Schur Decomposition Research Articles

Decay estimates of Green's matrices for discrete-time linear periodic systems

We study periodic Lyapunov matrix equations for a general discrete-time linear periodic system B p x p − A p x p − 1 = f p , where the matrix coefficients B p and A p can be singular. The block coefficients of the inverse operator of the system are referred to as the Green matrices. We derive new decay estimates of the Green matrices in terms of the spectral norms of special solutions to the periodic Lyapunov matrix equations. The study is based on the periodic Schur decomposition of matrices.

Secure and optimized satellite image sharing based on chaotic eπ map and Racah moments

Secure and optimized satellite image sharing based on chaotic eπ map and Racah moments

Robust watermark based on Schur decomposition and dynamic weighting factors

Since multimedia data are now more widely distributed in digital form and easier to copy and change, copyright protection has become an essential need. One of the most common copyright protection techniques is watermarking. Invisibility and robustness are crucial aspects of watermarking; however, many of the currently used techniques simply focus more on invisibility than how robust they are. Moreover, the trade-off between invisibility and robustness is challenging. To this end, this paper proposes a novel watermark technique that efficiently overcomes the idea of a trade-off between robustness and invisibility, thereby increasing both under most attacks. Schur decomposition and a dynamic weighting factors matrix are added to the embedding process to improve the robustness of the proposed technique. Besides that, the embedding function is improved to simultaneously maximize imperceptibility and robustness. Another key contribution of the proposed approach is its use of a trajectory-based optimization algorithm rather than the more prevalent population-based algorithms to determine the optimal scaling factors. Consequently, the proposed technique rapidly identifies the best scaling factors for the embedding function. Statistical analysis is performed using the Friedman test. Experimental results show that the proposed technique outperforms other existing techniques for different sizes, shapes, and types of watermarks.

A new deflation criterion for the QZ algorithm

AbstractThe QZ algorithm computes the generalized Schur form of a matrix pencil. It is an iterative algorithm and, at some point, it must decide when to deflate, that is when a generalized eigenvalue has converged and to move on to another one. Choosing a deflation criterion that makes this decision is nontrivial. If it is too strict, the algorithm might waste iterations on already converged eigenvalues. If it is not strict enough, the computed eigenvalues might not have full accuracy. Additionally, the criterion should not be computationally expensive to evaluate. There are two commonly used criteria: the elementwise criterion and the normwise criterion. This paper introduces a new deflation criterion based on the size of and the gap between the eigenvalues. We call this new deflation criterion the strict criterion. This new criterion for QZ is analogous to the criterion derived by Ahues and Tisseur for the QR algorithm. Theoretical arguments and numerical experiments suggest that the strict criterion outperforms the normwise and elementwise criteria in terms of accuracy. We also provide an example where the accuracy of the generalized eigenvalues using the elementwise or the normwise criteria is less than two digits whereas the strict criterion leads to generalized eigenvalues which are almost accurate to the working precision. Additionally, this paper evaluates some commonly used criteria for infinite eigenvalues.

Image watermarking techniques based on Schur decomposition and various image invariant moments: a review

Image watermarking techniques based on Schur decomposition and various image invariant moments: a review

Multi-Eigenvalue Demodulation Using Complex Moment-Based Eigensolver and Neural Network

Optical eigenvalues originating in optical solitons are the potential for becoming information carriers not affected by chromatic dispersion and nonlinear effects in optical fibers. They are obtained by attributing the associated eigenvalue equations deduced by solving the nonlinear Schrödinger equation with inverse scattering transform (IST) to the matrix eigenvalue problem, and maintaining constant values regardless of the transmission distance. The eigenvalue communication systems require to solve the eigenvalues in soliton-by-soliton. While effective eigenvalue solution methods have not been studied well in telecommunication systems, one of the most well-known eigenvalue solution methods is the QZ decomposition-based method. However, the QZ algorithm requires a large complexity. To reduce the complexity, a method to demodulate the optical eigenvalues using a complex-moment eigenvalue solver (CME) was investigated. CME is a parallelizable eigenvalue solver that can extract any eigenvalue. This paper proposes a novel optical eigenvalue demodulation method that combines CME and an artificial neural network (ANN) based on employing an on-off encoded discrete eigenvalue modulation scheme. The ANN is sensitive to the input order of the units; therefore, the eigenvalues must be sorted. A lightweight sorting algorithm is hence required. In addition to the proposed scheme, this study proposes partial sorting using CME and ANN. Here, 2000 km fiber-transmission experiments for an on-off-encoded four-discrete- eigenvalue were conducted. The experimental results indicated that the proposed demodulation method obtained bit error rate (BER) characteristics comparable to conventional methods by devising an extraction range of eigenvalues in CME.

Hydrodynamic Stability Analysis for MHD Casson Fluid Flow through a Restricted Channel

Flow instability is a major challenge experienced in medical, engineering and industrial settings globally. For instance, flow instability linked with irregular cardiac output of the heart leads to organ malfunctioning in the medical field, it also encourages mechanical vibrations in the case of fluctuating flow rate, and several other applications. In this study, linear stability analysis is conducted to monitor the behavior of a small disturbance that is imposed on hydromagnetic Casson fluid that flows steadily through a saturated porous medium. A new variant of the Orr-Sommerfield equation is obtained and solved numerically by using spectral point collocation weighted residual approach with eigenfunction expansion of the Chebyshev polynomial as the admissible trial function. Based on the QZ algorithm, numerical results are obtained for wave and Reynold’s numbers, wave velocity as functions of Magnetic field intensity and porosity shape parameters. Results are validated against previously released data. The biophysics of the heart, particularly in cardiac rhythm analysis, as well as several other medicinal and technical applications, is among the areas where the current work has applicability.

A Method for Robust Partial Quadratic Eigenvalue Assignment with Repeated Eigenvalues

Abstract A new method is proposed for robust partial quadratic eigenvalue assignment problem (RPQEAP) with repeated prescribed eigenvalues. We first derive a result on the solution of partial quadratic eigenvalue assignment problem, which leads to the partial Schur form of the close-loop system. Then by minimizing the normality departure of the partial Schur form of the close-loop system, we assign the prescribed eigenvalues step by step such that the close-loop system is as robust as possible. Our method allows the prescribed eigenvalues are repeated. Moreover, the proposed method does not use the unchanged eigenpairs of the open-loop system and avoids transforming the second-order control system to the first-order system. Numerical experiments show that the proposed method is efficient for solving the RPQEAP with repeated eigenvalues.

Perturbations of Tensor-Schur decomposition and its applications to multilinear control systems and facial recognitions

Perturbations of Tensor-Schur decomposition and its applications to multilinear control systems and facial recognitions

Novel schemes for the improvement of lifting wavelet transform-based image watermarking using Schur decomposition

Novel schemes for the improvement of lifting wavelet transform-based image watermarking using Schur decomposition

The triple decomposition of the velocity gradient tensor as a standardized real Schur form

The triple decomposition of a velocity gradient tensor provides an analysis tool in fluid mechanics by which the flow can be split into a sum of irrotational straining flow, shear flow, and rigid body rotational flow. In 2007, Kolář formulated an optimization problem to compute the triple decomposition [V. Kolář, “Vortex identification: New requirements and limitations,” Int. J. Heat Fluid Flow 28, 638–652 (2007)], and more recently, the triple decomposition has been connected to the Schur form of the associated matrix. We show that the standardized real Schur form, which can be computed by state of the art linear algebra routines, is a solution to the optimization problem posed by Kolář. We also demonstrate why using the standardized variant of the real Schur form makes computation of the triple decomposition more efficient. Furthermore, we illustrate why different structures of the real Schur form correspond to different alignments of the coordinate system with the fluid flow and may, therefore, lead to differences in the resulting triple decomposition. Based on these results, we propose a new, simplified algorithm for computing the triple decomposition, which guarantees consistent results.

Class-overlap undersampling based on Schur decomposition for Class-imbalance problems

Class-overlap undersampling based on Schur decomposition for Class-imbalance problems

Singular quadratic eigenvalue problems: linearization and weak condition numbers

The numerical solution of singular eigenvalue problems is complicated by the fact that small perturbations of the coefficients may have an arbitrarily bad effect on eigenvalue accuracy. However, it has been known for a long time that such perturbations are exceptional and standard eigenvalue solvers, such as the QZ algorithm, tend to yield good accuracy despite the inevitable presence of roundoff error. Recently, Lotz and Noferini quantified this phenomenon by introducing the concept of $$\delta $$ -weak eigenvalue condition numbers. In this work, we consider singular quadratic eigenvalue problems and two popular linearizations. Our results show that a correctly chosen linearization increases $$\delta $$ -weak eigenvalue condition numbers only marginally, justifying the use of these linearizations in numerical solvers also in the singular case. We propose a very simple but often effective algorithm for computing well-conditioned eigenvalues of a singular quadratic eigenvalue problems by adding small random perturbations to the coefficients. We prove that the eigenvalue condition number is, with high probability, a reliable criterion for detecting and excluding spurious eigenvalues created from the singular part.

Schur decomposition of several matrices

Schur decompositions and the corresponding Schur forms of a single matrix, a pair of matrices, or a collection of matrices associated with the periodic eigenvalue problem are frequently used and studied. These forms are upper-triangular complex matrices or quasi-upper-triangular real matrices that are equivalent to the original matrices via unitary or, respectively, orthogonal transformations. In general, for theoretical and numerical purposes we often need to reduce, by admissible transformations, a collection of matrices to the Schur form. Unfortunately, such a reduction is not always possible. In this paper we describe all collections of complex (real) matrices that can be reduced to the Schur form by the corresponding unitary (orthogonal) transformations and explain how such a reduction can be done. We prove that this class consists of the collections of matrices associated with pseudoforest graphs. In other words, we describe when the Schur form of a collection of matrices exists and how to find it.

B-essential spectra of 2 × 2 block operator matrix pencils

Abstract The paper is devoted to the study of the relative B-essential spectra of 2 × 2 {2\times 2} unbounded operator matrix pencils defined with nondiagonal domain. The main tool used to prove the results is the Frobenuis–Schur factorization associated with this kind of operator matrix which has proven very useful in dealing with stability analysis in the theory of B-Fredholm operators pencils. Therefore, the theoretical results are tested in pencil transport integro-differential equation with specific boundary conditions.

Structural backward stability in rational eigenvalue problems solved via block Kronecker linearizations

In this paper we study the backward stability of running a backward stable eigenstructure solver on a pencil $$S(\lambda )$$ that is a strong linearization of a rational matrix $$R(\lambda )$$ expressed in the form $$R(\lambda )=D(\lambda )+ C(\lambda I_\ell -A)^{-1}B$$ , where $$D(\lambda )$$ is a polynomial matrix and $$C(\lambda I_\ell -A)^{-1}B$$ is a minimal state-space realization. We consider the family of block Kronecker linearizations of $$R(\lambda )$$ , which have the following structure $$\begin{aligned} S(\lambda ):=\left[ \begin{array}{ccc} M(\lambda ) &{} {\widehat{K}}_2^T C &{} K_2^T(\lambda ) \\ B {\widehat{K}}_1 &{} A- \lambda I_\ell &{} 0\\ K_1(\lambda ) &{} 0 &{} 0 \end{array}\right] , \end{aligned}$$ where the blocks have some specific structures. Backward stable eigenstructure solvers, such as the QZ or the staircase algorithms, applied to $$S(\lambda )$$ will compute the exact eigenstructure of a perturbed pencil $$\widehat{S}(\lambda ):=S(\lambda )+\varDelta _S(\lambda )$$ and the special structure of $$S(\lambda )$$ will be lost, including the zero blocks below the anti-diagonal. In order to link this perturbed pencil with a nearby rational matrix, we construct in this paper a strictly equivalent pencil $$\widetilde{S}(\lambda )=(I-X)\widehat{S}(\lambda )(I-Y)$$ that restores the original structure, and hence is a block Kronecker linearization of a perturbed rational matrix $${{\widetilde{R}}}(\lambda ) = {{\widetilde{D}}}(\lambda )+ {{\widetilde{C}}}(\lambda I_\ell - {{\widetilde{A}}})^{-1} {{\widetilde{B}}}$$ , where $${{\widetilde{D}}}(\lambda )$$ is a polynomial matrix with the same degree as $$D(\lambda )$$ . Moreover, we bound appropriate norms of $${{\widetilde{D}}}(\lambda )- D(\lambda )$$ , $${{\widetilde{C}}} - C$$ , $${{\widetilde{A}}} - A$$ and $${{\widetilde{B}}} - B$$ in terms of an appropriate norm of $$\varDelta _S(\lambda )$$ . These bounds may be, in general, inadmissibly large, but we also introduce a scaling that allows us to make them satisfactorily tiny, by making the matrices appearing in both $$S(\lambda )$$ and $$R(\lambda )$$ have norms bounded by 1. Thus, for this scaled representation, we prove that the staircase and the QZ algorithms compute the exact eigenstructure of a rational matrix $${{\widetilde{R}}}(\lambda )$$ that can be expressed in exactly the same form as $$R(\lambda )$$ with the parameters defining the representation very near to those of $$R(\lambda )$$ . This shows that this approach is backward stable in a structured sense. Several numerical experiments confirm the obtained backward stability results.

Stationary Wavelet-Based Image Watermarking for E-Healthcare Applications

Watermarking is a technique for protecting multimedia data, mainly images, from malicious attacks by adding a signature into these images. Traditional watermarking techniques, unfortunately, have drawbacks when used on sensitive images like those used in medicine. A reliable and blind watermarking method is suggested in this work to protect medical picture transferred in telemedicine. Medical picture marking enables accurate patient identification, prevents scan confusion, and minimizes the risk of diagnostic mistakes that could have negative consequences. In this method, three distinct transforms are used to acquire the frequency content of the image. The low frequency subbands are subsequently subjected to Schur decomposition. In order to incorporate the watermark bits, the resultant upper triangular matrix values are modified. The proposed approaches effectively retain a considerable quality of watermarked images and are impressively resistant against numerous conventional attacks, according to imperceptibility and robustness experimental results. The average Peak signal-to-noise ratio (PSNR) which was obtained was 44.90 dB, demonstrating that the integration procedure produces very low distortion to the original image. The observed results for robustness demonstrate that the watermark is resilient to the most of the attacks performed in watermarking, with a normalized cross correlation rate higher than 0.9.

Application of Zero-Watermarking for Medical Image in Intelligent Sensor Network Security

The field of healthcare is considered to be the most promising application of intelligent sensor networks. However, the security and privacy protection of medical images collected by intelligent sensor networks is a hot problem that has attracted more and more attention. Fortunately, digital watermarking provides an effective method to solve this problem. In order to improve the robustness of the medical image watermarking scheme, in this paper, we propose a novel zero-watermarking algorithm with the integer wavelet transform (IWT), Schur decomposition and image block energy. Specifically, we first use IWT to extract low-frequency information and divide them into non-overlapping blocks, then we decompose the sub-blocks by Schur decomposition. After that, the feature matrix is constructed according to the relationship between the image block energy and the whole image energy. At the same time, we encrypt watermarking with the logistic chaotic position scrambling. Finally, the zero-watermarking is obtained by XOR operation with the encrypted watermarking. Three indexes of peak signal-to-noise ratio, normalization coefficient (NC) and the bit error rate (BER) are used to evaluate the robustness of the algorithm. According to the experimental results, most of the NC values are around 0.9 under various attacks, while the BER values are very close to 0. These experimental results show that the proposed algorithm is more robust than the existing zero-watermarking methods, which indicates it is more suitable for medical image privacy and security protection.

Robust Watermarking Algorithm for Medical Images Based on Non-Subsampled Shearlet Transform and Schur Decomposition

With the development of digitalization in healthcare, more and more information is delivered and stored in digital form, facilitating people's lives significantly. In the meanwhile, privacy leakage and security issues come along with it. Zero watermarking can solve this problem well. To protect the security of medical information and improve the algorithm's robustness, this paper proposes a robust watermarking algorithm for medical images based on Non-Subsampled Shearlet Transform (NSST) and Schur decomposition. Firstly, the low-frequency subband image of the original medical image is obtained by NSST and chunked. Secondly, the Schur decomposition of lowfrequency blocks to get stable values, extracting the maximum absolute value of the diagonal elements of the upper triangle matrix after the Schur decomposition of each low-frequency block and constructing the transition matrix from it. Then, the mean of the matrix is compared to each element's value, creating a feature matrix by combining perceptual hashing, and selecting 32 bits as the feature sequence. Finally, the feature vector is exclusive OR (XOR) operated with the encrypted watermark information to get the zero watermark and complete registration with a third-party copyright certification center. Experimental data show that the Normalized Correlation (NC) values of watermarks extracted in random carrier medical images are above 0.5, with higher robustness than traditional algorithms, especially against geometric attacks and achieve watermark information invisibility without altering the carrier medical image.

The effects of the Soret and slip boundary conditions on thermosolutal convection with a Navier–Stokes–Voigt fluid

In this paper, we study the problem of thermosolutal convection in a Navier–Stokes–Voigt fluid when the layer is heated from below and simultaneously salted from above or below. This problem is studied under the effects of Soret and slip boundary conditions. Both linear and nonlinear stability analyses are employed. When the layer is heated from below and salted from above, the boundaries exhibit great concordance, resulting in a very narrow region of probable subcritical instabilities. This proves that linear analysis is reliable enough to forecast the beginning of convective motion. The Chebyshev collocation technique and QZ algorithm have been used to solve systems of linear and nonlinear theories. For thermal convection in a dissolved salt field with a complex viscoelastic fluid of the Navier–Stokes–Voigt type, instability boundaries are computed. When the convection is of the oscillatory type, the Kelvin–Voigt parameter is observed to play a crucial role in functioning as a stabilizing agent. This effect's quantitative size is shown.