Automatic and adaptive approximation, optimization, or integration of functions in a cone with guarantee of accuracy is a relatively new paradigm. Our purpose is to create an open-source MATLAB package, Guaranteed Automatic Integration Library (GAIL)
, following the philosophy of reproducible research and sustainable practices of robust scientific software development. For our conviction that true scholarship in computational sciences are characterized by reliable reproducibility, we employ the best practices in mathematical research and software engineering known to us and available in MATLAB. This document describes the key features of functions in GAIL, which includes one-dimensional function approximation and minimization using linear splines, one-dimensional numerical integration using trapezoidal rule, and last but not least, mean estimation and multidimensional integration by Monte Carlo methods or Quasi Monte Carlo methods.
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.
