Highly effective polynomial root-finders have been recently designed based on eigen-solving for DPR1 (that is diagonal + rankone) matrices. We extend these algorithms to eigen-solving for the general matrix by reducing the problem to the case of the DPR1 input via intermediate transition to a TPR1 (that is triangular + rank-one) matrix. Our transforms use substantially fewer arithmetic operatio...

This paper proposes an efficient algorithm for simultaneous reduction of three matrices. The algorithm is a blocked version of the algorithm described by Miminis and Page (1982) which reduces A to the m-Hessenberg form, and B and E to the triangular form. The m-Hessenberg– triangular–triangular form of matrices A, B and E is specially suitable for solving multiple shifted systems. Such shifted ...

In this paper we show that Doolittle’s method to compute the LU factorization of Hessenberg matrices is mixed forward-backward stable and therefore, componentwise forward stable. We also conjecture that this algorithm for computing the LU factorization of dense matrices is forward stable.

We investigate the main phase space properties of QR-flow when restricted to upper Hessenberg matrices. A complete description linear behavior equilibrium matrices is given. The result classifies possible $ \alpha $- and \omega $-limits orbits for this system. Furthermore, we characterize set initial which there convergence towards an matrix. Several numerical examples show different limit illu...

Hessenberg varieties are a family of subvarieties of the flag variety, including the Springer fibers, the Peterson variety, and the entire flag variety itself. The seminal example arises from numerical analysis and consists, for a fixed linear operator M , of the full flags V1 ( V2 . . . ( Vn in GLn with MVi ⊆ Vi+1 for all i. In this paper, I show that all nilpotent Hessenberg varieties in type...

Regular semisimple Hessenberg varieties are a family of subvarieties of the flag variety that arise in number theory, numerical analysis, representation theory, algebraic geometry, and combinatorics. We give a “Giambelli formula” expressing the classes of regular semisimple Hessenberg varieties in terms of Chern classes. In fact, we show that the cohomology class of each regular semisimple Hess...

Regular semisimple Hessenberg varieties are a family of subvarieties of the flag variety that arise in number theory, numerical analysis, representation theory, algebraic geometry, and combinatorics. We give a “Giambelli formula” expressing the classes of regular semisimple Hessenberg varieties in terms of Chern classes. In fact, we show that the cohomology class of each regular semisimple Hess...

The usual QR algorithm for finding the eigenvalues of a Hessenberg matrix H is based on vector-vector operations, e.g. adding a multiple of one row to another. The opportunities for parallelism in such an algorithm are limited. In this report, we describe a reorganization of the QR algorithm to permit either matrix-vector or matrix-matrix operations to be performed, both of which yield more eff...