Design and analysis of efficient algorithms for solving nonlinear equations

The main topics of interest of the proposed project are: the design and development of new numerical algorithms of high efficiency for solving nonlinear equations with a special attention paid to the polynomial equations, the development of interval methods in complex interval arithmetic with the automatic error control, the study of the computational efficiency, the analysis of convergence rate, stating initial conditions for the fast and guaranteed convergence, the choice of initial approximations, the analysis of the numerical stability by the use of interval arithmetic, the acceleration of the convergence, and the implementation on parallel computers. These algorithms are most often of the iterative nature. The investigation will include the construction of very efficient methods with memory for solving nonlinear equations of the form f(x)=0, for the first time in the literature. A special attention will be paid to the determination of the zeros of algebraic polynomials and analytic functions, including the case of multiple roots. The majority of iterative methods finds polynomial roots simultaneously (parallel mod), which is very convenient for the application on parallel computers.