\(G\)-matrices for algebraically stable general linear methods (Q849279)

The aim of the paper is to find a \(G\)-matrix explicitly, in terms of the corresponding generalised eigenvectors, using an entirely different technique based on the theory of positive real control systems. This technique is feasible for a larger class of general linear methods (GLMs) than those that are algebraically stable. The first section is an introduction in nature. The second section details the steps connecting the condition of algebraical stability of a GLM \((A,U,B,V)\) with a generalized eigenproblem, in terms of the coefficient matrices \(A,U,B\) and \(V\). It is also shown that candidate \(G\)-matrices may be constructed from the eigenspaces of a generalized eigenproblem. The third section contains the details of the construction presented in the second section. For a simplified model problem, it is shown that such candidate \(G\)-matrices are Hermitian, provided that they are constructed from eigenspaces corresponding to eigenvalues inside the open unit disk. This choice of eigenspaces for the model problem suggests an outline algorithm for constructing candidate \(G\)-matrices for GLMs. Applications of the construction technique for several GLMs, including one method where the \(G\)-matrix is not unique, are within the fourth section.