Hessenberg Form Research Articles

Determinants of some Hessenberg matrices with generating functions

Abstract In this paper, we derive some relationships between the determinants of some special lower Hessenberg matrices whose entries are the terms of certain sequences and the generating functions of these sequences. Moreover, our results are generalizations of the earlier results from previous researches. Furthermore, interesting examples of the determinants of some special lower Hessenberg matrices are presented.

A Watermarking Optimization Method Based on Matrix Decomposition and DWT for Multi-Size Images

Image watermarking is a key technology for copyright protection, and how to better balance the invisibility and robustness of algorithms is a challenge. To tackle this challenge, a watermarking optimization method based on matrix decomposition and discrete wavelet transform (DWT) for multi-size images is proposed. The DWT, Hessenberg matrix decomposition (HMD), singular value decomposition (SVD), particle swarm optimization (PSO), Arnold transform and logistic mapping are combined for the first time to achieve an image watermarking optimization algorithm. The multi-level decomposition of DWT is used to be adapted to multi-size host images, the Arnold transform, logistic mapping, HMD and SVD are used to enhance the security and robustness, and the PSO optimized scaling factor to balance invisibility and robustness. The simulation results of the proposed method show that the PSNRs are higher than 44.9 dB without attacks and the NCs are higher than 0.98 under various attacks. Compared with the existing works, the proposed method shows high robustness against various attacks, such as noise, filtering and JPEG compression and in particular, the NC values are at least 0.44% higher than that in noise attacks.

The Jordan and Frobenius pairs of the inverse

Given a matrix there exists a nonsingular matrix such that , where J is a very sparse matrix with a diagonal block structure, known as the Jordan canonical form (JCF) of . Assume that is nonsingular and that and are given. How to obtain and such that and is the JCF of ? Curiously, the answer involves the Pascal matrix. For the Frobenius canonical form (FCF), where blocks are companion matrices, the analogous question has a very simple answer. Jordan blocks and companions are non-derogatory lower Hessenberg matrices. The answers to the two questions will be obtained by solving two linear matrix equations involving these matrices.

A simpler GMRES algorithm accelerated by Chebyshev polynomials for computing PageRank

PageRank is one of the most important ranking techniques in modern search engines. Many great interesting researches focus on developing efficient numerical methods to compute PageRank problems. In this paper, we consider a simpler generalized minimal residual (SGMRES) algorithm for computing PageRank. The main features of the SGMRES algorithm lie in that there is no need to factorize an upper Hessenberg matrix, and the residual vector is easily obtained at each iteration. To speed up the computation of PageRank problems, an accelerated technique based on Chebyshev polynomials is applied to improve the SGMRES algorithm, such that a new algorithm named SGMRES-Chebyshev is proposed here. The implementation and the convergence analysis of the new algorithm are discussed in detail. Numerical experiments are used to illustrate the efficiency of our proposed algorithm.

A stabilized GMRES method for singular and severely ill-conditioned systems of linear equations

Consider using the right-preconditioned GMRES (AB-GMRES) for obtaining the minimum-norm solution of inconsistent underdetermined systems of linear equations. Morikuni (Ph.D. thesis, 2013) showed that for some inconsistent and ill-conditioned problems, the iterates may diverge. This is mainly because the Hessenberg matrix in the GMRES method becomes very ill-conditioned so that the backward substitution of the resulting triangular system becomes numerically unstable. We propose a stabilized GMRES based on solving the normal equations corresponding to the above triangular system using the standard Cholesky decomposition. This has the effect of shifting upwards the tiny singular values of the Hessenberg matrix which lead to an inaccurate solution. We analyze why the method works. Numerical experiments show that the proposed method is robust and efficient, not only for applying AB-GMRES to underdetermined systems, but also for applying GMRES to severely ill-conditioned range-symmetric systems of linear equations.

Orthogonal Iterations on Companion-Like Pencils

We present a class of fast subspace algorithms based on orthogonal iterations for structured matrices/pencils that can be expressed as small rank perturbations of unitary matrices. The representation of the matrix by means of a new data-sparse factorization—named LFR factorization—using orthogonal Hessenberg matrices is at the core of these algorithms. The factorization can be computed at the cost of O(n k^2) arithmetic operations, where n and k are the sizes of the matrix and the small rank perturbation, respectively. At the same cost from the LFR format we can easily obtain suitable QR and RQ factorizations where the orthogonal factor Q is a product of orthogonal Hessenberg matrices and the upper triangular factor R is again given into the LFR format. The orthogonal iteration reduces to a hopping game where Givens plane rotations are moved from one side to the other side of these two factors. The resulting new algorithms approximate an invariant subspace of size s associated with a set of s leading or trailing eigenvalues using only O(nks) operations per iteration. The number of iterations required to reach an invariant subspace depends linearly on the ratio |lambda _{s+1}|/|lambda _s|. Numerical experiments confirm the effectiveness of our adaptations.

Determinantal and permanental representations of companion sequences associated to the r-Fibonacci sequence

Recently companion sequences of r-Fibonacci sequence were defined. The aim of this paper is to give some determinantal and permanental representations of these sequences via Hessenberg matrices. Several representations of classical sequences and polynomials are established. We conclude by using our representations to give n consecutive terms of companion sequences simultaneously.

Double-Matrix Decomposition Image Steganography Scheme Based on Wavelet Transform with Multi-Region Coverage.

On the basis of ensuring the quality and concealment of steganographic images, this paper proposes a double-matrix decomposition image steganography scheme with multi-region coverage, to solve the problem of poor extraction ability of steganographic images under attack or interference. First of all, the cover image is transformed by multi-wavelet transform, and the hidden region covering multiple wavelet sub-bands is selected in the wavelet domain of the cover image to embed the secret information. After determining the hidden region, the hidden region is processed by Arnold transform, Hessenberg decomposition, and singular-value decomposition. Finally, the secret information is embedded into the cover image by embedding intensity factor. In order to ensure robustness, the hidden region selected in the wavelet domain is used as the input of Hessenberg matrix decomposition, and the robustness of the algorithm is further enhanced by Hessenberg matrix decomposition and singular-value decomposition. Experimental results show that the proposed method has excellent performance in concealment and quality of extracted secret images, and secret information is extracted from steganographic images attacked by various image processing attacks, which proves that the proposed method has good anti-attack ability under different attacks.

Jacobian-Free Poincaré-Krylov Method to Determine the Stability of Periodic Orbits of Electric Power Systems

A first application of a time domain Poincaré-Krylov approach for the study of the all-important stability of periodic solutions of nonlinear power networks is reported in this work. Whilst a Newton method solves the nonlinear algebraic equations resulting from the Poincaré map method, an iterative Krylov-subspace method based on a GMRES algorithm is applied for solving the Newton correction equations. More importantly, a QR factorization of the Hessenberg matrix involved in the GMRES least square problem is implemented using Givens rotations in order to avoid the linear growth of the computational complexity in large-scale power networks. Further, the stability of periodic solutions is determined by computing the Floquet multipliers using Ritz values and the Hessenberg matrix. Numerical tests carried out on a modified three-phase version of the IEEE 118-node system with a hydro-turbine synchronous generator and a grid-tied power converter demonstrate that speedup factors up to 8 are attainable with the Krylov-Subspace approach with respect to the standard Poincaré map method. An outstanding outcome of the comparative study is that the incorporation of QR factorization to the Hessenberg least square problem outperformed the classic periodic steady-state solvers, providing further computational savings up to 50%.

An identity involving derangement numbers and Bell numbers

The aim of this paper is to establish a new identity between derangement numbers and Bell numbers. Two known identities can be recovered. We provide a combinatorial interpretation for the new identity and a representation of the derangement numbers in terms of the determinants of Hessenberg matrices.

Twofold of algebraic decomposition method used for a watermarking scheme with $LWT$ over medical images

Mathematics has always been of great importance in various sciences, especially computer science. The mechanism used to embed various types of information in a host medical images to safeguard the privacy of the patient including the patient's name, doctor's digital signature is called watermarking. There are a lot of improved watermark algorithms, however, this information is susceptible to attack when the data are transferred over universal internet channels. This paper proposed a robust watermark algorithm that uses a Lifting Wavelet Transform $(LWT)$ and two times of the Hessenberg Matrix Decomposition Method $(HMDM)$ to embed a watermark in a chosen channel of the host image after performing the transform. The experimental results demonstrate that the improvement appears (higher robustness against $JPEG$ compression attack) and good imperceptibility against some attacks, to evaluate the fineness of the original with watermarked images and the extracted watermark respectively.

Algorithm 1019: A Task-based Multi-shift QR/QZ Algorithm with Aggressive Early Deflation

The QR algorithm is one of the three phases in the process of computing the eigenvalues and the eigenvectors of a dense nonsymmetric matrix. This paper describes a task-based QR algorithm for reducing an upper Hessenberg matrix to real Schur form. The task-based algorithm also supports generalized eigenvalue problems (QZ algorithm) but this paper concentrates on the standard case. The task-based algorithm adopts previous algorithmic improvements, such as tightly-coupled multi-shifts and Aggressive Early Deflation (AED) , and also incorporates several new ideas that significantly improve the performance. This includes, but is not limited to, the elimination of several synchronization points, the dynamic merging of previously separate computational steps, the shortening and the prioritization of the critical path, and experimental GPU support. The task-based implementation is demonstrated to be multiple times faster than multi-threaded LAPACK and ScaLAPACK in both single-node and multi-node configurations on two different machines based on Intel and AMD CPUs. The implementation is built on top of the StarPU runtime system and is part of the open-source StarNEig library.

Determinantal representations for the number of subsequences without isolated odd terms

In this short note we propose two determinantal representations for the number of subsequences without isolated odd terms are presented. One is based on a tridiagonal matrix and other on a Hessenberg matrix. We also establish a new explicit formula for the terms of this sequence based on Chebyshev polynomials of the second kind.

Comparison theorems for splittings of M-matrices in (block) Hessenberg form

Some variants of the (block) Gauss–Seidel iteration for the solution of linear systems with M-matrices in (block) Hessenberg form are discussed. Comparison results for the asymptotic convergence rate of some regular splittings are derived: in particular, we prove that for a lower-Hessenberg M-matrix \(\rho (P_{GS})\ge \rho (P_S)\ge \rho (P_{AGS})\), where \(P_{GS}, P_S, P_{AGS}\) are the iteration matrices of the Gauss–Seidel, staircase, and anti-Gauss–Seidel method. This is a result that does not seem to follow from classical comparison results, as these splittings are not directly comparable. It is shown that the concept of stair partitioning provides a powerful tool for the design of new variants that are suited for parallel computation.

The Explicit Formulae of the Determinants of Specific Pentadiagonal Toeplitz Hessenberg Matrices

The pentadiagonal Toeplitz matrix is a special kind of sparse matrix widely used in linear algebra, combinatorics, computational mathematics, and has been attracted much attention. We use the determinants of two specific Hessenberg matrices to represent the recurrence relations to prove two explicit formulae to evaluate the determinants of specific pentadiagonal Toeplitz matrices proposed in a recent paper [3]. Further, four new results are established.

Characteristic Polynomials of Random Banded Hessenberg Matrices and Hermite–Padé Approximation

We consider a class of random banded Hessenberg matrices with independent entries having identical distributions along diagonals. The distributions may be different for entries belonging to different diagonals. For a sequence of \(n\times n\) matrices in the class considered, we investigate the asymptotic behavior of their empirical spectral distribution as n tends to infinity.

Sturm theorem for the generalized Frank matrix

One of the popular test matrices for eigenvalue routines is the Frank matrix due to its well-conditioned and poorly conditioned eigenvalues. All the eigenvalues of the Frank matrix are real, positive and different. Sturm Theorem is a very useful tool for computing the eigenvalues of tridiagonal symmetric matrices. In this paper, we apply Sturm Theorem to the generalized Frank matrix which is a special form of the Hessenberg matrix and examine its eigenvalues by using Sturm property. Moreover, we illustrate our results with an example.

On the constant coefficients of a certain recurrence relation: A simple proof

In this note, we provide a short proof for the explicit formulas of the coefficients of a particular 3-term recurrence relation derived from a k-periodic recurrence. Any of the recurrences can be naturally interpreted in terms of determinants of Hessenberg matrices families.

Spectral analysis of 2D outlier layout

Thompson’s partition of a cyclic subnormal operator into normal and completely non-normal components is combined with a non-commutative calculus for hyponormal operators for separating outliers from the cloud, in rather general point distributions in the plane. The main result provides exact transformation formulas from the power moments of the prescribed point distribution into the moments of the uniform mass carried by the cloud. The proposed algorithm solely depends on the Hessenberg matrix associated to the original data. The robustness of the algorithm is reflected by the insensitivity of the output under trace class, or by a theorem of Voiculescu, under certain Hilbert–Schmidt class, additive perturbations of the Hessenberg matrix.

STOCHASTIC PSEUDOSPIN NEURAL NETWORK WITH TRIDIAGONAL SYNAPTIC CONNECTIONS

Context. To reduce the computational resource time in the problems of diagnosing and recognizing distorted images based on a fully connected stochastic pseudospin neural network, it becomes necessary to thin out synaptic connections between neurons, which is solved using the method of diagonalizing the matrix of synaptic connections without losing interaction between all neurons in the network. Objective. To create an architecture of a stochastic pseudo-spin neural network with diagonal synaptic connections without loosing the interaction between all the neurons in the layer to reduce its learning time. Method. The paper uses the Hausholder method, the method of compressing input images based on the diagonalization of the matrix of synaptic connections and the computer mathematics system MATLAB for converting a fully connected neural network into a tridiagonal form with hidden synaptic connections between all neurons. Results. We developed a model of a stochastic neural network architecture with sparse renormalized synaptic connections that take into account deleted synaptic connections. Based on the transformation of the synaptic connection matrix of a fully connected neural network into a Hessenberg matrix with tridiagonal synaptic connections, we proposed a renormalized local Hebb rule. Using the computer mathematics system “WolframMathematica 11.3”, we calculated, as a function of the number of neurons N, the relative tuning time of synaptic connections (per iteration) in a stochastic pseudospin neural network with a tridiagonal connection Matrix, relative to the tuning time of synaptic connections (per iteration) in a fully connected synaptic neural network. Conclusions. We found that with an increase in the number of neurons, the tuning time of synaptic connections (per iteration) in a stochastic pseudospin neural network with a tridiagonal connection Matrix, relative to the tuning time of synaptic connections (per iteration) in a fully connected synaptic neural network, decreases according to a hyperbolic law. Depending on the direction of pseudospin neurons, we proposed a classification of a renormalized neural network with a ferromagnetic structure, an antiferromagnetic structure, and a dipole glass.