TY - JOUR
T1 - Finite precision behavior of stationary iteration for solving singular systems
AU - Higham, Nicholas J.
AU - Knight, Philip A.
PY - 1993/10/31
Y1 - 1993/10/31
N2 - A stationary iterative method for solving a singular system Ax=b converges for any starting vector if limi→∞Gi exists, where G is the iteration matrix, and the solution to which it converges depends on the starting vector. We examine the behavior of stationary iteration in finite precision arithmetic. A pertubation bound for singular systems is derived and used to define forward stability of a numerical method. A rounding error analysis enables us to deduce conditions under which a stationary iterative method is forward stable or backward stable. The component of the forward error in the null space of A can grow linearly with the number of iterations, but it is innocuous as long as the iteration converges reasonably quickly. As special cases, we show that when A is symmetric positive semidefinite the Richardson iteration with optimal parameter is forward stable, and if A also has unit diagonal and property A, then the Gauss-Seidel method is both forward and backward stable. Two numerical examples are given to illustrate the analysis.
AB - A stationary iterative method for solving a singular system Ax=b converges for any starting vector if limi→∞Gi exists, where G is the iteration matrix, and the solution to which it converges depends on the starting vector. We examine the behavior of stationary iteration in finite precision arithmetic. A pertubation bound for singular systems is derived and used to define forward stability of a numerical method. A rounding error analysis enables us to deduce conditions under which a stationary iterative method is forward stable or backward stable. The component of the forward error in the null space of A can grow linearly with the number of iterations, but it is innocuous as long as the iteration converges reasonably quickly. As special cases, we show that when A is symmetric positive semidefinite the Richardson iteration with optimal parameter is forward stable, and if A also has unit diagonal and property A, then the Gauss-Seidel method is both forward and backward stable. Two numerical examples are given to illustrate the analysis.
KW - stationary iterative method
KW - singular systems
KW - error analysis
UR - http://www.scopus.com/inward/record.url?scp=21144483536&partnerID=8YFLogxK
U2 - 10.1016/0024-3795(93)90242-G
DO - 10.1016/0024-3795(93)90242-G
M3 - Article
AN - SCOPUS:21144483536
SN - 0024-3795
VL - 192
SP - 165
EP - 186
JO - Linear Algebra and its Applications
JF - Linear Algebra and its Applications
ER -