# Update QR decomposition when one column is exchanged

I have got an input series of matrices

$A_1, A_2, A_3, \dots$

and the difference between $A_i$ and $A_{i+1}$ is the replacement of one single column. Before i get to know $A_{i+1}$, I have to compute $A_i^{-1} b$ for some fixed vector $b$, so I also have a series of vectors

$A_1^{-1}b, A_2^{-1}b, A_3^{-1}b, \dots$

to compute. For your interest, this appears in implementations of the simplex method, but I prefer to approach with the point of view of numerical linear algebra. Questions:

1. Where can I find a reference on updating the QR decomposition after replacing one column?
2. Since I am only interested in the QR decomposition in order to solve a sequence of linear problems, can this be exploited in the algorithms,too?

I use the QR decomposition only because this is a standard method - in the case that you have a better recommendation for the above setting, feel free to provide your advice.