The algebraic eigenvalue problem
The algebraic eigenvalue problem
Chasing algorithms for the eigenvalues problem
SIAM Journal on Matrix Analysis and Applications
Some perspectives on the eigenvalue problem
SIAM Review
Theory of Decomposition and Bulge-Chasing Algorithms for the Generalized Eigenvalue Problem
SIAM Journal on Matrix Analysis and Applications
Matrix computations (3rd ed.)
Applied numerical linear algebra
Applied numerical linear algebra
The symmetric eigenvalue problem
The symmetric eigenvalue problem
Bulge Exchanges in Algorithms of QR Type
SIAM Journal on Matrix Analysis and Applications
Matrix algorithms
The Multishift QR Algorithm. Part I: Maintaining Well-Focused Shifts and Level 3 Performance
SIAM Journal on Matrix Analysis and Applications
The Multishift QR Algorithm. Part II: Aggressive Early Deflation
SIAM Journal on Matrix Analysis and Applications
An Orthogonal Similarity Reduction of a Matrix into Semiseparable Form
SIAM Journal on Matrix Analysis and Applications
CMV matrices: Five years after
Journal of Computational and Applied Mathematics
SIAM Journal on Matrix Analysis and Applications
SIAM Review
The Matrix Eigenvalue Problem: GR and Krylov Subspace Methods
The Matrix Eigenvalue Problem: GR and Krylov Subspace Methods
Rational QR-iteration without inversion
Numerische Mathematik
Orthogonal Laurent polynomials on the unit circle and snake-shaped matrix factorizations
Journal of Approximation Theory
Structures preserved by the QR-algorithm
Journal of Computational and Applied Mathematics
Rank structures preserved by the QR-algorithm: The singular case
Journal of Computational and Applied Mathematics
Implicit double shift QR-algorithm for companion matrices
Numerische Mathematik
An Implicit Multishift $QR$-Algorithm for Hermitian Plus Low Rank Matrices
SIAM Journal on Scientific Computing
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The $QR$-algorithm is a renowned method for computing all eigenvalues of an arbitrary matrix. A preliminary unitary similarity transformation to Hessenberg form is indispensable for keeping the computational complexity of the subsequent $QR$-steps under control. When restraining computing time is the vital issue, we observe that the prominent role played by the Hessenberg matrix is sufficient but perhaps not necessary to fulfill this goal. In this paper, a whole new family of matrices, sharing the major qualities of Hessenberg matrices, will be put forward. This gives rise to the development of innovative implicit $QR$-type algorithms, pursuing rotations instead of bulges. The key idea is to benefit from the $QR$-factorization of the matrices involved. The prescribed order of rotations in the decomposition of the $Q$-factor uniquely characterizes several matrix types such as Hessenberg, inverse Hessenberg, and $CMV$ matrices. Loosening the fixed ordering of these rotations provides us the class of matrices under consideration. Establishing a new implicit $QR$-type algorithm for these matrices requires a generalization of diverse well-established concepts. We consider the preliminary unitary similarity transformation, a proof of uniqueness of this reduction, an extension of the $CMV$-decomposition to a double Hessenberg factorization, and an explicit and implicit $QR$-type algorithm. A detailed complexity analysis illustrates the competitiveness of the novel method with the traditional Hessenberg approach. The numerical experiments show comparable accuracy for a wide variety of matrix types, but disclose an intriguing difference between the average number of iterations before deflation can be applied.