The ODEON computer model for acoustics in large rooms is intended for use both in design (by predicting room acoustical indices quickly and easily) and in research (by forming the basis of an auralization system and allowing study of various room acoustical phenomena). These conflicting demands preclude the use of both ``pure'' image source and ``pure'' particle tracing methods. A hybrid model has been developed, in which rays discover potential image sources up to a specified order. Thereafter, the same ray tracing process is used in a different way to rapidly generate a dense reverberant decay. In this paper the computational model is described. Particular attention is paid to alternative methods of implementing the reverberant tail, and to the problems that arise when joining early and late parts of a reflectogram generated with different algorithms. A companion paper presents the features of the model as implemented.