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This paper introduces the VDR decomposition of Chebyshev-Vandermonde matrices, where V represents an ordinary Vandermonde matrix, D is diagonal, and R is upper triangular. Our motivation for this work stems from the study by Brubeck et al. [Vandermonde with Arnoldi. SIAM Rev. 2021;63(2):405–415]. We explore the VDR decomposition and combine it with a QR decomposition of V for Chebyshev-Vandermonde matrices. Furthermore, we demonstrate that the upper triangular factor in the QR decomposition can be recursively obtained from the upper Hessenberg matrix in the Arnoldi process. To compare our approach with the results obtained using the default QR code in Matlab, we present experimental results. Finally, we provide an algorithm for solving linear systems with Chebyshev-Vandermonde matrices as coefficient matrices, and experimental results for an upper bound on the relative error by applying an a posteriori error analysis for the computed solutions of the systems. [ABSTRACT FROM AUTHOR]

Solving the Perspective-n-Point (PnP) problem is difficult in low-power systems due to the high computing workload. To handle this challenge, we present an originally designed FPGA implementation of a PnP solver based on Vivado HLS. A matrix operation library and a matrix decomposition library based on QR decomposition have been developed, upon which the EPnP algorithm has been implemented. To enhance the operational speed of the system, we employed pipeline optimization techniques and adjusted the computational process to shorten the calculation time. The experimental results show that when the number of input data points is 300, the proposed system achieves a processing speed of 45.2 fps with a power consumption of 1.7 W and reaches a peak-signal-to-noise ratio of over 70 dB. Our system consumes only 3.9% of the power consumption per calculation compared to desktop-level processors. The proposed system significantly reduces the power consumption required for the PnP solution and is suitable for application in low-power systems. [ABSTRACT FROM AUTHOR]

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]

The two-sided Jacobi method for singular value decomposition (SVD) has the advantage of obtaining singular vectors quickly and accurately. In previous research, fast and accurate implementations of the two-sided Jacobi method have been achieved for real matrices. In this study, we implemented SVD for complex matrices using the two-sided Jacobi method. In SVD, given rectangular matrices can be converted into upper-triangular matrices by conducting QR decomposition as a preprocessing step. Then, the upper-triangular matrices are decomposed. In the case where the given matrices are complex, the upper-triangular matrices are complex. Two implementations are proposed: the first requires SVD for complex matrices, whereas the second requires SVD for only real matrices. [ABSTRACT FROM AUTHOR]

This paper presents a QR decomposition based low-rank approximation algorithm for training and prediction in Gaussian process regression. Conventional low-rank methods like FITC, Nyström, etc., rely on low-rank approximations to the full kernel matrix that are derived from a set of representative data points. Training with FITC involves the selection of these representative points and is computationally intensive when the dimension of the dataset is large. Prediction accuracy of Nyström suffers when the number of representative points is small or when the length scale is small. The algorithm proposed here uses the thin QR decomposition of the low-rank matrix used in the Nyström approximation. We show that the marginal likelihood and its derivatives needed for training can be split into the subspace belonging to that approximation and its orthogonal complement. During prediction, we further restrict the target vectors into this subspace and thus eliminate the numerical error caused by very low noise variance. Use of the QR factorization improves the training and prediction performance. Experiments on real and synthetic data show that our approach is more accurate and faster than conventional methods. Our algorithm performs well at low length scales and achieves the prediction accuracy comparable to the full kernel matrix approach even for low rank approximations, without the need to optimize the representative points. This results in a simpler and faster approximation that could be used to scale Gaussian process regression to large datasets. [ABSTRACT FROM AUTHOR]

The finite-difference frequency domain (FDFD) method is widely applied for simulating seismic wavefields, and a key to achieving successful FDFD simulation is to construct FDFD coefficients that can effectively suppress numerical dispersion. Among the existing FDFD coefficients for seismic wavefield simulation, adaptive FDFD coefficients that vary with the number of wavelengths per grid can suppress numerical dispersion to the maximum extent. The current methods for calculating adaptive FDFD coefficients involve numerical integration, conjugate gradient (CG) optimization, sequential initial value selection, and smooth regularization, which are difficult to implement and inefficient in calculations. To simplify the calculation of adaptive FDFD coefficients and improve the corresponding computational efficiency, this paper proposes a new method for calculating adaptive FDFD coefficients. First, plane-wave solutions with different discrete propagation angles are substituted in the FDFD scheme, and the corresponding least-squares problem is constructed. As this problem is ill-conditioned and obtaining smooth adaptive FDFD coefficients by the conventional solving method based on normal equations is difficult, this paper proposes solving the least-squares problem by solving the corresponding overdetermined linear system of equations through QR matrix decomposition. Compared with the existing methods for calculating adaptive FDFD coefficients based on numerical integration, CG optimization, and sequential initial value selection, the proposed method allows for a simplified computational process and considerably higher computational efficiency. Numerical wavefield simulation results show that the adaptive-coefficient FDFD method based on QR matrix decomposition can achieve the same accuracy as those based on numerical integration, CG optimization, and sequential initial value selection while requiring less computation time. [ABSTRACT FROM AUTHOR]

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The paper presents a rigorous perturbation analysis of the QR decomposition A = Q R of an n × m matrix A using the method of splitting operators. New asymptotic componentwise perturbation bounds are derived for the elements of Q and R and the subspaces spanned by the first p ≤ m columns of A. The new bounds are less conservative than the known bounds and are significantly better than the normwise bounds. An iterative scheme is proposed to determine global componentwise bounds in the case of perturbations for which such bounds are valid. Several numerical results are given that illustrate the analysis and the quality of the bounds obtained. [ABSTRACT FROM AUTHOR]

Dual number matrix decompositions have played an important role in fields such as kinematics and computer graphics in recent years. In this paper, we present a QR decomposition algorithm for dual number matrices. When dealing with large-scale problems, we present the thin QR decomposition of dual number matrices, along with its algorithm with column pivoting. In numerical experiments, we discuss the suitability of different QR algorithms when confronted with various large-scale dual matrices, providing their respective domains of applicability. Finally, we employ the QR decomposition of dual matrices to compute the DMPGI, attaining results of higher precision. [ABSTRACT FROM AUTHOR]

Matrix decomposition is one of the important operations in matrix inversion, which is widely used in neural networks, digital signal processing, wireless communication technology and other fields. Based on a column-vector optimized QR decomposition algorithm, this study proposes a one-dimensional linear matrix decomposition structure and completes the ASIC implementation of the structure to address the shortcomings of the traditional decomposition algorithm operations that are not conducive to hardware implementation. The matrix decomposer supports matrix decomposition operations of order 2～32 and operates at 700 MHz at TSMC 28 nm process. Simulation and FPGA test results show that the relative error between the decomposer and MATLAB results is less than 10-12. When performing matrix decomposition of more than 12-orders, the operation cycle of the decomposer has a speedup ratio of 2.3 times compared with the traditional one-dimensional linear structure. When performing 32-order matrix decomposition, the operation cycle of the decomposer has a speedup ratio of 22.8 times compared with NIVIDA RTX2070. [ABSTRACT FROM AUTHOR]

For protecting the digital copyright of color image, a color image watermarking method should have strong robustness and short running-time at same time. To achieve these two goals, an effective color image watermarking method is presented in this paper. Firstly, the element in the first row and first column of the R matrix of QR decomposition is obtained by the presented method in spatial domain. Secondly, the obtained element of R matrix is selected to protect the copyright of color image in spatial domain without the true QR decomposition. Finally, the processes of the presented watermarking technique are performed in spatial domain. In the compared experimentations, the imperceptibility of the presented method is about 40 dB, its average NC is 0.9435, its average running-time is 0.703613 s, its embedding capacity is 0.03125 bpp, and its key space is 2138, which proves the proposed new method has better performances in terms of the robustness and real-time feature. [ABSTRACT FROM AUTHOR]

An approach to eliminate multicollinearity problems in regression analysis using QR decomposition of rectangular matrices by Householder reflection has been proposed. The reliability of this computational procedure has been proved. [ABSTRACT FROM AUTHOR]

This paper proposes a novel observability analysis method with less computation, which is based on QR decomposition and is suitable for low-cost devices with limited computing power. On this basis, an adaptive information fusion mechanism (AIFM) based on the criteria from the degree of observability (DOO) for the traditional extended Kalman filter (EKF) is designed to improve the tolerance of the integrated navigation system against time-varying observation conditions. Theoretical analysis and experimental results illustrate that the proposed observability analysis method can effectively evaluate each state variable’s observability, whose computational complexity is significantly reduced by 93.07% (the size of stripped observability matrix is $105\times15$) compared with the conventional method based on singular value decomposition (SVD). Moreover, the simulation and real-world experiment results show that the AIFM makes the navigation system more robust by automatically updating the state variables’ weights according to their observability, which also enhances the reliability of the horizontal attitude angle estimation with excellent efficiency since they are given top priority in our fusion mechanism. In conclusion, our method is practical to improve the integrated navigation system’s accuracy, robustness, and fault-tolerance ability in complex and dynamic environments, suggesting that this technique is viable for multi-rotor navigation and control applications. [ABSTRACT FROM AUTHOR]

The subset selection problem of linear algebra is applied to identify independent patterns of COVID-19 evolution within Brazil. The data consist of a set of mortality curves in states of Brazil. A subset of the most independent curves is selected by using a functional version of the QR matrix decomposition technique with column pivoting. The selected subset is used next as a basis to represent the remaining curves filtering out any data redundancy. For each independent curve, an associated epidemiological region of influence is defined. The results show two main independent curves with a similar two-peak pattern and a 50-day shift between the patterns. Two main epidemiological regions are next identified: one encompassing most of the country from the center and northeast states to the south, an another one containing the Amazonian region at the northwest. [ABSTRACT FROM AUTHOR]

The outliers or impulsive noise is prominent in practical wireless sensor networks due to saturation effects, non-linearities, malfunction of sensors, environmental abnormalities, etc. The classical diffusion algorithms based on mean square error cost function are sensitive to outliers and their performance degrades in the presence of outliers in either desired data or the input data. A fast diffusion minimum generalized rank norm based on QR decomposition (FDGR-QR) is proposed, which is robust against outliers in both desired and input data and has faster convergence than the state of the art algorithms. Different outlier percentages distribution across the network is considered for the simulations based experiments. The proposed algorithm is validated for both stationary and non-stationary parameter estimation scenarios. [ABSTRACT FROM AUTHOR]

This study investigated the problem of QR decomposition for time-varying matrices. We transform the original QR decomposition problem into an equation system using its constraints. Then, we propose a continuous-time QR decomposition (CTQRD) model by applying zeroing neural network method, equivalent transformations, Kronecker product, and vectorization techniques. Subsequently, a high-precision ten-instant Zhang et al. discretization (ZeaD) formula is proposed. A ten-instant discrete-time QR decomposition model is also proposed by using the ten-instant ZeaD formula to discretize the CTQRD model. Moreover, three discrete-time QR decomposition models are proposed by applying three other ZeaD formulas, and three examples of QR decomposition are presented. The experimental results confirm the effectiveness and correctness of the proposed models for the QR decomposition of time-varying matrices. [ABSTRACT FROM AUTHOR]

The paper derives the Jacobians of the matrix decompositions such as SVD, spectral decomposition, QR decomposition, L ′ D L decomposition and so on, for singular random matrix. These results are established for the random matrix whose elements can be linearly dependent mutually. Furthermore, the density function is restated for the singular Wishart random matrix, and the density function of its M−P inverse is also derived. [ABSTRACT FROM AUTHOR]

The protection of sensitive and confidential data become a challenging task in the present scenario as more and more digital data is stored and transmitted between the end users. The privacy is vitally necessary in case of medical data, which contains the important information of the patients. In this article, a novel biometric inspired medical encryption technique is proposed based on newly introduced parameterized all phase orthogonal transformation (PR-APBST), singular value, and QR decomposition. The proposed technique utilizes the biometrics of the patient/owner to generate a key management system to obtain the parameters involved in the proposed technique. The medical image is then encrypted employing PR-APBST, QR and singular value decomposition and is ready for secure transmission or storage. Finally, a reliable decryption process is employed to reconstruct the original medical image from the encrypted image. The validity and feasibility of the proposed framework have been demonstrated using an extensive experiments on various medical images and security analysis. [ABSTRACT FROM AUTHOR]

Watermarking is the method by which a watermark image related to the original cover image is embedded into the cover image. Images have become an important part in digital media. Thus it is important to secure the image data and prevent its illegal duplication. Image watermarking aims at serving this purpose. Wavelet technique is commonly used technique to embed watermark on the cover image in transform domain. The watermark data and cover image are not distorted after applying transform; hence this technique is used widely in this field. The upgraded version of wavelet technique is shearlet transform, thus upgraded version improves the presentation of multidimensional data and transform is mainly based on multidirectional and multi-resolution. The shearlet transform overcome the drawback of wavelet's edge presentation of image which is the main characteristic of shearlet and hence the host image is modelled with different texture. There are two characteristics of watermark, namely, Imperceptibility and robustness. Imperceptibility is achieved using Shearlet transform, whereas robustness and security improvement is achieved using QR decomposition. Analyzing the experiments undergone on the various attack of host image with distinct texture characteristics and thus the result shows that this technique has a good resistance to attacks and provides a efficient transparency and high robustness. [ABSTRACT FROM AUTHOR]

The problem of solving overdetermined, underdetermined, singular, or ill conditioned SLAEs using matrix canonization is considered. A modification of an existing canonization algorithm based on matrix decomposition is proposed. Formulas using LU decomposition, QR decomposition, LQ decomposition, or singular value decomposition, depending on the properties of the given matrix, are obtained. A method for evaluating the condition number of the canonization problem is proposed. It is based on computing the norm of the matrices obtained as a result of canonization; this method does not require the original matrix to be inverted. A general step-by-step matrix canonization algorithm is described and implemented in MATLAB. The implementation is tested on a set of 100 000 randomly generated matrices. The testing results confirmed the validity and efficiency of the proposed algorithm. [ABSTRACT FROM AUTHOR]

The development of the Internet of Things (IoT) can be attributed to the sudden rise in miniature electronic devices, as well as their computing power and ability to make interconnections. These devices exchange large volumes of confidential information from diverse locations. Similar to the Internet, the IoT has also encountered various issues with information security. Due to limited computing and energy resources in the field of IoT, it is necessary to develop a scheme to ensure feasible and more effective concealment and security properties. This paper proposes a unique methodology that captures an image using IoT sensors, which are subjected to lighter cryptographic operations for conversion into a cipher image, and is then sent to a home server. At the home server, a combined cryptography and steganography approach is employed to conceal the cipher image in a cover image, camouflaging the presence of the secret image, which is then sent to the IoT-Cloud server for storage. During the embedding process, QR decomposition is performed on the RIWT transformed secret image and RIWT - DCT transformed cover image. Modification performed on the R matrix of QR decomposition does not affect the structural properties of the cover image. A block selection algorithm is used to select optimal blocks with high contrast areas to embed the secret image. The experimental results indicate that our scheme enhances imperceptibility, robustness, and resistance to steganalysis attacks. [ABSTRACT FROM AUTHOR]

While the proper orthogonal decomposition (POD) is optimal under certain norms, its also expensive to compute. For large matrix sizes, the QR decomposition provides a tractable alternative. Under the assumption that it is a rank-revealing QR (RRQR), the approximation error incurred is similar to the POD error; furthermore, the authors show the existence of an RRQR with exactly the same error estimate as POD. To numerically realize an RRQR decomposition, they discuss the (iterative) modified Gram Schmidt with pivoting and reduced basis methods. They show that these two seemingly different approaches are equivalent. They then describe an MPI/OpenMP parallel code that implements one of the QR-based model reduction algorithms analyzed, and document the codes scalability for large problems, such as gravitational waves, and demonstrate excellent scalability up to 32,768 cores and, for complex dense matrices, as large as 10,000x3,276,800. [ABSTRACT FROM AUTHOR]

Among the representative algorithms of manifold learning, Hessian locally linear embedding (HLLE) and local tangent space alignment (LTSA) algorithms haven been regarded as two different algorithms. However, in practice, the effects of these two algorithms are very similar and LTSA performs better than HLLE in some applications. This paper tries to account for this phenomenon from a mathematical point of view. There are only two differences between HLLE and LTSA. First, LTSA includes a data point into its neighborhood, while HLLE does not. Second, HLLE and LTSA use different methods to align the local coordinates of manifold. In this paper, we show that, the first difference between HLLE and LTSA is not essential. However, from the viewpoint of data utilization, LTSA does better than HLLE in the neighborhood construction. This may account for why LTSA can perform better than HLLE in some applications. As for the second difference between HLLE and LTSA, we first prove that, the alignment equations used by HLLE and LTSA are exactly the same. Second, we prove that, although HLLE and LTSA uses different methods to solve the alignment equation, their solutions are exactly the same, provided that HLLE adopts the same method as LTSA to construct the neighborhoods. Based on these arguments, we claim that HLLE and LTSA are equivalent to each other. This conclusion can also be verified experimentally by using manifold learning MATLAB demo (MANI), a widely-used experimental platform of manifold learning. When testing HLLE on MANI, if HLLE adopts the same method as LTSA to construct the neighborhoods, the experimental results presented by MANI will be the same as those of LTSA. [ABSTRACT FROM PUBLISHER]

We present normwise and componentwise perturbation bounds for the LU , the Cholesky, the L D L T and the QR decompositions by using a new approach. The explicit expressions of mixed and componentwise condition numbers for these matrix decompositions are derived. The condition numbers improve known results of the normwise and componentwise cases and reveal the characterizations of the structured perturbations. The exact explicit perturbation expressions are derived for the factors L and U of the LU decomposition, and the rigorous normwise and componentwise perturbation bounds are presented for the LU decomposition. [ABSTRACT FROM AUTHOR]

QR decomposition (QRD) has been a vital component in the transceiver processor of future multiple-input multiple-output (MIMO) systems, in which antenna configuration will be more and more flexible. Therefore, the QRD hardware architecture in the future MIMO systems should be more flexible to meet various antenna configurations. Unfortunately, the existing QRD hardware architectures mainly focus on the matrix of one or several fixed sizes. This paper presents a new triangular systolic array QRD hardware architecture based on parallel tiled QRD algorithm to decompose an 8\times 8 real matrix. The designed hardware architecture is flexible and can be used in various MIMO systems, in which the number of antennas is smaller than 4. This paper also proposes a modified algorithm for the bottleneck operations of parallel tiled QRD algorithm to reduce the hardware overhead. To further reduce the hardware overhead, the Newton–Raphson algorithm is adopted in the proposed algorithm. The implementation results show that the normalized processing latency performance and the normalized processing efficiency performance of the designed QRD hardware architecture both are better than most of the existing QRD hardware architectures. To the best of our knowledge, the hardware architecture presented in this paper achieves the superior normalized QRD rate performance to the existing QRD hardware architectures. [ABSTRACT FROM PUBLISHER]

Impulsive noise and interference are always present in the environment in addition to additive white Gaussian noise that corrupts the measured data. In such cases, conventional adaptive estimation algorithms based on least squares error cost function provides poor performance in estimating the optimum parameter. In order to alleviate this shortcoming, a robust diffusion strategy based on QR decomposition and the Wilcoxon norm is proposed. The proposed QR‐based diffusion minimum‐Wilcoxon‐norm (DMWN) provides faster convergence than the DMWN. To demonstrate the efficacy of the algorithm, simulations are carried out with different percentage of outliers in the desired data and found to be robust against traditional methods. Moreover, the condition for convergence in mean is analysed, and the algorithm is observed to be stable. [ABSTRACT FROM AUTHOR]

Adaptive beamforming methods for ultrasound imaging have been studied to improve image resolution and contrast. The most common approach is the minimum variance (MV) beamformer which minimizes the power of the beamformed output while maintaining the response from the direction of interest constant. The method achieves higher resolution and better contrast than the delay-and-sum (DAS) beamformer, but it suffers from high computational cost. This cost is mainly due to the computation of the spatial covariance matrix and its inverse, which requires O(L^3) computations, where L denotes the subarray size. In this study, we propose a computationally efficient MV beamformer based on QR decomposition. The idea behind our approach is to transform the spatial covariance matrix to be a scalar matrix \sigma \mathbfI and we subsequently obtain the apodization weights and the beamformed output without computing the matrix inverse. To do that, QR decomposition algorithm is used and also can be executed at low cost, and therefore, the computational complexity is reduced to O(L^2). In addition, our approach is mathematically equivalent to the conventional MV beamformer, thereby showing the equivalent performances. The simulation and experimental results support the validity of our approach. [ABSTRACT FROM PUBLISHER]