Lanczos method for solving a system of linear equations is well known. It is derived from a generalization of the method of moments and one of its main interests is that it provides the exact answer in at most n steps where n is the dimension of the system. Lanczos method can be implemented via several recursive algorithms known as Orthodir, Orthomin, Orthores, Biconjugate gradient,... In this paper, we show that all these procedures can be explained within the framework of formal orthogonal polynomials....
PERMON (Parallel, Efficient, Robust, Modular, Object-oriented, Numerical) is a newly emerging collection of software libraries, uniquely combining Quadratic Programming (QP) algorithms and Domain Decomposition Methods (DDM). Among the main applications are contact problems of mechanics. This paper gives an overview of PERMON and selected ingredients improving scalability, demonstrated by numerical experiments.
The class of sparse companion matrices was recently characterized in terms of unit Hessenberg matrices. We determine which sparse companion matrices have the lowest bandwidth, that is, we characterize which sparse companion matrices are permutationally similar to a pentadiagonal matrix and describe how to find the permutation involved. In the process, we determine which of the Fiedler companion matrices are permutationally similar to a pentadiagonal matrix. We also describe how to find a Fiedler...