Multistep Discretization Schemes for DAEs

We consider system (5.1.2) and an equidistant partition of the time axis with the step size h = tn — n-i Corresponding to Sec. 4 we denote the numerical approximation to the solution at time tn by Xn. We use the notation Xn for the numerical approximation to x(tn)- When we want to express its dependency on Xn we write Sh xn) instead. 

After normalizing = 1 we rewrite the implicit multistep discretization scheme (4.1.12) as 

This is a set of nonlinear equations for the unknowns yn and An, which can be solved by Newton s method. 

Viewing DAEs as the limiting case of singularly perturbed ODEs of the form 

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