Floating-point arithmetic is the fast way to perform scientific and engineering calculations. Today, individual floating-point operations are maximally accurate as a rule. However, after only two or just a few operations, the result can be completely wrong. Computers now carry out up to 1011 floating-point operations in a second. Thus, particular attention must be paid to the reliability of the computed results. In recent years, techniques have been developed in numerical analysis which make it possible for the computer itself to verify the correctness of computed results for numerous problems and applications. Moreover, the computer frequently establishes the existence and uniqueness of the solution in this way. For example, a verified solution of a system of ordinary differential equations is just as valid as a solution obtained by a computer algebra system, which as a rule still requires a valid formula evaluation. Furthermore, the numerical routine remains applicable even if the problem does not have a closed-form solution.
