Validated numerics, or rigorous computation, verified computation, reliable computation, numerical verification (German: Zuverlässiges Rechnen) is numerics including mathematically strict error (rounding error, truncation error, discretization error) evaluation, and it is one field of numerical analysis.
Validated numerics were used by Warwick Tucker in order to solve the 14th of Smale's problems,[1] and today it is recognized as a powerful tool for the study of dynamical systems.
[2] Computation without verification may cause unfortunate results.
Breuer–Plum–McKenna used the spectrum method to solve the boundary value problem of the Emden equation, and reported that an asymmetric solution was obtained.
This is a rare case, but it tells us that when we want to strictly discuss differential equations, numerical solutions must be verified.