QR Factorization

For the investigation of extrinsic pro-apoptotic signaling pathways, systems of stochastic differential equations are solved by Euler-Maruyama approximation. Forces and moments between particles are computed in parallel on GPGPU Density Functional Theory on GPGPU Development and parallelization of Density Functional Theory (DFT) methods for tightly coupled GPGPU many-core architectures. Focus: Parallelized computation of orbitals and density matrices. Matrix formulation allows the use of developed fault-tolerant matrix operations (e.g. DGEMM

For the investigation of extrinsic pro-apoptotic signaling pathways, systems of stochastic differential equations are solved by Euler-Maruyama approximation. Forces and moments between particles are computed in parallel on GPGPU Density Functional Theory on GPGPU Development and parallelization of Density Functional Theory (DFT) methods for tightly coupled GPGPU many-core architectures. Focus: Parallelized computation of orbitals and density matrices. Matrix formulation allows the use of developed fault-tolerant matrix operations (e.g. DGEMM

The sparse matrix-vector multiplication (SpMxV) is a kernel operation widely used in iterative linear solvers. The same sparse matrix is multiplied by a dense vector repeatedly in these solvers. Matrices with irregular sparsity patterns make it difficult to utilize cache locality effectively in SpMxV computations. In this work, we investigate single-and multiple-SpMxV frameworks for exploiting cache locality in SpMxV computations. For the single-SpMxV framework, we propose two cache-size-aware top-down row/column-reordering methods based on 1D and 2D sparse matrix partitioning by utilizing the column-net and enhancing the row-column-net hypergraph models of sparse matrices. The multiple-SpMxV framework depends on splitting a given matrix into a sum of multiple nonzero-disjoint matrices so that the SpMxV operation is performed as a sequence of multiple input-and outputdependent SpMxV operations. For an effective matrix splitting required in this framework, we propose a cachesize-aware top-down approach based on 2D sparse matrix partitioning by utilizing the row-column-net hypergraph model. For this framework, we also propose two methods for effective ordering of individual SpMxV operations. The primary objective in all of the three methods is to maximize the exploitation of temporal locality. We evaluate the validity of our models and methods on a wide range of sparse matrices using both cache-miss simulations and actual runs by using OSKI. Experimental results show that proposed methods and models outperform state-of-the-art schemes.

LU decomposition is a fundamental in linear algebra. Numerous tools exists that provide this important factorization. The authors present the conditions for a matrix to have none, one, or infinitely many LU factorizations. In the case where no factorization exists, the authors illustrate how to approximate an LU decomposition by considering LU factorization of nearby matrices.

LU decomposition is a fundamental in linear algebra. Numerous tools exists that provide this important factorization. The authors present the conditions for a matrix to have none, one, or infinitely many LU factorizations. In the case where no factorization exists, the authors illustrate how to approximate an LU decomposition by considering LU factorization of nearby matrices.

Discrete inverse problems correspond to solving a system of equations in a stable way with respect to noise in the data. A typical approach to enforce uniqueness and select a meaningful solution is to introduce a regularizer. While for most applications the regularizer is convex, in many cases it is not smooth nor strongly convex. In this paper, we propose and study two new iterative regularization methods, based on a primal-dual algorithm, to solve inverse problems efficiently. Our analysis, in the noise free case, provides convergence rates for the Lagrangian and the feasibility gap. In the noisy case, it provides stability bounds and early-stopping rules with theoretical guarantees. The main novelty of our work is the exploitation of some a priori knowledge about the solution set, i.e. redundant information. More precisely we show that the linear systems can be used more than once along the iteration. Despite the simplicity of the idea, we show that this procedure brings surprising advantages in the numerical applications. We discuss various approaches to take advantage of redundant information, that are at the same time consistent with our assumptions and flexible in the implementation. Finally, we illustrate our theoretical findings with numerical simulations for robust sparse recovery and image reconstruction through total variation. We confirm the efficiency of the proposed procedures, comparing the results with state-of-the-art methods.

International audienceGiven a full column rank matrix A ∈ R m×n (m ≥ n), we consider a special class of linear systems of the form A ⊤ Ax = A ⊤ b + c with x, c ∈ R n and b ∈ R m. The occurrence of c in the right-hand side of the equation prevents the direct application of standard methods for least squares problems. Hence, we investigate alternative solution methods that, as in the case of normal equations, take advantage of the peculiar structure of the system to avoid unstable computations, such as forming A ⊤ A explicitly. We propose two iterative methods that are based on specific reformulations of the problem and we provide explicit closed formulas for the structured condition number related to each problem. These formula allow us to compute a more accurate estimate of the forward error than the standard one used for generic linear systems, that does not take into account the structure of the perturbations. The relevance of our estimates is shown on a set of synthetic test prob...

Abstract. Given a full column rank matrix A ∈ R (m ≥ n), we consider a special class of linear systems of the form AAx = Ab+ c with x, c ∈ R and b ∈ R. The occurrence of c in the right-hand side of the equation prevents the direct application of standard methods for least squares problems. Hence, we investigate alternative solution methods that, as in the case of normal equations, take advantage of the peculiar structure of the system to avoid unstable computations, such as forming AA explicitly. We propose two iterative methods that are based on specific reformulations of the problem and we provide explicit closed formulas for the structured condition number related to each problem. These formula allow us to compute a more accurate estimate of the forward error than the standard one used for generic linear systems, that does not take into account the structure of the perturbations. The relevance of our estimates is shown on a set of synthetic test problems. Numerical experiments hi...

Many problems of industrial and scientific interest require the solving of tridiagonal linear systems. This paper presents several implementations for the parallel solving of large tridiagonal systems on multi-core architectures, using the OmpSs programming model. The strategy used for the parallelization is based on the combination of two different existing algorithms, PCR and Thomas. The Thomas algorithm, which cannot be parallelized, requires the fewest number of floating point operations. The PCR algorithm is the most popular parallel method, but it is more computationally expensive than Thomas. The method proposed in this paper starts applying the PCR algorithm to break down one large tridiagonal system into a set of smaller and independent ones. In a second step, these independent systems are concurrently solved using Thomas. The paper also contains an analytical study of which is the best point to switch from PCR to Thomas. Also, the paper addresses the main performance issues of combining PCR and Thomas proposing a set of alternative implementations, some of them even imply algorithmic changes. The performance evaluation shows that the best implementation achieves a peak speedup of 4 with respect to the Intel MKL counterpart routine and 2.5 with respect to a single-threaded Thomas.

The Partitioned Global Address Space (PGAS) programming model is one of the most relevant proposals to improve the ability of developers to exploit distributed memory systems. However, despite its important advantages with respect to the traditional message-passing paradigm, PGAS has not been yet widely adopted. We think that PGAS libraries are more promising than languages because they avoid the requirement to (re)write the applications using them, with the implied uncertainties related to portability and interoperability with the vast amount of APIs and libraries that exist for widespread languages. Nevertheless, the need to embed these libraries within a host language can limit their expressiveness and very useful features can be missing. This paper contributes to the advance of PGAS by enabling the simple development of arbitrarily complex task-parallel codes following a dataflow approach on top of the PGAS UPC++ library, implemented in C++. In addition, our proposal, called UPC++ DepSpawn, relies on an optimized multithreaded runtime that provides very competitive performance, as our experimental evaluation shows.

We present theory and algorithms for the equality constrained indefinite least squares problem, which requires minimization of an indefinite quadratic form subject to a linear equality constraint. A generalized hyperbolic QR factorization is introduced and used in the derivation of perturbation bounds and to construct a numerical method. An alternative method is obtained by employing a generalized QR factorization in combination with a Cholesky factorization. Rounding error analysis is given to show that both methods have satisfactory numerical stability properties and numerical experiments are given for illustration. This work builds on recent work on the unconstrained indefinite least squares problem by Chandrasekaran, Gu, and Sayed and by the present authors.

We show that a fast algorithm for the QR factorization of a Toeplitz or Hankel matrix A is weakly stable in the sense that R T R is close to A T A. Thus, when the algorithm is used to solve the semi-normal equations R T Rx = A T b, we obtain a weakly stable method for the solution of a nonsingular Toeplitz or Hankel linear system Ax = b. The algorithm also applies to the solution of the full-rank Toeplitz or Hankel least squares problem min Ax -b 2 .

Data hiding and watermarking are considered one of the most important topics in cyber security. This article proposes an optimized method for embedding a watermark image in a cover medium (color image). First, the color of the image is separated into three components (RGB). Consequently, the discrete wavelet transform is applied to each component to obtain four bands (high–high, high–low, low–high, and low–low), resulting in 12 bands in total. By omitting the low–low band from each component, a new square matrix is formed from the rest bands to be used for the hiding process after adding keys to it. These keys are generated using a hybrid approach, combining two chaotic functions, namely Gaussian and exponential maps. The embedding matrix is divided into square blocks with a specific length, each of which is converted using Hessenberg transform into two matrices, P and H. For each block, a certain location within the H-matrix is used for embedding a secret value; the updated blocks ...

Internet simplified digital data transferring. This data needs to be secured; so securing digital data becomes an important concern. Steganography provides security for data by inserting it into a cover and concealing it. In this paper, a steganography algorithm was introduced. This algorithm used the stationary wavelet transform (SWT) and اhybrid-matrix decomposition techniques, singular value decomposition (SVD) and QR factorization to conceal a video in another video. The algorithm performance was measured with reference to the peak signal to noise ratio (PSNR) and structural similarity index measure (SSIM) for the cover and the secret videos. The algorithm successfully hid a video in another cover video and both are of the same size; the hiding capacity was 100%. The algorithm achieved a SSIM that reached 0.97 and a high PSNR value that reached 68.8 proving that the imperceptibility of the proposed algorithm is very high. The comparative analysis shows that the suggested algorit...

This note reports an unexpected and rather erratic behavior of the LAPACK software implementation of the QR factorization with Businger-Golub column pivoting. It is shown that, due to finite precision arithmetic, software implementation of the factorization can catastrophically fail to produce triangular factor with the structure characteristic to the Businger-Golub pivot strategy. The failure of current state of the art software, and a proposed alternative implementations are analyzed in detail.

Simultaneous iteration methods are extensions of the power method that were devised for approximating several dominant eigenvalues of a matrix and the corresponding eigenvectors. The convergence analysis of these methods has been given for both hermitian and nonhermitian matrices. In this paper we concentrate on normal matrices which include also the hermitian matrices, and provide a rigorous analysis for a common version of simultaneous iteration methods. We improve on some of the known results and derive some new ones as well. We also present a detailed analysis concerning the formation of spurious eigenvalue approximations.

The m-th root of the diagonal of the upper triangular matrix R m in the QR decomposition of AX m B = Q m R m converges and the limit is given by the moduli of the eigenvalues of X with some ordering, where A, B, X ∈ C n×n are nonsingular. The asymptotic behavior of the strictly upper triangular part of R m is discussed. Some computational experiments are discussed.

CG, SYMMLQ, and MINRES are Krylov subspace methods for solving symmetric systems of linear equations. When these methods are applied to an incompatible system (that is, a singular symmetric least-squares problem), CG could break down and SYMMLQ's solution could explode, while MINRES would give a least-squares solution but not necessarily the minimum-length (pseudoinverse) solution. This understanding motivates us to design a MINRES-like algorithm to compute minimum-length solutions to singular symmetric systems. MINRES uses QR factors of the tridiagonal matrix from the Lanczos process (where R is uppertridiagonal). MINRES-QLP uses a QLP decomposition (where rotations on the right reduce R to lower-tridiagonal form). On ill-conditioned systems (singular or not), MINRES-QLP can give more accurate solutions than MINRES. We derive preconditioned MINRES-QLP, new stopping rules, and better estimates of the solution and residual norms, the matrix norm, and the condition number.

In this paper, an iterative method is proposed for solving matrix equation s j=1 A j X j B j = E. This method is based on the global least squares (GL-LSQR) method for solving the linear system of equations with the multiple right hand sides. For applying the GL-LSQR algorithm to solve the above matrix equation, a new linear operator, its adjoint and a new inner product are defined. It is proved that the new iterative method obtains the least norm solution of the mentioned matrix equation within finite iteration steps in the exact arithmetic, when the above matrix equation is consistent. Moreover, the optimal approximate solution (X * 1 , X * 2 , . . . , X * s ) to a given multiple matrices ( X1 , X2 , . . . , Xs ) can be derived by finding the least norm solution of a new matrix equation. Finally, some numerical experiments are given to illustrate the efficiency of the new method.

A quantum logic gate of particular interest to both electrical engineers and game theorists is the quantum multiplexer. This shared interest is due to the facts that an arbitrary quantum logic gate may be expressed, up to arbitrary accuracy, via a circuit consisting entirely of variations of the quantum multiplexer, and that certain one player games, the history dependent Parrondo games, can be quantized as games via a particular variation of the quantum multiplexer. However, to date all such quantizations have lacked a certain fundamental game theoretic property. The main result in this dissertation is the development of quantizations of history dependent quantum Parrondo games that satisfy this fundamental game theoretic property. Our approach also yields fresh insight as to what should be considered as the proper quantum analogue of a classical Markov process and gives the first game theoretic measures of multiplexer behavior. QUANTUM MULTIPLEXERS, PARRONDO GAMES, AND PROPER

We give an explicit formula for the solution to the initial value problem of the full symmetric Toda hierarchy. The formula is obtained by the orthogonalization procedure of Szegö, and is also interpreted as a consequence of the QR factorization method of Symes . The sorting property of the dynamics is also proved for the case of a generic symmetric matrix in the sense described in the text, and generalizations of tridiagonal formulae are given for the case of matrices with 2M + 1 nonzero diagonals.

We give an explicit formula for the solution to the initial value problem of the full symmetric Toda hierarchy. The formula is obtained by the orthogonalization procedure of Szegö, and is also interpreted as a consequence of the QR factorization method of Symes . The sorting property of the dynamics is also proved for the case of a generic symmetric matrix in the sense described in the text, and generalizations of tridiagonal formulae are given for the case of matrices with 2M + 1 nonzero diagonals.

In this paper, we compare with the inverse iteration algorithms on PowerXCell T M 8i processor, which has been known as a heterogeneous environment. When some of all the eigenvalues are close together or there are clusters of eigenvalues, reorthogonalization must be adopted to all the eigenvectors associated with such eigenvalues. Reorthogonalization algorithms need a lot of computational cost. The Classical Gram-Schmidt (CGS) algorithm, the modified Gram-Schmidt (MGS) algorithm, and the Householder orthogonalization algorithm in terms of the compact WY representation have been known as reorthogonalization algorithms. These algorithms can be computed using BLAS level-1 and level-2. Since synergistic processor elements in PowerXCell T M 8i processor archive the high performance of BLAS level-2 and level-3, the orthogonalization algorithms except the MGS algorithm can be computed high-speed on parallel computers.

Example with a memory of size 2 data. The graph of input data dependencies is shown on the left. The figure on the right corresponds to the partition and schedule produced by the scheduler ▸ Deque Model Data Aware Ready (DMDAR): Deque Model Data Aware Ready (DMDAR):

Recently neural style transfer (NST) has drawn a lot of interest of researchers, with notable advancements in color representation, texture, speed, and image quality. While previous studies focused on transferring artistic style across entire content images, a new approach proposes to transfer style specifically to objects within the content image based on the style image and maintain photorealism. Recent techniques have produced intriguing creative effects, but often only work with artificial effects, leaving real flaws visible in photographs used as references for styles. The suggested approach employs a two-dimensional wavelet transform (WT) to achieve style transfer by adjusting image structure with high-pass and low pass filters (LPF). Preserving the information content and numerical attributes of VGGNet19 through WT-based style transfer using the db5 WT at level 5, we can achieve a peak signal-to-noise ratio (PSNR) value of up to 96.76725. The qualitative result of the proposed methodology is compared with other existing algorithm. Also, the time complexity of the proposed methodology on different hardware platforms has been calculated and presented in the paper. The proposed methodology able to maintains appealing and precise quality of resultant image.

Recently, assuming ideal brick-wall transmit filtering, we proposed a frequency-domain block signal detection (FDBD) with maximum likelihood detection employing QR decomposition and M-algorithm (called QRM-MLD) for the reception of single-carrier (SC) signals transmitted over a frequency-selective fading channel. QR decomposition (QRD) is applied to a concatenation of the propagation channel and discrete Fourier transform (DFT). However, a large number of surviving paths is required in the M-algorithm to achieve sufficiently improved bit error rate (BER) performance. The introduction of filtering can achieve improved BER performance due to larger frequency diversity gain while keeping a lower peak-to-average power ratio (PAPR) than orthogonal frequency division multiplexing (OFDM). In this paper, we develop FDBD with QRM-MLD for filtered SC signal reception. QRD is applied to a concatenation of transmit filter, propagation channel, and DFT. We evaluate BER and throughput performances by computer simulation. From performance evaluation, we discuss how the filter roll-off factor affects the achievable BER and throughput performances and show that as the filter roll-off factor increases, the required number of surviving paths in the M-algorithm can be reduced.

The needs for predictive simulation of electronic structure in chemistry and materials science calls for fast/reduced-scaling formulations of quantum n-body methods that replace the traditional dense tensors with element-, block-, rank-, and block-rank-sparse (data-sparse) tensors. The resulting, highly irregular data structures are a poor match to imperative, bulk-synchronous parallel programming style due to the dynamic nature of the problem and to the lack of clear domain decomposition to guarantee a fair load-balance. TESSE runtime and the associated programming model aim to support performance-portable composition of applications involving irregular and dynamically changing data. In this paper we report an implementation of irregular dense tensor contraction in a paradigmatic electronic structure application based on the TESSE extension of PaRSEC, a distributed hybrid task runtime system, and analyze the resulting performance on a distributedmemory cluster of multi-GPU nodes. Unprecedented strong scaling and promising efficiency indicate a viable future for taskbased programming of complete production-quality reducedscaling models of electronic structure.

As the general purpose graphics processing units (GPGPU) are increasingly deployed for scientific computing for its raw performance advantages compared to CPUs, the fault tolerance issue has started to become more of a concern than before when they were exclusively used for graphics applications. The pairing of GPUs with CPUs to form a hybrid computing systems for better flexibility and performance creates a massive amounts of computations that have a higher possibility to be affected by transient error-a soft error that silently modifies data causing errors to pass unnoticed. This is despite the fact that the newest Fermi generation of GPUs from NVIDIA are equipped with error correcting units to protect their memories. This problem is particularly serious for applications that employ numerical linear algebra since large sections of data are often modified between steps, and therefore even a single error could eventually propagate into a large area of result. In order to give protection to dense linear algebra computations on such hybrid systems, we developed an algorithm that is resilient to soft errors. We chose the right-looking Householder QR factorization as a demonstration of our algorithm for a hybrid system that features both GPUs and CPUs. Algorithm based fault tolerance (ABFT) is used to protect from errors in the trailing matrix and the right factor, while a checkpointing method is used to ensure the left factor is error-free. This work is based on a previous study of fault tolerance in matrix factorizations. Our contribution includes (1) a stable multiple-error checkpointing and recovery mechanism for the left-factor, which is also scalable in performance in the hybrid execution environment and does not cause severe performance degradation. (2) optimized Givens rotation utilities on the GPU to efficiently reduce an upper Hessenberg matrix to upper triangular form, and (3) a recovery algorithm based on QR update inside a hybrid system. Experimental results show that, our fault tolerant QR factorization can successfully detect and correct data altered by soft errors in both the left and right factors and we observe a decreasing percentage of overhead as the matrix size grows.

Kepler is the newest GPU architecture from NVIDIA, and the GTX 680 is the first commercially available graphics card based on that architecture. Matrix multiplication is a canonical computational kernel, and often the main target of initial optimization efforts for a new chip. This article presents preliminary results of automatically tuning matrix multiplication kernels for the Kepler architecture using the GTX 680 card.

We consider Givens QR factorization of banded Hessenberg–Toeplitz matrices of large order and relatively small bandwidth. We investigate the asymptotic behaviour of the R factor and Givens rotation when the order of the matrix goes to infinity, and present some interesting convergence properties. These properties can lead to savings in the computation of the exact QR factorization and give insight into the approximate QR factorizations of interest in preconditioning. The properties also reveal the relation between the limit of the main diagonal elements of R and the largest absolute root of a polynomial. Copyright © 2005 John Wiley & Sons, Ltd.

In this modern era, it has become much simpler to replicate, sell, and copy the copyright owners' works without their permission as a result of the expansion of digitalization, and it is difficult to identify such violations, posing a threat to the creators' and copyright owners' rights. For many years, the internet has been regarded as one of the most serious threats to copyright, and the content available has varying levels of copyright protection. On the internet, there are numerous copyrighted works, including e-books, movies, news, and so on. Therefore, by using watermarking and steganography techniques, these issues can be solved, which are based on the author's signature information or logo. This paper concluded that the techniques of discrete cosine transform (DCT), discrete wavelet transform (DWT), one-time pad (OTP), and playfair are highly effective when used together to watermark an image or embed a secret message, our lab results validate that our algorithm scheme is robust against several sets of attacks, where the algorithm was assessed by computation of many evaluation metrics such as mean square error (MSE), signal-to-noise ratio (SNR), and peak signal-to-noise ratio (PSNR).

Patients' information and images transfer among medical institutes represent a major tool for delivering better healthcare services. However, privacy and security for healthcare information are big challenges in telemedicine. Evidently, even a small change in patients' information might lead to wrong diagnosis. This paper suggests a new model for hiding patient information inside magnetic resonance imaging (MRI) cover image based on Euclidean distribution. Both least signification bit (LSB) and most signification bit (MSB) techniques are implemented for the physical hiding. A new method is proposed with a very high level of security information based on distributing the secret text in a random way on the cover image. Experimentally, the proposed method has high peak signal to noise ratio (PSNR), structural similarity index metric (SSIM) and reduced mean square error (MSE). Finally, the obtained results are compared with approaches in the last five years and found to be better by increasing the security for patient information for telemedicine.

Sn, k _. Z'_jXn+j , j=o where the yj are determined as follows: (i) (ii) (2.1) (2.2) (2.3) (2.4) Use the least squares method to solve the overdetermined and, in general, inconsistent linear system where c = (Co,C t ..... ck-I)1". Set ck = 1, and compute the yj by U_Qt c =-u. +k , (2.5) cj Y/-k-, O<j<k, k assuming that _ci _: O. When this condition is not satisfied s,,.k does not exist. i=0 (2.6) 2.2 Definition of RRE For RRE the approximation s,.k to s, the desired limit or antilimit, is defined by k-I S.,k =X. + _, _iU.+i, (2.7) i=O where the _, are determined by solving the overdetermined and, in general, inconsistent linear system 3. Recipient's Catalog No. Efficient Implementation of Minimal Polynomial and Reduced Rank Extrapolation Methods 7. Author(s) Avram Sidi g.

Functional iterations such as Newton's are a popular tool for polynomial root-finding. We consider a realistic situation where some roots have already been approximated (we say tamed), and one would like to restrict further root-finding to the approximation of the remaining (wild) roots. A natural approach of applying explicit deflation has been much studied and recently advanced by one of the authors of this paper, but presently we consider the alternative of implicit deflation combined with mapping of the variable and reversion of an input polynomial. The hope is that the union of the sets of tame roots approximated in a number of such transformations can cover all roots of a polynomial. We also show another direction to substantial further progress in this 1 long and extensively studied area. Namely we dramatically increase the local efficiency of root-finding by means of the incorporation of fast algorithms for multipoint polynomial evaluation and the Fast Multipole Method.

Most recent HPC platforms have heterogeneous nodes composed of a combination of multi-core CPUs and accelerators, like GPUs. Scheduling on such architectures relies on a static partitioning and cost model. In this paper, we present a locality-aware work stealing scheduler for multi-CPU and multi-GPU architectures, which relies on the XKaapi runtime system. We show performance results on two dense linear algebra kernels, Cholesky (POTRF) and LU (GETRF) factorization, to evaluate our scheduler on a heterogeneous architecture composed of two hexa-core CPUs and eight NVIDIA Fermi GPUs. Our experiments show that an online locality-aware scheduling achieve performance results as good as static strategies, and in most cases outperform them.

In this paper, we present a comparison of scheduling strategies for heterogeneous multi-CPU and multi-GPU architectures. We designed and evaluated four scheduling strategies on top of XKaapi runtime: work stealing, data-aware work stealing, locality-aware work stealing, and Heterogeneous Earliest-Finish-Time (HEFT). On a heterogeneous architecture with 12 CPUs and 8 GPUs, we analysed our scheduling strategies with four benchmarks: a BLAS-1 AXPY vector operation, a Jacobi 2D iterative computation, and two linear algebra algorithms Cholesky and LU. We conclude that the use of work stealing may be efficient if task annotations are given along with a data locality strategy. Furthermore, our experimental results suggests that HEFT scheduling performs better on applications with very regular computations and low data locality.

The preservation of symplectic Gram Schmidt Jorthogonality is crucial for structure preserving methods for the eigenvalue problem. In this paper, we present the symplectic Gram-Schmidt algorithm with re-orthogonalization and we study the recapture of the J-orthogonality of vectors generated by the algorithm. Furthermore, the re-orthogonalization process is combined with a careful choice of the free parameters that was involved in the algorithm to ensure satisfactory recapture of the J-orthogonality. Some numerical experiments and computational aspects are given to confirm the superiority of the algorithm.

In this paper, we investigate some properties of eigenvalues and eigenvectors of Jacobi matrices. We propose a new algorithm for reconstructing a 2n-th order Jacobi matrix J 2n with a given n-th order leading principal submatrix J n and with all eigenvalues of J 2n. This algorithm needs to compute the eigenvalues of the n-th order tailing principal submatrix J n+1,2n and the first components of the unit eigenvectors of J n+1,2n. Our method needs not to reconstruct J n , and can avoid computing the coefficients of the characteristic polynomial for getting the eigenvalues of J n+1,2n. We also present some numerical results.

We present a probabilistic framework for overlapping community discovery and link prediction for relational data, given as a graph. The proposed framework has: (1) a deep architecture which enables us to infer multiple layers of latent features/communities for each node, providing superior link prediction performance on more complex networks and better interpretability of the latent features; and (2) a regression model which allows directly conditioning the node latent features on the side information available in form of node attributes. Our framework handles both (1) and (2) via a clean, unified model, which enjoys full local conjugacy via data augmentation, and facilitates efficient inference via closed form Gibbs sampling. Moreover, inference cost scales in the number of edges which is attractive for massive but sparse networks. Our framework is also easily extendable to model weighted networks with count-valued edges. We compare with various state-of-the-art methods and report ...

This article introduces new low cost algorithms for the adaptive estimation and tracking of principal and minor components. The proposed algorithms are based on the well-known OPAST method which is adapted and extended in order to achieve the desired MCA or PCA (Minor or Principal Component Analysis). For the PCA case, we propose efficient solutions using Givens rotations to estimate the principal components out of the weight matrix given by OPAST method. These solutions are then extended to the MCA case by using a transformed data covariance matrix in such a way the desired minor components are obtained from the PCA of the new (transformed) matrix. Finally, as a byproduct of our PCA algorithm, we propose a fast adaptive algorithm for data whitening that is shown to overcome the recently proposed RLS-based whitening method.

Internet simplified digital data transferring. This data needs to be secured; so securing digital data becomes an important concern. Steganography provides security for data by inserting it into a cover and concealing it. In this paper, a steganography algorithm was introduced. This algorithm used the stationary wavelet transform (SWT) and اhybrid-matrix decomposition techniques, singular value decomposition (SVD) and QR factorization to conceal a video in another video. The algorithm performance was measured with reference to the peak signal to noise ratio (PSNR) and structural similarity index measure (SSIM) for the cover and the secret videos. The algorithm successfully hid a video in another cover video and both are of the same size; the hiding capacity was 100%. The algorithm achieved a SSIM that reached 0.97 and a high PSNR value that reached 68.8 proving that the imperceptibility of the proposed algorithm is very high. The comparative analysis shows that the suggested algorit...

Steganography is one of the most important tools in the data security field as there is a huge amount of data transferred each moment over the internet. Hiding secret messages in an image has been widely used because the images are mostly used in social media applications. The proposed algorithm is a simple algorithm for hiding an image in another image. The proposed technique uses QR factorization to conceal the secret image. The technique successfully hid a gray and color image in another one and the performance of the algorithm was measured by PSNR, SSIM and NCC. The PSNR for the cover image was in the range of 41 to 51 dB. DWT was added to increase the security of the method and this enhanced technique increased the cover PSNR to 48 t0 56 dB. The SSIM is 100% and the NCC is 1 for both implementations. Which improves that the imperceptibility of the algorithm is very high. The comparative analysis showed that the performance of the algorithm is better than other state-of-the-art ...