Solution of linear and nonlinear algebraic problems with sharp, guaranteed bounds

In this paper new methods for solving algebraic problems with high accuracy are described. They deliver bounds for the solution of the given problem with an automatic verification of the correctness. Examples of such problems are systems of linear equations, algebraic eigenvalue problems, nonlinear systems, polynomial zeros, evaluation of arithmetic expressions, linear, quadratic and convex programming and others. The new methods apply for these problems over the space of real numbers, complex numbers as well as real intervals and complex intervals