68 Journal of Advances in Applied & Computational Mathematics, 2016, 3, 68-73 E-ISSN: 2409-5761/16 © 2016 Avanti Publishers Forward Stability of Iterative Refinement with a Relaxation for Linear Systems Alicja Smoktunowicz * , Jakub Kierzkowski and Iwona Wróbel Faculty of Mathematics and Information Science, Warsaw University of Technology, Koszykowa 75, 00-662 Warsaw, Poland Abstract: Stability analysis of Wilkinson’s iterative refinement method IR( !) with a relaxation parameter ! for solving linear systems is given. It extends existing results for ! =1 , i.e., for Wilkinson’s iterative refinement method. We assume that all computations are performed in fixed (working) precision arithmetic. Numerical tests were done in MATLAB to illustrate our theoretical results. A particular emphasis is given on convergence of iterative refinement method with a relaxation. A preliminary error analysis of the Algorithm IR( !) was given in [11]. Our opinion is opposite to that given in [11], since our experiments show that the choice ! =1 is the best choice from the point of numerical stability. Keywords: Iterative refinement, numerical stability, condition number. 1. INTRODUCTION We consider the system Ax = b , where A ! R n"n is nonsingular and b ! R n . Iterative refinement techniques for linear systems of equations are very useful in practice and the literature on this subject is very rich, see [1], [4]– [11]. The idea of relaxing the iterative refinement step is the following. We require a basic linear equation solver S for Ax = b which uses a factorization of A into simple factors (e.g., triangular, block-triangular etc.). Such factorization is used again in the next steps of iterative refinement. Wilkinson’s iterative refinement method with a relaxation IR( ! ) consists of three steps. Algorithm IR( ! ) Given ! >0 . Let x 0 be computed by the solver S . For k = 0,1, 2,… , the k th iteration consists of the three steps: 1. Compute r k = b ! Ax k . 2. Solve Ap k = r k for p k by the basic solution solver S . 3. Add the correction, x k+1 = x k + ! p k . Clearly, ! =1 corresponds to Wilkinson’s iterative refinement method [10]. Wu and Wang [11] proposed this method for ! = h h + 1 , where h >0 (i.e., for 0< ! <1 ). They developed the method as the *Address correspondence to this author at the Faculty of Mathematics and Information Science, Warsaw University of Technology, Koszykowa 75, 00-662 Warsaw, Poland; Tel: +48222347988; Fax: +48226257460; E-mail: smok@mini.pw.edu.pl numerical integration of a dynamic system with step size h . A preliminary error analysis of the Algorithm IR( !) was given in [11] for 0< ! <1 , assuming that the extended precision is used for computing the residual vectors r k . Wu and Wang considered only Gaussian elimination as a solver S . The purpose of this paper is to analyze the convergence of this method for 0< ! <2 and to show with examples that the choice ! =1 is the best choice from the point of numerical stability. Notice that for arbitrary ! >0 , the IR( ! ) method is a stationary method (in the theory) and we have p k = A !1 r k = x * ! x k , so x k+1 ! x * = (1 ! ")( x k ! x * ), k = 0,1, …, where x * is the exact solution to Ax = b . We see that the sequence {x k } is convergent for arbitrary initial x 0 if and only if 0< ! <2 . For ! =1 (Wilkinson’s iterative refinement) x 1 will be the exact solution x * . It is interesting to check the influence on the relaxation parameter ! on numerical properties of the algorithm IR( !) , assuming that all computations are performed only in the working (fixed) precision. Throughout the paper we use only the 2-norm and assume that all computations are performed in the working (fixed) precision. We use a floating point arithmetic which satisfies the IEEE floating point standard. For two floating point numbers a and b we have f !(a!b)=(a!b)(1 + "), | " | # $ M for results in the normalized range, where ! denotes any of the elementary scalar operations +, !,*,/ and ! M is machine precision.