Symmetric indefinite factorization of quasidefinite matrices

Matrices with special structures arise in numerous applications. In some cases, such as quasidefinite matrices or their generalizations, we can exploit this special structure. If the matrix H is quasidefinite, we propose a new variant of the symmetric indefinite factorization. We show that linear system Hz = b, H quasidefinite with a special structure, can be interpreted as an equilibrium system. So, even if some blocks in H are ill--conditioned, the important part of solution vector z can be accurately computed. In the case of a generalized quasidefinite matrix, we derive bounds on number of its positive and negative eigenvalues

Estimates for the spectral condition number of cardinal B-spline collocation matrices

The famous de Boor conjecture states that the condition of the polynomial B-spline collocation matrix at the knot averages is bounded independently of the knot sequence, i.e., it depends only on the spline degree. For highly nonuniform knot meshes, like geometric meshes, the conjecture is known to be false. As an effort towards finding an answer for uniform meshes, we investigate the spectral condition number of cardinal B-spline collocation matrices. Numerical testing strongly suggests that the conjecture is true for cardinal B-splines

Three-Level Parallel J-Jacobi Algorithms for Hermitian Matrices

The paper describes several efficient parallel implementations of the one-sided hyperbolic Jacobi-type algorithm for computing eigenvalues and eigenvectors of Hermitian matrices. By appropriate blocking of the algorithms an almost ideal load balancing between all available processors/cores is obtained. A similar blocking technique can be used to exploit local cache memory of each processor to further speed up the process. Due to diversity of modern computer architectures, each of the algorithms described here may be the method of choice for a particular hardware and a given matrix size. All proposed block algorithms compute the eigenvalues with relative accuracy similar to the original non-blocked Jacobi algorithm.Comment: Submitted for publicatio