27 research outputs found
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