The self-validating numerics-a new tool for computer assisted proofs of nonlinear problems

The self-validating numerical method is surveyed for application to nonlinear problems. By taking into account the effect of rounding error, this method provides a method of computer assisted proofs. The approach to this problem of L. V. Kantrovich and G. P. Akilov (1964) is surveyed. His method is based on his convergence theorem of Newton´s method and can be seen as an a posteriori error estimation method. M. Urabe´s (1965, 1966) approach to this problem is discussed. He treated practical nonlinear differential equations such as the Van der Pol equation and the Duffing equation and proved the existence of their periodic and quasi-periodic solutions by the self-validating numerics. Generalizations and abstraction of Urabe´s method to more general functional equations are also discussed. Methods for rigorous estimation of rounding errors are surveyed