# how to update QR decomposition like cholesky decomposition

suppose there is a linear equation:

$$\begin{bmatrix}J_0 \\J_1 \\J_2 \end{bmatrix}*x=\begin{bmatrix}e_0\\e_1\\e_2 \end{bmatrix}$$

one way to solve this equation is build Least Square $$Hx = b$$ while

$$H = J'_0J_0+J_1'J_1 +J_2'J_2$$

$$b = J'_0e_0+J'_1e_1+J'_2e_2$$

then use cholesky decomposition to solve this equation

but, after solve , I find $$J_2x=e_2$$ is a error constraint, I want to remove this constraint, in Least Square situation it's easy,there is no need to recompute this problem. we just need to subtract the error term in positive definite matrix H, like:

$$(H-J'_2J_2)*x = (b-J'_2e_2)$$

but again, for some reason I want to solve this equ by QR decomp , like :

$$(\begin{bmatrix} Q_0' \\ Q_1' \\ Q_2'\end{bmatrix},R) = qr(\begin{bmatrix}J_0 \\J_1 \\J_2 \end{bmatrix})$$

R is a up-triangulate matrix , so the equation become Rx=c, while

$$R = Q_0J_0+Q_1J_1+Q_2J_2$$

$$c =Q_0e_0+Q_1e_1+Q_2e_2$$

I know this approach is the same as Least Square upside. but how can I remove the error constraint $$J_2x=e_2$$ in this situation. is rank one update available?