Existing definitions of componentwise backward error and componentwise condi tion number for linear systems are extended to systems with multiple right-hand sides and to a general class of componentwise measure of perturbations involving Holder p-norms. It is shown that for a system of order n with r right-hand sides, the componentwise backward error can be computed by finding the minimum p-nor...

The technique that was used to build the eigCG algorithm for sparse symmetric linear systems is extended to the nonsymmetric case using the BiCG algorithm. We show that, similarly to the symmetric case, we can build an algorithm that is capable of computing a few smallest magnitude eigenvalues and their corresponding left and right eigenvectors of a nonsymmetric matrix using only a small window...

Sensitivity analysis provides useful information for equation-solving, optimization, and post-optimality analysis. However, obtaining useful sensitivity information for systems with nonsmooth dynamic systems embedded is a challenging task. In this article, for any locally Lipschitz continuous mapping between finitedimensional Euclidean spaces, Nesterov’s lexicographic derivatives are shown to b...

We present some new techniques for solving a family (or a sequence of families) of linear systems in which the coefficient matrices differ only by a scalar multiple of the identity (shifted systems). Our goal is to develop methods for shifted systems which have fewer restrictions usually associated with such methods (e.g., all residuals needing to be collinear). The systems are parameterized by i,

Projection techniques are developed for computing approximate solutions to linear systems of the form Ax n = b n , for a sequence n = 1; 2; :::, e.g., arising from time discretization of a partial diierential equation. The approximate solutions are based upon previous solutions, and can be used as initial guesses for iterative solution of the system, resulting in signiicantly reduced computatio...

In this paper, we propose a new method for solving general linear systems with several right-hand sides. This method is based on global least squares method and reduces the original matrix to the lower bidiagonal form. We derive a simple recurrence formula for generating the sequence of approximate solutions {Xk}. Some theoretical properties of the new method are discussed and we also show that...

A restarted nonsymmetric Lanczos algorithm is given for computing eigenvalues and both right and left eigenvectors. The restarting limits the storage so that finding eigenvectors is practical. Restarting also makes it possible to deal with roundoff error in new ways. We give a scheme for avoiding near-breakdown and discuss maintaining biorthogonality. A system of linear equations can be solved ...

We describe PP-BCG, a parallel iterative solver for the solution of dense and symmetric positive-definite linear systems with multiple right-hand sides suitable for MPPs. Such linear systems appear in the context of stochastic estimation of the diagonal of the matrix inverse in Uncertainty Quantification and the trace of matrix products in statistical analysis. We propose a novel numerical sche...

Many applications require the solution of large nonsymmetric linear systems with multiple right-hand sides. Instead of applying an iterative method to each of these systems individually, it is often more eecient to use a block version of the method that generates iterates for all the systems simultaneously. In this paper, we propose block versions of Galerkin/minimal residual pair of bidiagonal...