Block lower triangular matrices and block upper triangular matrices are popular preconditioners for 2×2 block matrices. In this note we show that a block lower triangular preconditioner gives the same spectrum as a block upper triangular preconditioner and that the eigenvectors of the two preconditioned matrices are related.

Block lower triangular matrices and block upper triangular matrices are popular preconditioners for 2×2 block matrices. In this note we show that a block lower triangular preconditioner gives the same spectrum as a block upper triangular preconditioner and that the eigenvectors of the two preconditioned matrices are related.

We present a new network diagram authoring tool, Dunnart, that provides continuous network layout. It continuously adjusts the layout in response to user interaction, while still maintaining the layout style and, where reasonable, the current layout topology. The diagram author uses placement constraints, such as alignment and distribution, to tailor the layout style and can guide the layout by...

High quality meshes are crucial for the solution of partial differential equations (PDEs) via the finite element method (or other PDE solvers). The accuracy of the PDE solution, and the stability and conditioning of the stiffness matrix depend upon the mesh quality. In addition, the mesh must be untangled in order for the finite element method to generate physically valid solutions. Tangled mes...

The efficiency of desktop publishing is severely limited by the lack of sophisticated automatic document layout systems. State-of-the-art algorithms either require the input to be written in a description language such as HTML and LATEX, or to be a manually designed layout template. However, description languages are ill-suited to express layout problems with weak semantics and layout templates...

In the fuzzy arithmetic, the definitions of addition and multiplication of fuzzy numbers are based on Zadeh’s extension principle. From theoretical and practical points of view, this multiplication of fuzzy numbers owns several unnatural properties. Recently, to avoid this shortcoming, a new multiplicative operation of product type is introduced, the so-called cross-product of fuzzy numbers. Th...