Suppose I have an overdetermined matrix A and I compute its QR in order to solve more than one system with it (Least Squares and reusing the QR multiple times). Then I add a row of ones and need to get my QR updated. This is solvable using QR updates. My question is, given the fact I know what the values for the extra row is always ones, is there a way to improve over the QR updates by exploiting this fact that I always append ones? My problem is in the context of an iterative algorithm that requires appending rows of ones per step (no other way around).
An interesting alternative solution/ view to the problem is ... given that I have an overdetermined matrix A and its QR so I can solve this system and get solution X, and now I append a row of ones to A (in linear regression an additional data point), is there a way to very quickly answer yes/no whether the solution X changes due to the new data point? a 'no' answer would spare the whole QR computation altogether.
Would there be some nice identity to take advantage of given this mutation pattern of appending one row with ones?
Many thanks in advance,
A = randn(10,5);
[Q0 R0] = qr(A);
A = [A; ones(1,5)];
[Q1 R1] = qr(A);
% would there be some sort of a linear mapping?
Q1 - [Q0 zeros(10,1); zeros(1,11)]
R1 - [R0; zeros(1,5)]