The QR algorithm for unitary Hessenberg matrices
Journal of Computational and Applied Mathematics
An implementation of a divide and conquer algorithm for the unitary eigen problem
ACM Transactions on Mathematical Software (TOMS)
Journal of Computational and Applied Mathematics - Special issue on computational complex analysis
On a class of Gauss-like quadrature rules
Numerische Mathematik
Matrix computations (3rd ed.)
Journal of Computational and Applied Mathematics - Special issue: dedicated to William B. Gragg on the occasion of his 60th Birthday
Polynomial zerofinders based on Szegő polynomials
Journal of Computational and Applied Mathematics - Special issue on numerical analysis 2000 Vol. V: quadrature and orthogonal polynomials
Convergence of the shifted QR algorithm for unitary Hessenberg matrices
Mathematics of Computation
A unitary Hessenberg QR-based algorithm via semiseparable matrices
Journal of Computational and Applied Mathematics
A unitary Hessenberg QR-based algorithm via semiseparable matrices
Journal of Computational and Applied Mathematics
Hi-index | 7.29 |
The QR-algorithm is a popular numerical method for the computation of eigenvalues of matrices. All eigenvalues of a general n × n upper Hessenberg matrix typically can be computed in O(n3) arithmetic floating point operations using O(n2) storage locations. When the upper Hessenberg matrix is Hermitian or unitary, then it can be represented by O(n) parameters, and there are variants of the QR-algorithm that reduce the operation count for computing all eigenvalues to O(n2) arithmetic floating point operations and the storage requirement to O(n) locations. However, for many structured matrices that can be represented with O(n) storage locations, available implementations of the QR-algorithm require O(n3) arithmetic floating point operations and O(n2) storage locations to determine all eigenvalues. This note shows that for some of the latter matrices, the operation count can be reduced to O(n2) arithmetic floating point operations and the memory requirement to O(n) storage locations by periodically restarting the QR-algorithm.