A hybrid Chebyshev Krylov subspace algorithm for solving nonsymmetric systems of linear equations
SIAM Journal on Scientific and Statistical Computing
GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems
SIAM Journal on Scientific and Statistical Computing
Least squares polynomials in the complex plane and their use for solving nonsymmetric linear systems
SIAM Journal on Numerical Analysis
A stable Richardson iteration method for complex linear systems
Numerische Mathematik
Analysis of some Krylov subspace approximations to the matrix exponential operator
SIAM Journal on Numerical Analysis
A Hybrid GMRES algorithm for nonsymmetric linear systems
SIAM Journal on Matrix Analysis and Applications
An adaptive semi-iterative method for symmetric semidefinite linear systems
Proceedings of the conference on Approximation and computation : a fetschrift in honor of Walter Gautschi: a fetschrift in honor of Walter Gautschi
Calculation of pseudospectra by the Arnoldi iteration
SIAM Journal on Scientific Computing - Special issue on iterative methods in numerical linear algebra; selected papers from the Colorado conference
Matrix algorithms
Large-Scale Computation of Pseudospectra Using ARPACK and Eigs
SIAM Journal on Scientific Computing
The Chebyshev iteration revisited
Parallel Computing - Parallel matrix algorithms and applications
An Implementation of the GMRES Method Using QR Factorization
Proceedings of the Fifth SIAM Conference on Parallel Processing for Scientific Computing
Iterative Methods for Sparse Linear Systems
Iterative Methods for Sparse Linear Systems
Modified Gram-Schmidt (MGS), Least Squares, and Backward Stability of MGS-GMRES
SIAM Journal on Matrix Analysis and Applications
Error Estimates and Evaluation of Matrix Functions via the Faber Transform
SIAM Journal on Numerical Analysis
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Many problems in scientific computing involving a large sparse square matrix A are solved by Krylov subspace methods. This includes methods for the solution of large linear systems of equations with A, for the computation of a few eigenvalues and associated eigenvectors of A, and for the approximation of nonlinear matrix functions of A. When the matrix A is non-Hermitian, the Arnoldi process commonly is used to compute an orthonormal basis for a Krylov subspace associated with A. The Arnoldi process often is implemented with the aid of the modified Gram-Schmidt method. It is well known that the latter constitutes a bottleneck in parallel computing environments, and to some extent also on sequential computers. Several approaches to circumvent orthogonalization by the modified Gram-Schmidt method have been described in the literature, including the generation of Krylov subspace bases with the aid of suitably chosen Chebyshev or Newton polynomials. We review these schemes and describe new ones. Numerical examples are presented.