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Recently a generalization of Francis's implicitly shifted QR algorithm was proposed, notably widening the class of matrices admitting low-cost implicit QR steps. This unifying framework covered the methods and theory for Hessenberg and inverse Hessenberg matrices and furnished also new, single-shifted, QR-type methods for, e.g., CMV matrices. Convergence of this approach was only suggested by numerical experiments. No theoretical claims supporting the results were presented. In this paper we present multishift variants of these new algorithms. We also provide a convergence theory that shows that the new algorithm performs nested subspace iterations on rational Krylov subspaces. Numerical experiments confirm the validity of the theory. [ABSTRACT FROM AUTHOR]

Algebraic multigrid (AMG) is a very efficient iterative solver and preconditioner for large unstructured sparse linear systems. Traditional coarsening schemes for AMG can, however, lead to computational complexity growth as problem size increases, resulting in increased memory use and execution time, and diminished scalability. Two new parallel AMG coarsening schemes are proposed that are based solely on enforcing a maximum independent set property, resulting in sparser coarse grids. The new coarsening techniques remedy memory and execution time complexity growth for various large three-dimensional (3D) problems. If used within AMG as a preconditioner for Krylov subspace methods, the resulting iterative methods tend to converge fast. This paper discusses complexity issues that can arise in AMG, describes the new coarsening schemes, and examines the performance of the new preconditioners for various large 3D problems. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to derive new computable convergence bounds for GMRES. The new bounds depend on the initial guess and are thus conceptually different from standard “worst-case” bounds. Most importantly, approximations to the new bounds can be computed from information generated during the run of a certain GMRES implementation. The approximations allow predictions of how the algorithm will perform. Heuristics for such predictions are given. Numerical experiments illustrate the behavior of the new bounds as well as the use of the heuristics. [ABSTRACT FROM AUTHOR]

The BR algorithm, a new method for calculating the eigenvalues of an upper Hes- senberg matrix, is introduced. It is a bulge-chasing algorithm like the QR algorithm, but, unlike the QR algorithm, it is well adapted to computing the eigenvalues of the narrow-band, nearly tridiagonal matrices generated by the look-ahead Lanczos process. This paper describes the BR algorithm and gives numerical evidence that it works well in conjunction with the Lanczos process. On the biggest problems run so far, the BR algorithm beats the QR algorithm by a factor of 30–60 in computing time and a factor of over 100 in matrix storage space. [ABSTRACT FROM AUTHOR]