[en] A general purpose algorithm for integrating nonlinear systems of ordinary differential equations is presented. A number of specific applications are discussed, such as mixed linear/nonlinear problems, mixed transient/steady-state problems, and substructuring. The problem of integrating past singularities is briefly discussed. It is concluded that while the algorithm is easily derived in an elementary form, its efficient use in some of the more complicated problems requires a number of special considerations in the derivation and in the programming. Each specific application can probably benefit from custom treatment. Nevertheless, each application ultimately distills to the following essence: (1) The incremental predictor and corrector phases employ substantially the same steps. (2) The predictor phase is a modified implicit extrapolation of the current rates. (3) The corrector phase is a modified Newton's method based on the residual of a vector function. (4) In partially linear problems one can save steps by matrix partitioning. (5) The frequency of updates of the tangent matrix is arbitrary, and the calculational efficiency can be optimized by simultaneously varying the size of the time step and the frequency of the matrix updates