A note on iterative refinement for seminormal equations

We present a roundoff error analysis of the method for solving the linear least squares problem minx‖b−Ax‖2 with full column rank matrix A, using only factors Σ and V   from the SVD decomposition of A=UΣVTA=UΣVT. This method (called SNESVDSNESVD here) is an analogue of the method of seminormal equations (SNEQRSNEQR), where the solution is computed from RTRx=ATbRTRx=ATb using only the factor R from the QR factorization of A. Such methods have practical applications when A is large and sparse and if one needs to solve least squares problems with the same matrix A   and multiple right-hand sides. However, in general both SNEQRSNEQR and SNESVDSNESVD are not forward stable. We analyze one step of fixed precision iterative refinement to improve the accuracy of the SNESVDSNESVD method. We show that, under the condition O(u)κ2(A)<1O(u)κ2(A)<1, this method (called CSNESVDCSNESVD) produces a forward stable solution, where κ(A)κ(A) denotes the condition number of the matrix A and u   is the unit roundoff. However, for problems with only O(u)κ(A)<1O(u)κ(A)<1 it is generally not forward stable, and has similar numerical properties to the corresponding CSNEQRCSNEQR method. Our forward error bounds for the CSNESVDCSNESVD are slightly better than for the CSNEQRCSNEQR since the terms O(u2)κ3(A)O(u2)κ3(A) are not present. We illustrate our analysis by numerical experiments.