Constructing Reliable Automatic Numerical Algorithms

Much modern research, development and operations rely on extensive computations. An important segment is the numerical solution of mathematical problems. When we turn to our calculators, or software packages like MATLAB or Mathematica, we expect the answers provided to be correct to within the error tolerances that we specify. Unfortunately, the numerical algorithms in these and other software packages can give substantially incorrect answers. Worse yet, the algorithms have no guarantees telling users which problems they will solve correctly. This talk describes a collaborative effort to construct automatic numerical algorithms with guarantees, starting with numerical algorithms for integration and function approximation. The theory arises by looking at cones of input functions. The implementation is the recently released Guaranteed Automatic Integration Library (GAIL).