We report on an extensive experiment to compare an iterative solver preconditioned by several versions of incomplete LU factorization with a sparse direct solver using LU factorization with partial pivoting. Our test suite is 24 nonsymmetric matrices drawn from benchmark sets in the literature. On a few matrices, the best iterative method is more than 5 times as fast and more than 10 times as m...

Recent research shows that structured matrices such as Toeplitz and Hankel matrices can be transformed into a diierent class of structured matrices called Cauchy-like matrices using the FFT or other trigonometric transforms. Gohberg, Kailath and Olshevsky demonstrate numerically that their fast variation of the straightforward Gaussian elimination with partial pivoting (GEPP) procedure on Cauch...

The simplex algorithm requires artificial variables for solving linear programs, which lack primal feasibility at the origin point. We present a new general-purpose solution algorithm, called Push-and-Pull, which obviates the use of artificial variables. The algorithm consists of preliminaries for setting up the initialization followed by two main phases. The Push Phase develops a basic variabl...

Kitahara and Mizuno (2011a) obtained an upper bound for the number of different solutions generated by the primal simplex method with Dantzig’s (the most negative) pivoting rule. In this paper, we obtain an upper bound with any pivoting rule which chooses an entering variable whose reduced cost is negative at each iteration. The bound is applied to special linear programming problems. We also g...

The use of threshold pivoting with the purpose to reduce fill-in during sparse Gaussian elimination has been generally acknowledged. Here we describe the application of threshold pivoting in dense Gaussian elimination for improving the performance of a parallel implementation. We discuss the effect on the numerical stability and conclude that the consequences are only of minor importance as lon...

In this paper we discuss the use of block principal pivoting and predictor-corrector methods for the solution of large-scale linear least squares problems with nonnegative variables (NVLSQ). We also describe two implementations of these algorithms that are based on the normal equations and corrected seminormal equations (CSNE) approaches. We show that the method of normal equations should be em...

In this work we present new kernels for the generation and application of block-Jacobi preconditioners that accelerate the iterative solution of sparse linear systems on graphics processing units (GPUs). Our approach departs from the conventional LU factorization and decomposes the diagonal blocks of the matrix using the Gauss-Huard method. When enhanced with column pivoting, this method is as ...

We introduce an efficient algorithm for computing a low-rank nonnegative CANDECOMP/PARAFAC (NNCP) decomposition. In text mining, signal processing, and computer vision among other areas, imposing nonnegativity constraints to low-rank factors has been shown an effective technique providing physically meaningful interpretation. A principled methodology for computing NNCP is alternating nonnegativ...

SYM-ILDL is a numerical software package that computes incomplete LDLT (or ‘ILDL’) factorizations of symmetric indefinite and skew-symmetric matrices. The core of the algorithm is a Crout variant of incomplete LU (ILU), originally introduced and implemented for symmetric matrices by [Li and Saad, Crout versions of ILU factorization with pivoting for sparse symmetric matrices, Transactions on Nu...

A standard method for solving the symmetric definite generalized eigenvalue problem Ax = λBx, where A is symmetric and B is symmetric positive definite, is to compute a Cholesky factorization B = LLT (optionally with complete pivoting) and solve the equivalent standard symmetric eigenvalue problem Cy = λy, where C = L−1AL−T . Provided that a stable eigensolver is used, standard error analysis s...