In this paper we study the performance of two classical dense linear algebra algorithms, the LU and the QR factorizations, on multilevel hierarchical platforms. We note that we focus on multilevel QR factorization, and give a brief description of the multilevel LU factorization. We first introduce a performance model called Hierarchical Cluster Platform (Hcp), encapsulating the characteristics ...

Complex symmetric matrices whose real and imaginary parts are positive definite are shown to have a growth factor bounded by 2 for LU factorization. This result adds to the classes of matrix for which it is known to be safe not to pivot in LU factorization. Block LDLT factorization with the pivoting strategy of Bunch and Kaufman is also considered, and it is shown that for such matrices only 1×...

AILU: A Preconditioner Based on the Analytic Factorization of the Elliptic Operator Martin J. Gander and Frederic Nataf Department of Mathematics, McGill University, Montreal, Canada and CMAP, CNRS UMR7641, Ecole Polytechnique, Palaiseau, France We investigate a new type of preconditioner for large systems of linear equations stemming from the discretization of elliptic symmetric partial differ...

We have recently developed a new program analysis strategy called fractal symbolic analysis that addresses some of limitations of techniques such as dependence analysis. In this paper, we show how fractal symbolic analysis can be used to convert between left-looking and right-looking versions of three kernels of central importance in computational science: Cholesky factorization, LU factorizati...

In this paper, the author gives a simpler alternative approach to the LU factorization and 1-banded factorization of the Vandermondematrix, and obtains explicit formulas of the triangular factors and 1-bandedmatrices by using symmetric functions. © 2004 Elsevier B.V. All rights reserved.

A singular matrix A may have more than one LU factorizations. In this work the set of all LU factorizations of A is explicitly described when the lower triangular matrix L is nonsingular. To this purpose, a canonical form of A under left multiplication by unit lower triangular matrices is introduced. This canonical form allows us to characterize the matrices that have an LU factorization and to...

Due to the evolution of massively parallel computers towards deeper levels of parallelism and memory hierarchy, and due to the exponentially increasing ratio of the time required to transfer data, either through the memory hierarchy or between different compute units, to the time required to compute floating point operations, the algorithms are confronted with two challenges. They need not only...