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...

We consider the solution of very large sparse systems of linear equations on parallel architectures. In this context, memory is often a bottleneck that prevents or limits the use of direct solvers, especially those based on the multifrontal method. This work focuses on memory and performance issues of the two memory and computationally intensive phases of direct methods, namely, the numerical f...

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...

I will review recent developments in matrix deflation methods, by Ronald Morgan/Walter Wilcox, Andreas Stathopoulos/Konstantinos Orginos, and Martin Lüscher, with application to lattice QCD fermion inversion. I will begin with a short review of deflation-related work in the field. The Morgan/Wilcox algorithms using GMRES and BiCGStab for deflation will be described. Typical results for quenched...

A parallel algorithm for solving a series of matrix equations with a constant tridiagonal matrix and different right-hand sides is proposed and studied. The process of solving the problem is represented in two steps. The first preliminary step is fixing some rows of the inverse matrix of SLAEs. The second step consists in calculating solutions for all right-hand sides. For reducing the communic...