Improved SOR method with orderings and direct methods

Abstract

A generalized SOR method with multiple relaxation parameters is considered for solving a linear system of equations. Optimal choices of the parameters are examined under the assumption that the coefficient matrix is tridiagonal and regular. It is shown that the spectral radius of the iterative matrix is reduced to zero for a pair of parameter values which is computed from the pivots of the Gaussian elimination applied to the system. A proper choice of orderings and starting vectors for the iteration is also proposed.When the system is well-conditioned, it is solved stably by Gaussian elimination; there is little advantage in using an iterative method. However, when the system is illconditioned, the direct method is not necessarily stable and it is often required to improve the numerical solution by a certain iterative algorithm. Some numerical examples are presented which show that the proposed method is more efficient than the standard iterative refinement method.It is also discussed how to apply our technique to a class of systems which includes Hessenberg systems.

Adaptive improved block SOR method with orderings

“…We also assume that all denominators are not zero (see Theorem 3.2 in [13]). For the vector solution z = (Ii, z2 i .. .…”

“…(1) where On the other hand, for Qk ordering, that is, Á = QkAQk , c = Qk^Qk and x ( n`) = Qkx(m) in (1) where we have e ( n`) _ [e ( "`) ] = x ( ') -x = 0 for m >_ max(k, n -k + 1), and we also have t-) ej =0,j=1,2,...,mforl<m<k-landj=n-m+l,n-m+2,...,n for 1<m<n-k. Proof. For Pk-ordering, A = UkLk, 2 < k < n -1 is UL factorization of A in [13], and we have For Case I l ), M R-'L R such that …”

“…q,1<j<kl,usingz( =zk +(1-wk (k) )( (7n) z k -(rn+l)) vk /II vk we modify z ( m +l) in (13) as z(^"+1) = W (k)z(-+1) + 1 - Then this modified method also satisfies (14) and is convergent.…”

“…In previous papers [11][12][13], we have clarified some fundamental properties of the "improved SOR method with orderings", which is an effective method to solve nonsymmetric linear equations derived from singular perturbation problems. In [13], for an n x n tridiagonal matrix A, we offered selections for the multiple relaxation parameters wi which correspond to the reciprocal numbers of the pivots in Gaussian elimination.…”

“…In [13], for an n x n tridiagonal matrix A, we offered selections for the multiple relaxation parameters wi which correspond to the reciprocal numbers of the pivots in Gaussian elimination. In addition, considering the starting vectors and orderings, we provided practical algorithms and some numerical experiments.…”