GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems
SIAM Journal on Scientific and Statistical Computing
The algebraic eigenvalue problem
The algebraic eigenvalue problem
Predicting the behavior of finite precision Lanczos and conjugate gradient computations
SIAM Journal on Matrix Analysis and Applications
Loss and recapture of orthogonality in the modified Gram-Schmidt algorithm
SIAM Journal on Matrix Analysis and Applications
Matrix computations (3rd ed.)
The symmetric eigenvalue problem
The symmetric eigenvalue problem
LSQR: An Algorithm for Sparse Linear Equations and Sparse Least Squares
ACM Transactions on Mathematical Software (TOMS)
Algorithm 583: LSQR: Sparse Linear Equations and Least Squares Problems
ACM Transactions on Mathematical Software (TOMS)
Accuracy and Stability of Numerical Algorithms
Accuracy and Stability of Numerical Algorithms
Accuracy of Two Three-term and Three Two-term Recurrences for Krylov Space Solvers
SIAM Journal on Matrix Analysis and Applications
Convergence properties of the conjugate gradient algorithm in exact and finite precision arithmetic
Convergence properties of the conjugate gradient algorithm in exact and finite precision arithmetic
On Stabilization and Convergence of Clustered Ritz Values in the Lanczos Method
SIAM Journal on Matrix Analysis and Applications
Modified Gram-Schmidt (MGS), Least Squares, and Backward Stability of MGS-GMRES
SIAM Journal on Matrix Analysis and Applications
The Lanczos and Conjugate Gradient Algorithms: From Theory to Finite Precision Computations (Software, Environments, and Tools)
On sensitivity of Gauss–Christoffel quadrature
Numerische Mathematik
A Useful Form of Unitary Matrix Obtained from Any Sequence of Unit 2-Norm $n$-Vectors
SIAM Journal on Matrix Analysis and Applications
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It is shown that a good implementation of the Hermitian matrix tridiagonalization process of Lanczos [J. Research Nat. Bur. Standards, 45 (1950), pp. 255-282] produces a tridiagonal matrix that is, at each step, the exact result for the process applied to a strange augmented problem. Since the process is not stable in the standard sense, this augmented stability result cannot be transformed to prove standard stability. The intent is to obtain an increased understanding of the Lanczos tridiagonalization process, and this result could later be used to analyze the many applications of the process to large sparse matrix problems, such as the solution of the eigenproblem, compatible linear systems, least squares, and the singular value decomposition.