水帖1号（别问我是什么 这只是一地水）

songce1026

clc;
clear all;
format shortEng

n=45;
H=hilb(n);
K=H(:,1:n-5);
x=ones(n-5,1);
b=K*x;
cond=cond(K)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Normal equations and the row oriented algorithm

M=K'*K;
B=K'*b;
% L=chol(M)';
[L,U]=lu(M);
X_Oriented=(L*U)\B

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%arguemented system

X=[eye(n),K;K',zeros(n-5,n-5)]\[b;zeros(n-5,1)];
X_Arguemented=X(n+1:n+n-5,


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Gram-Schmidt
r=zeros(n-5,n-5);
q=zeros(n,n-5);
for k=1:n-5
    r(k,k)=norm(K(:,k),2);
    if r(k,k)==0
        break
    else q(:,k)=K(:,k)/r(k,k);
    for j=k+1:n-5
        r(k,j)=q(:,k)'*K(:,j);
        K(:,j)=K(:,j)-r(k,j)*q(:,k);
    end
    end
end

Q=q;
R=r;
X_Gram=R\(Q'*b)

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Householder QR Factorization
K=H(:,1:n-5);
v=zeros(n,n-5);
a=K;
beta=zeros(n-5,1);
for k=1:n-5
    alpha=zeros(n-5,1);
    alpha(k,1)=-sign(a(k,k))*sqrt(sumsqr(a(k:n,k)));
    e=zeros(n,1);
    e(k,1)=1;
    v(:,k)=[zeros(k-1,1);a(k:n,k)]-alpha(k,1)*e;
    beta=v(:,k)'*v(:,k);
    if beta==0
        break
    end
    for j=k:n-5
        gama=v(:,k)'*a(:,j);
        a(:,j)=a(:,j)-(2*gama/beta)*v(:,k);
    end
end
bb=a/K*b;
triu(a);
x_House=triu(a)\bb


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%singular value decomposition
[U,S,V]=svd(K);

X_svd=zeros(n-5,1);
for i=1:n-5
    x=U(:,i)'*b*V(:,i)/S(i,i);
    X_svd=X_svd+x;
end
X_svd


Accuracy_Oriented=cond*norm(K,2)*norm(X_Oriented,2)/norm(b,2)
Accuracy_Arguement=cond*norm(K,2)*norm(X_Arguemented,2)/norm(b,2)
Accuracy_Gram=cond*norm(K,2)*norm(X_Gram,2)/norm(b,2)
Accuracy_Householder=cond*norm(K,2)*norm(x_House,2)/norm(b,2)
Accuracy_SVD=cond*norm(K,2)*norm(X_svd,2)/norm(b,2)

xinshou201632

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