数值分析课程设计大作业

数值分析大作业三四五六七数值分析大作业三四五六七Document number【SA80SAB-SAA9SYT-SAATC-SA6UT-SA18】大作业三1. 给定初值0x 及容许误差，编制牛顿法解方程f (x )=0的通用程序. 解：Matlab 程序如下：函数m 文件：fu.mfunction Fu=fu(x)Fu=x^3/3-x;end函数m 文件：dfu.mfunction Fu=dfu(x)Fu=x^2-1;end用Newton 法求根的通用程序Newton.mclear;x0=input('请输入初值x0:');ep=input('请输入容许误差:');flag=1;while flag==1x1=x0-fu(x0)/dfu(x0);if abs(x1-x0)<ep< p="">flag=0;endx0=x1;endfprintf('方程的一个近似解为：%f\n',x0);寻找最大δ值的程序：Find.mcleareps=input('请输入搜索精度：');ep=input('请输入容许误差：');flag=1;k=0;x0=0;while flag==1sigma=k*eps;x0=sigma;k=k+1;m=0;flag1=1;while flag1==1 && m<=10^3x1=x0-fu(x0)/dfu(x0);if abs(x1-x0)endm=m+1;x0=x1;endif flag1==1||abs(x0)>=epflag=0;endendfprintf('最大的sigma 值为：%f\n',sigma);2．求下列方程的非零根5130.6651()ln 05130.665114000.0918x x f x x +??=-= ?-解：Matlab 程序为：（1）主程序clearclcformat longx0=765;N=100;errorlim=10^(-5);x=x0-f(x0)/subs(df(),x0);n=1;while n<n< p="">x=x0-f(x0)/subs(df(),x0);if abs(x-x0)>errorlimn=n+1;elsebreak;endx0=x;enddisp(['迭代次数: n=',num2str(n)])disp(['所求非零根: 正根x1=',num2str(x),' 负根x2=',num2str(-x)])（2）子函数非线性函数ffunction y=f(x)y=log((513+0.6651*x)/(513-0.6651*x))-x/(1400*0.0918);end（3）子函数非线性函数的一阶导数dffunction y=df()syms x1y=log((513+0.6651*x1)/(513-0.6651*x1))-x1/(1400*0.0918);y=diff(y);end运行结果如下：迭代次数: n=5所求非零根: 正根x1=767.3861 负根x2=-767.3861大作业四试编写MATLAB 函数实现Newton 插值，要求能输出插值多项式. 对函数21()14f x x=+在区间[-5，5]上实现10次多项式插值.分析：（1）输出插值多项式。

数值分析大作业一一、算法设计方案1、求λ1和λ501的值：思路：采用幂法求出按模最大特征值λmax，该值必为λ1或λ501，若λmax小于0，则λmax=λ1；否则λmax=λ501。

再经过原点平移，使用幂法迭代出矩阵A-λmax I的特征值，此时求出的按模最大特征值即为λ1和λ501的另一个值。

2、求λs的值：采用反幂法求出按模最小的特征值λmin即为λs，其中的方程组采用LU分解法进行求解。

3、求与μk最接近的特征值：对矩阵A采用带原点平移的反幂法求解最小特征值，其中平移量为：μk。

4、A的条件数cond(A)=| λmax/λmin|；5、A的行列式的值：先将A进行LU分解，再求U矩阵对角元素的乘积即为A 行列式的值。

二、源程序#include<iostream>#include<iomanip>#include<math.h>#define N 501#define E 1.0e-12 //定义精度常量#define r 2#define s 2using namespace std;double a[N];double cc[5][N];void init();double mifa();double fmifa();int max(int aa,int bb);int min(int aa,int bb);int max_3(int aa,int bb,int cc);void LU();void main(){double a1,a2,d1,d501=0,ds,det=1,miu[39],lamta,cond;int i,k;init();/*************求λ1和λ501********************/a1=mifa();if(a1<0)d1=a1; //若小于0则表示λ1的值elsed501=a1; //若大于0则表示λ501的值for(i=0;i<N;i++)a[i]=a[i]-a1;a2=mifa()+a1;if(a2<0)d1=a2; //若小于0则表示λ1的值elsed501=a2; //若大于0则表示λ501的值cout<<"λ1="<<setiosflags(ios::scientific)<<setprecision(12)<<d1<<"\t";cout<<"λ501="<<setiosflags(ios::scientific)<<setprecision(12)<<d501<<endl;/**************求λs*****************/init();ds=fmifa();cout<<"λs="<<setiosflags(ios::scientific)<<setprecision(12)<<ds<<endl;/**************求与μk最接近的特征值λik**************/cout<<"与μk最接近的特征值λik:"<<endl;for(k=0;k<39;k++){miu[k]=d1+(k+1)*(d501-d1)/40;init();for(i=0;i<N;i++)a[i]=a[i]-miu[k];lamta=fmifa()+miu[k];cout<<"λi"<<k+1<<"\t\t"<<setiosflags(ios::scientific)<<setprecision(12)<<lamta<<en dl;}/**************求A的条件数**************/cout<<"矩阵A的条件式";cond=abs(max(abs(d1),abs(d501))/ds);cout<<"cond="<<setiosflags(ios::scientific)<<setprecision(12)<<cond<<endl;/**************求A的行列式**************/cout<<"矩阵A的行列式";init();LU();for(i=0;i<N;i++){det*=cc[2][i];}cout<<"det="<<setiosflags(ios::scientific)<<setprecision(12)<<det<<endl;system("pause");}/**************初始化函数，给a[N]赋值*************/void init(){int i;for(i=1;i<=501;i++)a[i-1]=(1.64-0.024*i)*sin((double)(0.2*i))-0.64*exp((double)(0.1/i)); }/**************幂法求最大绝对特征值**************/double mifa(){int i,k=0;double u[N],y[N]={0},b=0.16,c=-0.064,Beta_=0,error;for(i=0;i<501;i++)u[i]=1; //令u[N]=1for(k=1;k<2000;k++) //控制最大迭代次数为2000{/***求y(k-1)***/double sum_u=0,gh_sum_u;for(i=0;i<N;i++){sum_u+=u[i]*u[i]; }gh_sum_u=sqrt(sum_u);for(i=0;i<N;i++){y[i]=u[i]/gh_sum_u;}/****求新的uk****/u[0]=a[0]*y[0]+b*y[1]+c*y[2];u[1]=b*y[0]+a[1]*y[1]+b*y[2]+c*y[3]; //前两列和最后两列单独拿出来求中D间的循环求for(i=2;i<N-2;i++){u[i]=c*y[i-2]+b*y[i-1]+a[i]*y[i]+b*y[i+1]+c*y[i+2];}u[N-2]=c*y[N-4]+b*y[N-3]+a[N-2]*y[N-2]+b*y[N-1];u[N-1]=c*y[N-3]+b*y[N-2]+a[N-1]*y[N-1];/***求beta***/double Beta=0;for(i=0;i<N;i++){Beta+=y[i]*u[i];}//cout<<"Beta"<<k<<"="<<Beta<<"\t"; 输出每次迭代的beta /***求误差***/error=abs(Beta-Beta_)/abs(Beta);if(error<=E) //若迭代误差在精度水平内则可以停止迭代{return Beta;} //控制显示位数Beta_=Beta; //第个eta的值都要保存下来，为了与后个值进行误差计算 }if(k==2000){cout<<"error"<<endl;return 0;} //若在最大迭代次数范围内都不能满足精度要求说明不收敛}/**************反幂法求最小绝对特¬征值**************/double fmifa(){int i,k,t;double u[N],y[N]={0},yy[N]={0},b=0.16,c=-0.064,Beta_=0,error;for(i=0;i<501;i++)u[i]=1; //令u[N]=1for(k=1;k<2000;k++){double sum_u=0,gh_sum_u;for(i=0;i<N;i++){sum_u+=u[i]*u[i]; }gh_sum_u=sqrt(sum_u);for(i=0;i<N;i++){y[i]=u[i]/gh_sum_u;yy[i]=y[i]; //用重新赋值，避免求解方程组的时候改变y的值}/****LU分解法解方程组Au=y，求新的***/LU();for(i=2;i<=N;i++){double temp_b=0;for(t=max(1,i-r);t<=i-1;t++)temp_b+=cc[i-t+s][t-1]*yy[t-1];yy[i-1]=yy[i-1]-temp_b;}u[N-1]=yy[N-1]/cc[s][N-1];for(i=N-1;i>=1;i--){double temp_u=0;for(t=i+1;t<=min(i+s,N);t++)temp_u+=cc[i-t+s][t-1]*u[t-1];u[i-1]=(yy[i-1]-temp_u)/cc[s][i-1];}double Beta=0;for(i=0;i<N;i++){Beta+=y[i]*u[i];}error=abs(Beta-Beta_)/abs(Beta);if(error<=E){return (1/Beta);}Beta_=Beta;}if(k==2000){cout<<"error"<<endl;return 0;} }/**************求两数最大值的子程序**************/int max(int aa,int bb){return(aa>bb?aa:bb);}/**************求两数最小值的子程序**************/int min(int aa,int bb){return(aa<bb?aa:bb);}/**************求三数最大值的子程序**************/int max_3(int aa,int bb,int cc){ int tt;if(aa>bb)tt=aa;else tt=bb;if(tt<cc) tt=cc;return(tt);}/**************LU分解**************/void LU(){int i,j,k,t;double b=0.16,c=-0.064;/**赋值压缩后矩阵cc[5][501]**/for(i=2;i<N;i++)cc[0][i]=c;for(i=1;i<N;i++)cc[1][i]=b;for(i=0;i<N;i++)cc[2][i]=a[i];for(i=0;i<N-1;i++)cc[3][i]=b;for(i=0;i<N-2;i++)cc[4][i]=c;for(k=1;k<=N;k++){for(j=k;j<=min(k+s,N);j++){double temp=0;for(t=max_3(1,k-r,j-s);t<=k-1;t++)temp+=cc[k-t+s][t-1]*cc[t-j+s][j-1];cc[k-j+s][j-1]=cc[k-j+s][j-1]-temp;}//if(k<500){for(i=k+1;i<=min(k+r,N);i++){double temp2=0;for(t=max_3(1,i-r,k-s);t<=k-1;t++)temp2+=cc[i-t+s][t-1]*cc[t-k+s][k-1];cc[i-k+s][k-1]=(cc[i-k+s][k-1]-temp2)/cc[s][k-1];}}}}三、程序结果。

数值分析上机作业（一）一、算法的设计方案1、幂法求解λ1、λ501幂法主要用于计算矩阵的按模最大的特征值和相应的特征向量，即对于|λ1|≥|λ2|≥.....≥|λn|可以采用幂法直接求出λ1，但在本题中λ1≤λ2≤……≤λ501，我们无法判断按模最大的特征值。

但是由矩阵A的特征值条件可知|λ1|和|λ501|之间必然有一个是最大的，通过对矩阵A使用幂法迭代一定次数后得到满足精度ε=10−12的特征值λ0，然后在对矩阵A做如下的平移：B=A-λ0I由线性代数(A-PI)x=(λ-p)x可得矩阵B的特征值为：λ1-λ0、λ2-λ0…….λ501-λ0。

对B矩阵采用幂法求出B矩阵按模最大的特征值为λ∗=λ501-λ0，所以λ501=λ∗+λ0，比较λ0与λ501的大小，若λ0>λ501则λ1=λ501，λ501=λ0；若λ0<λ501，则令t=λ501，λ1=λ0，λ501=t。

求矩阵M按模最大的特征值λ的具体算法如下：任取非零向量u0∈R nηk−1=u T(k−1)∗u k−1y k−1=u k−1ηk−1u k=Ay k−1βk=y Tk−1u k(k=1,2,3……)当|βk−βk−1||βk|≤ε=10−12时，迭终终止，并且令λ1=βk2、反幂法计算λs和λik由已知条件可知λs是矩阵A 按模最小的特征值，可以应用反幂法直接求解出λs。

使用带偏移量的反幂法求解λik，其中偏移量为μk=λ1+kλ501−λ140(k=1,2,3…39)，构造矩阵C=A-μk I，矩阵C的特征值为λik−μk，对矩阵C使用反幂法求得按模最小特征值λ0，则有λik=1λ0+μk。

求解矩阵M按模最小特征值的具体算法如下：任取非零向量u 0∈R n ηk−1= u T (k−1)∗u k−1y k−1=u k−1ηk−1 Au k =y k−1βk =y T k−1u k (k=1,2,3……)在反幂法中每一次迭代都要求解线性方程组Au k =y k−1，当K 足够大时，取λn =1βk 。

《数值分析》大作业四一、算法设计方案：复化梯形积分法，选取步长为1/500=0.002，迭代误差控制在E ≤1.0e-10①复化梯形积分法：11()[()()2()]2n bak hf x dx f a f b f a kh -=⎰≈+++∑，截断误差为：322()''()''(),[,]1212T b a b a R f h f a b n ηηη--=-=-∈其中。

复化Simpson 积分法，选取步长为1/50=0.02，迭代误差控制在E ≤1.0e-10②Simpson 积分法：121211()[()()4()2()]3m m bi i a i i hf x dx f a f b f x f x --==≈+++∑∑⎰， 截断误差为：4(4)(),[,]180s b a R h f a b ηη-=-∈。

③Guass积分法选用Gauss-Legendre 求积公式：111()()ni i i f x dx A f x -=≈∑⎰截断误差为：R= ()()n 2n 422n!2×(2[2!]2n 1f n n ⨯（2）η（））＋ η∈(1,1)。

选择9个节点：-0.9681602395,-0.8360311073,-0.6133714327,-0.3242534234,0,0.3242534234,0.6133714327,0.8360311073,0.9681602395， 对应的求积系数依次为：0.0812743884,0.1806481607,0.2606106964,0.3123470770,0.3302393550,0.3123470770,0.2606106964,0.1806481607,0.0812743884。

二、程序源代码：#include<stdio.h>#include<math.h>#include<stdlib.h>#define E 1.0e-10/****定义函数g和K*****/double g(double a){double b;b=exp(4*a)+(exp(a+4)-exp(-a-4))/(a+4);return b;}double K(double a,double b){double c;c=exp(a*b);return c;}/******复化梯形法******/void Tixing( ){double u[1001],x[1001],h,c[1001],e;int i,j,k;FILE *fp;fp=fopen("f:/result0. xls ","w");h=1.0/1500;for(i=0;i<3001;i++){x[i]=i*h-1;u[i]=g(x[i]);}for(k=0;k<100;k++){e=0;for(i=0;i<1001;i++){for(j=1,c[i]=0;j<N-1;j++)c[i]+=K(x[i],x[j])*u[j];u[i]=g(x[i])-h*c[i]-h/2*(K(x[i],x[0])*u[0]+K(x[i],x[N-1])*u[N-1]);e+=h*(exp(4*x[i])-u[i])*(exp(4*x[i])-u[i]);}if(e<=E) break;}for(i=0;i<1001;i++)fprintf(fp,"%.12lf,%.12lf\n",x[i],u[i]);fclose(fp);}/******复化Simpson法******/void simpson( ){double u[101],x[101],h,c[101],d[101],e;int i,j,k;FILE *fp;fp=fopen("f:/result1.xls","w");h=1.0/50;for(i=0;i<101;i++){x[i]=i*h-1;u[i]=g(x[i]);}for(k=0;k<50;k++){e=0;for(i=0;i<101;i++){for(j=1,c[i]=0,d[i]=0;j<51;j++){c[i]+=K(x[i],x[2*j-1])*u[2*j-1];if(j<50)d[i]+=K(x[i],x[2*j])*u[2*j];}u[i]=g(x[i])-4*h/3*c[i]-2*h/3*d[i]-h/3*(K(x[i],x[0])*u[0]+K(x[i],x[M-1])*u[M-1]);e+=h*(exp(4*x[i])-u[i])*(exp(4*x[i])-u[i]);}if(e<=E) break;}for(i=0;i<101;i++)fprintf(fp,"%.12lf,%.12lf\n",x[i],u[i]);fclose(fp);}/******Gauss积分法******/void gauss( ){double x[9]={-0.9681602395,-0.8360311073,-0.6133714327,-0.3242534234,0,\0.3242534234,0.6133714327,0.8360311073,0.9681602395},A[9]={0.0812743884,0.1806481607,0.2606106964,0.3123470770,0.3302393550,\0.3123470770,0.2606106964,0.1806481607,0.0812743884},u[9],c[9],e;int i,j,k;FILE *fp;fp=fopen("f:/result2. xls ","w");for(i=0;i<9;i++)u[i]=g(x[i]);for(k=0;k<50;k++){e=0;for(i=0;i<9;i++){for(j=0,c[i]=0;j<9;j++)c[i]+=A[j]*K(x[i],x[j])*u[j];u[i]=g(x[i])-c[i];e+=A[i]*(exp(4*x[i])-u[i])*(exp(4*x[i])-u[i]);}if(e<=E) break;}for(i=0;i<9;i++)fprintf(fp,"%.12lf,%.12lf\n",x[i],u[i]);fclose(fp);}/******主函数******/main(){Tixing ( );Simpson( );Gauss( );return 0;}三、运算结果复化梯形数据-10.018323-0.920.02523-0.9980.018471-0.9180.025433-0.9960.018619-0.9160.025637-0.9940.018768-0.9140.025843-0.9920.018919-0.9120.026051-0.990.019071-0.910.02626-0.9880.019224-0.9080.026471-0.9860.019378-0.9060.026683-0.9840.019534-0.9040.026897-0.9820.019691-0.9020.027113-0.980.019849-0.90.027331-0.9780.020008-0.8980.02755-0.9760.020169-0.8960.027772-0.9740.020331-0.8940.027995-0.9720.020494-0.8920.028219-0.970.020658-0.890.028446-0.9680.020824-0.8880.028674-0.9660.020992-0.8860.028905-0.9640.02116-0.8840.029137-0.9620.02133-0.8820.029371-0.960.021501-0.880.029607-0.9580.021674-0.8780.029844-0.9560.021848-0.8760.030084-0.9540.022023-0.8740.030326-0.9520.0222-0.8720.030569-0.950.022378-0.870.030815-0.9480.022558-0.8680.031062-0.9460.022739-0.8660.031311-0.9440.022922-0.8640.031563-0.9420.023106-0.8620.031816-0.940.023291-0.860.032072-0.9380.023478-0.8580.032329-0.9360.023667-0.8560.032589-0.9340.023857-0.8540.032851-0.9320.024048-0.8520.033114-0.930.024241-0.850.03338-0.9280.024436-0.8480.033648-0.9260.024632-0.8460.033918-0.9240.02483-0.8440.034191-0.9220.025029-0.8420.034465-0.840.034742-0.760.047841-0.8380.035021-0.7580.048225-0.8360.035302-0.7560.048613 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0.74219.452890.82226.78914 0.74419.609140.82427.00431 0.74619.766640.82627.22121 0.74819.925410.82827.43985 0.7520.085450.8327.66025 0.75220.246780.83227.88242 0.75420.409410.83428.10638 0.75620.573340.83628.33213 0.75820.738580.83828.5597 0.7620.905160.8428.78909 0.76221.073070.84229.02033 0.76421.242330.84429.25342 0.76621.412950.84629.48839 0.76821.584940.84829.72524 0.7721.758310.8529.964 0.77221.933080.85230.20467 0.77422.109250.85430.44728 0.77622.286830.85630.69184 0.77822.465840.85830.93836 0.7822.646290.8631.18686 0.78222.828190.86231.43735 0.78423.011550.86431.68986 0.78623.196380.86631.9444 0.78823.382690.86832.20098 0.7923.570510.8732.45962 0.79223.759830.87232.72034 0.79423.950670.87432.98315 0.79624.143040.87633.24807 0.79824.336960.87833.51513 0.824.532440.8833.78432 0.80224.729490.88234.05568 0.80424.928110.88434.32922 0.80625.128340.88634.60496 0.80825.330170.88834.882910.8935.163090.94643.99154 0.89235.445520.94844.344880.89435.730220.9544.701070.89636.017210.95245.060110.89836.306510.95445.422040.936.598120.95645.786870.90236.892080.95846.154630.90437.188410.9646.525350.90637.487110.96246.899050.90837.788210.96447.275750.9138.091730.96647.655470.91238.397680.96848.038240.91438.70610.9748.424090.91639.016990.97248.813040.91839.330380.97449.205110.9239.646280.97649.600330.92239.964720.97849.998720.92440.285720.9850.400320.92640.60930.98250.805140.92840.935480.98451.213210.9341.264280.98651.624560.93241.595720.98852.039210.93441.929820.9952.45720.93642.26660.99252.878540.93842.606090.99453.303270.9442.948310.99653.73140.94243.293270.99854.162980.94443.64101154.59802复化Simpson数据：-1 0.018319929 -0.34 0.256658088 0.32 3.596641805 -0.98 0.0198445 -0.32 0.278035042 0.34 3.896195298-0.96 0.021494322 -0.3 0.301192133 0.36 4.220697765-0.94 0.023283225 -0.28 0.326278124 0.38 4.572227037-0.92 0.025220379 -0.26 0.353453177 0.4 4.95303418-0.9 0.027320224 -0.24 0.382891765 0.42 5.365557596-0.88 0.029594431 -0.22 0.41478194 0.44 5.812438891-0.86 0.032059069 -0.16 0.527292277 0.54 8.671138204-0.84 0.034728638 -0.14 0.571209036 0.56 9.39333156-0.82 0.037621263 -0.12 0.61878367 0.58 10.17567433-0.8 0.040754615 -0.1 0.670320427 0.6 11.02317608-0.78 0.044149394 -0.08 0.726149698 0.62 11.94126383-0.76 0.047826844 -0.06 0.78662861 0.64 12.93581634-0.74 0.051810827 -0.04 0.85214479 0.66 14.01320231-0.72 0.056126648 -0.02 0.92311742 0.68 15.1803205-0.7 0.060802006 0 1.0000013 0.7 16.44464467 -0.68 0.065866854 0.02 1.083288424 0.72 17.81427057 -0.66 0.071353499 0.04 1.173512427 0.74 19.29796874 -0.64 0.077297255 0.06 1.271250748 0.76 20.90523965 -0.62 0.083735917 0.08 1.377129533 0.78 22.64637562 -0.6 0.090711017 0.1 1.491826493 0.8 24.53252554 -0.58 0.098266855 0.12 1.616076341 0.82 26.57576756 -0.56 0.106452202 0.14 1.750674449 0.84 28.78918506 -0.54 0.11531904 0.16 1.896482943 0.86 31.18695183 -0.52 0.12492459 0.18 2.054435268 0.88 33.78442141 -0.5 0.135329888 0.2 2.225543071 0.9 36.59822683 -0.48 0.14660204 0.22 2.410901825 0.92 39.64638571 -0.46 0.158812728 0.24 2.611698647 0.94 42.94841704 -0.44 0.17204064 0.26 2.829219145 0.96 46.52546475 -0.42 0.18636997 0.28 3.064856356 0.98 50.40043451 -0.4 0.201892977 0.3 3.320119013 1 54.59813904 -0.38 0.218708553 0.46 6.296539601-0.36 0.236924875 0.48 6.820959636-0.2 0.449328351 0.5 7.389057081-0.18 0.486751777 0.52 8.0044696750102030405060四、讨论①在满足相同精度要求的情况下复化梯形积分法比复化Simpson 积分法计算所需节点数多，计算量大。

数值分析大作业二一、算法设计方案(一)算法流程1.将要求解的矩阵进行Householder变换，进行拟上三角化，并输出拟上三角化的结果；2.将拟上三角化后的矩阵进行带双步位移的QR分解(最大迭代次数1000次)，逐个求出特征值，输出QR法结束后得到的Q、R、RQ阵，输出求出的特征值（用实部和虚部表示）3.对于求出来的实特征值，使用带原点平移的反幂法求出对应的特征向量，输出这些特征向量。

（二）程序设计流程1. 定义精度和最大迭代次数；2. 使用容器vector进行编程（方便增减元素），使用传引用调用（提高执行效率）；3. 将各个步骤用到的数学算法封装成函数，方便调用。

具体需要的函数如下：double VectMod(vector< double > &p)：求向量的模vector< double > NumbMultiVect(vector< double > &vect, double a)：数乘向量double VectMultiVect(vector< double > &y, vector< double > &u)：求两个向量的积vector< double > ConveArraMultiVect(vector< vector< double > > &B, vector<double > &u)：矩阵的转置乘向量vector< double > ArraMultiVect(vector< vector< double > > &B, vector< double > &u)：矩阵乘向量vector< vector< double > > VectMultiCovVect(vector< double > &a, vector< double >&b)：向量乘向量转置得到矩阵void ArraSubs(vector< vector< double > > &C, vector< vector < double > > D) ：两个矩阵相减Vector< double > VectSubs(vector< double > &a, vector< double > &b)：两个向量相减vector<vector<double>> GetM(vector<vector<double>> &A)：求解矩阵M （用于双部位移QR迭代用）double GetMode(vector < vector < double > > &B, const int r)：求解矩阵B的r列向量的模double GetModeH(vector < vector < double > > &B, const int r)：求解矩阵B的第r列向量的模，用于拟对角化vector< vector< double > > NumbMultiArra(vector< vector< double > > &D, double a)：一个实数乘矩阵bool IsBirZeroH(vector< vector< double > > &B, const int r)：判断B[i][r]对角线下是否为零void GausElim(vector< vector< double > > a)：列主元高斯消元法求齐次方程解向量void Stop(vector< vector< double > > &Ar)：停止，结束程序void SolutS1S2(complex< double > &s1, complex< double > &s2, vector< vector<double > > &A)：求解二阶子阵的特征值s1，s2；void Save2(complex< double > &s1, complex< double > &s2)：保存两个特征值void Save1(complex< double > &s)：保存一个特征值void JudgemBelow2(vector< vector< double > > &A, vector< vector< double > > Abk)：对于m == 1 及m == 0 的处理void Hessenberg(vector< vector< double > > &A)：矩阵拟上三角化void QRMethod(vector< vector< double > > A)：矩阵QR分解void CalculatAk(vector< vector < double > > &Ak)：带双步位移QR迭代法二、源程序#include "stdafx.h"#include <vector>#include <iostream>#include <math.h>#include <complex>#include <fstream>using namespace std;const double epsion = 1e-12;const int L = 1000;int m,n;int k = 1;vector< vector< double > > I;vector< complex< double > > Lambda;///////////////////////////以下为自定义的算法流程中用到的函数double VectMod(vector< double > &p) //求向量的模{double value = 0.0;vector< double >::size_type i,j;j = p.size();for (i=1; i<j; i++){value += p[i] * p[i];}value = sqrt(value);return value;}vector< double > NumbMultiVect(vector< double > &vect, double a) //数乘向量{int j = vect.size();vector< double > b(j, 0);for (int i=1; i<j; i++){b[i] = a * vect[i];}return b;}double VectMultiVect(vector< double > &y, vector< double > &u)//两个向量相乘{vector< double >::size_type a = y.size();double value = 0;for (vector< double >::size_type i=1; i<a; i++){value += y[i] * u[i];}return value;}vector< double > ConveArraMultiVect(vector< vector< double > > &B, vector< double > &u)//矩阵的转置乘向量{int a = B.size();int b = u.size();vector< double > vec(a, 0);if (a != b){cerr << "Array and Vector not match in size!";}else{for (int i=1; i<a; i++){for (int j=1; j<a; j++){vec[i] += B[j][i] * u[j];}}return vec;}}vector< double > ArraMultiVect(vector< vector< double > > &B,vector< double > &u) //矩阵乘向量{int a = B.size();int b = u.size();vector< double > vec(a, 0);if (a != b){cerr << "Array and Vector not match in size!";}else{for (int i=1; i<a; i++){for (int j=1; j<a; j++){vec[i] += B[i][j] * u[j];}}return vec;}}vector< vector<double> > GetM(vector< vector <double> > &A){int a = A.size();double s = A[a-2][a-2] + A[a-1][a-1];double t = A[a-2][a-2] * A[a-1][a-1] - A[a-1][a-2] * A[a-2][a-1];vector<vector<double>> D(a, vector< double >(a, 0));for (int i=1; i<a; i++){{double sum = 0;for (int k=1; k<a; k++){sum += A[i][k] * A[k][j];}D[i][j] = sum - s * A[i][j] + t *(i==j ? 1.0 : 0);}}return D;}double GetMode(vector < vector < double > > &B, const int r){double value = 0;int a = B.size();for (int k=r; k<a; k++){value += B[k][r] * B[k][r];}value = sqrt(value);return value;}double GetModeH(vector < vector < double > > &B, const int r){double value = 0;int a = B.size();for (int k=r+1; k<a; k++){value += B[k][r] * B[k][r];}value = sqrt(value);return value;}vector< vector< double > > NumbMultiArra(vector< vector< double > > &D, double a)//数乘//向量{int b = D.size();vector< vector< double > > U(b, vector< double >(b, 0));for (int i=1; i<b; i++){{U[i][j] = a *D[i][j];}}return U;}bool IsBirZero(vector< vector< double > > &B, const int r){bool b = true;int a = B.size();for (int i=r+1; i<a; i++){if(abs(B[i][r]) > epsion){b = false;}}return b;}bool IsBirZeroH(vector< vector< double > > &B, const int r){bool b = true;int a = B.size();for (int i=r+2; i<a; i++){if(abs(B[i][r]) > epsion){b = false;}}return b;}vector< vector< double > > VectMultiCovVect(vector< double > &a,vector< double > &b)//向量乘向量的转置得到矩阵{int s1 = a.size();int s2 = b.size();if (s1 != s2){cerr << "Vectors not match in size ! ";}else{vector< vector< double > > U(s1, vector< double >(s1, 0));for (int i=1; i<s1; i++){for (int j=1; j<s1; j++){U[i][j] = a[i] * b[j];}}return U;}}void ArraSubs(vector< vector< double > > &C,vector< vector < double > > D){int a = C.size();int b = D.size();int c = C[0].size();int d = D[0].size();if (a!=b || c!=d){cerr << "Vectors not match in size !";}else{for (int i=1; i<a; i++){for (int j=1; j<a; j++){C[i][j] -= D[i][j];}}}}vector< double > VectSubs(vector< double > &a,vector< double > &b){int s1 = a.size();int s2 = b.size();if(s1 != s2){cerr << "Vectors not match in size !";}else{vector< double > value(s1,0);for (int i=1;i<s1;i++){value[i] = a[i] - b[i];}return value;}}void GausElim(vector< vector< double > > a) //高斯消元{int sz = a.size();vector< int > fx, ufx,ufxp, record;vector< double > x(sz, 0);vector< vector< double > > ret;//*****消元过程********for (int k = 1; k < sz-1; k++){double max = a[k][k];int p=0;for (int i=k+1; i<sz; i++){if (abs(a[i][k]) > abs(max)){p =i;max = a[i][k];}}//选出主元行if (abs(max) >= epsion){if (p != 0){for (int j=k; j<sz; j++){double temp = a[k][j];a[k][j] = a[p][j];a[p][j] = temp;}//交换主元行}for (int i=k+1; i<sz; i++){double tt = a[i][k] / a[k][k];for (int j=k+1; j<sz; j++){a[i][j] = a[i][j] - tt * a[k][j];}}}// end of if (abs(max) >= epsion)}// end of for (int k = 1; k < sz; k++)for (int i=1; i<sz; i++){int p=0;for (int j=1; j<=i;j++){if(abs(a[j][i]) >= 10*epsion)//不为零{p=j;}}record.push_back(p);}for (int i=1; i<sz; i++){int p=0;for (int j=sz-2; j>=0; j--){if (record[j] == i){p=j + 1;}}if (p != 0){ufxp.push_back(p);ufx.push_back(i);}}for (int i=1; i<sz; i++){int p = 0;for (int j=0; j < ufxp.size(); j++){if (ufxp[j] == i){p = 1;}}if (p == 0){fx.push_back(i);}}//end of for (int i=1; i<sz; i++)int c = fx.size();//***************************************************** for (int i=0; i<c; i++){for (int j=0; j<c; j++){if (i == j){x[fx.at(j)] = 1.0 + epsion;}else{x[fx.at(j)] = 0;}}int b = ufxp.size();for (int s=b-1; s>=0; s--){double temp=0;for(int t=ufxp.at(s)+1; t<sz; t++){temp += x[t] * a[ufx.at(s)][t];}x[ufxp.at(s)] = (0 - temp) / a[ufx.at(s)][ufxp.at(s)];}ret.push_back(x);}//end of for (int i=0; i<c; i++)//******************************************************* int sz1 = ret.size();int sz2 = ret[0].size();ofstream result2("CharactVector .txt", ios::app);result2.precision(12);for (int i=0; i<sz1; i++){for (int j=1;j<sz2; j++){result2 << scientific << ret[i][j] << endl;}result2 << endl << endl;}result2 << "下一个特征值的特征向量！" << endl;}void Stop(vector< vector< double > > &Ar){int a = Lambda.size();for (int i=0; i<a; i++){if (abs(Lambda[i].imag())<=epsion){vector< vector< double > > An=Ar;ArraSubs(An,NumbMultiArra(I,Lambda[i].real()));GausElim(An);}}ofstream result("result.txt");result.precision(12);vector<complex< double > >::iterator i,j;j = Lambda.end();for (i = Lambda.begin(); i<j; i++){result << scientific <<(*i) << endl;}exit(0);}void SolutS1S2(complex< double > &s1, complex< double > &s2,vector< vector< double > > &A){double s = A[m-1][m-1] + A[m][m];double t = A[m-1][m-1] * A[m][m] - A[m][m-1] * A[m-1][m];double det = s *s - 4.0 * t;if (det > 0){s1.imag(0);s1.real ((s + sqrt(det)) / 2);s2.imag (0);s2.real((s - sqrt(det)) / 2);}else{s1.imag(sqrt(0 - det) / 2);s1.real(s / 2);s2.imag(0 - sqrt(0 - det) / 2);s2.real(s / 2);}}void Save2(complex< double > &s1, complex< double > &s2)//存储特征值s1，s2 {Lambda.push_back(s1);Lambda.push_back(s2);}void Save1(complex< double > &s){Lambda.push_back(s);}void JudgemBelow2(vector< vector< double > > &A,vector< vector< double > > Abk){if (m == 1){complex< double > lbd;lbd.imag(0);lbd.real(A[1][1]);Save1(lbd);Stop(Abk);}else if (m == 0){Stop(Abk);}}//拟上三角化void Hessenberg(vector< vector< double > > &A){int a = A.size();vector< double > u(a,0);vector< double > p(a,0);vector< double > q(a,0);vector< double > w(a,0);double d, c, h, t;for (int r=1; r<a-2; r++){if (!IsBirZeroH(A, r)){d = GetModeH(A,r);c = (A[r+1][r] > 0) ?(0-d) : d;h = c * c - c * A[r+1][r];for (int j=1; j<a; j++){if (j < r+1){u[j] = 0;}else if (j == r+1){u[j] = A[j][r] - c;}else{u[j] = A[j][r];}}// end of for (int j=1; j<=m; j++)p = NumbMultiVect(ConveArraMultiVect(A, u), 1 / h);q = NumbMultiVect(ArraMultiVect(A, u), 1 / h);t = VectMultiVect(p, u) / h;w = VectSubs(q, NumbMultiVect(u, t));ArraSubs(A, VectMultiCovVect(w, u));ArraSubs(A, VectMultiCovVect(u, p));}// end of if (!IsBirZero(B, r))}// end of for (int r=1; r<m; r++)ofstream result("NISHANGDANJIAO.txt");result.precision(12);for (int i=1; i<a; i++){for (int j=1; j<a; j++){result << scientific << A[i][j] << " ";}result << endl;}}void QRMethod(vector< vector< double > > A){int a = A.size();vector< vector< double > > Q(a,vector < double > (a,0));for (int i=1; i<a; i++){Q[i][i] = 1.0;}vector< vector< double > > RQ(A);//vector< double > u(a,0);vector< double > v(a,0);vector< double > p(a,0);vector< double > q(a,0);vector< double > w(a,0);vector< double > l(a,0);double d, c, h, t;for (int r=1; r<a-1; r++){if (!IsBirZero(A, r)){d = GetMode(A,r);c = (A[r][r] > 0) ?(0-d) : d;h = c * c - c * A[r][r];for (int j=1; j<a; j++){if (j < r){u[j] = 0;}else if (j == r){u[j] = A[j][j] - c;}else{u[j] = A[j][r];}}// end of for (int j=1; j<=m; j++)l = ArraMultiVect(Q, u);ArraSubs(Q, VectMultiCovVect(l,NumbMultiVect(u, 1/h)));v = NumbMultiVect(ConveArraMultiVect(A, u), 1 / h);ArraSubs(A, VectMultiCovVect(u, v));p = NumbMultiVect(ConveArraMultiVect(RQ, u), 1 / h);q = NumbMultiVect(ArraMultiVect(RQ, u), 1 / h);t = VectMultiVect(p, u) / h;w = VectSubs(q, NumbMultiVect(u, t));ArraSubs(RQ, VectMultiCovVect(w, u));ArraSubs(RQ, VectMultiCovVect(u, p));}// end of if (!IsBirZero(B, r))}// end of for (int r=1; r<m; r++)ofstream result("QR.txt");result.precision(12);for (int i=1; i<a; i++){for (int j=1; j<a; j++){result << scientific << Q[i][j] << " ";}result << endl;}result << endl << endl;for (int i=1; i<a; i++){for (int j=1; j<a; j++){result << scientific << A[i][j] << " ";}result << endl;}result << endl << endl;for (int i=1; i<a; i++){for (int j=1; j<a; j++){result << scientific << RQ[i][j] << " ";}result << endl;}}void CalculatAk(vector< vector < double > > &Ak){int a = Ak.size();vector< vector < double > > B(a,vector < double > (a,0));vector< double > u(a,0);vector< double > v(a,0);vector< double > p(a,0);vector< double > q(a,0);vector< double > w(a,0);double d, c, h, t;B = GetM(Ak);for (int r=1; r<a-1; r++){if (!IsBirZero(B, r)){d = GetMode(B,r);c = (B[r][r] > 0) ?(0-d) : d;h = c * c - c * B[r][r];for (int j=1; j<a; j++){if (j < r){u[j] = 0;}else if (j == r){u[j] = B[j][j] - c;}else{u[j] = B[j][r];}}// end of for (int j=1; j<=m; j++)v = NumbMultiVect(ConveArraMultiVect(B, u), 1 / h);ArraSubs(B, VectMultiCovVect(u, v));p = NumbMultiVect(ConveArraMultiVect(Ak, u), 1 / h);q = NumbMultiVect(ArraMultiVect(Ak, u), 1 / h);t = VectMultiVect(p, u) / h;w = VectSubs(q, NumbMultiVect(u, t));ArraSubs(Ak, VectMultiCovVect(w, u));ArraSubs(Ak, VectMultiCovVect(u, p));}// end of if (!IsBirZero(B, r))}// end of for (int r=1; r<m; r++)}int _tmain(int argc, _TCHAR* argv[]){vector< vector< double > > A(11, vector< double >(11,0));vector< vector< double > > Abk;I=A;for (int i=1; i<11; i++){vector< double > temp;for (int j=1; j<11; j++){if(i != j){A[i][j] = sin(0.5 * i + 0.2 * j);I[i][j] = 0;}else{A[i][j] = 1.5 * cos(i + 1.2 *j);I[i][j] = 1.0;}}}Abk = A;n = A.size() - 1;m = n;//初始化问题Hessenberg(A);QRMethod(A);while (1){if (abs(A[m][m-1]) <= epsion){complex< double > lbdm;lbdm.imag(0);lbdm.real(A[m][m]);Save1(lbdm);m -= 1;//对A进行降维处理！！！！A.pop_back();int a = A.size();for (int i=0; i<a; i++){A[i].pop_back();}JudgemBelow2(A, Abk);}else{complex< double > va1, va2;SolutS1S2(va1, va2, A);if (m == 2){Save2(va1,va2);Stop(Abk);}//end of if (m == 2)if ( abs(A[m-1][m-2]) <= epsion){Save2(va1,va2);m = m - 2;//矩阵降维A.pop_back();A.pop_back();int a = A.size();for (int i=0; i<a; i++){A[i].pop_back();A[i].pop_back();}JudgemBelow2(A, Abk);}else{if (k == L){cerr << "Stop without solution";exit(-1);}else{CalculatAk(A);k += 1;}}}// end of if (abs(A[m][m-1]) >= epsion)}}三、实验结果（1）A经过拟上三角化后得到的矩阵-8.827516758830e-001 -9.933136491826e-002 -1.103349285994e+000-7.600443585637e-001 1.549101079914e-001 -1.946591862872e+000-8.782436382928e-002 -9.255889387184e-001 6.032599440534e-0011.518860956469e-001-2.347878362416e+000 2.372370104937e+000 1.819290822208e+0003.237804101546e-001 2.205798440320e-001 2.102692662546e+0001.816138086098e-001 1.278839089990e+000 -6.380578124405e-001-4.154075603804e-0011.0556********e-016 1.728274599968e+000 -1.171467642785e+000-1.243839262699e+000 -6.399758341743e-001 -2.002833079037e+0002.924947206124e-001 -6.412830068395e-001 9.783997621285e-0022.557763574160e-001-5.393383812774e-017 -3.165873865181e-017 -1.291669534130e+000-1.111603513396e+000 1.171346824096e+000 -1.307356030021e+0001.803699177750e-001 -4.246385358369e-001 7.988955239304e-0021.608819928069e-0011.533464996622e-017 5.963321406181e-017 0.000000000000e+0001.560126298527e+000 8.125049397515e-001 4.421756832923e-001-3.588616128137e-002 4.691742313671e-001 -2.736595050092e-001 -7.359334657750e-0021.300562737229e-016 -3.097060010889e-017 0.000000000000e+0000.000000000000e+000 -7.707773755194e-001 -1.583051425742e+000-3.042843176799e-001 2.528712446035e-001 -6.709925401449e-0012.544619929082e-0011.610216724767e-016 -2.211571837369e-016 0.000000000000e+0000.000000000000e+000 6.483933100712e-017 -7.463453456938e-001-2.708365157019e-002 -9.486521893682e-001 1.195871081495e-0011.929265617952e-0021.368550186199e-016 7.151513190805e-017 0.000000000000e+0000.000000000000e+000 -2.522454384246e-017 1.072074718358e-016-7.701801374364e-001 -4.697623990618e-001 4.988259468008e-0011.137691603776e-001-2.780851300718e-017 -6.708630788363e-017 0.000000000000e+0000.000000000000e+000 -3.282796821065e-017 4.879774287475e-0171.854829286220e-016 7.013167092107e-001 1.582180688475e-0013.862594614233e-001-2.124604440055e-017 -1.707979758930e-016 0.000000000000e+0000.000000000000e+000 1.013496750890e-016 -4.153758325241e-0171.222621691887e-016 -5.551115123126e-017 4.843807602783e-0013.992777995177e-001（2）Q-3.519262579534e-001 4.427591982245e-001 -6.955982513607e-0016.486200753651e-002 3.709718861896e-001 1.855847143605e-001-1.628942319628e-002 -1.181053169648e-001 -5.255375383724e-002 -5.486582943568e-002-9.360277287361e-001 -1.664679186545e-001 2.615299548560e-001 -2.438671728934e-002 -1.394774360893e-001 -6.977585391241e-0026.124472142963e-003 4.440505443139e-002 1.975907909728e-0022.062836970533e-002-4.208697095111e-017 -8.810520554692e-001 -3.989762796959e-0013.720308728479e-002 2.127794064090e-001 1.064463557221e-001-9.343171079758e-003 -6.774200464527e-002 -3.014340698675e-002 -3.146955080444e-002-2.150178169911e-017 4.009681353156e-017 -5.371806806439e-001 -1.234945854205e-001 -7.063151608719e-001 -3.533456368496e-0013.101438948264e-002 2.248676491598e-001 1.000601783527e-0011.044622748702e-0016.113458775639e-018 -3.721194648197e-0177.068175586628e-0189.892235468621e-001 -1.239411731211e-001 -6.200358589825e-0025.442272839461e-003 3.945881637235e-002 1.755813350011e-0021.833059462907e-0025.184948268593e-017 -4.198303559531e-017 3.255071298412e-017-1.527665297025e-017 5.323610690264e-001 -6.733900344896e-0015.910581205868e-002 4.285425323867e-001 1.906901343193e-0011.990794495295e-0016.419444583601e-017 4.121668945107e-017 3.096964582901e-017-5.050360323651e-017 -7.078737686239e-017 -6.0597********e-001 -9.165783032818e-002 -6.645586508974e-001 -2.957110877580e-001 -3.0872********e-0015.455993559780e-017 -9.724896332186e-017 3.956603694870e-0171.913857001710e-018 -4.316583456405e-0172.797376665557e-0179.933396625117e-001 -9.690440311939e-002 -4.311990584470e-002-4.501694411183e-002-1.108640877071e-017 4.655237056115e-017 -1.0722********e-017 -9.470236121745e-018 4.277993227986e-017 8.866601870176e-017 -2.516725033065e-016 5.410088006061e-001 -5.817540838226e-001 -6.0734********e-001-8.470152033092e-018 9.650816729410e-017 -1.429362211392e-017 -2.789222052040e-017 -3.690793770141e-017 -8.090765462521e-017 -1.964050295697e-016 -6.772274683437e-017 -7.221591336735e-0016.917269588876e-001（3）R2.508342744917e+000 -2.185646885493e+000 -1.314609070786e+000-3.558787493836e-002 -2.609857850388e-001 -1.283121847090e+000 -1.390878610606e-001 -8.712897972161e-001 3.849367902971e-0013.353802899665e-0012.100627753398e-016 -1.961603277854e+000 2.407523727633e-0017.054714572823e-001 5.957204318279e-001 5.526978774676e-001-3.268209924413e-001 -5.769498668364e-002 2.871129330189e-001 -8.895128754189e-002-3.300197935770e-016 -1.872873410421e-016 2.404534601993e+0001.706758096328e+000 -4.239566704091e-001 3.405332305815e+000-1.050017655852e-001 1.462257102734e+000 -6.684487469283e-001 -4.027*********e-0013.0773********e-017 1.746386351950e-017 0.000000000000e+0001.577122080722e+000 6.399535133956e-001 3.468127872427e-001-5.701786649768e-002 4.014788054433e-001 -2.222476176311e-001 -6.317059236442e-0021.760039865880e-016 9.988285339980e-017 0.000000000000e+0000.000000000000e+000 -1.447846997770e+000 -1.415724007744e+000-2.806139044665e-001 -2.817910521892e-001 -4.611434881851e-0021.996629079956e-0018.804885435596e-017 4.996802050991e-017 0.000000000000e+0000.000000000000e+000 8.831633340975e-017 1.231641451542e+0001.619701003419e-001 1.962638275504e-001 5.350035621760e-001-1.509273424767e-001-7.728357669400e-018 -4.385868928757e-018 0.000000000000e+0000.000000000000e+000 -7.751835246841e-018 -1.279231078565e-017-7.753441914209e-001 -3.464514508821e-001 4.312226803504e-0011.234643696237e-001-5.603391361152e-017 -3.179943413314e-017 0.000000000000e+0000.000000000000e+000 -5.620413613517e-017 -9.274974944455e-0170.000000000000e+000 1.296312940612e+000 -4.288053318338e-0012.737334158165e-001-2.493361499851e-017 -1.414991023727e-017 0.000000000000e+0000.000000000000e+000 -2.500935953597e-017 -4.127119443934e-0170.000000000000e+000 0.000000000000e+000 -6.707396440648e-001-4.842320121884e-001-2.603055667460e-017 -1.477242832192e-017 0.000000000000e+0000.000000000000e+000 -2.610963355436e-017 -4.308689959101e-0170.000000000000e+000 0.000000000000e+000 -1.110223024625e-0167.168323926323e-002（4）RQ1.163074414164e+0002.632670934508e+000 -1.772796003272e+000-8.668899138521e-002 3.300503471047e-001 1.455162371214e+000 -9.730650448593e-001 -4.873031174655e-001 -7.756411630489e-001 -3.249201979113e-0011.836115060851e+000 1.144286420080e-001 -9.880381403133e-0015.589725694767e-001 4.694190067101e-002 -2.978478237007e-0011.617130577649e-002 6.936977702522e-001 1.367670571405e-0011.419099231519e-0025.598200524418e-016 -2.118520153533e+000 -1.876189745783e+000-5.407071940597e-001 1.171538359721e+000 -2.550323020223e+0001.691577936540e+000 1.229951613262e+000 1.387947777212e+0008.667502917242e-001-3.179059303049e-017 -5.261714527977e-017 -8.471995127808e-0014.382910468318e-001 -1.008632199185e+000 -7.959374261495e-0014.769258865577e-001 4.072683083890e-001 4.096390493527e-0013.363378940862e-001-3.514195999269e-016 3.391949386044e-017 -2.160938214545e-016 -1.432244342447e+000 -5.742284908055e-001 1.213151477723e+000 -3.457508625575e-001 -4.749853573124e-001 -3.176158274191e-001 -4.294507015032e-002-3.704735750656e-017 -1.0998********e-016 -1.953241363386e-0178.269089741494e-017 6.556779598004e-001 -9.275250974463e-0012.529079844053e-001 6.905949216976e-001 -2.374430675823e-002-2.429781119781e-001-6.420051822287e-017 3.865817713597e-017 -3.806718328740e-0172.129734126613e-017 7.853*********e-017 4.698400884876e-001-2.730776009527e-001 7.821296259798e-001 -9.580964936399e-0027.846239841323e-0021.478496020372e-016 -8.383952267131e-017 9.577540382744e-017-7.120338911078e-017 -1.220876056602e-016 -1.854471832043e-0161.287679058937e+000 -3.576058900348e-001 -4.116725408807e-0033.914268216423e-0014.477158378610e-017 -6.204017427568e-017 3.360322998507e-017-1.133882337819e-017 -2.759056827456e-017 -9.217582575819e-0172.214639994444e-016 -3.628760503545e-001 7.398980975354e-0017.241608309576e-0023.408891482791e-017 2.353495494255e-017 1.932283926688e-017-3.456910153081e-017 -2.015177337156e-017 -8.084422691634e-017 -5.839476980893e-017 5.551115123126e-017 -5.176670596524e-0024.958522909877e-002（5）特征值(6.360627875745e-001,0.000000000000e+000)(5.650488993501e-002,0.000000000000e+000)(9.355889078188e-001,0.000000000000e+000)(-9.805309562902e-001,1.139489127430e-001)(-9.805309562902e-001,-1.139489127430e-001)(1.577548557113e+000,0.000000000000e+000)(-1.484039822259e+000,0.000000000000e+000)(-2.323496210212e+000,8.930405177200e-001)(-2.323496210212e+000,-8.930405177200e-001)(3.383039617436e+000,0.000000000000e+000)（6）实特征值的特征向量4.745031936539e+0003.157868541750e+0001.729946912420e+001-1.980049331455e+000-3.187521973524e+0017.794009603201e+000-1.004255685845e+0011.670757770480e+0011.310524273054e+0011.000000000001e+000下一个实特征值对应的特征向量：-5.105003830625e+000-4.886319842360e+0009.505161576151e+000-6.788331788214e-001-9.604334110499e+000-3.0457********e+0001.574873885602e+001-7.395037126277e+000-7.109654943661e+0001.000000000001e+000下一个实特征值对应的特征向量：2.792418944529e+0001.598236841512e+000-5.207507040911e-001-1.667886451702e+000-1.225708535859e+0017.241214790777e+000-5.398214501433e+0002.841008912974e+001-1.216518754416e+0011.000000000001e+000下一个实特征值对应的特征向量：6.217350824581e-001-1.115111815226e-001-2.483773580804e+000-1.306860840421e+000-3.815605442533e+0008.117305509395e+000-1.239170883675e+000-6.800309586197e-0012.691900144840e+0001.000000000001e+000下一个实特征值对应的特征向量：-1.730784582112e+0012.408192534967e+0014.133852365119e-001-8.572044074538e+0009.287334657928e-002-7.832726042776e-002-6.374274025716e-001-3.403204760832e-001。

迭代格式的比较1、自定义函数：function y=fun1(x)y=(3*x+1)/x^2;主程序：x0=input('Please input the initial value£ºx0=');k=input('Please input the number of iterations£ºk='); x(1)=x0;i=1;while i<=kx(i+1)=fun1(x(i));i=i+1;xk=x(i);endfprintf(' x=%f\n',x);运行结果：Please input the initial value：x0=1Please input the number of iterations：k=5x=1.000000x=4.000000x=0.812500x=5.207101x=0.613018x=7.554877结论：发散2、自定义函数：function y=fun2(x)y=(x^3-1)/3;主程序：x0=input('Please input the initial value£ºx0=');k=input('Please input the number of iterations£ºk='); x(1)=x0;i=1;while i<=kx(i+1)=fun2(x(i));i=i+1;xk=x(i);endfprintf(' x=%f\n',x);运行结果：Please input the initial value：x0=1Please input the number of iterations：k=8x=1.000000x=0.000000x=-0.333333x=-0.345679x=-0.347102x=-0.347273x=-0.347294x=-0.347296x=-0.347296结论：收敛3、自定义函数：function y=fun3(x)y=(3*x+1)^(1/3);主程序：x0=input('Please input the initial value£ºx0=');k=input('Please input the number of iterations£ºk='); x(1)=x0;i=1;while i<=kx(i+1)=fun3(x(i));i=i+1;xk=x(i);endfprintf(' x=%f\n',x);运行结果：Please input the initial value：x0=1Please input the number of iterations：k=11x=1.000000x=1.587401x=1.792790x=1.854542x=1.872325x=1.877384x=1.878819x=1.879225x=1.879340x=1.879372x=1.879382x=1.879384结论：收敛4、自定义函数：function y=fun4(x)y=1/(x^2-3);主程序：x0=input('Please input the initial value£ºx0=');k=input('Please input the number of iterations£ºk='); x(1)=x0;i=1;while i<=kx(i+1)=fun4(x(i));i=i+1;xk=x(i);endfprintf(' x=%f\n',x);运行结果：Please input the initial value：x0=1Please input the number of iterations：k=9x=1.000000x=-0.500000x=-0.363636x=-0.348703x=-0.347414x=-0.347306x=-0.347297x=-0.347296x=-0.347296x=-0.347296结论：收敛5、自定义函数：function y=fun5(x)y=sqrt(3+1/x);主程序：x0=input('Please input the initial value£ºx0=');k=input('Please input the number of iterations£ºk='); x(1)=x0;i=1;while i<=kx(i+1)=fun5(x(i));i=i+1;xk=x(i);endfprintf(' x=%f\n',x);运行结果：Please input the initial value：x0=1Please input the number of iterations：k=6x=1.000000x=2.000000x=1.870829x=1.880033x=1.879336x=1.879389x=1.879385结论：收敛6、自定义函数：function y=fun6(x)y=x-1/3*((x^3-3*x-1)/(x^2-1));主程序：x0=input('Please input the initial value£ºx0=');k=input('Please input the number of iterations£ºk='); x(1)=x0;i=1;while i<=kx(i+1)=fun6(x(i));i=i+1;xk=x(i);endfprintf(' x=%f\n',x);运行结果：Please input the initial value：x0=3Please input the number of iterations：k=6x=3.000000x=2.291667x=1.965507x=1.884402x=1.879404x=1.879385x=1.879385结论：收敛要求2，输出迭代次数：自定义函数：interation.m%x0为初始值，k为最大迭代次数，e为控制精度function interation(x0,k,e)x(1)=x0;i=1;while i<=kx(i+1)=fun6(x(i));if abs(x(i+1)-x(i))<efprintf(' the number of iterations: n=%f\n',i);breakendi=i+1;endx=x(i);fprintf(' a root of the original equation: x=%8.6f\n',x);取形式6为例，运行结果：>>interation(5,24,0.00001)the number of iterations: n=7.000000a root of the original equation: x=1.879385。

数值分析第二次大作业1（1）用Lagrange插值法程序：function f=Lang(x,y,x0)syms t;f=0;n=length(x);for(i=1:n)l=y(i);for(j=1:i-1)l=l*(t-x(j))/(x(i)-x(j));end;for(j=i+1:n)l=l*(t-x(j))/(x(i)-x(j));endf=f+l;simplify(f);if(i==n)if(nargin==3)f=subs(f,'t',x0);elsef=collect(f);f=vpa(f,6);endendendx=[1,2,3,-4,5];y=[2,48,272,1182,2262]; t=[-1];disp('插值结果')yt=Lang(x,y,t)disp('插值多项式')yt=Lang(x,y)ezplot(yt,[-1,5]);运行结果：插值结果：Yt= 12.0000插值多项式：yt =4.0*t^4 - 2.0*t^3 + t^2 - 3.0*t + 2.0（2）构造arctan x在[1,6]基于等距节点的10次插值多项式程序：function f=New(x,y,x0)syms t;if(length(x)==length(y))n=length(x);c(1:n)=0.0;elsedisp('xºÍyάÊý²»µÈ£¡');return;endf=y(1);y1=0;xx=linspace(x(1),x(n),(x(2)-x(1)));for(i=1:n-1)for(j=1:n-i)y1(j)=y(j+1)-y(j);endc(i)=y1(1);l=t;for(k=1:i-1)l=l*(t-k);end;f=f+c(i)*l/factorial(i);simplify(f);y=y1;if(i==n-1)if(nargin==3)f=subs(f,'t',(x0-x(1))/(x(2)-x(1)));elsef=collect(f);f=vpa(f,6);endendend>>x=[1,1.5,2,2.5,3,3.5,4,4.5,5,5.5,6];y=[atan(1),atan(1.5),atan(2),atan(2.5),atan(3),atan(3.5),atan(4),atan(4.5),atan(5),atan (5.5),atan(6)];disp('插值多项式')yt=New(x,y)ezplot(yt,[1,6]);hold onezplot('atan(t)',[1,6])grid on运行结果：插值多项式yt =1.34684*10^(-10)*t^10 - 6.61748*10^(-9)*t^9 + 1.28344*10^(-7)*t^8 - 0.00000104758*t^7 - 0.00000243837*t^6 + 0.000149168*t^5 - 0.00176296*t^4 + 0.0125826*t^3 - 0.0640379*t^2 + 0.250468*t + 0.7853982（1）用MATLAB自带spline函数用于进行三次样条插值程序：>>x=[-5,-4,-3,-2,-1,0,1,2,3,4,5];y=[0.03846,0.05882,0.10000,0.20000,0.50000,1.00000,0.50000,0.20000,0.10000,0.0 5882,0.03846];xi=linspace(-5,5)yi=spline(x,y,xi);plot(x,y,'rp',xi,yi);hold on;syms xfx=1/(1+x^2);ezplot(fx);grid on运行结果：由图可知，三次样条插值多项式图像与原函数图像基本一致。

数值分析大作业数值分析大作业姓名：黄晨晨学号：S1*******学院：储运与建筑工程学院学院班级：储建研17-2实验3.1 Gauss消去法的数值稳定性实验实验目的：理解高斯消元过程中出现小主元即很小时引起方程组解数值不定性实验内容：求解方程组Ax=b，其中（1）A1=0.3×10?1559.14315.291?6.130?1211.29521211，b1=59.1746.7812;（2）A2=10?7013 2.099999999999625?15?10102，b2=85.90000000000151;实验要求：(1)计算矩阵的条件数，判断系数矩阵是良态的还是病态的(2)用Gauss列主元消去法求得L和U及解向量x1,x2∈R4(3)用不选主元的高斯消去法求得L和U及解向量x1,x2∈R4(4)观察小主元并分析对计算结果的影响（1）计算矩阵的条件数，判断系数矩阵是良态的还是病态的代码：format longeformat compactA1=[0.3*10^-15,59.14,3,1;5.291,-6.130,-1,2;11.2,9,5,2;1,2,1,1] b1=[59.17;46.78;1;2]n=4C1=cond(A1,1) %C1为A1矩阵1范数下的条件数C2=cond(A1,2) %C2为A1矩阵2范数下的条件数C3=cond(A1,inf) %C3为1矩阵谱范数下的条件数结果：C1 =1.362944708720448e+02C2 =6.842955771253409e+01C3 =8.431146*********e+01显然A1矩阵为病态矩阵将矩阵A2，b2输入上述代码中求得A2矩阵的条件数为：C1 =1.928316831682894e+01C2 =8.993938090170119e+00C3 =1.835643564356072e+01显然A2矩阵也为病态矩阵(2)用Gauss列主元消去法求得L和U及解向量x1,x2∈R4代码：for k=1:n-1a=max(abs(A1(k:n,k)))if a==0returnendfor i=k:nif abs(A1(i,k))==ay=A1(i,:)A1(i,:)=A1(k,:)A1(k,:)=yx=b1(i,:)b1(i,:)=b1(k,:)b1(k,:)=xbreakendendif A1(k,k)~=0A1(k+1:n,k)=A1(k+1:n,k)/A1(k,k)A1(k+1:n,k+1:n)=A1(k+1:n,k+1:n)-A1(k+1:n,k)*A1(k,k+1:n) elsebreakendendL=tril(A1,0);for i=1:nL(i,i)=1;endLU=triu(A1,0)y1=L\b1x1=U\y1得到如下结果：x1 =3.845714853511634e+001.609517394778522e+00-1.547605454206655e+011.041130489899787e+01将A2=[10,-7,0,1;-3,2.0999********,6,2;5,-1,5,-1;0,1,0,2]b2=[8;5.900000000001;5;1]代入上述代码求得结果如下：X2 =4.440892098500626e-16-9.999999999999993e-019.999999999999997e-011.000000000000000e+00(3)用不选主元的高斯消去法求得L和U及解向量x1,x2∈R4代码：format longeformat compactA1=[0.3*10^-15,59.14,3,1;5.291,-6.130,-1,2;11.2,9,5,2;1,2,1,1] b1=[59.17;46.78;1;2][L,U]=lu(A1)y1=L\b1x1=U\y1求得如下结果：x1=3.845714853511634e+001.609517394778522e+00-1.547605454206655e+011.041130489899787e+01将A2=[10,-7,0,1;-3,2.0999********,6,2;5,-1,5,-1;0,1,0,2] b2=[8;5.900000000001;5;1]代入上述代码，求得结果如下：x 2 =4.440892098500626e-16 -9.999999999999993e-01 9.999999999999997e-01 9.999999999999999e-01（2）（3）求得结果相同，可验证结果正确。

数值分析大作业一、算法设计方案1、矩阵初始化矩阵[]501501⨯=ij a A 的下半带宽r=2,上半带宽s=2，设置矩阵[][]5011++s r C ，在矩阵C 中检索矩阵A 中的带内元素ij a 的方法是：j s j i ij c a ,1++-=。

这样所需要的存储单元数大大减少，从而极大提高了运算效率。

2、利用幂法求出5011λλ，幂法迭代格式：0111111nk k k k kk T k k k u R y u u Ay y u ηηβ------⎧∈⎪⎪=⎪=⎨⎪=⎪⎪=⎩非零向量 当1210/-≤-k k βββ时，迭代终止。

首先对于矩阵A 利用幂法迭代求出一个λ，然后求出矩阵B ，其中I A B λ-=(I 为单位矩阵），对矩阵B 进行幂法迭代，求出λ'，之后令λλλ+'=''，比较的大小与λλ''，大者为501λ，小者为1λ。

3、利用反幂法求出ik s λλ,反幂法迭代格式：0111111nk k k k kk T k k k u R y u Au y y u ηηβ------⎧∈⎪⎪=⎪=⎨⎪=⎪⎪=⎩非零向量 当1210/-≤-k k βββ时，迭代终止，1s k λβ=。

每迭代一次都要求解一次线性方程组1-=k k y Au ，求解过程为：（1）作分解LU A =对于n k ,...,2,1=执行[][]s k n r k k k i c c c c c n s k k k j c cc c k s ks k t k s k r i t t s t i k s k i k s k i js j t k s j r k t t s t k j s j k j s j k <+++=-=++=-=+++----=++-++-++-++----=++-++-++-∑∑);,min(,...,2,1/)(:),min(,...,1,:,1,11),,1max(,1,1,1,11),,1max(,1,1,1（2）求解y Ux b Ly ==,（数组b 先是存放原方程组右端向量，后来存放中间向量y))1,...,2,1(/)(:/:),...,3,2(:,1),min(1.1.11),1max(,1--=-===-=+++-++-+--=++-∑∑n n i c x c b x c b x n i b c b b i s t n s i i t t s t i i i ns n n ti r i t t s t i i i使用反幂法，直接可以求得矩阵按模最小的特征值s λ。

数值分析上机作业第 1 章1.1计算积分，n=9。

（要求计算结果具有6位有效数字）程序：n=1:19;I=zeros(1,19);I(19)=1/2*((exp(-1)/20)+(1/20));I(18)=1/2*((exp(-1)/19)+(1/19));for i=2:10I(19-i)=1/(20-i)*(1-I(20-i));endformat longdisp(I(1:19))结果截图及分析：在MATLAB中运行以上代码，得到结果如下图所示：当计算到数列的第10项时，所得的结果即为n=9时的准确积分值。

取6位有效数字可得.1.2分别将区间[-10.10]分为100,200,400等份，利用mesh或surf命令画出二元函数z=的三维图形。

程序：>> x = -10:0.1:10;y = -10:0.1:10;[X,Y] = meshgrid(x,y);Z = exp(-abs(X))+cos(X+Y)+1./(X.^2+Y.^2+1);subplot(2,2,1);mesh(X,Y,Z);title('步长0.1')>> x = -10:0.2:10;y = -10:0.2:10;[X,Y] = meshgrid(x,y);Z = exp(-abs(X))+cos(X+Y)+1./(X.^2+Y.^2+1);subplot(2,2,1);mesh(X,Y,Z);title('步长 0.2')>>x = -10:0.05:10;y = -10:0.05:10;[X,Y] = meshgrid(x,y);Z = exp(-abs(X))+cos(X+Y)+1./(X.^2+Y.^2+1);subplot(2,2,1);mesh(X,Y,Z);title('步长0.05')结果截图及分析：由图可知，步长越小时，绘得的图形越精确。