# How can I compute recursive QR-factorization?

I wonder if it's possible to find the $$Q$$ and $$R$$ matrices from this QR-equation with only compute QR at one time only:

$$A = QR$$

if, the first column of $$A$$ got removed and then a new column got added to $$A$$.

I take it again: Assume that we first got our data matrix $$A$$ and we compute $$Q$$ and $$R$$. Then we change our data matrix $$A$$ by remove first column and add new data column to $$A$$. That means $$Q$$ and $$R$$ are going to change. Can we compute the new $$Q$$ and $$R$$ if we know the new data column of $$A$$ and the past $$Q$$ and $$R$$?

If you wonder what I got this question from. I got this from the paper Recursive Subspace Identification Algorithm using the Propagator Based Method, 2017.

• Do you mean that the first column gets removed, such that the old second column of $A$ becomes the new first column, the old third column becomes the new second column ect. and that new last column of $A$ becomes the new column, or do you mean that the first column of $A$ gets replaced with the new column? – Kwin van der Veen Jan 29 at 3:52
• Yes! I mean this one "Do you mean that the first column gets removed, such that the old second column of A becomes the new first column, the old third column becomes the new second column ect. and that new last column of A becomes the new column". – Daniel Mårtensson Jan 29 at 12:10