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A class of linear implicit methods for numerical solution of stiff ODE's is presented. These require only occasional calculation of the Jacobian matrix while maintaining stability. Especially, an effective second order stable algorithm with automatic stepsize control is designed and tested. 1. Introduction. During the last decade there has been a considerable amount of research on the numerical integration of stiff systems of ODE's. This work indicates that all efficient integration methods for such problems are implicit in character. This is due to the fact that only such methods have the required stability properties. Thus, the practical problem is not the stability restrictions, but the implicitness the need to avoid these give rise to. The relevant question is now, what is the cheapest type of implicitness we have to require. Mainly, two different approaches to the implicitness can be found in the litera- ture. The first approach involves the numerical solution of nonlinear algebraic equations by the simplified Newton iteration. The simplification consists of treating the iteration matrix as piecewise constant (which means the use of an approximate Jacobian matrix). Examples of such an approach are semi-implicit Runge-Kutta formulas in Ne'rsett (4) and the formulas based on backward-differences in Gear (3). Among recent methods proposed for numerical solution of stiff ODE's are the class of modified Rosenbrock methods introduced in Wolfbrandt (6). When solving the system of equations

An attempt to avoid exact Jacobian and nonlinear equations in the numerical solution of stiff differential equations