Exact arithmetic as a tool for convergence assessment of the IRM-CG method

Abstract: Using exact computer arithmetic, it is possible to determine the (exact) solution of a numerical model without rounding error. For such purposes, a corresponding system of equations should be exactly defined, either directly or by rationalisation of numerically given input data. In the latter case there is an initial round off error, but this does not propagate during the solution process. If this system is first exactly solved, then by the floating-point arithmetic, convergence of the numerical method is easily followed. As one example, IRM-CG, a special case of the more general Iterated Ritz method and interesting replacement for a standard or preconditioned CG, is verified. Further, because the computer demands and execution time grow enourmously with the number of unknowns using this strategy, the possibilities for larger systems are also provided.