This paper presents a complete algorithm for the evaluation and control of error in radiosity calculations. Providing such control is both extremely important for industrial applications and one of the most challenging issues remaining in global illumination research. In order to control the error, we need to estimate the accuracy of the calculation while computing the energy exchanged between two objects. Having this information for each radiosity interaction allows to allocate more resources to refine interactions with greater potential error, and to avoid spending more time to refine interactions already represented with sufficient accuracy. Until now, the accuracy of the computed energy exchange could only be approximated using heuristic algorithms. This paper presents the first exhaustive algorithm to compute fully reliable upper and lower bounds on the energy being exchanged in each interaction. This is accomplished by computing first and second derivatives of the radiosity function where appropriate, and making use of two concavity conjectures. These bounds are then used in a refinement criterion for hierarchical radiosity, resulting in a global illumination algorithm with complete control of the error incurred. Results are presented, demonstrating the possibility to create radiosity solutions with guaranteed precision. We then extend our algorithm to consider linear bounding functions instead of constant functions, thus creating simpler meshes in regions where the function is concave, without loss of precision. Our experiments show that the computation of radiosity derivatives along with the radiosity values only requires a modest extra cost, with the advantage of a much greater precision.