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 procedure65 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 < Na, 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 Rm = c is thus also an LS solution to Am = b.