On the numerical solution of equations of the Abel type

Equations of this type arise naturally in the study of multidimensional branching processes; see [6], [7]. If X(t) is a 5 X 5 matrix, we are required to store 25 functions (i.e., the elements %a(t), i,j = 1, 2, ■ • • , 5) if we proceed in the usual fashion. If high order accuracy were required—say, intervals of 1Q-3 over 0 | i g 5—we would find that rapidaccess storage capacity would be exceeded. On the other hand, if we use the foregoing technique, differential approximation of order 5 would lead to the task of solving about 250 simultaneous differential equations plus those required to determine F(t). This is a simple matter for a modern computer. Furthermore, it is clear that we could use an approximation of order 10 without coming close to the storage capacity.