A necessary and sufficient condition for semiconvergence and optimal parameter of the SSOR method for solving the rank deficient linear least squares problem

摘要

Let A is an element of C-r(mxn) be partitioned as [GRAPHICS] where A(11) is an element of C-r(rxr). Write B = A(21)A(11)(-1) and C=A(11)(-1)A(12). Suppose that B not equal 0. For finding the minimum norm least squares solution A(+)b of the linear systems Ax =b, many authors studied the SOR, AOR, and SSOR methods for solving the augmented systems (A) over capz = (b) over cap, and obtained many results. In this paper we deeply study the SSOR method, whose iteration matrix is written as J(omega), and prove the following new conclusions: (1) If vertical bar vertical bar B vertical bar < 1, then J(omega) is semiconvergent double left right arrow omega is an element of (0, 2). If vertical bar vertical bar B vertical bar >= 1, then J omega is semiconvergent omega is an element of (0, omega(2)) boolean OR (omega(1), 2), where [GRAPHICS] (2) The optimal parameters of J(omega) are [GRAPHICS] min(omega)delta(J(omega)) =min(omega) max{vertical bar lambda vertical bar : lambda is an element of sigma(J(omega)),lambda not equal 1} = (1 - (omega) over tilde (1))(2) = (1 - (omega) over tilde (2))(2) = [GRAPHICS] In addition, we obtain other results concerning the SOR, AOR and SSOR methods. (c) 2006 Elsevier Inc. All rights reserved.