the next point it is sufficient to get the derivatives of x and u, that is, the system to solve is now This is a semi-explicit DAE of index 1. Another set of similar equations may be obtained starting from and a sign for x. DAEs also naturally occur in the modelling of circuits with non-linear devices. Modified nodal analysis employing DAEs is used for example in the ubiquitous SPICE family of numeric circuit simulators. Similarly, Fraunhofer's Analog Insydes Mathematica package can be used to derive DAEs from a netlist and then simplify or even solve the equations symbolically in some cases. It is worth noting that the index of a DAE (of a circuit) can be made arbitrarily high by cascading/coupling via capacitors operational amplifiers with positive feedback. Semi-explicit DAE of index 1 DAE of the form are called semi-explicit. The index-1 property requires that g is solvable for y. In other words, the differentiation index is 1 if by differentiation of the algebraic equations for t an implicit ODE system results, which is solvable for if Every sufficiently smooth DAE is almost everywhere reducible to this semi-explicit index-1 form. Numerical treatment of DAE and applications Two major problems in solving DAEs are index reduction and consistent initial conditions. Most numerical solvers require ordinary differential equations and algebraic equations of the form It is a non-trivial task to convert arbitrary DAE systems into ODEs for solution by pure ODE solvers. Techniques which can be employed include Pantelides algorithm and dummy derivative index reduction method. Alternatively, a direct solution of high-index DAEs with inconsistent initial conditions is also possible. This solution approach involves a transformation of the derivative elements through orthogonal collocation on finite elements or direct transcription into algebraic expressions. This allows DAEs of any index to be solved without rearrangement in the open equation form Once the model has been converted to algebraic equation form, it is solvable by large-scale nonlinear programming solvers (see APMonitor). Tractability Several measures of DAEs tractability in terms of numerical methods have developed, such as differentiation index, perturbation index, tractability index, geometric index, and the Kronecker index''. Structural analysis for DAEs We use the -method to analyze a DAE. We construct for the DAE a signature matrix , where each row corresponds to each equation and each column corresponds to each variable . The entry in position is , which denotes the highest