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PROCEEDINGS OF THE SEVENTH INTERNATIONAL CONFERENCE ON ENGINEERING COMPUTATIONAL TECHNOLOGY
Numerical Methods for Singular Linear Differential Systems
N. Thome, D. Ginestar, E. Sánchez and C. Coll
Institute of Multidisciplinary Mathematics, Universidad Politécnica de Valencia, Spain
, "Numerical Methods for Singular Linear Differential Systems", in , (Editors), "Proceedings of the Seventh International Conference on Engineering Computational Technology", Civil-Comp Press, Stirlingshire, UK, Paper 24, 2010. doi:10.4203/ccp.94.24
Keywords: singular systems, numerical methods, shuffle algorithm, Drazin inverse.
Singular systems arise naturally in many fields, robotics, neural delay systems, network theory, etc. Specifically, they are particularly well suited to describe large-scale systems because they preserve physical sparsity. For more information on singular systems and their applications see  and the references therein.
We consider an initial-value problem given by a singular system and an admissible initial condition. Then, we obtain the solution of this problem using numerical techniques. An important requisite of a numerical method for the solution of singular systems is that it preserves the structure of the system and takes advantage of sparsity. This solution involves Drazin inverses of the coefficient matrices of the system. So, we propose numerical techniques which use as the main operation the product Drazin inverse by a vector. This operation can be efficiently computed by several algorithms based on the product matrix by vector. This methodology preserves the sparsity of the matrices and this is important for large systems of equations.
In this paper, two different strategies are proposed. Using the explicit solutions for a solvable singular system, we propose a method of approximation of these solutions by standard explicit numerical techniques. An alternative way to solve the initial-value problem is to use an associated ordinary differential problem and apply a numerical algorithm using the discretization approach. The performance of these methods is studied obtaining the numerical solution of a singular system modeling an electric circuit associated with an ideal transformer. The obtained results show that the proposed methods work well and the same methodology can be applied to obtain higher order numerical methods.
The outline of the paper is as follows. Section 2 presents some preliminaries of singular systems and introduces the concept of admissible initial conditions for a solvable singular system. In section 3, a numerical method is presented for the solution of an initial value problem associated with a singular system and a possible implementation of the product Drazin inverse by vector is given. In section 4, numerical results are presented for an electric circuit associated with an ideal transformer. Finally, the main conclusions of the paper are given in section 4.
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