The increasing interest in product networks (PNs) as a method of combining desirable properties of component networks, has prompted a need for the general study of the algorithmic issues related to this important class of interconnection networks. In this paper we present unified parallel algorithms for Gaussian elimination, with partial and complete pivoting, on product networks. A parallel al...

Low-rank matrices arise in many scientific and engineering computations. Both computational and storage costs of manipulating such matrices may be reduced by taking advantages of their low-rank properties. To compute a low-rank approximation of a dense matrix, in this paper, we study the performance of QR factorization with column pivoting or with restricted pivoting on multicore CPUs with a GP...

Consider the Gaussian Elimination algorithm with the well-known Partial Pivoting strategy for improving numerical stability (GEPP). Vavasis proved that the problem of determining the pivot sequence used by GEPP is log space-complete for P, and thus inherently sequential. Assuming P 6 = NC, we prove here that either the latter problem cannot be solved in parallel time O(N 1=2?) or all the proble...

Introduction In this paper we present a modification of Gauss-Huard's method for solving dense linear systems that allows an efficient implementation on machines with a hierarchical memory structure. GaussHuard's method resembles Gauss-Jordan's method in the fact that it reduces the given system by elementary transformations to a diagonal system and it resembles regular Gaussian elimination in ...

An active area of research for linear programming is to construct initial Simplex tableau that is close to the optimal solution, and to improve its pivoting selection strategies efficiently. In this paper, we present a new approach to the problem of initialization and pivoting rules: the algorithm is free from any artificial variables and artificial constraints, known as the big-M methods. By s...

We describe the implementation and performance of a novel fill-minimization ordering technique for sparse LU factorization with partial pivoting. The technique was proposed by Gilbert and Schreiber in 1980 but never implemented and tested. Like other techniques for ordering sparse matrices for LU with partial pivoting, our new method preorders the columns of the matrix (the row permutation is c...

The simplex algorithm is among the most widely used algorithms for solving linear programs in practice. Most pivoting rules are known, however, to need an exponential number of steps to solve some linear programs. No non-polynomial lower bounds were known, prior to this work, for Cunningham’s Least Recently Considered rule [5], which belongs to the family of history-based rules. Also known as t...

abstract This note is concerned with the accuracy of the solution of nearly uncoupled Markov chains by a direct method based on the LU decomposition. It is shown that plain Gaussian elimination may fail in the presence of rounding errors. A modiication of Gaussian elimination with diagonal pivoting as well as corrections of small pivots by sums of oo-diagonal elements in the pivoting columns is...

The QR decomposition with column pivoting (QRP) of a matrix is widely used for rank revealing. The performance of LAPACK implementation (DGEQP3) of the Householder QRP algorithm is limited by Level 2 BLAS operations required for updating the column norms. In this paper, we propose an implementation of the QRP algorithm using a distribution of the matrix columns in a round-robin fashion for bett...