QR分解

September 29th, 2019

修正施密特正交

算法：

代码如下：

function [q,r] = mgs(a)
[m,n] = size(a);
v = a;
r = zeros(n);
q = zeros(m,n);
for i = 1:n
    r(i,i) = norm(v(:,i),2);
    q(:,i) = v(:,i)/r(i,i);
    for j  = (i+1) : n
        r(i,j) = q(:,i)'*v(:,j);
        v(:,j) = v(:,j) - r(i,j)*q(:,i);
    end
end
end

基于Householder的QR分解

推理过程：

算法：

题目：

代码如下：

function [W,R] = house(A)
[m,n] = size(A);
W = zeros(m,n);
R = A;
for i = 1 : n 
    x = R(i:m,i);
    s = sign(x(1));
    if(s == 0)
        s = 1;
    end
    v = x;
    v(1) = s*norm(x) + v(1);
    v = v/norm(v);
    W(i:m,i) = v;
    R(i:m,i:n) = R(i:m,i:n) - 2*v*(v'*R(i:m,i:n));
end
R = R(1:n,1:n);

function Q = formQ(W)
[m,n] = size(W);
Q = eye(m);
for i = 1 : m
    for j = n : -1: 1
        Q(j:m,i) = Q(j:m,i) - 2*W(j:m,j)*(W(j:m,j)'*Q(j:m,i));
    end
end