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 any rounding error. For such purposes, a corresponding system of equations should be exactly defined, either directly or by rationalising the 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 exactly solved first and then using floating-point arithmetic, the convergence of the numerical method easily follows. As an example, the IRM–CG, which is an alternative to the Conjugate Gradient (CG) method and a special case of the more general Iterated Ritz Method (IRM), is verified. The method is not based on conjugacy; therefore, restarting strategies are not required, while an overrelaxation factor and preconditioning like techniques could be easily adopted. The exact arithmetic approach is introduced by means of a simple example and is then applied to small structural engineering problems. The perturbation of the displacement increment and the different condition numbers of the system matrix are used to check the stability of the algorithm. Interestingly, a large difference in the number of steps between the exact and numerical approaches is detected, even for well-conditioned systems. According to the tests, the IRM-CG may be considered to be stable and useful for not well-posed or well-posed but ill-conditioned models. Because the computer demands and execution time grow enormously with the number of unknowns using this strategy, three possibilities for larger systems are also provided.