Hello,
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?
Matlab code:
Code:
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)]
Many thanks in advance,
Best regards,
Giovanni