Assessment of Two Task Frameworks with Dependencies for Matrix Factorizations on a Multicore Architecture

In this study, we evaluate two task frameworks with dependencies for important application kernels coming from the numerical linear algebra. In this approach, the algorithms of the matrix factorization are considered, namely the tiled LU and the WZ factorizations both without pivoting. In tiled algorithms, the operations are represented as a sequence of small tasks which operate on square blocks (tiles) of the data. The dependencies among tasks are expressed as a direct acyclic graph and the runtime system runs the graph on a multicore architecture. The performance of applications based on the task dependencies is related to efficient compilers and the runtime systems. We report the performance and the scalability of two task frameworks with dependencies on the multicore architecture for the matrix factorizations. Namely, we compare OpenMP and Intel Thread Building Blocks. Our results show that the number of tiles in both factorizations always have an impact on the performance and the speedup. Both the frameworks show their suitability for efficient parallelization of such applications, although both have their own merits and flaws

Twisted factorization of a banded matrix

The twisted factorization of a tridiagonal matrix T plays an important role in inverse iteration as featured in the MRRR algorithm. The twisted structure simplifies the computation of the eigenvector approximation and can also improve the accuracy. A tridiagonal twisted factorization is given by T=M k Δ k N k where Δ k is diagonal, M k ,N k have unit diagonals, and the k-th column of M k and the k-th row of N k correspond to the k-th column and row of the identity, that is $M_{k}e_{k}=e_{k},\;e_{k}^{t}N_{k}=e_{k}^{t}$ . This paper gives a constructive proof for the existence of the twisted factorizations of a general banded matrix A. We show that for a given twist index k, there actually are two such factorizations. We also investigate the implications on inverse iteration and discuss the role of pivotin