householder ls solution

if (Ain mathbb{R}^{mtimes n}) has full column rank and (b in mathbb{R}^m), then the following algorithm computes a vector (x_{LS}in mathbb{R}^n) such that (|Ax_{LS}-b|_2) is minimum.

Overwrite (A) with its QR factorization.
for (j=1:n)
 (v=left[
begin{array}{c}
1\
A(j+1:m,j)
end{array}
right])
 (beta = 2/v^Tv)
 (b(j:m)=b(j:m) - beta (v^Tb(j:m))v)
end
Solve (R(1:n,1:n)cdot x_{LS} = b(1:n))