In this paper we present a novel matrix method for polynomial rootfinding. We approximate the roots by computing the eigenvalues of a permuted version of the companion matrix associated with the polynomial. This form, referred to as a lower staircase form of the companion matrix in the literature, has a block upper Hessenberg shape with possibly nonsquare subdiagonal blocks. It is shown that th...

Abstract: In this paper, we investigated relationships between the Fibonacci, Lucas, Padovan numbers and 1-factors of some bipartite graphs with upper Hessenberg adjacency matrix. We calculated permanent of these upper Hessenberg matrices by contraction method and show that their permanents are equal to elements of the Fibonacci, Lucas and Padovan numbers. At the end of the paper, we give some ...

Smallto medium-sized polynomial eigenvalue problems can be solved by linearizing the matrix polynomial and solving the resulting generalized eigenvalue problem using the QZ algorithm. The QZ algorithm, in turn, requires an initial reduction of a matrix pair to Hessenberg– triangular form. In this paper, we discuss the design and evaluation of high-performance parallel algorithms and software fo...

This paper describes LAPACK-based Fortran 77 subroutines for the reduction of a Hamiltonian matrix to square-reduced form and the approximation of all its eigenvalues using the implicit version of Van Loan's method. The transformation of the Hamiltonian matrix to a square-reduced form transforms a Hamiltonian eigenvalue problem of order 2n to a Hessenberg eigenvalue problem of order n. The eige...

In this paper we consider discrete-time multidimensional Markov chains having a block transition probability matrix which is the sum of a matrix with repeating block rows and a matrix of upper-Hessenberg, quasi-Toeplitz structure. We derive sufficient conditions for the existence of the stationary distribution, and outline two algorithms for calculating the stationary distribution.

The induced dimension reduction (IDR) method, which has been introduced as a transpose-free Krylov space method for solving nonsymmetric linear systems, can also be used to determine approximate eigenvalues of a matrix or operator. The IDR residual polynomials are the products of a residual polynomial constructed by successively appending linear smoothing factors and the residual polynomials of...

In this paper we consider fast numerical algorithms for solving certain modified matrix eigenvalue problems associated with algebraic equations. The matrices under consideration have the form A = T + uv , where u, v ∈ Rn×n and T = (ti,j) ∈ Rn×n is a tridiagonal matrix such that tj+1,j = ±tj,j+1, 1 ≤ j ≤ n−1. We show that the DQR approach proposed in [Uhlig F., Numer. Math. 76 (1997), no. 4, 515...

We perform a backward error analysis of the inexact shift-and-invert Arnoldi algorithm. We consider inexactness in the solution of the arising linear systems, as well as in the orthonormalization steps, and take the non-orthonormality of the computed Krylov basis into account. We show that the computed basis and Hessenberg matrix satisfy an exact shift-and-invert Krylov relation for a perturbed...

Solution of large-scale dense nonsymmetric eigenvalue problem is required in many areas of scientific and engineering computing, such as vibration analysis of automobiles and analysis of electronic diffraction patterns. In this study, we focus on the Hessenberg reduction step and consider accelerating it in a hybrid CPU-GPU computing environment. Considering that the Hessenberg reduction algori...

Recently, an extension of the class of matrices admitting a Francis type of multishift QR algorithm was proposed by the authors. These so-called condensed matrices admit a storage cost identical to that of the Hessenberg matrix and share all of the properties essential for the development of an effective implicit QR type method. This article continues along this trajectory by discussing the gen...