Suppose we begin with a matrix V and create a new matrix L by appending A*V and V as columns, i.e., $$L = [AV, V],$$where A has dimensions (N,N), and V of dim (N,r).
Afterward, we perform a truncated QR decomposition on L, resulting in Q with dimensions (N, 2r) and R with dimensions (2r, 2*r), where r is the rank of V. We concatenate two such matrices.
Now, the objective is to compute the modified QR decomposition of subsets of the first column of V, such as $$L_1 = [A*V(:,1), V(:,1)],$$ $$L_2 = [A*V(:,1:2), V(:,1:2)],$$ and so on, until a certain condition is met.
My question is: What is the most efficient method to compute the modified QR decomposition of L_1, L_2, and so forth? MATLAB provides an algorithm called "qrdelete" that updates the R matrix by removing the corresponding columns to be deleted and applies Givens rotations to eliminate subdiagonal elements one at a time. Is there a more optimized approach for this task?
I appreciate your assistance in advance.