A reduction algorithm for matrices depending on a parameter

International audienceIn this article, we study square matrices perturbed by a parameter $\epsilon$. An efficient algorithm computing the $\epsilon$-expansion of the eigenvalues in formal Laurent-Puiseux series is provided, for which the computation of the characteristic polynomial is not required. We show how to reduce the initial matrix so that the Lidskii-Edelman-Ma perturbation theory can be applied. We also explain why this approach may simplify the perturbed eigenvector problem. The implementation of the algorithm in the computer algebra system Maple has been used in a quantum mechanics context to diagonalize some perturbed matrices and is available

Higher-Order linear differential systems with truncated coefficients

International audienceWe consider the following problem: given a linear differential system with formal Laurent series coefficients, we want to decide whether the system has non-zero Laurent series solutions, and find all such solutions if they exist. Let us also assume we need only a given positive integer number l of initial terms of these series solutions. How many initial terms of the coefficients of the original system should we use to construct what we need? Supposing that the series coefficients of the original systems are represented algorithmically, we show that these questions are undecidable in general. However, they are decidable in the scalar case and in the case when we know in advance that a given system has an invertible leading matrix. We use our results in order to improve some functionality of the Maple [17] package ISOLDE [11]