Schur Decomposition Research Articles

Computing the Mittag-Leffler function of a matrix argument

It is well-known that the two-parameter Mittag-Leffler (ML) function plays a key role in Fractional Calculus. In this paper, we address the problem of computing this function, when its argument is a square matrix. Effective methods for solving this problem involve the computation of higher order derivatives or require the use of mixed precision arithmetic. In this paper, we provide an alternative method that is derivative-free and works entirely using IEEE standard double precision arithmetic. If certain conditions are satisfied, our method uses a Taylor series representation for the ML function; if not, it switches to a Schur-Parlett technique that will be combined with the Cauchy integral formula. A detailed discussion on the choice of a convenient contour is included. Theoretical and numerical issues regarding the performance of the proposed algorithm are discussed. A set of numerical experiments shows that our novel approach is competitive with the state-of-the-art method for IEEE double precision arithmetic, in terms of accuracy and CPU time. For matrices whose Schur decomposition has large blocks with clustered eigenvalues, our method far outperforms the other. Since our method does not require the efficient computation of higher order derivatives, it has the additional advantage of being easily extended to other matrix functions (e.g., special functions).

Robust image tamper detection and recovery with self-embedding watermarking using SPIHT and LDPC

In today’s digital landscape, the pervasive use of digital images across diverse domains has led to growing concerns regarding their authenticity and reliability. The potential for malicious manipulation of these images underscores the critical need to develop robust methods for detecting tampering and ensuring their integrity. Fragile watermarking has been found to have extensive applications for tamper detection and recovery. An image watermarking technique for tamper detection, correction, and recovery is presented in this study. The proposed method employs a self-embedding method to generate the reference watermark from the original image, which has the advantage of superior tamper detection, localization, recovery capabilities, and robustness against attacks. Set Partitioning in the Hierarchical Tree (SPIHT) algorithm is applied to generate a reference watermark of the original image. Low-Density Parity Check (LDPC) is employed for error correction, providing higher-quality reconstruction to recover the original image. Schur decomposition processed the watermarked image blocks to generate authentication bits for each block to enhance tampering detection. The proposed watermarking method was evaluated using PSNR, SSIM, BER, and NC metrics for grayscale and colored images. The technique demonstrated high robustness in tamper detection and recovery against various malicious attacks. Comparative analysis with existing methods shows the efficacy of the proposed method.

Matrix decompositions in quantum optics: Takagi/Autonne, Bloch–Messiah/Euler, Iwasawa, and Williamson

In this tutorial, we summarize four important matrix decompositions commonly used in quantum optics, namely the Takagi/Autonne, Bloch–Messiah/Euler, Iwasawa, and Williamson decompositions. The first two of these decompositions are specialized versions of the singular-value decomposition when applied to symmetric or symplectic matrices. The third factors any symplectic matrix in a unique way in terms of matrices that belong to different subgroups of the symplectic group. The last one instead gives the symplectic diagonalization of real, positive definite matrices of even size. While proofs of the existence of these decompositions exist in the literature, we review explicit constructions to implement these decompositions using standard linear algebra packages and functionalities such as singular-value, polar, Schur, and QR decompositions, and matrix square roots and inverses.

Analysis of eigenvalue condition numbers for a class of randomized numerical methods for singular matrix pencils

The numerical solution of the generalized eigenvalue problem for a singular matrix pencil is challenging due to the discontinuity of its eigenvalues. Classically, such problems are addressed by first extracting the regular part through the staircase form and then applying a standard solver, such as the QZ algorithm, to that regular part. Recently, several novel approaches have been proposed to transform the singular pencil into a regular pencil by relatively simple randomized modifications. In this work, we analyze three such methods by Hochstenbach, Mehl, and Plestenjak that modify, project, or augment the pencil using random matrices. All three methods rely on the normal rank and do not alter the finite eigenvalues of the original pencil. We show that the eigenvalue condition numbers of the transformed pencils are unlikely to be much larger than the δ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\delta $$\\end{document}-weak eigenvalue condition numbers, introduced by Lotz and Noferini, of the original pencil. This not only indicates favorable numerical stability but also reconfirms that these condition numbers are a reliable criterion for detecting simple finite eigenvalues. We also provide evidence that, from a numerical stability perspective, the use of complex instead of real random matrices is preferable even for real singular matrix pencils and real eigenvalues. As a side result, we provide sharp left tail bounds for a product of two independent random variables distributed with the generalized beta distribution of the first kind or Kumaraswamy distribution.

Robust zero-watermarking algorithm for multi-medical images based on FFST-Schur and Tent mapping

Robust zero-watermarking algorithm for multi-medical images based on FFST-Schur and Tent mapping

Deflating subspaces of T-palindromic pencils and algebraic T-Riccati equations

By exploiting the connection between solving algebraic ⊤ -Riccati equations and computing certain deflating subspaces of ⊤ -palindromic matrix pencils, we obtain theoretical and computational results on both problems. Theoretically, we introduce conditions to avoid the presence of modulus-one eigenvalues in a ⊤ -palindromic matrix pencil and conditions for the existence of solutions of a ⊤ -Riccati equation. Computationally, we improve the palindromic QZ algorithm with a new ordering procedure and introduce new algorithms for computing deflating subspaces of the ⊤ -palindromic pencil, based on quadraticizations of the pencil or on an integral representation of the projector on the sought deflating subspace.

Robust Cooperative Fault-Tolerant Control for Uncertain Multi-Agent Systems Subject to Actuator Faults.

This article investigates the robust cooperative fault-tolerant control problem of multi-agent systems subject to mismatched uncertainties and actuator faults. During the design process of the intermediate variable estimator, there is no need to satisfy fault estimation matching conditions, and this overcomes a crucial constraint of traditional observers and estimators. The feedback term of the designed estimator contains the centralized estimation errors and the distributed estimation errors of the agent, and this further improves the design freedom of the proposed estimator. A novel fault-tolerant control protocol is designed based on the fault estimation information. In this work, the bounds of the fault and its derivatives are unknown, and the considered method is applicable to both directed and undirected multi-agent systems. Furthermore, the parameters of the estimator are determined through the resolution of a linear matrix inequality (LMI), which is decoupled by employing coordinate transformation and Schur decomposition. Lastly, a numerical simulation result is used to demonstrate the effectiveness of the proposed method.

A symmetric optical cryptosystem based on QZ decomposition and Hermite Gaussian beam speckles

A symmetric optical cryptosystem based on QZ decomposition and Hermite Gaussian beam speckles

Novel Schur Decomposition Orthogonal Exponential DLPP With Mixture Distance for Fault Diagnosis

Novel Schur Decomposition Orthogonal Exponential DLPP With Mixture Distance for Fault Diagnosis

Slip topology of steady flows around a critical point: Taking the linear velocity field as an example

The flow of viscous fluids is considered as the aggregation of the motion of fluid particles when the fluid is conceived to be made up by an infinite number of particles. As an alternative of this conventional model, fluid motion could be understood as the slip of fluid layers with a molecular scale over each other, where the slip structures of fluid and their associated small-scale motion are characterized by an axial-vector-valued differential 1-form, called the vortex field. In this paper, in the case of steady flows we define the swirling degree of the velocity field at a point, and further the swirl field of the steady flow, to study the slip topology of fluid or the local streamline pattern around the critical point. The linear velocity field in the right real Schur form is used to carry out detailed analyses around the isolated critical point. Theoretical deduction and numerical test unveil the connection between the swirling degree and the swirl field, greatly make clear the topological property of slip structures of fluid in steady flows, especially in three-dimensional space.

Biometric watermarking schemes based on QR decomposition and Schur decomposition in the RIDWT domain

Biometric watermarking schemes based on QR decomposition and Schur decomposition in the RIDWT domain

Spectral clustering of Markov chain transition matrices with complex eigenvalues

The Robust Perron Cluster Analysis (PCCA+) has become a popular spectral clustering algorithm for coarse-graining transition matrices of nearly decomposable Markov chains with transition states. Originally developed for reversible Markov chains, the algorithm only worked for transition matrices with real eigenvalues. In this paper, we therefore extend the theoretical framework of PCCA+ to Markov chains with a complex eigen-decomposition. We show that by replacing a complex conjugate pair of eigenvectors by their real and imaginary components, a real representation of the same subspace is obtained, which is suitable for the cluster analysis. We show that our approach leads to the same results as the generalized PCCA+ (GPCCA), which replaces the complex eigen-decomposition by a conceptually more difficult real Schur decomposition. We apply the method on non-reversible Markov chains, including circular chains, and demonstrate its efficiency compared to GPCCA. The experiments are performed in the Matlab programming language and codes are provided.

Positivity of Schur forms for strongly decomposably positive vector bundles

Abstract In this paper, we define two types of strongly decomposable positivity, which serve as generalizations of (dual) Nakano positivity and are stronger than the decomposable positivity introduced by S. Finski. We provide the criteria for strongly decomposable positivity of type I and type II and prove that the Schur forms of a strongly decomposable positive vector bundle of type I are weakly positive, while the Schur forms of a strongly decomposable positive vector bundle of type II are positive. These answer a question of Griffiths affirmatively for strongly decomposably positive vector bundles. Consequently, we present an algebraic proof of the positivity of Schur forms for (dual) Nakano positive vector bundles, which was initially proven by S. Finski.

Distributed Robust Fault Estimation for Multiagent Systems Based on Transition Variable Estimator.

This article investigates the distributed robust fault estimation for a class of multiagent systems with actuator faults and nonlinear uncertainties. To estimate the actuator faults and system states simultaneously, a novel transition variable estimator is constructed. Compared with existing similar results, the fault estimator existing condition is not necessary for designing the transition variable estimator. Furthermore, the bounds of the faults and their derivatives can be unknown in designing the estimator for each agent in the system. The parameters of the estimator are calculated by using Schur decomposition and linear matrix inequality algorithm. Finally, the performance of the proposed method is demonstrated through experiments of wheeled mobile robots.

Multiple single-channel cryptosystem based on QZ decomposition, CMYK color space fusion and wavelength multiplexing

Multiple single-channel cryptosystem based on QZ decomposition, CMYK color space fusion and wavelength multiplexing

Real block-circulant matrices and DCT-DST algorithm for transformer neural network

In the encoding and decoding process of transformer neural networks, a weight matrix-vector multiplication occurs in each multihead attention and feed forward sublayer. Assigning the appropriate weight matrix and algorithm can improve transformer performance, especially for machine translation tasks. In this study, we investigate the use of the real block-circulant matrices and an alternative to the commonly used fast Fourier transform (FFT) algorithm, namely, the discrete cosine transform–discrete sine transform (DCT-DST) algorithm, to be implemented in a transformer. We explore three transformer models that combine the use of real block-circulant matrices with different algorithms. We start from generating two orthogonal matrices, U and Q. The matrix U is spanned by the combination of the reals and imaginary parts of eigenvectors of the real block-circulant matrix, whereas Q is defined such that the matrix multiplication QU can be represented in the shape of a DCT-DST matrix. The final step is defining the Schur form of the real block-circulant matrix. We find that the matrix-vector multiplication using the DCT-DST algorithm can be defined by assigning the Kronecker product between the DCT-DST matrix and an orthogonal matrix in the same order as the dimension of the circulant matrix that spanned the real block circulant. According to the experiment's findings, the dense-real block circulant DCT-DST model with largest matrix dimension was able to reduce the number of model parameters up to 41%. The same model of 128 matrix dimension gained 26.47 of BLEU score, higher compared to the other two models on the same matrix dimensions.

A novel efficient Rank-Revealing QR matrix and Schur decomposition method for big data mining and clustering (RRQR-SDM)

A novel efficient Rank-Revealing QR matrix and Schur decomposition method for big data mining and clustering (RRQR-SDM)

Blind watermarking scheme for medical and non-medical images copyright protection using the QZ algorithm

Blind watermarking scheme for medical and non-medical images copyright protection using the QZ algorithm

Lead Green's functions from quadratic eigenvalue problems without mode velocity calculations.

In quantum transport calculations, the proper handling of incoming and outgoing modes for retarded Green's functions is achieved via the lead self-energies. Computationally efficient and accurate methods to calculate the self-energies are thus very important. Here we present an alternative method for calculating lead self-energies which improves on a standard approach to solving quadratic eigenvalue problems that arise in quantum transport modeling. The method is based on a perturbative analysis of the generalized Schur decomposition to determine the relevant set of eigenvalues for transmitting modes. This allows us to circumvent finding the velocities of the modes (left- or right-moving) that are needed in order to calculate the lead Green's function from translationally invariant Green's functions. This saves computational time irrespective of the value of the imaginary part added to the energy. We compare our method with two existing methods-a popular iterative method and a standard eigenvalue method that explicitly calculates the velocities of the propagating modes. Our comparison shows that both eigenvalue methods are more robust than the iterative method. Furthermore, the comparison also shows that above a small threshold of propagating modes, the standard eigenvalue method requires extra computation time over our perturbation method. This excess of computation time grows linearly with the number of propagating modes.

The Schur decomposition of discrete Sine and Cosine transformations of type IV

The Schur decomposition of discrete Sine and Cosine transformations of type IV