Chasing bulges or rotations? A new family of matrices admitting linear time QR-steps

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. 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, for example, Hessenberg, inverse Hessenberg and CM V 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 explicit and implicit QR-type algorithm and; a convergence analysis of this novel method. 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 discloses an intriguing difference between the average number of iterations before deflation can be applied.