Matlab QR householder factorization incorrect output

Question

I'm trying to write a program that gets an mxn matrix and QR decomposes it.

It's not finished yet, but I've encountered a problem. I've tried running my program with the example shown at wikipedia http://en.wikipedia.org/wiki/QR_decomposition

A=[12,-51,4;6,167,-68;-4,24,-41]

What they called Q1, Q2...I called Qtemp. and everytime i calculate Qtemp, i print it in order to see that I get same result as wikipedia. For Q1 I do, but for Q2 I don't.

Their Q2 and mine are of the same value, but different signs. where they have a + i have a -, where they have a -, i have a +.

this is my code:

    Q=eye(m);
R=A;
for i=1:min(m-1,n)
    ei=zeros(n,1);
    ei(i,1)=1;
    x=A(:,i);
    for j=1:i-1
        x(j,1)=0;
    end
    u=x-norm(x)*ei;
    v=u/norm(u);
    Qtemp=eye(m)-2*(v*v');
    A=Qtemp*A;
    disp(Qtemp);
end

I literally just copied their algorithm and translated it to code, but still bad output for second Qtemp.

Answer 1

It looks like you're not reducing the size of the blocks on each iteration. Everything seems to be a function of the same m and n (which you didn't define in your code). See the line on the Wikipedia page where they define A′ and use it to build Q₂ (just the lower two thirds). Below is some code of mine adapted to perform QR-decomposition of 3-by-3 matrices that might help. Note particularly that the second block only works on A(:,2) and q(2:3,:):

function [q,r]=qr3(A)

u = A(:,1);
u(1) = u(1)-(1-2*(u(1)<0))*norm(u); % Flip < to > to match sign convention of qr
u = u/norm(u);
u(~isfinite(u)) = sqrt(3)/3;
q = -2*(u*u');
q([1 5 9]) = q([1 5 9])+1;

u = q(2:3,:)*A(:,2);
u(1) = u(1)-(1-2*(u(1)<0))*norm(u); % Flip < to > to match sign convention of qr
u = u/norm(u);
u(~isfinite(u)) = sqrt(2)/2;
q(:,2:3) = q(:,2:3)*[1-2*u(1)^2 -2*u(1)*u(2);
                     -2*u(1)*u(2) 1-2*u(2)^2];
r = triu(q'*A);

The above code and the method detailed on Wikipedia use a different sign convention from Matlab's qr function. See my comments in the code for how to flip the sign.