A simple alternative to the conjugate gradient (CG) method is presented; this method is developed as a special case of the more general iterated Ritz method (IRM) for solving a system of linear equations. This novel algorithm is not based on conjugacy, i.e. it is not necessary to maintain overall orthogonalities between various vectors from distant steps. This method is more stable than CG, and restarting techniques are not required. As in CG, only one matrix-vector multiplication is required per step with appropriate transformations. The algorithm is easily explained by energy considerations without appealing to the A-orthogonality in n-dimensional space. Finally, relaxation factor and preconditioning-like techniques can be adopted easily.
Exact arithmetic as a tool for convergence assessment of the IRM-CG method This is an author produced version of a paper submitted to a peer-review Exact arithmetic as a tool for convergence assessment of the IRM-CG method
AbstractUsing 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 roundoff 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.
On smoothness of solutions to structural problems and physical interpretation of weak formulation Typical structural modelling solutions are classified and described in the paper based on an approach that is more interpretative than formal. A special emphasis is placed on weak formulation and its physical interpretation which, in the authors' opinion, is lacking in the literature. The attention is also drawn to approximation errors, caused by insisting on excessive smoothness of solutions. These considerations are backed by examples. It is hoped that the paper will contribute to the understanding of the essence of approximation of practical models in civil engineering, while clearly demonstrating the power of weak formulation-the foundation of approximation procedures.
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