Automatic numerical algorithms attempt to provide approximate solutions that differ from exact solutions by no more than a user-specified error tolerance. The computational cost is often determined emph{adaptively} by the algorithm based on the funct
ion values sampled. While adaptive, automatic algorithms are widely used in practice, most lack emph{guarantees}, i.e., conditions on input functions that ensure that the error tolerance is met. This article establishes a framework for guaranteed, adaptive, automatic algorithms. Sufficient conditions for success and two-sided bounds on the computational cost are provided in Theorems ref{TwoStageDetermThm} and ref{MultiStageThm}. Lower bounds on the complexity of the problem are given in Theorem ref{complowbd}, and conditions under which the proposed algorithms have optimal order are given in Corollary ref{optimcor}. These general theorems are illustrated for univariate numerical integration and function recovery via adaptive algorithms based on linear splines. The key to these adaptive algorithms is performing the analysis for emph{cones} of input functions rather than balls. Cones provide a setting where adaption may be beneficial.
