A combined direct-iterative approach for solving large scale singular and rectangular consistent systems of linear equations

AbstractWe construct a certain iterative scheme for solving large scale consistent systems of linear equations Ax=b, (*) where A is a complex m X n matrix of rank r, m⩾n, and where A is assumed reasonably well conditioned. The iterative method is obtained through a careful exploitation of an LU-decomposition of A, and, disregarding roundoff errors, it converges to a solution to (*), though not necessarily the minimal l2-norm one, from any starting vector in r iterations. Moreover, once the LU-decomposition of A is complete, only about r3/2 arithmetic operations (multiplications and divisions) are needed to execute the r iterations