# Block-wise QR decomposition

Although the sparse storage technique can be utilized, the insufficient-memory problem remains when too many subapertures are involved. A block-wise QR decomposition procedure^{65} is effective for solving it. For example, each block can be the data set of a subaperture and applied with QR decomposition one after another. It finally leads to a smaller-scale LS problem.

Suppose the matrix A is divided into Na blocks. The number of columns in each block is identical and equals L. The vector b is also divided into Na blocks. Denote the block i as A,- and b,, then we can get an LS solution to the linear LS problem by applying the following procedure.

Step 0: Let i = 1, [Я,-__{ь} с,_:] is an empty matrix. Denote the augmented matrix by

Step 1: (a) Triangulate the augmented matrix by QR decomposition

(b) If *i < N _{a},* let i

*=*i + 1. Update the augmented matrix and then return to step 1(a). Otherwise, we obtain the following after triangulation:

where e is a scalar. The LS solution to R*m =* c is thus also an LS solution to Am *=* b.