北航数值分析作业第一题题解.doc

北航数值分析作业第一题：一、算法设计方案1. 要求计算矩阵的最大最小特征值，通过幂法求得模最大的特征值，进行一定判断即得所求结果；2. 求解与给定数值接近的特征值，可以该数做漂移量，新数组特征值倒数的绝对值满足反幂法的要求，故通过反幂法即可求得；3. 反幂法计算时需要方程求解中间过渡向量，需设计Doolite分解求解；4. |A|=|B|C|，故要求解矩阵的秩，只需将Doolite分解后的U矩阵的对角线相乘即为矩阵的Det。算法编译环境：vlsual c+6.0需要编译函数：幂法，反幂法，Doolite分解及方程的求解二、源程序如下：#include#include#include#includeint Max(int value1,int value2); int Min(int value1,int value2); void Transform(double A5501); double mifa(double A5501);void daizhuangdoolite(double A5501,double x501,double b501);double fanmifa(double A5501);double Det(double A5501);/*定义2个判断大小的函数，便于以后调用*/int Max(int value1,int value2) return(value1value2)?value1:value2); int Min(int value1,int value2) return (value1value2)?value1:value2); /*/*将矩阵值转存在一个数组里，节省空间*/ void Transform(double A5501,double b,double c) int i=0,j=0;Aij=0,Aij+1=0;for(j=2;j=500;j+)Aij=c;i+;j=0;Aij=0;for(j=1;j=500;j+)Aij=b;i+;for(j=0;j=500;j+)Aij=(1.64-0.024*(j+1)*sin(0.2*(j+1)-0.64*exp(0.1/(j+1);i+;for(j=0;j=499;j+)Aij=b;Aij=0;i+;for(j=0;j=498;j+)Aij=c;Aij=0,Aij+1=0; /*转存结束*/用于求解模最大的特征值，幂法double mifa(double A5501)int s=2,r=2,m=0,i,j;double b2,b1=0,sum,u501,y501;for (i=0;i=500;i+) ui = 1.0; dosum=0;if(m!=0)b1=b2;m+;for(i=0;i=500;i+)sum+=ui*ui;for(i=0;i=500;i+)yi=ui/sqrt(sum);for(i=0;i=500;i+)ui=0;for(j=Max(i-r,0);j=Min(i+s,500);j+)ui=ui+Ai-j+sj*yj;b2=0;for(i=0;i=exp(-12);return b2;/带状DOOLITE分解，并且求解出方程组的解void daizhuangdoolite(double A5501,double x501,double b501)int i,j,k,t,s=2,r=2;double B5501,c501;for(i=0;i=4;i+)for(j=0;j=500;j+)Bij=Aij;for(i=0;i=500;i+)ci=bi;for(k=0;k=500;k+)for(j=k;j=Min(k+s,500);j+)for(t=Max(0,Max(k-r,j-s);t=k-1;t+)Bk-j+sj=Bk-j+sj-Bk-t+st*Bt-j+sj;for(i=k+1;i=Min(k+r,500);i+)for(t=Max(0,Max(i-r,k-s);t=k-1;t+)Bi-k+sk=Bi-k+sk-Bi-t+st*Bt-k+sk;Bi-k+sk=Bi-k+sk/Bsk;for(i=1;i=500;i+)for(t=Max(0,i-r);t=0;i-)xi=ci;for(t=i+1;t=Min(i+s,500);t+)xi=xi-Bi-t+st*xt;xi=xi/Bsi;/用于求解模最大的特征值，反幂法double fanmifa(double A5501)int s=2,r=2,m=0,i;double b2,b1=0,sum=0,u501,y501;for (i=0;i=500;i+)ui = 1.0; doif(m!=0)b1=b2;m+;sum=0;for(i=0;i=500;i+)sum+=ui*ui;for(i=0;i=500;i+)yi=ui/sqrt(sum);daizhuangdoolite(A,u,y);b2=0;for(i=0;i=fabs(b1)*exp(-12);return 1/b2;/行列式的LU分解，U的主线乘积即位矩阵的DETdouble Det(double A5501)int i,j,k,t,s=2,r=2;for(k=0;k=500;k+)for(j=k;j=Min(k+s,500);j+)for(t=Max(0,Max(k-r,j-s);t=k-1;t+)Ak-j+sj=Ak-j+sj-Ak-t+st*At-j+sj;for(i=k+1;i=Min(k+r,500);i+)for(t=Max(0,Max(i-r,k-s);t=k-1;t+)Ai-k+sk=Ai-k+sk-Ai-t+st*At-k+sk;Ai-k+sk=Ai-k+sk/Ask;double det=1;for(i=0;i0) /判断求得最大以及最小的特征值.如果K0，则它为最大特征值值， /并以它为偏移量再用一次幂法求得新矩阵最大特征值，即为最大 /与最小的特征值的差lamda501=k;for(i=0;i=500;i+)A2i=A2i-k;lamda1=mifa(A)+lamda501;for(i=0;i=500;i+)A2i=A2i+k;else /如果K=0，则它为最小特征值值，并以它为偏移量再用一次幂法 /求得新矩阵最大特征值，即为最大与最小的特征值的差 lamda1=k;for(i=0;i=500;i+)A2i=A2i-k;lamda501=mifa(A)+lamda1;for(i=0;i=500;i+)A2i=A2i+k;lamdas=fanmifa(A);FILE *fp=fopen(result.txt,w);fprintf(fp,1=%.12en,lamda1);fprintf(fp,501=%.12en,lamda501);fprintf(fp,s=%.12enn,lamdas);fprintf(fp,t要求接近的值ttt实际求得的特征值n);for(i=1;i=39;i+) /反幂法求得与给定值接近的特征值p=lamda1+(i+1)*(lamda501-lamda1)/40;for(j=0;j=500;j+)A2j=A2j-p;q=fanmifa(A)+p;for(j=0;j=500;j+)A2j=A2j+p;fprintf(fp,%d: %.12e i%d: %.12en,i,p,i,q);double cond=fabs(mifa(A)/fanmifa(A);double det=Det(A);fprintf(fp,ncond(A)=%.12en,cond);fprintf(fp,ndetA=%.12en,det);三、程序运行结果1=-1.069936345952e+001501=9.722283648681e+000s=-5.557989086521e-003要求接近的值实际求得的特征值1: -9.678281104107e+000 i1: -9.585702058251e+0002: -9.167739926402e+000 i2: -9.172672423948e+0003: -8.657198748697e+000 i3: -8.652284007885e+0004: -8.146657570993e+000 i4: -8.093482780052e+0005: -7.636116393288e+000 i5: -7.659405420574e+0006: -7.125575215583e+000 i6: -7.119684646576e+0007: -6.615034037878e+000 i7: -6.611764337314e+0008: -6.104492860173e+000 i8: -6.066103126985e+0009: -5.593951682468e+000 i9: -5.585101045269e+00010: -5.083410504763e+000 i10: -5.114083539196e+00011: -4.572869327058e+000 i11: -4.578872177367e+00012: -4.062328149353e+000 i12: -4.096473385708e+00013: -3.551786971648e+000 i13: -3.554211216942e+00014: -3.041245793943e+000 i14: -3.041090018133e+00015: -2.530704616238e+000 i15: -2.533970334136e+00016: -2.020163438533e+000 i16: -2.003230401311e+00017: -1.509622260828e+000 i17: -1.503557606947e+00018: -9.990810831232e-001 i18: -9.935585987809e-00119: -4.885399054182e-001 i19: -4.870426734583e-00120: 2.200127228676e-002 i20: 2.231736249587e-00221: 5.325424499917e-001 i21: 5.324174742068e-00122: 1.043083627697e+000 i22: 1.052898964020e+00023: 1.553624805402e+000 i23: 1.589445977158e+00024: 2.064165983107e+000 i24: 2.060330427561e+00025: 2.574707160812e+000 i25: 2.558075576223e+00026: 3.085248338516e+000 i26: 3.080240508465e+00027: 3.595789516221e+000 i27: 3.613620874136e+00028: 4.106330693926e+000 i28: 4.091378506834e+00029: 4.616871871631e+000 i29: 4.603035354280e+00030: 5.127413049336e+000 i30: 5.132924284378e+00031: 5.637954227041e+000 i31: 5.594906275501e+00032: 6.148495404746e+000 i32: 6.080933498348e+00033: 6.659036582451e+000 i33: 6.680354121496e+00034: 7.169577760156e+000 i34: 7.293878467852e+00035: 7.680118937861e+000 i35: 7.717111851857e+00036: 8.190660115566e+000 i36: 8.225220016407e+00037: 8.701201293271e+000 i37: 8.648665837870e+00038: 9.211742470976e+000 i38: 9.254