GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems
SIAM Journal on Scientific and Statistical Computing
The dynamics of Rayleigh quotient iteration
SIAM Journal on Numerical Analysis
On a block implementation of Hessenberg multishift QR iteration
International Journal of High Speed Computing
A note on one-sided Jacobi algorithm
Numerische Mathematik
LAPACK's user's guide
Implicit application of polynomial filters in a k-step Arnoldi method
SIAM Journal on Matrix Analysis and Applications
Journal of Computational Physics
A completed theory of the unsymmetric Lanczos process and related algorithms, part I
SIAM Journal on Matrix Analysis and Applications
Jacobi's method is more accurage than QR
SIAM Journal on Matrix Analysis and Applications
Forward Stability and Transmission of Shifts in the $QR$ Algorithm
SIAM Journal on Matrix Analysis and Applications
QMRPACK: a package of QMR algorithms
ACM Transactions on Mathematical Software (TOMS)
A Jacobi--Davidson Iteration Method for Linear EigenvalueProblems
SIAM Journal on Matrix Analysis and Applications
Matrix computations (3rd ed.)
Applied numerical linear algebra
Applied numerical linear algebra
Computing an Eigenvector with Inverse Iteration
SIAM Review
The symmetric eigenvalue problem
The symmetric eigenvalue problem
Rational Krylov: A Practical Algorithm for Large Sparse Nonsymmetric Matrix Pencils
SIAM Journal on Scientific Computing
A Truncated RQ Iteration for Large Scale Eigenvalue Calculations
SIAM Journal on Matrix Analysis and Applications
Jacobi--Davidson Style QR and QZ Algorithms for the Reduction of Matrix Pencils
SIAM Journal on Scientific Computing
Structured Backward Error and Condition of Generalized Eigenvalue Problems
SIAM Journal on Matrix Analysis and Applications
On Modern Matrix Iteration Processes of Bernoulli and Graeffe Type
Journal of the ACM (JACM)
Unitary Triangularization of a Nonsymmetric Matrix
Journal of the ACM (JACM)
The Jacobi Method for Real Symmetric Matrices
Journal of the ACM (JACM)
Remarks on the Unitary Triangularization of a Nonsymmetric Matrix
Journal of the ACM (JACM)
On Pre-Conditioning of Matrices
Journal of the ACM (JACM)
Journal of the ACM (JACM)
An Algorithm for Numerical Computation of the Jordan Normal Form of a Complex Matrix
ACM Transactions on Mathematical Software (TOMS)
Templates for the solution of algebraic eigenvalue problems: a practical guide
Templates for the solution of algebraic eigenvalue problems: a practical guide
The rational Krylov algorithm for nonlinear eigenvalue problems
Computational Mechanics for the Twenty-First Century
Matrix algorithms
Accuracy and Stability of Numerical Algorithms
Accuracy and Stability of Numerical Algorithms
Parallel Algorithms and Matrix Computation
Parallel Algorithms and Matrix Computation
A Survey of Numerical Mathematics
A Survey of Numerical Mathematics
SIAM Journal on Scientific Computing
Numerical Computation of Deflating Subspaces of Skew-Hamiltonian/Hamiltonian Pencils
SIAM Journal on Matrix Analysis and Applications
Performance of the QZ Algorithm in the Presence of Infinite Eigenvalues
SIAM Journal on Matrix Analysis and Applications
Thick-Restart Lanczos Method for Symmetric Eigenvalue Problems
IRREGULAR '98 Proceedings of the 5th International Symposium on Solving Irregularly Structured Problems in Parallel
Computing smallest singular triplets with implicitly restarted Lanczos bidiagonalization
Applied Numerical Mathematics - Numerical algorithms, parallelism and applications
Handbook for Automatic Computation: Linear Algebra (Grundlehren Der Mathematischen Wissenschaften, Vol 186)
Google's PageRank and Beyond: The Science of Search Engine Rankings
Google's PageRank and Beyond: The Science of Search Engine Rankings
Rational krylov for large nonlinear eigenproblems
PARA'04 Proceedings of the 7th international conference on Applied Parallel Computing: state of the Art in Scientific Computing
Hi-index | 0.00 |
This paper sketches the research developments in the area of computational methods for solving the eigenvalue problems and how the methods developed relate to each other as they evolve over centuries. This is an attempt to write a complete overview on the research on computational aspects of eigenvalue problem, emphasize the history of methods that still play a role and some of those that no longer are considered to be on the main track but are somehow related to the present techniques in some smaller steps. This contribution brings out the state-of-the-art of the algorithms for solving large-scale eigenvalue problems for both symmetric and nonsymmetric matrices separately, thereby clearly drawing a comparison between the differences in the algorithms in practical use for the two. Some of the methods or modifications to the earlier methods that have been notable at the turn of the 21st century will also be covered through this paper under "Modem Approaches". Also as the standard eigenvalue problem has become better understood, in a numerical sense, progress has been made in the generalized eigenvalue problem and this will briefly cover developments in solving these problems too.