21 research outputs found
GR decompositions and their relations to Cholesky-like factorizations
For a given matrix, we are interested in computing GR decompositions ,
where is an isometry with respect to given scalar products. The orthogonal
QR decomposition is the representative for the Euclidian scalar product. For a
signature matrix, a respective factorization is given as the hyperbolic QR
decomposition. Considering a skew-symmetric matrix leads to the symplectic QR
decomposition. The standard approach for computing GR decompositions is based
on the successive elimination of subdiagonal matrix entries. For the hyperbolic
and symplectic case, this approach does in general not lead to a satisfying
numerical accuracy. An alternative approach computes the QR decomposition via a
Cholesky factorization, but also has bad stability. It is improved by repeating
the procedure a second time. In the same way, the hyperbolic and the symplectic
QR decomposition are related to the and a skew-symmetric Cholesky-like
factorization. We show that methods exploiting this connection can provide
better numerical stability than elimination-based approaches
Novel Modifications of Parallel Jacobi Algorithms
We describe two main classes of one-sided trigonometric and hyperbolic
Jacobi-type algorithms for computing eigenvalues and eigenvectors of Hermitian
matrices. These types of algorithms exhibit significant advantages over many
other eigenvalue algorithms. If the matrices permit, both types of algorithms
compute the eigenvalues and eigenvectors with high relative accuracy.
We present novel parallelization techniques for both trigonometric and
hyperbolic classes of algorithms, as well as some new ideas on how pivoting in
each cycle of the algorithm can improve the speed of the parallel one-sided
algorithms. These parallelization approaches are applicable to both
distributed-memory and shared-memory machines.
The numerical testing performed indicates that the hyperbolic algorithms may
be superior to the trigonometric ones, although, in theory, the latter seem
more natural.Comment: Accepted for publication in Numerical Algorithm
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