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In this article, we determine probabilistic approximations of the entries of random perturbation matrices implementing the Markoff inequality. These approximations are used to derive with prescribed probability asymptotic componentwise perturbation bounds of some orthogonal and unitary matrix decompositions. We show that the probabilistic asymptotic bounds are significantly less conservative than the corresponding deterministic perturbation bounds. As case studies, we consider the determining of probabilistic perturbation bounds of the QR decomposition, the singular value decomposition and the Schur decomposition of a matrix using an unified method for asymptotic componentwise perturbation analysis of these decompositions. It is demonstrated that the probability bounds of the orthogonal transformations, singular values and eigenvalues are much tighter than the corresponding deterministic asymptotic bounds. The probabilistic bounds derived are appropriate for perturbation analysis of large‐order matrix problems. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In order to protect the copyright of color digital images in the current Internet environment, this paper proposes a blind color image watermarking scheme based on nonsubsampled contourlet transform (NSCT) and Schur decomposition. In the stage of watermark embedding, NSCT is applied to acquire low-frequency coefficients of host images, and non-overlapping coefficient blocks of sized 4 × 4 are selected from the layered low-frequency coefficients by using Hash pseudo-random algorithm based on MD5. The watermark information encrypted by Lorenz chaotic mapping is embedded into the maximum eigenvalue of the matrix by quantization index modulation. In the extraction process, watermark information is extracted from the attacked watermarked image without using any original information. The simulation results show that all values of peak signal-to-noise ratio of the proposed watermarking scheme are more than 40 dB under the payload of 1/16 bit/pixel, and the values of Structural Similarity Index Metric are more than 0.97; when subjected to various common attacks, the mean of normalized cross-correlation is above 0.96, and the bit error rate is about 0.05; the key space is more than 2624; the running time is about 1.5 s. Compared with other schemes, on the premise of ensuring invisibility and embedding capacity, the proposed scheme has been significantly improved in terms of security and robustness. [ABSTRACT FROM AUTHOR]

Imperceptibility, robustness, and reliability are essential characteristics of image watermarking. In this paper, two image watermarking schemes, first Lifting wavelet transform (LWT)–Discrete cosine transform (DCT) with zigzag scanning-based embedding in Schur domain and second LWT-Schur decomposition-based embedding scheme in Schur domain itself, are proposed. In this paper, two schemes are proposed in which a pre-processed watermark image is generated using LWT and Schur decomposition, where the principle diagonal coefficients of the upper triangular matrix are embedded into the cover image in LWT-Schur domain. The remaining matrix, after removing diagonal components and other unitary matrix of Schur decomposition of LL sub-band of watermark image are used as keys which provide better security to the watermarking scheme. In the LWT-Schur-based scheme principle diagonal coefficients of upper triangular matrix obtained from the Schur decomposition of the LL sub-band of the cover image are modified with the pre-processed watermark in Schur domain. In the LWT–DCT-based scheme, the low and mid-band frequency coefficients are used to embed the watermark using zigzag scanning to avoid any artifacts like diagonal line problem present in the SVD-based watermarking scheme, especially for larger value of the embedding gain factor. The Schur decomposition has lesser computation complexity as compared to SVD and both the schemes are also free from False positive problem. The LWT–DCT with zigzag scanning-based scheme offers higher robustness against filtering and noise attacks as Normalized correlation coefficient (NCC) value is close to 1.0000. Both the proposed schemes provide better imperceptibility and higher robustness as NCC value of more than 0.9800 against various checkmark attacks. Performance analysis shows that these schemes perform better in terms of Peak signal-to-noise ratio, and NCC under most of the attacks and also have lesser computational time than existing schemes. [ABSTRACT FROM AUTHOR]

The Schur decomposition of a square matrix A is an important intermediate step of state-of-the-art numerical algorithms for addressing eigenvalue problems, matrix functions, and matrix equations. This work is concerned with the following task: Compute a (more) accurate Schur decomposition of A from a given approximate Schur decomposition. This task arises, for example, in the context of parameter-dependent eigenvalue problems and mixed precision computations. We have developed a Newton-like algorithm that requires the solution of a triangular matrix equation and an approximate orthogonalization step in every iteration. We prove local quadratic convergence for matrices with mutually distinct eigenvalues and observe fast convergence in practice. In a mixed low-high precision environment, our algorithm essentially reduces to only four high-precision matrix-matrix multiplications per iteration. When refining double to quadruple precision, it often needs only 3–4 iterations, which reduces the time of computing a quadruple precision Schur decomposition by up to a factor of 10–20. [ABSTRACT FROM AUTHOR]

The high-speed train bogie is a typical underactuated mechanical system. It has been difficult to design controls to increase the stability against hunting motion with limited sensor measurements and actuation hardware. This paper develops a control design for the stability improvement of the bogie by making use of the balanced truncation reduced order model of the bogie. When the train travels at speed higher than the critical speed, the bogie has unstable modes. The balanced truncation model reduction (BTMR) is usually applied to stable systems. In this paper, we first apply the real Schur decomposition to separate the stable and unstable modes of the bogie. The BTMR is then carried out for the stable subsystem. An augmented X 0 -balanced truncation algorithm is considered to deal with the dependence of the BTMR on initial conditions. The reduced order subsystem is then combined with the unstable subsystem. A linear quadratic regulator control together with a Luenberger observer for the state estimation is designed for the bogie with non-zero initial conditions. Simulation results show that the control designed with the balanced truncation reduced order model performs very well in terms of the stability improvement by significantly increasing the critical train speed. [ABSTRACT FROM AUTHOR]

We propose a simple technique that, if combined with algorithms for computing functions of triangular matrices, can make them more efficient. Basically, such a technique consists in a specific scaling similarity transformation that reduces the departure from normality of a triangular matrix, thus decreasing its norm and in general its function condition number. It can easily be extended to non-triangular matrices, provided that it is combined with algorithms involving a prior Schur decomposition. Situations where the technique should be used or not will be discussed in detail. Special attention is devoted to particular algorithms like the inverse scaling and squaring to the matrix logarithm or inverse cosine and the scaling and squaring to the matrix exponential. The advantages of our proposal are supported by theoretical results and illustrated with numerical experiments, involving matrices of small, medium and large size. [ABSTRACT FROM AUTHOR]

Medical image security is a critical concern in the healthcare domain, and various watermarking techniques have been explored to embed imperceptible and secure data within medical images. This paper introduces an innovative frequency-based watermarking technique for medical images, utilizing the Fractional Discrete Cosine Transform (FDCT) and Schur decomposition to ensure robust and secure watermark embedding. The watermark bits are integrated by modulating the obtained Schur coefficients, thereby ensuring robust and secure watermarking without significantly altering the visual quality of the medical images. The experiments conducted on the ocular database demonstrate the capacity, imperceptibility, and robustness of the proposed method. This approach achieved a favorable trade-off between imperceptibility and information embedding capacity for ensuring the authenticity and integrity of medical images during transmission. [ABSTRACT FROM AUTHOR]

The mathematical modeling of the real-time system results in large-dimensional ordinary or partial differential equations, which are challenging to employ for investigation and control synthesis. Finding a comparable system of the same kind in a lower-order dimension that can maintain the core characteristics of the higher-order system (HOS) is essential. This article discusses a novel technique for HOS’s reduced order approximation and control design. The proposed method modifies the Schur method for balanced truncation. The method circumvents the requirement of balancing transformation and steady-state gain (SSG) deviation as required in the truncated model reduction algorithm. The offered technique eliminates the SSG deviation without altering the dynamical behavior compared to the HOS by appending a gain enhancement factor to the numerator polynomial of the reduced system transfer matrix. The offered technique ensures that the reduced system retains the HOS stability, passivity, SSG, and all critical characteristics. The recommended method’s findings are compared to the outcomes of recently published work’s lower-order models. Further, using the proposed method, the reduced model has been used to design a PID controller for the flexible-missile control model and an automatic voltage regulator system. The controller produced using the simplified model provides almost the exact time domain specifications (TDS) as the controller made using the HOS, and its design is also relatively more straightforward. Step response and TDS assess lower- and higher-order controller performance. [ABSTRACT FROM AUTHOR]

Medical data is transferred between hospitals and healthcare providers via telemedicine to improve patient care. This transfer exposes medical data to a number of security risks. While most existing security solutions, such as cryptographic techniques, protect data from unauthorized access, this protection is only effective when the data is encrypted. In this context, we present a frequency-domain watermarking method for hiding electronic patient records in their associated ECG signals in this paper. The signal is transformed into a 2D image in this method, and the frequency content of the image is extracted using the integer wavelet transform. Finally, the acquired coefficients are subjected to Schur decomposition, and the watermark bits are integrated by altering the least significant bit of the generated Eigen values. We used the ECG data from the MIT-BIH Arrhythmia Database to test the suggested method. Based on the outcomes of the studies, we can conclude that using the Integer wavelet transform allows for the generation of a watermarked signal that is nearly identical to the host signal. The watermark's resistance to various attacks commonly used in watermarking is also confirmed by the robustness results. [ABSTRACT FROM AUTHOR]

This paper proposes an effective optimal blind colour image watermarking based on Triangular Vertex Transform (TVT), Lifting Wavelet Transform (LWT) and Schur decomposition. Firstly, the RGB channels of the colour host image are processed with TVT to attain U , V , and W coefficients. The image quality is preserved by applying LWT on the W coefficients. As the middle frequency (HL and LH) components of the LWT wavelet domain achieve much robustness, these two components are split into 4 × 4 nonoverlapping blocks further it is processed with Schur decomposition for improving perceptual transparency. The optimal solution attained by an Adaptive Chaotic Grasshopper Optimization Algorithm (ACGOA) is used as the embedding factor to ensure the embedding strength of the watermark. The confidentiality of the watermark is improved by 2D Logistic-Modulated-Sine-Coupling-Logistic Chaotic Map (LSMCL). Subsequently, each bit of encrypted watermark image is inserted in the highest eigenvalue of upper triangular matrix of Schur decomposition using the embedding factor β obtained by ACGOA to accomplish the watermarked image. The watermark recovery is performed using the embedding factor β and the decryption process is done using LSMCL. The efficiency of the ACGOA based scheme is estimated and the highest Peak signal-to-noise ratio (PSNR) value achieved is 54.2987 dB. The Normalized Cross-Correlation (NCC) results show better reliability for different attacks including geometric attacks and the NCC value obtained is close to 1. [ABSTRACT FROM AUTHOR]

A matrix A is said to possess an UV-displacement structure if rank (A U − V A) is small compared with the rank of A. Estimates for the rank A ◯ # V − U A ◯ # are presented, where A ◯ # is the core inverse of A. The computation of the displacement is also considered. [ABSTRACT FROM AUTHOR]

We discuss the use of a matrix‐oriented approach for numerically solving the dense matrix equation AX + XAT + M1XN1 + ... + MℓXNℓ = F, with ℓ ≥ 1, and Mi, Ni, i = 1, ... , ℓ of low rank. The approach relies on the Sherman–Morrison–Woodbury formula formally defined in the vectorized form of the problem, but applied in the matrix setting. This allows one to solve medium size dense problems with computational costs and memory requirements dramatically lower than with a Kronecker formulation. Application problems leading to medium size equations of this form are illustrated and the performance of the matrix‐oriented method is reported. The application of the procedure as the core step in the solution of the large‐scale problem is also shown. In addition, a new explicit method for linear tensor equations is proposed, that uses the discussed matrix equation procedure as a key building block. [ABSTRACT FROM AUTHOR]

In order to enhance the security of exchanged medical images in telemedicine, we propose in this paper a blind and robust approach for medical image protection. This approach consists in embedding patient information and image acquisition data in the image. This imperceptible integration must generate the least possible distortion. The watermarked image must present the same clinical reading as the original image. The proposed approach is applied in the frequency domain. For this purpose, four transforms were used: discrete wavelets transform, non-subsampled contourlet transform, non-subsampled shearlet transform and discreet cosine transform. All these transforms was combined with Schur decomposition and the watermark bits were integrated in the upper triangular matrix. To obtain a satisfactory compromise between robustness and imperceptibility, the integration was performed in the medium frequencies of the image. Imperceptibility and robustness experimental results shows that the proposed methods maintain a high quality of watermarked images and are remarkably robust against several conventional attacks. [ABSTRACT FROM AUTHOR]

This work proposes a novel frequency domain despeckling technique pertaining to the enhancement of the quality of medical ultrasound images. The results of the proposed method have been validated in comparison to both the time-domain and the frequency-domain projections of the schur decomposition as well as with several other benchmark schemes such as frost, lee, probabilistic non-local means (PNLM) and total variation filtering (TVF). The proposed algorithm has shown significant improvements in edge detection and signal to noise ratio (SNR) levels when compared with the performance of the other techniques. Both real and simulated medical ultrasound images have been used to evaluate the numerical and visual effects of each algorithm used in this work. [ABSTRACT FROM AUTHOR]

With the continuous development of the computer network information technique, the problem of copyright protection of color images is increasingly getting prominent. Based on above issue, this paper presents a quaternary watermarking of color image using Schur decomposition. The superiorities of the presented scheme contains three points: Firstly, the algorithm obtains the maximum eigenvalue of the pixel block using Schur decomposition that have a high efficiency, and accomplishes the hiding and blind detection of color watermark by quantifying the maximum eigenvalue to four intervals. Secondly, the algorithm uses affine transform that have high security to encrypt the location of the color watermark pixels, which improves the security of the watermarking algorithm. Thirdly, the new idea of four intervals quantization is proposed to apply to this scheme. Simulation consequences indicate that the presented scheme can satisfy the needs of visual imperceptibility, and also has large watermark capacity, strong robustness and high security. [ABSTRACT FROM AUTHOR]

The Schur--Parlett algorithm, implemented in MATLAB as funm, evaluates an analytic function f at an n X n matrix argument by using the Schur decomposition and a block recurrence of Parlett. The algorithm requires the ability to compute f and its derivatives, and it requires that / have a Taylor series expansion with a suitably large radius of convergence. We develop a version of the Schur--Parlett algorithm that requires only function values and not derivatives. The algorithm requires access to arithmetic of a matrix-dependent precision at least double the working precision, which is used to evaluate f on the diagonal blocks of order greater than 2 (if there are any) of the reordered and blocked Schur form. The key idea is to compute by diagonalization the function of a small random diagonal perturbation of each diagonal block, where the perturbation ensures that diagonalization will succeed. Our algorithm is inspired by Davies's randomized approximate diago-nalization method, but we explain why that is not a reliable numerical method for computing matrix functions. This multiprecision Schur--Parlett algorithm is applicable to arbitrary analytic functions / and, like the original Schur--Parlett algorithm, it generally behaves in a numerically stable fashion. The algorithm is especially useful when the derivatives of f are not readily available or accurately computable. We apply our algorithm to the matrix Mittag--Leffler function and show that it yields results of accuracy similar to, and in some cases much greater than, the state-of-the-art algorithm for this function. [ABSTRACT FROM AUTHOR]

We use the procedures of Blanchard-Khan (1980) and Klein (2000) to numerically solve a New Keynesian model of rational expectations. With respect to this model, we show how to decouple the linear system of rational expectations depending on the state (predetermined) or control (non-predetermined) variables. The solution is plausible if we know the parameters of the model, it is also possible to extract the impulse-response functions to plot the time path of the endogenous variables driven by disturbances of the only stochastic exogenous variable, namely the natural product. [ABSTRACT FROM AUTHOR]

The objective of this paper is to contribute to the sharing, transfer, and secured indexing of large color images. Watermarking and digital encryption have emerged as alternative and complementary solutions to ensure authorized access, facilitate authentication of content, or prevent illegal reproduction. For this reason, we suggest a new system based on four techniques derived from various transformation and decomposition methods, such as the discrete orthogonal Racah moment transformations (DORMT), which allows the information capacity to be reduced to a number of coefficients, as well as the discrete wavelet transform (DWT), which makes it possible to decompose into a certain number of multilevel sub-bands, to have the sub-band that represents most of the image information. In addition, Schur decomposition (SD) and singular value decomposition (SVD) have been used to increase security. The proposed system consists of two main phases. The first phase is the integration of a color watermark, first applying DORMT-SD-SVD to a color host image and then applying DWT-SD-SVD to achieve a compromise between capacity, invisibility and robustness in terms of quality and privacy. The second phase is the encryption of the watermarked image with an eπ map to increase security for sharing, transfer and indexing. This system uses an Aquila Optimization Algorithm (AOA) as an optimization procedure to find the best set of DORMT parameters and provide a dynamic and adaptive scale factor. The experimental results, in which the proposed system's properties are tested and compared to those of other systems, make it clear that the proposed system is useful in terms of its level of security, capacity, invisibility, and resistance to attacks that use signal processing. [ABSTRACT FROM AUTHOR]

The aim of this paper is to investigate the rebinding effect, a phenomenon describing a "short-time memory" which can occur when projecting a Markov process onto a smaller state space. For guaranteeing a correct mapping by the Markov state model, we assume a fuzzy clustering in terms of membership functions, assigning degrees of membership to each state. The macro states are represented by the membership functions and may be overlapping. The magnitude of this overlap is a measure for the strength of the rebinding effect, caused by the projection and stabilizing the system. A minimal bound for the rebinding effect included in a given system is computed as the solution of an optimization problem. Based on membership functions chosen as a linear combination of Schur vectors we are thus able to compute the minimal rebinding effect for nonreversible processes. [ABSTRACT FROM AUTHOR]

In order to protect the color image copyright protection of the multimedia big data, it is necessary to design a color image watermarking algorithm. To achieve this purpose, an improved color image watermarking algorithm based on Schur decomposition is proposed in this paper. First, the watermark information is, respectively, embedded into the upper triangular matrix and the unitary matrix of Schur decomposition by two different methods, and two temporary watermarked image blocks are obtained. Then, the proposed improved method is used to select the final watermarked image block from these temporary watermarked image blocks. The highlight of the proposed method is that the final watermarked block has less visual distortion. Meanwhile, the embedded flag is created and uploaded to the cloud service provider with the watermarked image. When extracting watermark, the original host image or the watermark image is not needed. Experimental results show that the proposed watermarking algorithm has better performance; in particular, the watermark invisibility has been obviously improved than other methods considered in this paper. [ABSTRACT FROM AUTHOR]

In this paper, a blind audio watermarking technique using discrete cosine transform (DCT), Hadamard transform (HT) and Schur decomposition (SD) is presented. The embedding of watermark is carried out by modifying the DCT coefficients through HT and SD. Watermark is embedded in the selected high-energy low-frequency frames of audio signal. The watermark bits are adaptively embedded in the unitary matrix obtained from SD. To make the proposed watermarking technique fully blind, embedding of watermark is performed by effectively utilizing the ratio between the first column elements of unitary matrix. Experimental results confirm that the proposed technique is robust to various signal processing attacks, such as re-sampling, re-quantization, noise addition, filtering, scaling and compression. The comparison of payload and robustness reveals that the proposed watermarking technique outperforms the existing techniques. [ABSTRACT FROM AUTHOR]

Matrix functions have a major role in science and engineering. One of the fundamental matrix functions, which is particularly important due to its connections with certain matrix differential equations and other special matrix functions, is the matrix gamma function. This research article focus on the numerical computation of this function. Well-known techniques for the scalar gamma function, such as Lanczos, Spouge and Stirling approximations, are extended to the matrix case. This extension raises many challenging issues and several strategies used in the computation of matrix functions, like Schur decomposition and block Parlett recurrences, need to be incorporated to make the methods more effective. We also propose a fourth technique based on the reciprocal gamma function that is shown to be competitive with the other three methods in terms of accuracy, with the advantage of being rich in matrix multiplications. Strengths and weaknesses of the proposed methods are illustrated with a set of numerical examples. Bounds for truncation errors and other bounds related with the matrix gamma function will be discussed as well. [ABSTRACT FROM AUTHOR]

Highlights • A novel robust HDR color image watermarking method is presented. • The Tucker decomposition is used to divide the HDR color image along the three color channels so that feature maps are extracted for watermark embedding to balance the distortion of each channel. • The first feature map consists of RGB correlations, which is transformed by using Schur to build the stable size relation between pair-wise coefficients to obtain robustness. • The experimental results show the proposed method can resist most of TMOs effectively. Abstract In order to protect the copyright of high dynamic range (HDR) images, a novel watermarking method based on feature map extraction is proposed. Since the HDR image provides wide dynamic ranges of visible luminance, the correlations of the three RGB channels are much stronger than those of neighbor pixels. In order to preserve strong relationships of the three color channels, the Tucker decomposition is used to divide the HDR color image along the three color channels so that three feature maps are extracted. The first feature map of the three feature maps is employed to embed watermark for robustness, because it contains most of the energies in the HDR image. The first feature map is divided into non-overlapping blocks, and the Schur decomposition is operated on each block to generate a unitary matrix. Since the texture or edge remains after the tone mapping operation (TMO) of the HDR image, the stable size relation between the pair-wise coefficients in the unitary matrix is established to embed watermark. The experimental results show that the proposed method can efficiently resist different TMOs and common image attacks, outperforming other existing HDR image watermarking methods. [ABSTRACT FROM AUTHOR]

Combining redundant discrete wavelet transform (RDWT> with matrix Schur composition, we propose a new digital watermarking algorithm for gray image blind detection. The algorithm firstly performs secondary RDWT on the carrier image and extracts the low frequency part of the approximate sub-graph. Then Schur decomposition is performed on the low frequency coefficients of each block after blocking procession, and the maximum energy elements on the diagonal of the upper triangular matrix are modified to complete the embedding of the watermark image which is encrypted by the Fibonacci transformation. Therefore, the watermark extraction process can achieve blind extraction. Experimental results show that the watermark embedding time of the proposed watermarking algorithm is only 0.97s. and the average watermark extraction time is 0. 211s, Compared with similar blind watermarking algorithms, the PSNR value of invisibility is increased by 14.12%, and the resistance toGauss-ian noise, salt and pepper noise attack and shear geometric attack by 21. 32%. 17. 31% and 27. 83% respectively. [ABSTRACT FROM AUTHOR]

Perturbation analysis has been primarily considered to be one of the main issues in many fields. The Schur decomposition can factor a square matrix as the product of a unitary matrix and an upper triangular matrix, which contains eigenvalues of the square matrix. In view of the importance of T-eigenvalue problems, this paper discusses a tensor Schur decomposition (T-Schur), the T-Schur decomposition = U ∗ T ∗ U ∗ , which is based on T-product multiplication of third-order tensors. We present the normwise and componentwise perturbation analysis for the unitary tensor U , the upper triangular tensor T and T-eigenvalues of the tensor A. We explore some applications of the T-Schur decomposition and perform the T-Schur form to solve the tensor pole assignment. We give an algorithm to solve the tensor based T-Sylvester equation and present its perturbation bound and the backward error. The T-Sylvester equation can also be used to estimate the condition of the T-sign function. We apply the T-Schur decomposition to facial recognation and compare it with other types of tensor decompositions. [ABSTRACT FROM AUTHOR]

The Dynamic Mode Decomposition (DMD) is a tool of the trade in computational data driven analysis of uid ows. More generally, it is a computational device for Koopman spectral analysis of nonlinear dynamical systems, with a plethora of applications in applied sciences and engineering. Its exceptional performance triggered developments of several modifications that make the DMD an attractive method in data driven framework. This work offers improvements of the DMD to make it more reliable, and to enhance its functionality. In particular, data driven formula for the residuals allows selection of the Ritz pairs, thus providing more precise spectral information of the underlying Koopman operator, and the well-known technique of refining the Ritz vectors is adapted to data driven scenarios. Further, the DMD is formulated in a more general setting of weighted inner product spaces, and the consequences for numerical computation are discussed in detail. Numerical experiments illustrate the advantages of the proposed method, designated as DDMD_RRR (Refined Rayleigh{Ritz Data Driven Modal Decomposition). [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

It is shown that the matrix sequence generated by Euler's method starting from the identity matrix converges to the principal pth root of a square matrix, if all the eigenvalues of the matrix are in a region including the one for Newton's method given by Guo in 2010. The convergence is cubic if the matrix is invertible. A modification version of Euler's method using the Schur decomposition is developed. Numerical experiments show that the modified algorithm has the overall good numerical behavior. [ABSTRACT FROM AUTHOR]

• Investigation of the general perturbation formula related to the group inverse. • We give refined perturbation bounds for the group inverse and its oblique projection. • In addition, we partially answered the the open question stated by Wei. • The classical perturbation formula is expanded to the general case by the Schur decomposition. This paper investigates the refined perturbation formulae and perturbation bounds for the group inverse and its oblique projection by the Schur decomposition. In addition, relative perturbation formulae and bounds of some rational expressions that involve group inverses of initial and perturbed matrix are considered. The obtained perturbation limit is sharper than the existing ones derived using the Jordan canonical form. [ABSTRACT FROM AUTHOR]

The class-imbalance problem is an important area that plagues machine learning and data mining researchers. It is ubiquitous in all areas of the real world. At present, many methods have been developed to deal with the class-imbalance problem. For the class-imbalance problem, many researchers believe that the class distribution imbalance is not the main factor affecting the performance of the classification model. When the class distribution imbalance coexists with problems such as class overlap, small disjuncts, and noise, the model performance will be severely affected. For the problem of class-distribution imbalance and class-overlap, the existing methods mainly use nearest neighbors to obtain the local similarity of instances in the local domain, and find the overlapping domains in the data set. To the best of our knowledge, no researchers have considered global similarity. In this regard, to find the global similarity of datasets, a novel Schur decomposition class-overlap undersampling method (SDCU) is proposed. SDCU attempts to obtain potentially overlapping instances on global similarity, and is the first to use matrix decomposition to deal with the problem of class-overlap on class-imbalanced data. We conduct comparative experiments on 46 publicly available real datasets. The experimental results show that when using AUC as the performance evaluation metric, the performance of SDCU has obvious advantages compared with other state-of-the-art methods on three different types of classifiers: SVM, CART, and 3NN. In addition, the test results of Friedman ranking and Holm's post-hoc test also confirmed the conclusions obtained by the experiments. [ABSTRACT FROM AUTHOR]

In view of the present most of the color image digital watermarking techniques used binary or gray images as the watermarks and these techniques were non-blind extraction .this paper proposed a blind digital watermarking algorithm to embed a color watermark image into a color host image. The algorithm analyzed essential features of Schur decomposition, by modifying the largest eigenvalue element on the diagonal of the upper triangular matrix to embed the watermark. The watermark might be extracted from watermarked image without original host image and the original watermark image to achieve the goal of blind extraction. The experimental results show that the watermarking scheme has fast speed in embedding and extraction. In addition, the value of peak signal-to-noise ratio more than 40 dB embodies the good invisibility. The solution also has a good robustness in terms of resistance of common image attack. [ABSTRACT FROM AUTHOR]

Finding metastable sets as dominant structures of Markov processes has been shown to be especially useful in modeling interesting slow dynamics of various real world complex processes. Furthermore, coarse graining of such processes based on their dominant structures leads to better understanding and dimension reduction of observed systems. However, in many cases, e.g., for nonreversible Markov processes, dominant structures are often not formed by metastable sets but by important cycles or a mixture of both. This paper aims at understanding and identifying these different types of dominant structures for reversible as well as nonreversible ergodic Markov processes. Our algorithmic approach generalizes spectral-based methods for reversible processes by using Schur decomposition techniques which can also tackle nonreversible cases. We illustrate the mathematical construction of our new approach by numerical experiments. [ABSTRACT FROM AUTHOR]

Given the square matrices A,B,D,E and the matrix C of conforming dimensions, we consider the linear matrix equation AXE + DXB = C in the unknown matrix X. Our aim is to provide an overview of the major algorithmic developments that have taken place over the past few decades in the numerical solution of this and related problems, which are producing reliable numerical tools in the formulation and solution of advanced mathematical models in engineering and scientific computing. [ABSTRACT FROM AUTHOR]

An algorithm for computing an analytic function of a matrixis described. The algorithm is intended for the case wherehas some close eigenvalues, and clusters (subsets) of close eigenvalues are separated from each other. This algorithm is a modification of some well known and widely used algorithms. A novel feature is an approximate calculation of divided differences for the Newton interpolating polynomial in a special way. This modification does not require to reorder the Schur triangular form and to solve Sylvester equations. [ABSTRACT FROM AUTHOR]

For computing logarithms of a nonsingular matrix A, one well known algorithm named the inverse scaling and squaring method, was proposed by Kenney and Laub [14] for use with a Schur decomposition. In this paper we further consider (the computation of) the logarithms of matrices with central symmetry.We first investigate the structure of the logarithms of these matrices and then give the classifications of their logarithms. Then, we develop several algorithms for computing the logarithms. We show that our algorithms are four times cheaper than the standard inverse scaling and squaring method for matrices with central symmetry under certain conditions. [ABSTRACT FROM AUTHOR]

• The torus geometry was used in the ICA optimization problem for complex signals. • The Lie group structure of the optimization space was used, which ensured good stability of the proposed algorithms. • The simulation results carried out on communication signals showed a high speed of the proposed algorithm. • Quick response to signal changes in gradient methods makes it possible to use the algorithm in on-line applications. The article introduces the new gradient descent/ascent algorithm for independent component analysis for complex valued data (cICA). This algorithm uses toral decomposition of skew-Hermitian gradient matrix and very simple and computationally inexpensive method of projection based on Lie structure of optimization landscape. Toral decomposition is the diagonalization of the gradient matrix. In this stage a very fast Schur decomposition was used. The algorithm was tested for complex communication signals and compared to several classes of algorithms as gradient descent, quasi-Newton and complex JADE. The simulations showed very good results in terms of speed of operation and separation quality, as well as the highest versatility of applications in the context of the cost functions used among all the tested algorithms. The specificity of gradient methods characterized by quick response to changes in the signal indicates the practical usage potential of this method in on-line applications. [ABSTRACT FROM AUTHOR]

SUMMARY We analyze an algorithm for computing a skew-Hermitian logarithm of a unitary matrix and also skew-Hermitian approximate logarithms for nearly unitary matrices. This algorithm is very easy to implement using standard software, and it works well even for unitary matrices with no spectral conditions assumed. Certain examples, with many eigenvalues near − 1, lead to very non-Hermitian output for other basic methods of calculating matrix logarithms. Altering the output of these algorithms to force skew-Hermitian output creates accuracy issues, which are avoided by the considered algorithm. A modification is introduced to deal properly with the J-skew-symmetric unitary matrices. Applications to numerical studies of topological insulators in two symmetry classes are discussed. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Abstract: A new asymmetric cryptosystem using Schur decomposition in gyrator transform (GT) domain is introduced. In this approach, the color image to be encoded is first dissociated into red, green and blue channels and then modulated by multiplying three different random phase masks. The three modulated color channels are combined into one gray image by using convolution. The convoluted image is gyrator transformed. The gyrator spectrum is then amplitude- and phase-truncated. The asymmetric decryption phase keys for red, green and blue channels are generated digitally during the encryption process. The phase-truncated image is divided into U and T parts by Schur decomposition. Finally, U and T parts are independently gyrator transformed to obtain two encoded images, which can be assigned to a joint authorized user (or two authorized users). GT evades optical alignment problems. The Schur decomposition represents an intermediate step in SVD decomposition, and thus requires less number of computations. The multiplication order of Schur decomposition, three transformation angles of GT, and three asymmetric keys are regarded as decryption keys, which boost the degree of security against existing attacks. The proposed encryption process can be carried out digitally, while the decryption process can be realized optically. Numerical simulations prove the validity and security of the proposed approach. [Copyright &y& Elsevier]

Abstract: We propose a hyperbolic counterpart of the Schur decomposition, with the emphasis on the preservation of structures related to some given hyperbolic scalar product. We give results regarding the existence of such a decomposition and research the properties of its block triangular factor for various structured matrices. [Copyright &y& Elsevier]

The Schur--Padé algorithm [N. J. Higham and L. Lin, SIAM J. Matrix Anal. Appl., 32 (2011), pp. 1056--1078] computes arbitrary real powers At of a matrix A C n x n using the building blocks of Schur decomposition, matrix square roots, and Padé approximants. We improve the algorithm by basing the underlying error analysis on the quantities |(I-A)k|1/k, for several small k, instead of |I-A|. We extend the algorithm so that it computes along with At one or more Fréchet derivatives, with reuse of information when more than one Fréchet derivative is required, as is the case in condition number estimation. We also derive a version of the extended algorithm that works entirely in real arithmetic when the data is real. Our numerical experiments show the new algorithms to be superior in accuracy to, and often faster than, the original Schur--Padé algorithm for computing matrix powers and more accurate than several alternative methods for computing the Fréchet derivative. They also show that reliable estimates of the condition number of At are obtained by combining the algorithms with a matrix norm estimator. [ABSTRACT FROM AUTHOR]

In this paper, we first investigate the structures of the logarithms of circulant matrices, quasi-skew circulant matrices and Hermitian circulant matrices. Then give the classifications of their logarithms. In the end, we develop several algorithms for computing the logarithms. We show that our algorithms are more efficient than the standard inverse scaling and squaring method which is based on the Schur decomposition. [ABSTRACT FROM AUTHOR]

In this paper, we first investigate the structures of the logarithms of circulant matrices, quasi-skew circulant matrices and Hermitian circulant matrices. Then give the classifications of their logarithms. In the end, we develop several algorithms for computing the logarithms. We show that our algorithms are more efficient than the standard inverse scaling and squaring method which is based on the Schur decomposition. [ABSTRACT FROM AUTHOR]

Abstract: We consider the solution of the ★-Sylvester equations AX ± X ★ B ★ = C, for ★= T, H and , and the related linear matrix equations AXB ★ ± X ★ = C, AXB ★ ± CX ★ D ★ = E and AX ± X ★ A ★ = C. Solvability conditions and numerical methods are considered, in terms of the (generalized and periodic) Schur and QR decompositions. We emphasize the square cases where m = n but the rectangular cases will be considered. [Copyright &y& Elsevier]

Abstract: In this paper, a blind dual color image watermarking scheme based on Schur decomposition is introduced. This is the first time to use Schur decomposition to embed color image watermark in color host image, which is different from using the binary image as watermark. By analyzing the 4×4 unitary matrix U via Schur decomposition, we can find that there is a strong correlation between the second row first column element and the third row first column element. This property can be explored for embedding watermark and extracting watermark in the blind manner. Since Schur decomposition is an intermediate step in SVD decomposition, the proposed method requires less number of computations. Experimental results show that the proposed scheme is robust against most common attacks including JPEG lossy compression, JPEG 2000 compression, low-pass filtering, cropping, noise addition, blurring, rotation, scaling and sharpening et al. Moreover, the proposed algorithm outperforms the closely related SVD-based algorithm and the spatial-domain algorithm. [Copyright &y& Elsevier]

Abstract: Iterative procedure is described to generate patterns of dominant Schur vectors of the system dynamics. Their roles in estimating the filter gain is study. These patterns are produced by several integrations of the model from a set of perturbations. This approach is motivated by a number of interesting results on stability of the filter whose gain is approximated in a subspace of dominant Schur vectors. A simple method for the filter design is presented which is aimed at overcoming the most serious drawback of advanced filtering algorithms for high dimensional systems related to very high computational cost in evaluation of the filter gain. The resulting filter will be compared with the existing ones, showing its relevance from a practical point of view. In order to demonstrate its efficiency, the new filter is tested on various experiments. These experiments include the much studied problem of estimating the solution of the Lorenz system as well as that of assimilating sea surface height observations in a high dimensional oceanic model. It is shown that significant increases in efficiency can be obtained by using this filter and that the proposed filter is very promising for solving realistic assimilation problems in meteorology and oceanography. [Copyright &y& Elsevier]