Analiza procesnih i računskih iteracija primenom savremenih računarskih aritmetika

Abstract

The proposed theme relates to the field of application of modern computer arithmetics in the analysis of process performance and computational iterations, where the concept of the modern computer arithmetic applies to multiple-precision arithmetic and interval arithmetic involved in the new standard IEEE 754 in 2008. Advance computer arithmetic, first of all interval arithmetic and multi-precision arithmetic, are employed in the the dissertation for the analysis of matrix models in the design iteration process software and iterative numerical computations. These are important and current research topics from the point at which the application is working in the world. Research in this area have led to the development and analysis of algorithms to control the accuracy, optimality, rate calculations and other performance aspects of various processes such as designing software and hardware, designing industrial products, transportation optimization, modeling systems, the implementation of numerical algorithms and others. The first part of dissertation is devoted to engineering design and development of new products, which are either industrial products, technical innovations, hardware or software, often contain a very complex set of relationships among many coupled tasks. Controlling, redesigning and identifying features of these tasks can be usefully performed by a suitable model based on the design structure matrix in an iteration procedure. The proposed interval matrix model of design iteration controls and predicts slow and rapid convergence of iteration work on tasks within a project. A new model is based on Perron- Frobenius theorem and interval linear algebra where intervals and interval matrices are employed instead of real numbers and real matrices. In this way a more relaxed quantitative estimation of tasks is achieved and the presence of undetermined quantities is allowed to a certain extent. The presented model is demonstrated in the example of simplified software development process. An additional contribution in this dissertation is the ranking of tasks within a design mode using components of eigenvalue vector corresponding to the spectral radius of design structure matrix. The second part deals with computational efficiency of numerical iterative algorithms. Computational cost of iterative procedures for the implementation of basic arithmetic operations in multi-precision arithmetic are studied and later applied for the analysis of computational efficiency of the iterative methods for solving nonlinear equations. A new approach that deals with the weights of employed arithmetic operations, involved in the realization of these algorithms, is proposed. This enables a precise ranking of considered root-finding algorithms of different structure, especially in a regime of variable (dynamic) precision of applied multi-precision arithmetic.