Both of these methods require the symmetry of A^T * A. When these methods are combined with preconditioning, e.g. a block Gauss-Seidel method for the normal equations, then this symmetry is lost. There are two approaches to overcome this issue:

- Replace the block Gauss-Seidel method by a symmetric block Gauss-Seidel method, which is done for instance in an SSOR-preconditioned LSQR method.
- Replace the iterative solver by another one which is capable of dealing with non-symmetric matrices. We did this by combining a block Gauss-Seidel preconditioner (i.e. Kaczmarz method) with IDR(s)biortho. The resulting least-squares solver we call LSIDR(s).

Figure 1: LSIDR(10) compared to LSQR on small dimensioned dense systems with elements from a normal distribution.

back to main page: Back