[SADE] A Maple package for the Symmetry Analysis of Differential Equations

We present the package SADE (Symmetry Analysis of Differential Equations) for the determination of symmetries and related properties of systems of differential equations. The main methods implemented are: Lie, nonclassical, Lie-B\"acklund and potential symmetries, invariant solutions, first-integrals, N\"other theorem for both discrete and continuous systems, solution of ordinary differential equations, reduction of order or dimension using Lie symmetries, classification of differential equations, Casimir invariants, and the quasi-polynomial formalism for ODE's (previously implemented in the package QPSI by the authors) for the determination of quasi-polynomial first-integrals, Lie symmetries and invariant surfaces. Examples of use of the package are given

A domain decomposing parallel sparse linear system solver

The solution of large sparse linear systems is often the most time-consuming part of many science and engineering applications. Computational fluid dynamics, circuit simulation, power network analysis, and material science are just a few examples of the application areas in which large sparse linear systems need to be solved effectively. In this paper we introduce a new parallel hybrid sparse linear system solver for distributed memory architectures that contains both direct and iterative components. We show that by using our solver one can alleviate the drawbacks of direct and iterative solvers, achieving better scalability than with direct solvers and more robustness than with classical preconditioned iterative solvers. Comparisons to well-known direct and iterative solvers on a parallel architecture are provided.Comment: To appear in Journal of Computational and Applied Mathematic

An Optimized and Scalable Eigensolver for Sequences of Eigenvalue Problems

In many scientific applications the solution of non-linear differential equations are obtained through the set-up and solution of a number of successive eigenproblems. These eigenproblems can be regarded as a sequence whenever the solution of one problem fosters the initialization of the next. In addition, in some eigenproblem sequences there is a connection between the solutions of adjacent eigenproblems. Whenever it is possible to unravel the existence of such a connection, the eigenproblem sequence is said to be correlated. When facing with a sequence of correlated eigenproblems the current strategy amounts to solving each eigenproblem in isolation. We propose a alternative approach which exploits such correlation through the use of an eigensolver based on subspace iteration and accelerated with Chebyshev polynomials (ChFSI). The resulting eigensolver is optimized by minimizing the number of matrix-vector multiplications and parallelized using the Elemental library framework. Numerical results show that ChFSI achieves excellent scalability and is competitive with current dense linear algebra parallel eigensolvers.Comment: 23 Pages, 6 figures. First revision of an invited submission to special issue of Concurrency and Computation: Practice and Experienc

Factorization of Z-homogeneous polynomials in the First (q)-Weyl Algebra

We present algorithms to factorize weighted homogeneous elements in the first polynomial Weyl algebra and $q$-Weyl algebra, which are both viewed as a $\mathbb{Z}$-graded rings. We show, that factorization of homogeneous polynomials can be almost completely reduced to commutative univariate factorization over the same base field with some additional uncomplicated combinatorial steps. This allows to deduce the complexity of our algorithms in detail. Furthermore, we will show for homogeneous polynomials that irreducibility in the polynomial first Weyl algebra also implies irreducibility in the rational one, which is of interest for practical reasons. We report on our implementation in the computer algebra system \textsc{Singular}. It outperforms for homogeneous polynomials currently available implementations dealing with factorization in the first Weyl algebra both in speed and elegancy of the results.Comment: 26 pages, Singular implementation, 2 algorithms, 1 figure, 2 table