The block GMRES (BGMRES) method is a natural generalization of the GMRES algorithm for solving large nonsymmetric systems with multiple right-hand sides. Unfortunately, its cost increases significantly per iteration, frequently rendering the method impractical. In this paper we propose a hybrid block GMRES method which offers significant performance improvements over BGMRES. This method uses th...

Many applications require the solution of multiple linear systems that have the same coeecient matrix, but diier in their right-hand sides. Instead of applying an iterative method to each of these systems individually, it is more eecient to employ a block version of the method that generates iterates for all the systems simultaneously. In this paper, we propose a block version of Freund and Nac...

In this paper, we present the block least squares method for solving nonsymmetric linear systems with multiple righthand sides. This method is based on the block bidiagonalization. We first derive two algorithms by using two different convergence criteria. The first one is based on independently minimizing the 2-norm of each column of the residual matrix and the second approach is based on mini...

We analyze a class of Krylov projection methods but mainly concentrate on a specific conjugate gradient (CG) implementation by Smith, Peterson, and Mittra [IEEE Transactions on Antennas and Propogation, 37 (1989), pp. 1490–1493] to solve the linear system AX = B, where A is symmetric positive definite and B is a multiple of right-hand sides. This method generates a Krylov subspace from a set of...

A new approach is discussed for solving large nonsymmetric systems of linear equations with multiple right-hand sides. The first system is solved with a deflated GMRES method that generates eigenvector information at the same time that the linear equations are solved. Subsequent systems are solved by combining restarted GMRES with a projection over the previously determined eigenvectors. This a...

In this work we consider the simultaneous solution of large linear systems of the form Ax(j) = b(j); j = 1; : : : ; K where A is sparse and non-Hermitian. Our single-seed approach uses QMR to solve the seed system j and generate biorthogonal Krylov subspaces. Approximate solutions to the non-seed systems are simulanteously generated by minimizing their appropriately projected residuals. After t...

In a wide number of applications in computational science and engineering the solution of linear systems of equations with several right-hand sides given at once is required. Direct methods based on Gaussian elimination are known to be popular in that setting. Nevertheless if the dimension of the problem is very large or if matrix-free algorithms are preferred, preconditioned block Krylov space...

We consider two-stage quadratic integer programs with stochastic right-hand sides, and present an equivalent reformulation using value functions. We first derive some basic properties of value functions of quadratic integer programs. We then propose a two-phase solution approach. The first phase constructs the value functions of quadratic integer programs in both stages. The second phase solves...

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...