The Cauchy problem for a stiff system of ODEs unresolved with respect to the derivative often arises in chemical kinetics, mechanical engineering, and other important applications. The two-stage L-stable Rosenbrock-type method is derived. An integration algorithm of alternating stepsize is designed based on the method, aimed at
solving implicit stiff systems of ODEs. Numerical results confi rming the effi ciency of the new algorithm are given.
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