linear systems interval parameters nonlinear dependencies automatic result verification structural steel frames Department of Biomathematics
Institute of Mathematics and Informatics at the Bulgarian Academy of Sciences
In this paper we investigate the application of a self-verified general-purpose parametric fixed-point iteration method to linear systems involving nonlinear dependencies. The inclusion method is combined with a simple interval arithmetic technique providing inner and outer bounds for the range of monotone rational functions. The arithmetic on proper and improper intervals is considered as an intermediate computational tool for eliminating the dependency problem in range computation and for obtaining inner estimations by outwardly rounded interval arithmetic. Therefore the target problems to be solved are restricted to linear systems whose input data are rational functions of uncertain parameters varying within given intervals. Supporting software tools with result verification, developed in the environment of Mathematica are presented. The discussed methodology and software tools can be applied to a wide range of practical problems leading to linear systems with dependent uncertain data, in particular applications to FEM models of mechanical structures. Beside some illustrative examples, an advanced practical application from structural engineering mechanics is solved by the discussed parametric iteration, as well as by a combination of interval techniques based on the parametric solver; the results are compared to literature data produced by other methods. A comparison of different measures of overestimation is done. AMS subject classification: 15A06, 65G20.