In the numerical solution of large stiff systems of ordinary differential equations, matrix operations associated with the solution of linear equations often dominate the solution time. A matrix factorization is suggested that will allow efficient updating after a change in stepsize or order. This updating technique is shown to be applicable to a wide variety of methods for stiff systems including multistep methods, Runge-Kutta methods, and methods using a rational function of a matrix The technique is particularly useful if the system is large and the Jacobian is dense Numerical results are included to illustrate the use of the technique.