Hessenberg Form Research Articles

Surjective Additive Rank-1 Preservers on Hessenberg Matrices

Let $H_{n}(\mathbb{F})$ be the space of all $n\times n$ upper Hessenberg matrices over a field~$\mathbb{F}$, where $n$ is a positive integer greater than two. In this paper, surjective additive maps preserving rank-$1$ on $H_{n}(\mathbb{F})$ are characterized.

Harmonic numbers associated with inversion numbers in terms of determinants

It has been known that some numbers, including Bernoulli, Cauchy, and Euler numbers, have such corresponding numbers in terms of determinants of Hessenberg matrices. There exist inversion relations between the original numbers and the corresponding numbers. In this paper, we introduce the numbers related to harmonic numbers in determinants. We also give several of their arithmetical and/or combinatorial properties and applications. These concepts can be generalized in the case of hyperharmonic numbers.

Some permanents of Hessenberg matrices

In this paper, we present new relationships between the terms of sequence {Rn} and {Sn} and permanents of some upper Hessenberg matrices.

Observability of Linear Time-Varying Systems with Quasiderivative Coefficients

On the basis of quasiderivatives and the upper triangular Hessenberg form, an approach to investigating the observability problem for linear time-varying systems is proposed. The quasiderivatives are defined for some lower triangular matrix, and the simplest rules of the quasidifferentiation are described. Conditions for linear independence of continuous quasidifferentiable functions are established in terms of the Wronski matrix. The necessary and sufficient conditions for an upper triangular Hessenberg form in the orbit of linear time-varying systems are obtained, and the method to find such systems is described. New efficient conditions for various types of observability (complete, differential, uniform) are obtained.

Positive functionals and Hessenberg matrices

Not every positive functional defined on bi-variate polynomials of a prescribed degree bound is represented by the integration against a positive measure. We isolate a couple of conditions filling this gap, either by restricting the class of polynomials to harmonic ones, or imposing the vanishing of a defect indicator. Both criteria offer constructive cubature formulas and they are obtained via well known matrix analysis techniques involving either the dilation of a contractive matrix to a unitary one or the specific structure of the Hessenberg matrix associated to the multiplier by the underlying complex variable.

Explicit inversion formulas for properly Hessenberg and Toeplitz–Hessenberg matrices

ABSTRACT The nice inversion properties of Toeplitz–Hessenberg matrices from a Hessenbergian representation of Catalan's numbers encourage us to provide explicit inversion formulas (in terms of basic arithmetical operations involving entries from the original matrix) for the non-singular Toeplitz–Hessenberg matrices. Our approach is based on an elementary matrix inflation method, joint with the nested sums formulas. These explicit inversion formulas are then extended to apply to the entries of the inverse of every non-singular properly Hessenberg matrix.

On Certain Hessenberg Matrices Related with Linear Recurrences

In this paper, we present the permanents and determinants of some Hessenbergmatrices. Also, some special cases for permanents are given.

Another orthogonal matrix, revisited

A 2006 paper by Parlett and Barszcz [4] proposed the following problem: Given an unit vector q, compute the orthogonal Hessenberg matrix A with first column q. In complex form, this translates to completing the unitary Hessenberg matrix U with first column q. Looking at the matrix I−qq⁎ in the special form I−qq⁎=LD2L⁎, where L is n×(n−1) and lower triangular with 1's on its main diagonal and D2=diag(μ12,…,μn−12) is positive definite, Parlett observed that for i>j the entries of L˜=LD can be written as l˜ij=−qiq‾jμj/ρj, where q‾j denotes the complex conjugate of qj and ρi=∑j=i+1n|qj|2, ρn=0, and μi=ρi/ρi−1, for i=1,…,n. Furthermore, one solution to this problem is U=[qL˜]. Section 1 provides some background, as well as details on the derivation of Parlett's formula. Section 2 contains the main result, where Parlett's method is extended to “tall thin” matrices as suggested by Parlett in his original paper. In other words, using a repeated application of Parlett's method, a solution is given to the problem of completing the unitary k-Hessenberg matrix given its first k columns.

On the Darboux transform and the solutions of some integrable systems

The relation between the Darboux transformation and the solutions of the full Kostant Toda lattice is analyzed. The discrete Korteweg de Vries equation is used to obtain such solutions and the main result of [1] is extended to the case of general (p + 2)-banded Hessenberg matrices.

An algebraic method for calculating PageRank

An algebraic method is proposed for finding PageRank estimates for pages of websites. The amount of calculation in the proposed method does not depend on the value of the damping coefficient, which allows obtaining more accurate estimates of the rankings of PageRank in comparison with analogues. A distinctive feature of the proposed method is a step-by-step performance of calculations simultaneously with the work of the graph traversal algorithm. The comparative analysis of algorithms for traversing graphs has shown that, in contrast to the depth search algorithm, the breadth search algorithm gives a more orderly arranged matrix of transitions, which has the blockwise Hessenberg form. The use of this circumstance makes it possible to reduce significantly the amount of calculations by the proposed method. The resulting equations describing the proposed method have a block structure that allows efficient distribution of the entire volume of operations to parallel computational threads . Proceeding from the fact that the bulk of the calculations can be performed while the graph traversal algorithm is running, the study has determined the conditions under which the proposed method makes it possible to obtain PageRank values faster than the known iterative algorithms. The applicability area of the developed method is, first of all, its use in direct verification of the reliability of posting advertising materials on a relevant web resource; therefore, it is limited to specific Internet sites or segments with no more than 10 4 or 10 5 pages.

A new real structure-preserving quaternion QR algorithm

New real structure-preserving decompositions are introduced to develop fast and robust algorithms for the (right) eigenproblem of general quaternion matrices. Under the orthogonally JRS-symplectic transformations, the Francis JRS-QR step and the JRS-QR algorithm are firstly proposed for JRS-symmetric matrices and then applied to calculate the Schur form of quaternion matrices. A novel quaternion Givens matrix is defined and utilized to compute the QR factorization of quaternion Hessenberg matrices. An implicit double shift quaternion QR algorithm is presented with a technique for automatically choosing shifts and within real operations. Numerical experiments are provided to demonstrate the efficiency and accuracy of newly proposed algorithms.

On two generalized inverse eigenvalue problems for Hessenberg-upper triangular pencils and their application to the study of GMRES convergence

We discuss two generalized inverse eigenvalue problems. The first one: For a given unreduced upper Hessenberg matrix H, find a nonsingular upper triangular matrix T such that all the pencils Hk−λTk have prescribed eigenvalues, where Hk and Tk are the leading k×k principal submatrices of H and T, respectively. The second one: For a given unitary unreduced upper Hessenberg matrix Q, find a nonsingular upper triangular matrix T such that all the pencils Tk−θQk⁎ have prescribed eigenvalues, where Tk is the leading k×k principal submatrix of T, and Qk⁎ is the conjugate transpose of the leading k×k principal submatrix of Q. We present the necessary and sufficient conditions for the solvability of the two problems. Our results lead to an alternative proof for the statement that any admissible Ritz value set or admissible harmonic Ritz value set is possible for the prescribed GMRES residual norms. Here, the term “admissible” means there are some restrictions on the sets if GMRES stagnates at some iterations.

Two-sided orthogonal reductions to condensed forms on asymmetric multicore processors

Abstract We investigate how to leverage the heterogeneous resources of an Asymmetric Multicore Processor (AMP) in order to deliver high performance in the reduction to condensed forms for the solution of dense eigenvalue and singular-value problems. The routines that realize this type of two-sided orthogonal reductions (TSOR) in LAPACK are especially challenging, since a significant fraction of their floating-point operations are cast in terms of memory-bound kernels while the remaining part corresponds to efficient compute-bound kernels. To deal with this scenario: (1) we leverage implementations of memory-bound and compute-bound kernels specifically tuned for AMPs; (2) we select the algorithmic block size for the TSOR routines via a practical model; and (3) we adjust the type and number of cores to use at each step of the reduction. Our experiments validate the model and assess the performance of our asymmetry-aware TSOR routines, using an ARMv7 big.LITTLE AMP, for three key operations: the reduction to tridiagonal form for symmetric eigenvalue problems, the reduction to Hessenberg form for non-symmetric eigenvalue problems, and the reduction to bidiagonal form for singular-value problems.

Uniform Global Attainability and Global Lyapunov Reducibility of Linear Control Systems in the Hessenberg Form

For a linear control system in the Hessenberg form, we obtain new sufficient conditions for Lyapunov reducibility to a system in the Frobenius canonical form, for uniform global attainability, and for global Lyapunov reducibility.

Sequential Linear Interpolation Correction Methods Used for Finding Zeros of Functions and Characteristic Polynomials of Matrices of a Special Form

Computational schemes of the method of "correction of sequential linear interpolation (MCSLI)" are considered in this paper. MCSLI are used for finding zeros of nonlinear (including transcendental) functions, as well as zeros of characteristic polynomials of matrices of a special form, such as almost triangular (Hessenberg form), tridiagonal and others forms of matrices obtained, for example, by Givens or Householder methods from matrices of general form. The proposed computational schemes of MCSLI for cases of simple and multiple roots (including pathologically close roots) have a structural and functional similarity. MCSLI schemes designed to localize and improve multiple roots can also be used to localize a group of closely related roots consisting of simple roots and roots of different multiplicity (including pathologically close roots). The schemes of MCSLI have computational stability and a high convergence rate (the order of the convergence rate is approximately equal to two). Based on the results of computational experiments for MCSLI and other effective methods, the dependences of diagonalization time of matrices of a special form on the order of these matrices are obtained.DOI 10.14258/izvasu(2018)1-16

Complex Orthogonal Polynomials and Numerical Quadrature via Hyponormality

By interpreting a truncated moment problem in two real variables in complex terms one exploits the hyponormality of the coordinate multiplier to obtain characterizations of: the moments of a positive measure, the associated complex orthogonal polynomials and the structure of the Hessenberg matrices. All reflecting the intricate structure of multivariate positive polynomials versus sums of squares or sums of hermitian squares of polynomials. The principal step in the proposed algorithm is stated in terms of a linear matrix inequality, making our approach prone for numerical applications.

On computing: the inverse of centrosymmetric matrix

Centrosymmetric matrix has special structure and symmetry on its centre. By orthogonal matrix, centrosymmetric matrix can be formed as block matrix. One of the block matrices of centrosymmetric matrix is lower Hessenberg form. Based on the algorithm of lower Hessenberg matrix’s inverse, the computation on inverse of centrosymmetric matrix is computed. Numerical example is presented for simple understanding.

Refined interlacing properties for zeros of paraorthogonal polynomials on the unit circle

The purpose of this note is to extend in a simple and unified way the known results on interlacing of zeros of paraorthogonal polynomials on the unit circle. These polynomials can be regarded as the characteristic polynomials of any matrix similar to an unitary upper Hessenberg matrix with positive subdiagonal elements.

Reduction of Matrix Polynomials to Simpler Forms

A square matrix can be reduced to simpler form via similarity transformations. Here “simpler form” may refer to diagonal (when possible), triangular (Schur), or Hessenberg form. Similar reductions exist for matrix pencils if we consider general equivalence transformations instead of similarity transformations. For both matrices and matrix pencils, well-established algorithms are available for each reduction, which are useful in various applications. For matrix polynomials, unimodular transformations can be used to achieve the reduced forms but we do not have a practical way to compute them. In this work we introduce a practical means to reduce a matrix polynomial with nonsingular leading coefficient to a simpler (diagonal, triangular, Hessenberg) form while preserving the degree and the eigenstructure. The key to our approach is to work with structure preserving similarity transformations applied to a linearization of the matrix polynomial instead of unimodular transformations applied directly to the matrix polynomial. As an application, we illustrate how to use these reduced forms to solve parameterized linear systems.

The $QR$ Steps with Perfect Shifts

In this paper we revisit the problem of performing a $QR$ step on an unreduced Hessenberg matrix $H$ when we know an “exact” eigenvalue $\lambda_0$ of $H$. In exact arithmetic, this eigenvalue will...