41283006840e-0019.78399762128e-0022.55776357416e-0010.00000000000e+0000.00000000000e+000-1.29166953413e+000-1.11160351340e+0001+000-1.30735603002e+0001.80369917775e-001-4.24638535837e-0017.988955239

5、30e-0021.60881992807e-0010.00000000000e+0000.00000000000e+0000.00000000000e+0001.56012629853e+0008.12504939751e-0014.42175683292e-001-3.58861612814e-0024.69174231367e-001-2.73659505009e-001-7.35933465775e-0020.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+000-7.70777375519e-001-

6、1.58305142574e+000-3.04284317680e-0012.52871244603e-001-6.70992540145e-0012.54461992908e-0010.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+000-7.46345345694e-001-2.70836515702e-002-9.48652189368e-0011.19587108150e-0011.92926561795e-0020.00000000000e+0000.00000

7、000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+000-7.70180137436e-001-4.69762399062e-0014.98825946801e-0011-0010.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0007.01316709211e-00

8、11.58218068848e-0013.86259461423e-0010.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0004.84380760278e-0013.99277799518e-0012) 3位小数位的形式-0.883-0.099-1.103-0.7600.155-1.947-0.088-0.9260.6030.152-2.3482.3721.81

9、90.3240.2212.1030.1821.279-0.638-0.4150.0001.728-1.171-1.244-0.640-2.0030.292-0.6410.0980.2560.0000.000-1.292-1.1121.171-1.3070.180-0.4250.0800.1610.0000.0000.0001.5600.8130.442-0.0360.469-0.274-0.0740.0000.0000.0000.000-0.771-1.583-0.3040.253-0.6710.2540.0000.0000.0000.0000.000-0.746-0.027-0.9490.1

10、200.0190.0000.0000.0000.0000.0000.000-0.770-0.4700.4990.1140.0000.0000.0000.0000.0000.0000.0000.7010.1580.3860.0000.0000.0000.0000.0000.0000.0000.0000.4840.3993. a(n-1)进行qr分解得到q，r，以及rq1) 12位有效数字的e型输出：拟上三角a,qr之后的q：-3.51926257953e-001-9.36027728736e-0010.00000000000e+0000.00000000000e+0000.00000000000

11、e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+000-7.39595535350e-001-3.31120858752e+000-8.81052055469e-0010.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0002.40254545121e-001-4.4

12、8034907538e+000-5+000-5.37180680644e-0010.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+000-1.43125202505e-0012.66904781503e+0006.54274505882e+0001.19520089248e+0009.89223546862e-0010.00000000000e+0000.00000000000e+0000.0000000000

13、0e+0000.00000000000e+0000.00000000000e+000-1.96077644897e-0013.65652310377e+0008.96338324661e+0003.85510755662e-0012.70771861608e+0005.32361069026e-001-0.00000000000e+000-0.00000000000e+000-0.00000000000e+000-0.00000000000e+000-5.63148539010e-0011.05017869060e+0012.57434557752e+0011.10721351706e+000

14、4.11588263550e+0009.26346777492e-001-6.05976150575e-0010.00000000000e+0000.00000000000e+0000.00000000000e+000-3-0015.87640573131e+0001.44050714811e+0016.19555121015e-0012.30309341878e+0003.78744079911e-002-7.93763324139e-0019.93339662512e-0010.00000000000e+0000.00000000000e+000-9.181777

15、90891e-0021.71224940381e+0004.19730634392e+0001.80524105226e-0016.71068424056e-0011.10357309328e-002-1.60433215213e+0002.40658785801e+0005.41008800606e-0010.00000000000e+000-1.56690148242e-0012.92201157088e+0007.16284536362e+0003.08070496692e-0011+0001.88328484197e-002-2.73784712766e+00

16、02.07736354786e+0002.51193105972e-001-7.22159133673e-001-8.85680693195e-0021.65164770259e+0004.04875093812e+0001.74134809446e-0016.47317276562e-0011.06451429335e-002-1.54754996986e+0001+000-2.51782156703e-0016.91726958888e-001拟上三角a,qr之后的r：2.50834274492e+000-2+000-1.31460907

17、079e+000-3.55878749384e-002-2.60985785039e-001-1.28312184709e+000-1.39087861061e-001-8.71289797216e-0013.84936790297e-0013.35380289967e-0010.00000000000e+000-1.96160327785e+0002.40752372763e-0017.05471457282e-0015.95720431828e-0015.52697877468e-001-3.26820992441e-001-5.76949866836e-0022.87112933019e

18、-001-8.89512875419e-0020.00000000000e+0000.00000000000e+0002.40453460199e+0001.70675809633e+000-4.23956670409e-0013.40533230581e+000-1.05001765585e-0011.46225710273e+000-6.68448746928e-001-4.02764620966e-0010.00000000000e+0000.00000000000e+0000.00000000000e+0001.57712208072e+0006.39953513396e-0013.4

19、6812787243e-001-5.70178664977e-0024.01478805443e-001-2.22247617631e-001-6.31705923644e-0020.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+000-1.44784699777e+000-1.41572400774e+000-2.80613904467e-001-2.81791052189e-001-4.61143488185e-0021.99662907996e-0010.00000000000e+0000.00000

20、000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0001.23164145154e+0001.61970100342e-0011.96263827550e-0015.35003562176e-001-1.50927342477e-0010.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+000-7.75344191421e-001-3.46451450882e-0

21、014.31222680350e-0011.23464369624e-0010.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0001.29631294061e+000-4.28805331834e-0012.73733415816e-0010.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.0000000

22、0000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+000-6.70739644065e-001-4.84232012188e-0010.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0007.16832392632e-002拟上三角a,qr之后的rq：1.163

23、07441416e+0006.54021266522e+0001.71698760912e+001-1.50944630535e+001-2.16705228879e+001-6.04262699693e+001-3.39980110592e+001-1+001-1.81045579997e+001-1.00161647810e+0011.83611506085e+0006.28316224606e+0007+000-2.22795075154e+000-2.83544695057e+000-1.04594830181e+001-6.0270

24、0244190e+000-1.27420738622e+000-2+000-1.44657481541e+0000.00000000000e+000-2.11852015353e+000-1.32735235180e+0011.73527877647e+0012.28756497487e+0016.52639670964e+0013.63833511946e+0011.34796269561e+0012.07757966695e+0011+0010.00000000000e+0000.00000000000e+000-8.4719951278

25、1e-0012.51803480278e+0002.52544119293e+0004.73600481425e+0002.90818562795e+0001.65542017861e+0002.20518743642e+0001.26449870642e+0000.00000000000e+0000.00000000000e+0000.00000000000e+000-1.43224434245e+000-4.67403861531e+000-7.10057435575e+000-3.44531980370e+000-1.24013323626e+000-1.65761440713e+000

26、-6.99183495559e-0010.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0006.55677959800e-0011.04277707175e+0001-0015-0012.30840029271e-001-2.46194560930e-0010.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0004.69840088

27、488e-0012.71295815438e-0016.43439025273e-0011.42222749450e+0007.69904500551e-0010.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0001.28767905894e+0002.88770352479e+0002.38752121986e+0001.81946570709e+0000.00000000000e+0000.00000000000e+0000.00

28、000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+000-3.62876050354e-0011.81207395928e-001-1.66076063019e-0010.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+000-5/p>

29、e-0024.95852290988e-0022) 3位小数形式输出拟上三角a,qr之后的q：-0.352-0.9360.0000.0000.0000.0000.0000.0000.0000.000-0.740-3.311-0.8810.0000.0000.0000.0000.0000.0000.0000.240-4.480-5.139-0.5370.0000.0000.0000.0000.0000.000-0.1432.6696.5431.1950.9890.0000.0000.0000.0000.000-0.1963.6578.9630.3862.7080.532-0.000-0.000-

30、0.000-0.000-0.56310.50225.7431.1074.1160.926-0.6060.0000.0000.000-0.3155.87614.4050.6202.3030.038-0.7940.9930.0000.000-0.0921.7124.1970.1810.6710.011-1.6042.4070.5410.000-0.1572.9227.1630.3081.1450.019-2.7382.0770.251-0.722-0.0891.6524.0490.1740.6470.011-1.5481.174-0.2520.692拟上三角a,qr之后的r：2.508-2.186

31、-1.315-0.036-0.261-1.283-0.139-0.8710.3850.3350.000-1.9620.2410.7050.5960.553-0.327-0.0580.287-0.0890.0000.0002.4051.707-0.4243.405-0.1051.462-0.668-0.4030.0000.0000.0001.5770.6400.347-0.0570.401-0.222-0.0630.0000.0000.0000.000-1.448-1.416-0.281-0.282-0.0460.2000.0000.0000.0000.0000.0001.2320.1620.1

32、960.535-0.1510.0000.0000.0000.0000.0000.000-0.775-0.3460.4310.1230.0000.0000.0000.0000.0000.0000.0001.296-0.4290.2740.0000.0000.0000.0000.0000.0000.0000.000-0.671-0.4840.0000.0000.0000.0000.0000.0000.0000.0000.0000.072拟上三角a,qr之后的rq：1.1636.54017.170-15.094-21.671-60.426-33.998-11.352-18.105-10.0161.8

33、366.2837.172-2.228-2.835-10.459-6.027-1.274-2.186-1.4470.000-2.119-13.27417.35322.87665.26436.38313.48020.77611.5640.0000.000-0.8472.5182.5254.7362.9081.6552.2051.2640.0000.0000.000-1.432-4.674-7.101-3.445-1.240-1.658-0.6990.0000.0000.0000.0000.6561.0430.1130.5160.231-0.2460.0000.0000.0000.0000.0000

34、.4700.2710.6431.4220.7700.0000.0000.0000.0000.0000.0001.2882.8882.3881.8190.0000.0000.0000.0000.0000.0000.000-0.3630.181-0.1660.0000.0000.0000.0000.0000.0000.0000.000-0.0520.0504. 特征值和特向量1) 12位有效数字e型特征值：实部：3.38303961744e+000-2.32349621021e+000-2.32349621021e+000-1.48403982226e+000-9.80530956290e-001

35、-9.80530956290e-0011.57754855711e+0009.35588907819e-0015.65048899350e-0026.36062787575e-001虚部：0.00000000000e+000-8.93040517720e-0018.93040517720e-0010.00000000000e+000-1-0011-0010.00000000000e+0000.00000000000e+0000.00000000000e+0000.00000000000e+000特向量：-1.05221573436e-0010

36、.00000000000e+0000.00000000000e+000-5.60118116886e-0010.00000000000e+0000.00000000000e+0006.24962559918e-0028.05651548268e-002-2.09017518582e-0011.07037468397e-001-2-0010.00000000000e+0000.00000000000e+0007.79341508753e-0010.00000000000e+0000.00000000000e+000-1.12090045149e-0024.6111346

37、8803e-002-2.00063796685e-0017.12345583256e-002-4.73017877776e-0010.00000000000e+0000.00000000000e+0001.33780114833e-0020.00000000000e+0000.00000000000e+000-2.49666705179e-001-1.50243791979e-0023.89176061817e-0013.90237917138e-001-2.60887478870e-0010.00000000000e+0000.00000000000e+000-2.77409287770e-

38、0010.00000000000e+0000.00000000000e+000-1.31364486150e-001-4.81208346183e-002-2.77939118707e-002-4.46655513756e-002-3.05805625140e-0010.00000000000e+0000.00000000000e+0003.00557587803e-0030.00000000000e+0000.00000000000e+000-3.83541256120e-001-3.53633892069e-001-3.93236547906e-001-7.19034744275e-001

39、-2.58379783221e-0010.00000000000e+0000.00000000000e+000-2.53483409185e-0030.00000000000e+0000.00000000000e+0008-0012.08919077811e-001-1.24703810952e-0011.75815688440e-0018.73379363951e-0020.00000000000e+0000.00000000000e+000-2.06284849136e-0020.00000000000e+0000.00000000000e+000-1.24560

40、352068e-001-1.55745966395e-0016.44810939558e-001-2.26537961544e-0014.05454033853e-0010.00000000000e+0000.00000000000e+000-1.10134829109e-0020.00000000000e+0000.00000000000e+000-6.83561054714e-0028.19670427271e-001-3.02779853108e-0013.76886150502e-0015.09013113720e-0010.00000000000e+0000.00000000000e

41、+000-1.22486166509e-0020.00000000000e+0000.00000000000e+0002.70587401777e-001-3.50982512819e-001-2.91095263314e-0012.95625408505e-0012.40925044562e-0010.00000000000e+0000.00000000000e+0003.23620929756e-0020.00000000000e+0000.00000000000e+0001.00519108146e-0012.88513852782e-0024.09436555812e-0022.255

42、77972559e-0022) 3位小数特征值（上一行为实部，下一行为虚部）：3.383-2.323-2.323-1.484-0.981-0.9811.5780.9360.0570.6360.000-0.8930.8930.000-0.1140.1140.0000.0000.0000.000特向量：-0.1050.0000.000-0.5600.0000.0000.0620.081-0.2090.107-0.2180.0000.0000.7790.0000.000-0.0110.046-0.2000.071-0.4730.0000.0000.0130.0000.000-0.250-0.0150

43、.3890.390-0.2610.0000.000-0.2770.0000.000-0.131-0.048-0.028-0.045-0.3060.0000.0000.0030.0000.000-0.384-0.354-0.393-0.719-0.2580.0000.000-0.0030.0000.0000.8160.209-0.1250.1760.0870.0000.000-0.0210.0000.000-0.125-0.1560.645-0.2270.4050.0000.000-0.0110.0000.000-0.0680.820-0.3030.3770.5090.0000.000-0.01

44、20.0000.0000.271-0.351-0.2910.2960.2410.0000.0000.0320.0000.0000.1010.0290.0410.0235. 源程序1) 主函数#include#include#define size 10#define deta 0.0001#define maxstep 1000 /*注意这里的最大步数指m qr 分解并且qaq 的一次过程，有效的控制运算时间#define a(a,b) *(double *)a+(a)*(size)+b) /a 的宏定义，以方便操作二维数组#define a_reg(a,b) *(double *)a_reg

45、+(a)*(size)+b) /a 的宏定义，以方便操作二维数组#define m(a,b) *(double *)m+(a)*(size)+b) /a 的宏定义，以方便操作二维数组#define l(a,b) *(double *)l+(a)*(size)+b) /l u 的宏定义，以方便操作二维数组#define u(a,b) *(double *)u+(a)*(size)+b)#define v(a,b) *(double *)v+(a)*(size)+b) /存储特向量/#define u(a,b) *(double *)u+(a)*(3)+b)int main()double asi

46、zesize=0;double a_regsizesize=0;double lsizesize=0;double usizesize=0;double msizesize=0;double vsizesize=0;double realsize=0; /特征值的实部虚部double imagsize=0;double real_temp=0;double imag_temp=0;double sum_temp=0;double s,t;file *fd;int ii,jj,kk,m,counter=0;kk=0;s=0;t=0;m=0;counter=0; /计数器for(ii=0;iisi

47、ze;ii+) for(jj=0;jjsize;jj+) if(ii=jj)a(ii,jj)=1.5*cos(ii+1.2*jj+2.2);elsea(ii,jj)=sin(0.5*ii+0.2*jj+0.7); for(ii=0;iisize;ii+)for(jj=0;jjsize;jj+)a_reg(ii,jj)=a(ii,jj);m(ii,jj)=a(ii,jj);liketriangle(double *)a); /拟上三角化fd = fopen(tmptmp.txt, w);fprintf(fd,拟上三角的a：n);for(ii=0;iisize;ii+)for(jj=0;jjsiz

48、e;jj+)fprintf(fd,%.3f,a(ii,jj);fprintf(fd,n);fprintf(fd,n);for(ii=0;iisize;ii+)for(jj=0;jjsize;jj+)/a_reg(ii,jj)=a(ii,jj);m(ii,jj)=a(ii,jj);qr_only(double *)m);for(ii=0;iisize;ii+)for(jj=0;jj0)&(counter2if(a(m-1),(m-2)=0)realm-1=a(m-1),(m-1);m=m-1;/continue;else if(a(m-1),(m-2)!=0)&(a(m-2),(m-3)=0)i

49、f(eig2dim(double *)a, m,&real_temp,&imag_temp)=1)/realm-1=real_temp+imag_temp;realm-2=real_temp-imag_temp;elserealm-1=real_temp;realm-2=real_temp;imagm-1=imag_temp;imagm-2=-imag_temp;m=m-2;/continue;elsecounter+;s=a(size-1,size-1)+a(size-2,size-2); /根据右下角元素求出s tt=a(size-1,size-1)*a(size-2,size-2)-a(size-1,size-2)*a(size-2,size-1);m_out(double *)a,(double *)m,size,s,t,m); /求出mqr(double *)m,(double *)a,m);/qr分解m并且得到新的a/ 生成特向量for(kk=0;kksize;kk+)if(imagkkdeta)continue;else;for(ii=0;iisize;ii+)a_reg(ii,ii)=a_reg(ii,ii)-realkk;lu(d