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In this paper, we compute paradeterminants and parapermanents of some triangular matrices that give bihyperbolic numbers of the Fibonacci type. Using connections between the paradeterminant of triangular matrix and the lower Hessenberg determinant, we also obtain the general term of these sequences. [ABSTRACT FROM AUTHOR]

Hessenberg matrices arise in various mathematical and engineering applications due to their special properties and efficient algorithms for solving linear systems. This study aims to compute the determinants and inverses of generalized lower Hessenberg matrices composed of conditional polynomial sequences which are defined based on specific conditions and exhibit intriguing behavior. Moreover, we present more generalized results compared to some earlier works using the factorization properties of some special matrices and some analytical methods. [ABSTRACT FROM AUTHOR]

We present some accelerated variants of fixed point iterations for computing the minimal non-negative solution of the unilateral matrix equation associated with an M/G/1-type Markov chain. These variants derive from certain staircase regular splittings of the block Hessenberg M-matrix associated with the Markov chain. By exploiting the staircase profile, we introduce a two-step fixed point iteration. The iteration can be further accelerated by computing a weighted average between the approximations obtained at two consecutive steps. The convergence of the basic two-step fixed point iteration and of its relaxed modification is proved. Our theoretical analysis, along with several numerical experiments, shows that the proposed variants generally outperform the classical iterations. [ABSTRACT FROM AUTHOR]

In view of a general formula for higher order derivatives of the ratio of two differentiable functions, the authors establish the first form for the Maclaurin power series expansion of a logarithmic expression in term of determinants of special Hessenberg matrices whose elements involve the Bernoulli numbers. On the other hand, for comparison, the authors recite and revise the second form for the Maclaurin power series expansion of the logarithmic expression in terms of the Bessel zeta functions and the Bernoulli numbers. [ABSTRACT FROM AUTHOR]

We consider the Gaussian Leonardo numbers and investigate some of their amazing characteristic properties, including their generating function, the associated Binet formula and Cassini identity, and their matrix representation. Then, we define the hybrid Gaussian Leonardo numbers and obtain some of their particular properties. Furthermore, we define n n Hessenberg matrices whose permanents yield the Leonardo and Gaussian Leonardo sequences. [ABSTRACT FROM AUTHOR]

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.

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. [ABSTRACT FROM AUTHOR]

In view of a general formula for higher order derivatives of the ratio of two differentiable functions, the authors establish the first form for the Maclaurin power series expansion of a logarithmic expression in term of determinants of special Hessenberg matrices whose elements involve the Bernoulli numbers. On the other hand, for comparison, the authors recite and revise the second form for the Maclaurin power series expansion of the logarithmic expression in terms of the Bessel zeta functions and the Bernoulli numbers.

In this paper, we give locations of the zeros of polynomials satisfying particular (d + 1) -order recurrence relations and we establish some representations of the smallest and largest real zeros of these polynomials in terms of the parameters in the (d + 1) -order recurrence relation. Particular interest is devoted to (d + 1) -fold symmetric polynomials. [ABSTRACT FROM AUTHOR]

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 ρ (P GS) ≥ ρ (P S) ≥ ρ (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. [ABSTRACT FROM AUTHOR]

We consider the Gaussian Leonardo numbers and investigate some of their amazing characteristic properties, including their generating function, the associated Binet formula and Cassini identity, and their matrix representation. Then, we define the hybrid Gaussian Leonardo numbers and obtain some of their particular properties. Furthermore, we define nn Hessenberg matrices whose permanents yield the Leonardo and Gaussian Leonardo sequences.

Let Sk(n) denote the sum of the k-th powers of the first n positive integers. In this article, starting from two explicit formulas for Sk(n) involving the Stirling numbers of the second kind, we derive a pair of corresponding recursive formulas for Sk(n) involving the Stirling numbers of the first kind. Our derivation makes use of the orthogonality relations for the Stirling numbers. Furthermore, we obtain yet another recursive formula for Sk(n) in the form of the so-called Newton-Girard identities. [ABSTRACT FROM AUTHOR]

With a proper function theoretic definition of the cloud of a positive measure with compact support in the real plane, a computational scheme of transforming the moments of the original measure into the moments of the uniformly distributed mass on the cloud is described. The main limiting operation involves exclusively truncated Christoffel-Darboux kernels, while error bounds depend on the spectral asymptotics of a Hankel kernel belonging to the Hilbert-Schmidt class. [ABSTRACT FROM AUTHOR]

The convolved (u, v)-Lucas first kind p-polynomials are defined using the generating function of the (u, v)-Lucas first kind p-polynomials. The determinantal and permanental representations of the convolved (u, v)-Lucas first kind p-polynomials are used to derive some identities of these polynomials.

In frequency analysis an often appearing problem is the reconstruction of a signal from given samples. Since the samples are usually noised, pure interpolating approaches are not recommended and appropriate approximation methods are more suitable as they can be interpreted as a kind of denoising. Two approaches are widely used. One uses the reflection coefficients of a finite sequence of Szegő polynomials and the other one the zeros of the so called Prony polynomial. We show that both approaches are closely related. As a kind of inverse problem, it’s not surprising that they have in common that both methods depend very sensitive on sampling errors. We use known properties of the signal to estimate the positions of the zeros of the corresponding Szegő- or Prony-like polynomials and construct adaptive algorithms to calculate these ones. Hereby, we get the corresponding parameters in the exponential parts of the signal, too. Then, the coefficients of the signal (as a linear combination of such exponential functions) can be obtained from a system of linear equations by minimizing the residuals with respect to a suitable norm as a kind of denoising. [ABSTRACT FROM AUTHOR]

In this paper, we evaluate determinants of several families of Hessenberg matrices having various subsequences of the Jacobsthal sequence as their nonzero entries. These identities may be written equivalently as formulas for certain linearly recurrent sequences expressed in terms of sums of products of Jacobsthal numbers with multinomial coefficients. Among the sequences that arise in this way include the Mersenne, Lucas and Jacobsthal-Lucas numbers as well as the squares of the Jacobsthal and Mersenne sequences. These results are extended to Hessenberg determinants involving sequences that are derived from two general families of linear second-order recurrences. Finally, combinatorial proofs are provided for several of our determinant results which make use of various correspondences between Jacobsthal tilings and certain restricted classes of binary words. [ABSTRACT FROM AUTHOR]

We characterize Hessenberg matrices D associated with measures in the unit circle ν, which are matrix representations of compact and actually Hilbert Schmidt perturbations of the forward shift operator as those with recursion coefficients {αn}n=0∞ verifying ∑n=0∞|αn|2<∞, ie, associated with measures verifying Szegö condition. As a consequence, we obtain the following dichotomy result for Hessenberg matrices associated with measures in the unit circle: either D=SR+K2 with K2, a Hilbert Schmidt matrix, or there exists an unitary matrix U and a diagonal matrix Λ such that D=U∗ΛU+K2 with K2, a Hilbert Schmidt matrix. Moreover, we prove that for 1 ≤ p ≤ 2, if ∑n=0∞|αn|p<∞, then D=SR+Kp with Kp an absolutely p summable matrix inducing an operator in the p Schatten class. Some applications are given to classify measures on the unit circle. [ABSTRACT FROM AUTHOR]

The Combined matrix of a nonsingular matrix A is defined by ϕ (A) = A ∘ A - 1 T where ∘ means the Hadamard (entrywise) product. If the matrix A describes the relation between inputs and outputs in a multivariable process control, ϕ (A) describes the "relative gain array" (RGA) of the process and it defines the Bristol method (IEEE Trans Autom Control 1:133–134, 1966) often used for Chemical processes (McAvoy in Interaction analysis: principles and applications. Instrument Society of America, Pittsburgh, 1983; Papadourakis et al. in Ind Eng Chem Res 26(6):1259–1262, 1987; Wang et al. in Chem Eng Technol, 10.1002/ceat.201500202, 2016; Kariwala et al. in Ind Eng Chem Res 45(5):1751–1757, 10.1021/ie050790r, 2006; Golender et al. in J Chem Inf Comput Sci 21(4):196-204, 10.1021/ci00032a004, 1981). The combined matrix has been studied in several works such as Bru et al. (J Appl Math, 10.1155/2014/182354, 2014), Fiedler and Markham (Linear Algebra Appl 435:1945–1955, 2011) and Johnson and Shapiro (SIAM J Algebraic Discrete Methods 7:627–644, 1986). Since ϕ (A) = (c ij) has the property of ∑ k c ik = ∑ k c kj = 1 , ∀ i , j , when ϕ (A) ≥ 0 , ϕ (A) is a doubly stochastic matrix. In certain chemical engineering applications a diagonal of the RGA in wchich the entries are near 1 is used to determine the pairing of inputs and outputs for further design analysis. Applications of these matrices can be found in Communication Theory, related with the satellite-switched time division multiple-access systems, and about a doubly stochastic automorphism of a graph. In this paper we present new algorithms to generate doubly stochastic matrices with the Combined matrix using Hessenberg matrices in Sect. 3 and orthogonal/unitary matrices in Sect. 4. In addition, we discuss what kind of doubly stochastic matrices are obtained with our algorithms and the possibility of generating a particular doubly stochastic matrix by the map ϕ . [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This paper proposes a combination of a hybrid CPU{GPU and a pure GPU software implementation of a direct algorithm for solving shifted linear systems (A -- σI)X = B with a large number of complex shifts σ and multiple right-hand sides. Such problems often appear, e.g., in control theory when evaluating the transfer function, or as a part of an algorithm performing interpolatory model reduction, as well as when computing pseudospectra and structured pseudospectra, or solving large linear systems of ordinary differential equations. The proposed algorithm first jointly reduces the general full n x n matrix A and the n x m full right-hand side matrix B to the controller Hessenberg canonical form that facilitates efficient solution: A is transformed to a so-called m-Hessenberg form, and B is made upper triangular. This is implemented as a blocked highly parallel CPU{GPU hybrid algorithm; individual blocks are reduced by the CPU, and the necessary updates of the rest of the matrix are split among the cores of the CPU and the GPU. To enhance parallelization, the reduction and the updates are overlapped. In the next phase, the reduced m-Hessenberg{triangular systems are solved entirely on the GPU, with shifts divided into batches. The benefits of such load distribution are demonstrated by numerical experiments. In particular, we show that our proposed implementation provides an excellent basis for efficient implementations of computational methods in systems and control theory, from evaluation of transfer function to the interpolatory model reduction. [ABSTRACT FROM AUTHOR]

In this note we study the connection between orthogonal polynomials on an ellipse and orthogonal Laurent polynomials on the unit circle relative to some multiplicative measures and then establish the recurrence relations for orthogonal polynomials on an ellipse. The matrix representation of the operator of multiplication by coordinate function is obtained. [ABSTRACT FROM AUTHOR]

In this note we study the connection between orthogonal polynomials on an ellipse and orthogonal Laurent polynomials on the unit circle relative to some multiplicative measures and then establish the recurrence relations for orthogonal polynomials on an ellipse. The matrix representation of the operator of multiplication by coordinate function is obtained

We describe subspaces of generalized Hessenberg matrices where the determinant is convertible into the permanent by affixing ± signs. An explicit characterization of convertible Hessenberg-type matrices is presented. We conclude that convertible matrices with the maximum number of nonzero entries can be reduced to a basic set. [ABSTRACT FROM AUTHOR]

In this paper a hybrid of efficient techniques for obtaining the transient response vectors of a linear time-invariant system is presented. The proposed method is called the restrictive Padé approximation and is used for approximating the resulting exponential matrix which arises when solving the linear time-invariant system in the Hessenberg form. The suggested method does not require the knowledge of eigen-values of the matrix. Some numerical examples are given to validate the accuracy of the proposed method. [ABSTRACT FROM AUTHOR]

It is a challenging work to design a blind color image watermarking scheme for protecting copyright, which is different from the existing schemes used binary image or grayscale image as watermark and is also different from other non-blind watermarking schemes. In this paper, we analyze the feature of the upper Hessenberg matrix, and propose a blind color image watermarking scheme using upper Hessenberg matrix of Hessenberg transform. Arnold transform is used to improve the security, and the MD5-based Hash pseudo-random algorithm is also used to improve the robustness. In the process of watermark embedding, the encrypted watermark information is embedded into the biggest energy element of the Hessenberg matrix by quantization technique. In the process of watermark extraction, the watermark is extracted from the attacked host image with blind manner. Simulation results show that the proposed scheme outperforms other related methods in the aspects of the invisibility, robustness, capacity and computational complexity. [ABSTRACT FROM AUTHOR]

For a Markov chain described by an irreducible stochastic matrix A of order n , the mean first passage time m i , j measures the expected time for the Markov chain to reach state j for the first time given that the system begins in state i , thus quantifying the short-term behaviour of the chain. In this article, a lower bound for the maximum mean first passage time is found in terms of the stationary distribution vector of A , and some matrices for which equality is attained are produced. The main objective of this article is to characterise the directed graphs for which any stochastic matrix A respecting this directed graph attains equality in this lower bound, producing a class of Markov chains with optimal short-term behaviour. [ABSTRACT FROM AUTHOR]

In this paper, a novel blind color image watermarking based on Contourlet transform and Hessenberg decomposition is proposed to protect digital copyright of color image. Firstly, each color channel of the host image is transformed by Contourlet transform and its low frequency sub-band is divided into 4 × 4 non-overlap coefficient block. Secondly, the coefficient block selected by MD5-based Hash pseudo-random algorithm is decomposed by Hessenberg decomposition. Thirdly, the watermark information permuted by Arnold transform is embedded into the biggest energy element of the upper Hessenberg matrix by quantization technique. In extraction process, the quantization strength is used for blindly extracting watermark information from the attacked host image without the help of any original image. The results show that the proposed scheme has higher imperceptibility and robustness against most common image attacks in comparison with other related methods. [ABSTRACT FROM AUTHOR]

There are many relations between number theory and matrix theory. In this paper, our aim is to obtain some relationships between the Padovan, Perrin and Tribonacci numbers and upper Hessenberg matrices. [ABSTRACT FROM AUTHOR]