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The modified Newton method in the solution of stiff ordinary differential equations. (English) Zbl 0734.65060
When studying a stiff system of ordinary differential equations (1) \(y'=f(t,y),\) \(f: R\times R^ n\to R,\) \(f\in C^ 1\) by means of either implicit multistep methods or implicit Runge-Kutta methods a system of nonlinear equations must be solved. It is proved that near a smooth solution of (1) the modified Newton method converges to the locally unique solution. At the same time the Jacobian is supposed to be essentially negative dominant and slowly varying.
These assumptions are weaker than the ones before (e.g. the validity of a Lipschitz condition or the existence of a globally unique solution of the system of nonlinear equations).
Reviewer: M.Bartušek (Brno)

MSC:
65L05 Numerical methods for initial value problems involving ordinary differential equations
65H10 Numerical computation of solutions to systems of equations
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
34E13 Multiple scale methods for ordinary differential equations
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