The stable ATA-orthogonal s-step Orthomin(k) algorithm with the CADNA library

Abstract

The major drawback of the s-step iterative methods for nonsymmetric linear systems of equations is that, in the floating-point arithmetic, a quick loss of orthogonality of s-dimensional direction subspaces can occur, and consequently slow convergence and instability in the algorithm may be observed as s gets larger than 5. In [18], Swanson and Chronopoulos have demonstrated that the value of s in the s-step Orthomin(k) algorithm can be increased beyond s=5 by orthogonalizing the s direction vectors in each iteration, and have shown that the ATA-orthogonal s-step Orthomin(k) is stable for large values of s (up to s=16). The subject of this paper is to show how by using the CADNA library, it is possible to determine a good value of s for ATA-orthogonal s-step Orthomin(k), and during the run of its code to detect the numerical instabilities and to stop the process correctly, and to restart the ATA-orthogonal s-step Orthomin(k) in order to improve the computed solution. Numerical examples are used to show the good numerical properties.