Miodrag Petkovic Boeken

The aim of this book is to present formulas and methods developed using complex interval arithmetic. While most of numerical methods described in the literature deal with real intervals and real vectors, there is no systematic study of methods in complex interval arithmetic. The book fills this gap. Several main subjects are considered: outer estimates for the range of complex functions, especially complex centered forms, the best approximations of elementary complex functions by disks, iterative methods for the inclusion by polynomial zeros including their implementation on parallel computers, the analysis of numerical stability of iterative methods by using complex interval arithmetic and numerical computation of curvilinear integrals with error bounds. Mainly new methods are presented developed over the last years, including a lot of very recent results by the authors some of which have not been published before.

The challenge of solving nonlinear equations and systems is significant in applied mathematics and various engineering fields, physics, computer science, astronomy, and finance. The extensive bibliography and contributions from prominent mathematicians highlight the contemporary interest in this area. While digital computers have enabled effective implementation of numerous numerical methods, practical applications face challenges such as computational efficiency, the development of iterative methods with rapid convergence in the presence of multiple solutions, control of rounding errors, and establishing error bounds for approximate solutions. Additionally, it is crucial to determine computationally verifiable initial conditions that ensure reliable convergence. This study focuses on identifying and analyzing initial conditions that guarantee the convergence of iterative methods for equations of the form f(z) = 0. The traditional approach relies on asymptotic convergence analysis, which typically requires strong assumptions about differentiability and derivative bounds across a broad domain.