Schur Decomposition Research Articles

An imperceptible, robust, and high payload capacity audio watermarking scheme based on the DCT transformation and Schur decomposition

This paper proposes a novel audio watermarking scheme based on the discrete cosine transform (DCT) and Schur decomposition. The proposed scheme uses the DCT transformation to increase robustness and the Schur decomposition to achieve perceptual transparency. The proposed scheme first applies the DCT transformation to the original audio signal and then applies the Schur decomposition to the mid-frequency band of the DCT coefficients that generate two matrices (U and S). The watermark bits are embedded into the diagonal elements of the triangular matrix S. The Schur decomposition increases the perceptual transparency and the DCT transformation increases robustness of the proposed audio watermarking scheme by effectively resisting several types of audio signal attacks. The imperceptibility of the proposed watermarking scheme is measured subjectively using subjective difference grades (SDG) and objectively using signal-to-noise ratio (SNR) and objective difference grades (ODG) metrics. Its robustness is evaluated against several types of attacks in terms of NC and BER for different types of audio. The resulting of payload capacity, SNR, NC, and BER are as high as 516.26 bps, 77.95, 0.05326, and 0.9727, respectively. Experimental results confirm the proposed scheme is efficient, imperceptible, and robust with a high payload capacity and no effecting audio signal.

Spectral clustering for non-reversible Markov chains

Spectral clustering methods are based on solving eigenvalue problems for the identification of clusters, e.g., the identification of metastable subsets of a Markov chain. Usually, real-valued eigenvectors are mandatory for this type of algorithms. The Perron Cluster Analysis (PCCA+) is a well-known spectral clustering method of Markov chains. It is applicable for reversible Markov chains, because reversibility implies a real-valued spectrum. We also extend this spectral clustering method to non-reversible Markov chains and give some illustrative examples. The main idea is to replace the eigenvalue problem by a real-valued Schur decomposition. By this extension non-reversible Markov chains can be analyzed. Furthermore, the chains do not need to have a positive stationary distribution. In addition to metastabilities, dominant cycles and sinks can also be identified. This novel method is called GenPCCA (i.e., generalized PCCA), since it includes the case of non-reversible processes. We also apply the method to real-world eye-tracking data.

Tensor inversion and its application to the tensor equations with Einstein product

ABSTRACTRecently, the inverse of an even-order square tensor has been put forward in [Brazell M, Li N, Navasca C, Tamon C. Solving multilinear systems via tensor inversion. SIAM J Matrix Anal Appl. 2013;34(2):542–570] by means of the tensor group consisting of even-order square tensors equipped with the Einstein product. In this paper, several necessary and sufficient conditions for the invertibility of a tensor are obtained, and some approaches for calculating the inverse (if it exists) are proposed. Furthermore, the Cramer's rule and the elimination method for solving the tensor equations with the Einstein product are derived. In addition, the tensor eigenvalue problem mentioned in [Qi L-Q. Theory of tensors (hypermatrices). Hong Kong: Department of Applied Mathematics, The Hong Kong Polytechnic University; 2014] can also be addressed by using the elimination method mentioned above. By the way, the LU decomposition and the Schur decomposition of matrices are extended to tensor case. Numerical examples are provided to illustrate the main results.

The onset of penetrative convection stimulated by internal heating in a magnetic nanofluid saturating a rotating porous medium

In this work, we investigate the effect of rotation on the onset of penetrative convection stimulated by internal heating in a thin layer of magnetic nanofluid saturating a porous medium. A model that includes the effect of Brownian diffusion, thermophoresis, and magnetophoresis is considered, while the Brinkman model is used for the porous medium. The following three boundary conditions are considered: rigid–rigid, rigid–free, and free–free. We discretized the partial differential equations by applying the Chebyshev pseudospectral method and used the QZ algorithm to solve the resulting eigenvalue problem for water and ester-based magnetic nanofluids. The nature of stability is determined by using the numerical method and is found to be stationary. The results indicate that the onset of convection is advanced with an increase in the Lewis number Le, concentration Rayleigh number Rn, and modified diffusivity ratio NA but the opposite is true in the case with an increase in the width of magnetic nanofluid layer d, Langevin parameter αL, porosity ε, Darcy number Da, modified diffusivity ratio [Formula: see text], and Taylor number TA. Moreover, the parameter NB does not affect the stability of the system significantly.

Non-Hermitian dynamics of slowly varying Hamiltonians

We develop a theoretical description of non-Hermitian time evolution that accounts for the break- down of the adiabatic theorem. We obtain closed-form expressions for the time-dependent state amplitudes, involving the complex eigen-energies as well as inter-band Berry connections calculated using basis sets from appropriately-chosen Schur decompositions. Using a two-level system as an example, we show that our theory accurately captures the phenomenon of "sudden transitions", where the system state abruptly jumps from one eigenstate to another.

The Schur decomposition of the velocity gradient tensor for turbulent flows

The velocity gradient tensor for turbulent flow contains crucial information on the topology of turbulence, vortex stretching and the dissipation of energy. A Schur decomposition of the velocity gradient tensor (VGT) is introduced to supplement the standard decomposition into rotation and strain tensors. Thus, the normal parts of the tensor (represented by the eigenvalues) are separated explicitly from non-normality. Using a direct numerical simulation of homogeneous isotropic turbulence, it is shown that the norm of the non-normal part of the tensor is of a similar magnitude to the normal part. It is common to examine the second and third invariants of the characteristic equation of the tensor simultaneously (the$\unicode[STIX]{x1D64C}{-}\unicode[STIX]{x1D64D}$diagram). With the Schur approach, the discriminant function separating real and complex eigenvalues of the VGT has an explicit form in terms of strain and enstrophy: where eigenvalues are all real, enstrophy arises from the non-normal term only. Re-deriving the evolution equations for enstrophy and total strain highlights the production of non-normality and interaction production (normal straining of non-normality). These cancel when considering the evolution of the VGT in terms of its eigenvalues but are important for the full dynamics. Their properties as a function of location in$\unicode[STIX]{x1D64C}{-}\unicode[STIX]{x1D64D}$space are characterized. The Schur framework is then used to explain two properties of the VGT: the preference to form disc-like rather than rod-like flow structures, and the vorticity vector and strain alignments. In both cases, non-normality is critical for explaining behaviour in vortical regions.

Representations for the image-kernel (p,q)-inverses of block matrices in rings

Abstract We present the conditions for a block matrix of a ring to have the image-kernel ( p , q ) {(p,q)} -inverse in the generalized Banachiewicz–Schur form. We give representations for the image-kernel inverses of the sum and the product of two block matrices. Some characterizations of the image-kernel ( p , q ) {(p,q)} -inverse in a ring with involution are investigated too.

Block Kronecker linearizations of matrix polynomials and their backward errors

We introduce a new family of linearizations of matrix polynomials---which we call Kronecker pencils---and perform a backward stability analysis of complete polynomial eigenproblems. These problems are solved by applying any backward stable algorithm to a block Kronecker pencil, such as the staircase algorithm for singular pencils or the QZ algorithm for regular pencils. This analysis allows us to identify those block Kronecker pencils that yield a computed complete eigenstructure which is exactly that of a slightly perturbed matrix polynomial. These favorable pencils include the famous Fiedler linearizations, which are just a very particular case of block Kronecker pencils. Thus, our analysis offers the first proof available in the literature of global backward stability for Fiedler pencils. In addition, the theory developed for block Kronecker pencils is much simpler than the theory available for Fiedler especially in the case of rectangular matrix polynomials. The global backward error analysis in this work presents for the first time the following key properties: it is a rigorous analysis valid for finite perturbations (i.e., it is not a first order analysis), it provides precise bounds, it is valid simultaneously for a large class of linearizations, and it establishes a framework that may be generalized to other classes of linearizations. These features are related to the fact that block Kronecker pencils are a particular case of the new family of strong block minimal bases pencils, which include certain perturbations of block Kronecker pencils; this allows us to extend the results in this paper to other contexts.

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.

Performance comparisons of the DOM, CCSM, and hybrid CCS-DOM for thermal radiation analysis in one-dimensional cylindrical coordinates

Recently, the Chebyshev collocation spectral method (CCSM) was developed to solve radiative transfer equation (RTE) directly in one-dimensional cylindrical medium (IJHMT, 189 (2017) 206–220). However, the direct solver is memory consuming and difficult to solve the problems with large grid number. In this paper, we develop a less memory consumed iterative solver based on the Schur decomposition. To simplify the solution procedure, a hybrid method of the CCSM and discrete ordinates method (DOM), named the CCS-DOM, is also investigated. In the CCS-DOM, the angular dependence is approximated by the DOM instead to avoid the coupling of radial and azimuthal directions as that in the pure CCSM. Performances of the CCSM and CCS-DOM are assessed by comparing to the DOM. In addition, the angular quadrature scheme and pole condition for the DOM are optimized in our study. The comparative results show that the CCS-DOM fails to obtain the radiative information accurately in cylindrical coordinates. Its performance is worse than that of the DOM. The CCSM presents a fifth order convergence, which is much higher than the second one of DOM, thus has great advantage on acquiring highly accurate results. It spends less CPU time and costs less memory than the DOM to achieve the same accuracy. Finally, results for the DOM and CCSM are compared and shown for various parameters. It is found that two methods have the same iterative convergence rate and present similar behaviors for the considered parameters. All these results support the CCSM to be a good alternative for thermal radiation calculation in the one-dimensional cylindrical medium with the newly developed iterative solver.

Convection in a magnetic nanofluid saturating a porous medium under the influence of a variable gravity field

This paper presents a numerical investigation of the onset of instability in a thin magnetic nanofluid (or ferrofluid) layer saturating a porous medium, subject to a gravity field which varies with distance across the layer. A model that accounts for the effect of Brownian diffusion, thermophoresis, magnetophoresis and Darcy’s law is considered. We employed the Chebyshev pseudospectral method and QZ algorithm to obtain the numerical solutions. The impact of important parameters, which affect the instability of the system has been analyzed at the onset of convection for water-based and ester-based magnetic nanofluids. The results indicate that increase in the width of magnetic nanofluid layer (d), Langevin parameter (αL), gravity coefficient (δ), permeability parameter (K); and decrease in the Lewis number (Le), concentration Rayleigh number (Rn), modified diffusivity ratio (NA), nonlinearity of fluid magnetization (M3) is to delay the onset of convection.

Some remarks on the complex J-symmetric eigenproblem

The eigenproblem for complex J-symmetric matrices HC=[ACD−AT], A,C=CT, D=DT∈Cn×n is considered. A proof of the existence of a transformation to the complex J-symmetric Schur form proposed in [21] is given. The complex symplectic unitary QR decomposition and the complex symplectic SR decomposition are discussed. It is shown that a QR-like method based on the complex symplectic unitary QR decomposition is not feasible here. A complex symplectic SR algorithm is presented which can be implemented such that one step of the SR algorithm can be carried out in O(n) arithmetic operations. Based on this, a complex symplectic Lanczos method can be derived. Moreover, it is discussed how the 2n×2n complex J-symmetric matrix HC can be embedded in a 4n×4n real Hamiltonian matrix.

Speckle suppression in medical ultrasound images through Schur decomposition

A technique based on Schur decomposition to supress the multiplicative (speckle) noise from medical ultrasound images is presented in this study. An image which carries the speckle noise is divided into small overlapping segments, size of these segments depends on the nature of speckle carried by the image and a global covariance matrix is calculated for the whole image by averaging the covariances of all segments. The global covariance matrix is decomposed through Schur decomposition to obtain the orthogonal vectors. A subset of these orthogonal vectors that correspond to largest magnitudes of eigenvalues are selected to filter out the speckle noise from the image. The proposed approach is compared with four benchmark filtering techniques, homomorphic wavelet despeckling, Wiener, Frost and Gamma. Two types of simulated ultrasound images and five types of real ultrasound images of foetal neck, left kidney, right kidney, musculo skeletal nerve and lymph node are tested. The proposed approach performed maximum suppression of speckle noise in all types of the images with optimal resolution and edge detection. The despeckling performance of the proposed approach is even better compared with the benchmark schemes once the speckle noise is rough, which is usually the case for soft tissue.

Effect of settling particles on the stability of a particle-laden flow in a vertical plane channel

The stability of a viscous particle-laden flow in a vertical plane channel in the presence of the gravity force is studied. The flow is described using a two-fluid “dusty-gas” model with negligibly small volume fraction of fines and two-way coupling of the phases. Two different profiles of the particle number density in the main flow are considered: homogeneous and non-homogeneous in the form of two layers symmetric about the channel axis. The novel element of the linear-stability problem formulation is a particle velocity slip in the main flow caused by the gravity-induced settling of the dispersed phase. The eigenvalue problem for a linearized system of governing equations is solved using the orthonormalization and QZ algorithms. For a uniform particle number density distribution, it is found that there exists a domain in the plane of Froude and Stokes numbers, in which the two-phase flow in a vertical channel is stable for an arbitrary Reynolds number. This stability domain corresponds to relatively sm...

Rortex—A new vortex vector definition and vorticity tensor and vector decompositions

A vortex is intuitively recognized as the rotational/swirling motion of the fluids. However, an unambiguous and universally accepted definition for vortex is yet to be achieved in the field of fluid mechanics, which is probably one of the major obstacles causing considerable confusions and misunderstandings in turbulence research. In our previous work, a new vector quantity that is called vortex vector was proposed to accurately describe the local fluid rotation and clearly display vortical structures. In this paper, the definition of the vortex vector, named Rortex here, is revisited from the mathematical perspective. The existence of the possible rotational axis is proved through real Schur decomposition. Based on real Schur decomposition, a fast algorithm for calculating Rortex is also presented. In addition, new vorticity tensor and vector decompositions are introduced: the vorticity tensor is decomposed to a rigidly rotational part and a non-rotationally anti-symmetric part, and the vorticity vector is decomposed to a rigidly rotational vector which is called the Rortex vector and a non-rotational vector which is called the shear vector. Several cases, including the 2D Couette flow, 2D rigid rotational flow, and 3D boundary layer transition on a flat plate, are studied to demonstrate the justification of the definition of Rortex. It can be observed that Rortex identifies both the precise swirling strength and the rotational axis, and thus it can reasonably represent the local fluid rotation and provide a new powerful tool for vortex dynamics and turbulence research.

Vorticity and helicity decompositions and dynamics with real Schur form of the velocity gradient

The real Schur form (RSF) of a generic velocity gradient field ∇u is exploited to expose the structures of flows, in particular, our field decomposition resulting in two vorticities with only mutual linkage as the topological content of the global helicity (accordingly decomposed into two equal parts). The local transformation to the RSF may indicate alternative (co)rotating frame(s) for specifying the objective argument(s) of the constitutive equation. When ∇u is uniformly of RSF in a fixed Cartesian coordinate frame, i.e., ux = ux(x, y) and uy = uy(x, y), but uz = uz(x, y, z), the model, with the decomposed vorticities both frozen-in to u, is for two-component-two-dimensional-coupled-with-one-component-three-dimensional flows in between two-dimensional-three-component (2D3C) and fully three-dimensional-three-component ones and may help curing the pathology in the helical 2D3C absolute equilibrium, making the latter effectively work in more realistic situations.

A Hybrid Video Watermarking Technique based on DWT, SVD and SCHUR Decomposition

A Hybrid Video Watermarking Technique based on DWT, SVD and SCHUR Decomposition

A Householder-Based Algorithm for Hessenberg-Triangular Reduction

The QZ algorithm for computing eigenvalues and eigenvectors of a matrix pencil $A - \lambda B$ requires that the matrices first be reduced to Hessenberg-triangular (HT) form. The current method of ...

Stability of natural convection in a vertical non-Newtonian fluid layer with an imposed magnetic field

A numerical study has been conducted to analyze the influence of a uniform horizontal magnetic field on the stability of buoyancy driven parallel shear flow in a differentially heated vertical layer of an electrically conducting couple stress fluid; a type of non-Newtonian fluid. Within the framework of linear stability theory, the resulting complex generalized eigenvalue problem is solved numerically using the Chebyshev collocation method with QZ algorithm. The critical Grashof number $$G_{c}$$ and the corresponding wave number $$\alpha_{c}$$ and wave speed $$c_{c}$$ are computed for a wide range of couple stress parameter $$\varLambda_{c}$$ , Prandtl number $$Pr$$ and Hartmann number $$M$$ . It is found that the value of $$Pr$$ at which the instability switches over from stationary to travelling-wave mode increases with increasing $$M$$ and decreasing $$\varLambda_{c}$$ . The effect of magnetic field is to delay the onset of instability while an opposite kind of behavior is observed with increasing $$\varLambda_{c}$$ . The streamlines presented herein demonstrate the development of complex dynamics at the transition mode.

Role of slip on the linear stability of a liquid flow through a porous channel

The linear stability of a liquid flow bounded by slippery and porous walls is studied for infinitesimal disturbances of arbitrary wavenumbers. The Orr-Sommerfeld type eigenvalue problem is formulated by using the normal mode decomposition and resolved based on the Chebyshev spectral collocation method along with the QZ algorithm. The results are computed numerically in detail for various values of the flow parameters. The presence of an upper wall slip shows a destabilizing effect on the fluid layer mode, but it shows a stabilizing effect on the porous layer mode. On the other hand, the decreasing value of the depth ratio has a stabilizing effect on the fluid layer mode but it has a destabilizing effect on the porous layer mode. In fact, there occurs a competition between the most unstable porous layer mode and the most unstable fluid layer mode to control the primary instability. The most unstable porous layer mode triggers the primary instability unless the upper wall slip dominates the effect of the porous layer otherwise the most unstable fluid layer mode triggers the primary instability. A new phase boundary is detected in the plane of the depth ratio and slip length, which separates the domain of the most unstable porous layer mode from the domain of the most unstable fluid layer mode.