Hi all,
I'm trying to implement the QR method for solving the linear system Ax = b. The QR factorization is achieved using Householder method.
The main function is
Code:
function x = lin_solve(A,b)
[R,v] = householder(A);
y = Qt_times_b(v,b);
x = R\y;
Here are the individual functions:
Code:
function [R,v] = householder(A)
[m,n] = size(A);
if m>=n,
NumberOfReflections = n;
else
NumberOfReflections = m-1;
end
R = A;
v = cell(NumberOfReflections,1);
for k = 1:NumberOfReflections,
x = R(k:m,k);
xnorm = norm(x);
if xnorm>0,
% Compute the normal vector of the reflector
v{k} = -x;
v{k}(1) = v{k}(1) - sign(x(1))*xnorm;
v{k} = (sqrt(2)/norm(v{k}))*v{k};
% Update R
for j = k:NumberOfReflections,
R(k:m,j) = R(k:m,j) - (v{k}'*R(k:m,j))*v{k};
end
else
v{k} = zeros(m-k+1,1);
end
end
Code:
function [Q,y] = Qt_times_b(v,b)
NumberOfReflections = length(v);
y = b;
mv = length(v{1});
Q=eye(mv);
for k = 1:NumberOfReflections,
F = eye(mv-k+1)-2*(v{k}*v{k}')/norm(v{k})^2;
Qk = [eye(k-1) zeros(k-1,mv-k+1);zeros(mv-k+1,k-1) F];
y = Qk*y;
end
What I don't understand is the function works for square matrices, i.e. A that are n-by-n, but if m < n, or m > n, then I get the incorrect solution x.
I tried to check my R and Qk against the qr factorization function provided by matlab. Say [QR, RR] = qr(A).
If m < n, then my R is not equal to RR; if m > n, R = RR. In both cases, Q' = QR, where Q' = transpose of matrix Q.
What is wrong with my program?
Thank you.
Regards,
Rayne