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1、上机习题1先用你所熟悉的的计算机语言将不选主元和列主元Gauss消去法编写成通用的子程序；然后用你编写的程序求解84阶方程组；最后将你的计算结果与方程的精确解进行比较，并就此谈谈你对Gauss消去法的看法。Sol :(1) 先用matlab 将不选主元和列主元 Gauss消去法编写成通用的子程序，得到L,U,P：不选主元Gauss消去法：L,U GaussLA(A)得到L,U满足A LU列主元 Gauss 消去法：L,U,P GaussCol(A)得到 L,U ,P 满足 PA LU(2) 用前代法解Ly b or Pb，得y用回代法解Ux y，得x求解程序为x Gauss A,b, L,U,
2、P ( P可缺省，缺省时默认为单位矩阵)(3) 计算脚本为ex1_1代码%算法(计算三角分解：Gauss消去法)fun ctio nL,U=GaussLA(A)n=len gth(A);for k=1:n-1A(k+1: n,k)=A(k+1: n,k)/A(k,k);A(k+1: n,k+1: n)=A(k+1: n,k+1: n)-A(k+1: n,k)*A(k,k+1: n);endU=triu(A);L=tril(A);L=L-diag(diag(L)+diag( on es(1, n);end%算法122(计算列主元三角分解：列主元 Gauss消去法)fun ctio nL,U,P=
3、GaussCol(A)n=len gth(A);for k=1:n-1s,t=max(abs(A (k:n, k);p=t+k-1;temp=A(k,1: n);A(k,1: n)=A(p,1: n);A(p,1: n)=temp;u(k)=p;if A(k,k)=0A(k+1: n,k)=A(k+1: n,k)/A(k,k);A(k+1: n,k+1: n)=A(k+1: n,k+1: n)-A(k+1: n,k)*A(k,k+1: n);elsebreak ;endendL=tril(A);U=triu(A);L=L-diag(diag(L)+diag(o nes(1, n);P=eye(
4、 n);for i=1:n-1temp=P(i,:);P(i,:)=P(u(i),:);P(u(i),:)=temp;endend%高斯消去法解线性方程组fun ctio nx=Gauss(A,b,L,U,P)if nargin5P=eye(le ngth(A);endn=len gth(A);b=P*b;for j=1:n-1b(j)=b(j)/L(j,j);b(j+1: n)=b(j+1: n)-b(j)*L(j+1: n,j);endb(n )=b( n)/L( n,n);y=b；for j=n:-1:2y(j)=y(j)/u(j,j)； y(1:j-1)=y(1:j-1)-y(j)*U
5、(1:j-1,j);endy(1)=y(1)/U(1,1);x=y;end ex1_1clc;clear;%第一题A=6*eye(84)+diag(8*o nes(1,83),-1)+diag(o nes(1,83),1);b=7;15*o nes(82,1);14;%不选主元 Gauss消去法L,U=GaussLA(A);x1_1=Gauss(A,b,L,U);%列主元 Gauss消去法L,U,P=GaussCol(A);x1_2=Gauss(A,b,L,U,P);%解的比较o- );title( Gauss);subplot(1,3,1);plot(1:84,x1_1.subplot(1,
6、3,2);plot(1:84,x1_2,.- );title( PGauss);subplot(1,3,3);plot(1:84,ones(1,84),*- );title(精确解);结果为(其中Gauss表示不选主元的 Gauss消去法，PGauss表示列主元 Gauss消去法，精确解为1,11 84）：由图，显然列主元消去法与精确解更为接近，不选主元的Gauss消去法误差比列主元消去法大，且不如列主元消去法稳定。Gauss消去法重点在于 A的分解过程，无论 A如何分解，后面两步的运算过程不变。2先用你所熟悉的的计算机语言将平方根法和改进的平方根法编写成通用的子程序；然后用你编写的程序求解对
7、称正定方程组Ax=b。Sol :（1 ）先用matlab将平方根法和改进的平方根法编写成通用的子程序，得到L，（D）:平方根法：L=Cholesky(A)改进的平方根法：L,D=LDLt(A)(2 )求解得Ly b求解得 LTxy or DLTxy求解程序为x=Gauss(A,b,L,U,P) ( ULT or U DLT , P此时缺省，缺省时默认为单位矩阵)(3)计算脚本为ex1_2代码%算法(计算Cholesky 分解：平方根法)fun ctio nL=Cholesky(A)n=len gth(A);for k=1:nA(k,k)=sqrt(A(k,k);A(k+1: n,k)=A(k+
8、1: n,k)/A(k,k);for j=k+1:nA(j: n,j)=A(j: n,j)-A(j: n,k)*A(j,k);endendL=tril(A);end%计算LDL 分解：改进的平方根法fun ctio nL,D=LDLt(A)n=len gth(A);for j=1:nfor i=1:nv(i,1)=A(j,i)*A(i,i);endA(j,j)=A(j,j)-A(j,1:j-1)*v(1:j-1,1);A(j+1: n,j)=(A(j+1: n,j)-A(j+1: n,1:j-1)*v(1:j-1,1)/A(j,j);endL=tril(A);D=diag(diag(A);L=
9、L-diag(diag(L)+diag( on es(1, n);end%高斯消去法解线性方程组fun ctio nx=Gauss(A,b,L,U,P)if nargin5P=eye(le ngth(A);endn=len gth(A);b=P*b;for j=1:n-1b(j)=b(j)/L(j,j);b(j+1: n)=b(j+1: n)-b(j)*L(j+1: n,j); endb(n )=b( n)/L( n,n);y=b;for j=n:-1:2y(j)=y(j)/u(j,j);y(1:j-1)=y(1:j-1)-y(j)*U(1:j-1,j); endy(1)=y(1)/U(1,1
10、);x=y;end ex1_2%第二题%第一问A=10*eye(100)+diag(o nes(1,99),-1)+diag(o nes(1,99),1);b=ro un d(100*ra nd(100,1);%平方根法L=Cholesky(A);x1_2_1_ 仁Gauss(A,b,L,L);%改进的平方根法L,D=LDLt(A); x1_2_1_2=Gauss(A,b,L,D*L);%第二问A=hilb(40);b=sum(A);b=b;%平方根法L=Cholesky(A);x1_2_2_1=Gauss(A,b,L,L);%改进的平方根法L,D=LDLt(A);x1_2_2_2=Gauss
11、(A,b,L,D*L);结果分别为x1_2_1_1 =7.25868.4143-0.40138.59845.41770.22492.33364.43898.27728.7890-0.16678.87848.38243.29787.64010.30143.34578.24186.23688.39065.8575-0.96567.79817.98425.36016.41436.49662.62016.30240.35637.1342-0.69852.85060.19270.22277.58015.97621.65839.4409-1.06774.23622.70606.70377.25700.72
12、654.47783.49595.56375.86756.76141.51806.05825.90020.93970.70314.02949.00321.93825.61500.91207.26521.43604.37495.81467.47918.39424.57890.81691.25231.66038.14480.89157.94010.70758.98492.44371.57771.77905.63193.90182.35067.59254.72454.16278.64831.35436.80876.55892.60275.41400.25770.00904.65226.46858.66
13、26-0.09485.28564.2385-0.67063.4671x1_2_1_2 =7.25868.4143-0.40138.59845.41770.22492.33364.43898.27728.7890-0.16678.87848.38243.29787.64010.30143.34578.24186.23688.39065.8575-0.96567.79817.98425.36016.41436.49662.62016.30240.35637.1342-0.69852.85060.19270.22277.58015.97621.65839.4409-1.06774.23622.706
14、06.70377.25700.72654.47783.49595.56375.86756.76141.51806.05825.90020.93970.70314.02949.00321.93825.61500.91207.26521.43604.37495.81467.47918.39424.57890.81691.25231.66038.14480.89157.94010.70758.98492.44371.57771.77905.63193.90182.35067.59254.72454.16278.64831.35436.80876.55892.60275.41400.25770.00904.6
