Certification of a Numerical Result: Use of Interval Arithmetic and Multiple Precision

Using floating-point arithmetic to solve a numerical problem yields a computed result, which is an approximation of the exact solution because of roundoff errors. In this paper, we present an approach to certify the computed solution. Here, ”certify” means computing a guaranteed enclosure of the error between the computed, approximate, result and the exact, unknown result. We discuss an iterative refinement method: classically, such methods aim at computing an approximation of the error and they add it to the previous result to improve its accuracy. We add two ingredients: interval arithmetic is used to get an enclosure of the error instead of an approximation, and multiple precision is used to reach higher accuracy. We exemplify this approach on the certification of the solution of a linear system.