QR Algorithm Research Articles

A Deep Learning-based Approach for Channel Estimation in Multi-access Multi-antenna Systems

This paper studies estimating the channel state information at the end of receiver (CSIR) for multiple transmitters communicating with only one receiver so that the latter can decode the incoming signal more efficiently. The transmitters and the receiver are all equipped with multi-antennas and using orthogonal space-time block codes (OSTBC). An algorithm is developed based on deep learning for estimating multi-user multiple-input multiple-output (MU-MIMO) channels. The algorithm could estimate the CSIR using a single pilot block. The proposed convolutional neural network (CNN) architecture designed for this task begins with an input layer that accepts grayscale images, followed by six convolutional blocks for feature extraction and processing. The network concludes with a fully connected layer to output the estimated channel information. It is trained using a regression loss function to map input images to accurate channel information accurately. The performance of the proposed method is compared with classical methods like least square and subspace-based methods, including Capon and rank revealing QR (RRQR) methods. CNN achieved better performance in comparison with the reference. Computer simulations are included to validate the proposed method.

A Modified High-Selective Frequency Selective Surface Designed by Multilevel Green’s Function Interpolation Method

A compact high-selective band-pass frequency selective surface (FSS) with the unit cell less than λ/7 is presented. For the simulation of the structure, the multilevel Green’s function interpolation method (MLGFIM) using Floquet theory is adopted to accelerate the calculation of the complex unit cell. The radial basis function (RBF)-QR method is used in the interpolation, which makes the shape parameter in the RBF function not required to be retested for different periodicity. In this design, with an aperture coupling structure between the top and bottom layers patterned by triangular patches and meander lines, the FSS has two transmission zeros (TZs) on both sides of the pass-band and achieves a steep roll-off rate of 192 dB/GHz. Consequently, the FSS has high selectivity and out-of-band suppression, besides profiting from the low profile and symmetric geometry, this FSS exhibits good angular and polarization stabilities. The prototype of the proposed FSS is fabricated and good performance is obtained.

QR decomposition of dual matrices and its application

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.

The Algorithm of Gu and Eisenstat and D-Optimal Design of Experiments

This paper addresses the following problem: given m potential observations to determine n parameters, m>n, what is the best choice of n observations. The problem can be formulated as finding the n×n submatrix of the complete m×n observation matrix that has maximum determinant. An algorithm by Gu and Eisenstat for a determining a strongly rank-revealing QR factorisation of a matrix can be adapted to address this latter formulation. The algorithm starts with an initial selection of n rows of the observation matrix and then performs a sequence of row interchanges, with the determinant of the current submatrix strictly increasing at each step until no further improvement can be made. The algorithm implements rank-one updating strategies, which leads to a compact and efficient algorithm. The algorithm does not necessarily determine the global optimum but provides a practical approach to designing an effective measurement strategy. In this paper, we describe how the Gu–Eisenstat algorithm can be adapted to address the problem of optimal experimental design and used with the QR algorithm with column pivoting to provide effective designs. We also describe implementations of sequential algorithms to add further measurements that optimise the information gain at each step. We illustrate performance on several metrology examples.

Robust residual-guided iterative reconstruction for sparse-view CT in small animal imaging

Objective. We introduce a robust image reconstruction algorithm named residual-guided Golub–Kahan iterative reconstruction technique (RGIRT) designed for sparse-view computed tomography (CT), which aims at high-fidelity image reconstruction from a limited number of projection views. Approach. RGIRT utilizes an inner-outer dual iteration framework, with a flexible least square QR (FLSQR) algorithm implemented in the inner iteration and a restarted iterative scheme applied in the outer iteration. The inner FLSQR employs a flexible Golub–Kahan bidiagonalization method to reduce the size of the inverse problem, and a weighted generalized cross-validation method to adaptively estimate the regularization hyper-parameter. The inner iteration efficiently yields the intermediate reconstruction result, while the outer iteration minimizes the residual and refines the solution by using the result obtained from the inner iteration. Main results. The reconstruction performance of RGIRT is evaluated and compared to other reference methods (FBPConvNet, SART-TV, and FLSQR) using projection data from both numerical phantoms and real experimental Micro-CT data. The experimental findings, from testing various numbers of projection views and different noise levels, underscore the robustness of RGIRT. Meanwhile, theoretical analysis confirms the convergence of residual for our approach. Significance. We propose a robust iterative reconstruction algorithm for x-ray CT scans with sparse views, thereby shortening scanning time and mitigating excessive ionizing radiation exposure to small animals.

Fast Eigenvalue Decomposition of Arrowhead and Diagonal-Plus-Rank-k Matrices of Quaternions

Quaternions are a non-commutative associative number system that extends complex numbers, first described by Hamilton in 1843. We present algorithms for solving the eigenvalue problem for arrowhead and DPRk (diagonal-plus-rank-k) matrices of quaternions. The algorithms use the Rayleigh Quotient Iteration with double shifts (RQIds), Wielandt’s deflation technique and the fact that each eigenvector can be computed in O(n) operations. The algorithms require O(n2) floating-point operations, n being the order of the matrix. The algorithms are backward stable in the standard sense and compare well to the standard QR method in terms of speed and accuracy. The algorithms are elegantly implemented in Julia, using its polymorphism feature.

Türkiye'de CoronaVac ile Kovid-19 Aşılama Başlangıcında Sars-Cov-2 Yayılımının Matematiksel Modellenmesi

The Sars-CoV-2 virus, first detected in Wuhan, China, became a global crisis that affected the entire world and was declared a pandemic by the World Health Organization (WHO) in March 2020. The most basic protective measure in the fight against pandemics facing humanity is vaccination. From this point of view, data is collected between January 13 and February 11, 2021 by taking the number of daily cases, deaths and recovered patients in Türkiye. During this period, vaccination against Covid-19 with Sinovac's CoronaVac vaccine is started in Türkiye. Mathematical predictive models of the observed values are constructed and compared using polynomial regression (up to the 3rd degree) and nonlinear regression, i.e., curve fitting methods, and SIR (Susceptible-Infected-Removed), which is a system of ordinary differential equations (ODEs). The efficiencies of these prediction models are tested, validated, and the most effective mathematical prediction models are proposed. The values of root mean square error (RMSE) and mean absolute percentage error (MAPE) are used as performance measures to compare the methods. The proposed prediction models are also used for forecasting. The number of new cases occurring each day is predicted using the time-dependent equations of the SIR method, which are solved using the Euler method. It is found that the SIR method is quite successful in predicting the observed values compared to the other methods, but the QR method are given more successful results in predicting the total number of deaths

A hybrid short-term wind power point-interval prediction model based on combination of improved preprocessing methods and entropy weighted GRU quantile regression network

Accurate wind power prediction is significant to the stability of power system. Existing deterministic prediction methods unable to describe the uncertainty of wind power while both the point and probabilistic models have difficulty in tracking the abrupt changes in wind power accurately. To settle these problems, a point-interval prediction model based on combination of improved preprocessing methods and entropy weighted GRU quantile regression network (QR-EGRU) is proposed. Firstly, an improved wavelet threshold denoising (IWTD) is applied to reduce noise interference. An optimized variational mode decomposition (OVMD) based on sparrow search algorithm (SSA) is proposed to decompose the series into subsequences. Secondly, two update gate matrices based on information entropy (IE) are introduced to replace the traditional update gate matrix of the GRU to construct the EGRU. Point prediction results are obtained by using the EGRU model. Furthermore, the QR algorithm with nonlinear loss function is derived to realize the interval prediction of the EGRU. Finally, the proposed model is validated on real wind power data from the Kaggle competition. Experimental results demonstrate that the proposed model performs well in both point and interval prediction. It can track the mutation series more precisely than other models and improve the prediction accuracy.

Optimal sensor placement for reconstructing wind pressure field around buildings using compressed sensing

Deciding how to optimally deploy sensors in a large, complex, and spatially extended structure is critical to ensure that the surface pressure field is accurately captured for subsequent analysis and design. In some cases, reconstruction of missing data is required in downstream tasks such as the development of digital twins. This paper presents a data-driven sparse sensor selection algorithm, aiming to provide the most information contents for reconstructing aerodynamic characteristics of wind pressures over tall building structures parsimoniously. The algorithm first fits a set of basis functions to the training data, then applies a computationally efficient QR algorithm that ranks existing pressure sensors in order of importance based on the state reconstruction to this tailored basis. The findings of this study show that the proposed algorithm successfully reconstructs the aerodynamic characteristics of tall buildings from sparse measurement locations, generating stable and optimal solutions across a range of conditions. As a result, this study serves as a promising first step toward leveraging the success of data-driven and machine learning algorithms to supplement traditional genetic algorithms currently used in wind engineering.

A new deflation criterion for the QZ algorithm

AbstractThe QZ algorithm computes the generalized Schur form of a matrix pencil. It is an iterative algorithm and, at some point, it must decide when to deflate, that is when a generalized eigenvalue has converged and to move on to another one. Choosing a deflation criterion that makes this decision is nontrivial. If it is too strict, the algorithm might waste iterations on already converged eigenvalues. If it is not strict enough, the computed eigenvalues might not have full accuracy. Additionally, the criterion should not be computationally expensive to evaluate. There are two commonly used criteria: the elementwise criterion and the normwise criterion. This paper introduces a new deflation criterion based on the size of and the gap between the eigenvalues. We call this new deflation criterion the strict criterion. This new criterion for QZ is analogous to the criterion derived by Ahues and Tisseur for the QR algorithm. Theoretical arguments and numerical experiments suggest that the strict criterion outperforms the normwise and elementwise criteria in terms of accuracy. We also provide an example where the accuracy of the generalized eigenvalues using the elementwise or the normwise criteria is less than two digits whereas the strict criterion leads to generalized eigenvalues which are almost accurate to the working precision. Additionally, this paper evaluates some commonly used criteria for infinite eigenvalues.

Spectral computation with third-order tensors using the t-product

The tensor t-product is a powerful tool for the analysis of and computation with third-order tensors. This paper discusses properties and the computation of eigentubes and eigenslices of third-order tensors under the t-product; the eigentubes and eigenslices are analogues of eigenvalues and eigenvectors for matrices. The computational methods considered and analysed include the tensor power method, tensor subspace iteration, and the tensor QR algorithm. Computed examples illustrate the performance of these methods.

Algorithms for modifying recurrence relations of orthogonal polynomial and rational functions when changing the discrete inner product

Often, polynomials or rational functions, orthogonal for a particular inner product are desired. In practical numerical algorithms these polynomials are not constructed, but instead the associated recurrence relations are computed. Moreover, also typically the inner product is changed to a discrete inner product, which is the finite sum of weighted functions evaluated in specific nodes. For particular applications it is beneficial to have an efficient procedure to update the recurrence relations when adding or removing nodes from the inner product. The construction of the recurrence relations is equivalent to computing a structured matrix (polynomial) or pencil (rational) having prescribed spectral properties. Hence the solution of this problem is often referred to as solving an Inverse Eigenvalue Problem. In [34] we proposed updating techniques to add nodes to the inner product while efficiently updating the recurrences. To complete this study we present in this article manners to efficiently downdate the recurrences when removing nodes from the inner product. The link between removing nodes and the QR algorithm to deflate eigenvalues is exploited to develop efficient algorithms. We will base ourselves on the perfect shift strategy and develop algorithms, both for the polynomial case and the rational function setting. Numerical experiments validate our approach.

Constraint-matrix-based method for reaction and driving forces uniqueness analysis in overconstrained or overactuated multibody systems

Redundantly constrained mechanisms have – in general – non-uniquely calculated reactions when modeled as rigid multibody systems (MBSs). However, some of the reactions may be unique. An analogous problem of indeterminacy is also present in overactuated MBSs. This paper discusses the constraint-matrix-based method for the uniqueness analysis of the reactions and driving forces (torques) for MBSs with nonholonomic constraints. Four approaches are studied: The rank comparison, SVD, QR, and nullspace methods. The uniqueness criteria are written in a new way. The equivalence of the SVD, QR, and nullspace methods is shown. It is also presented how to check the uniqueness of the selected elements (reactions, driving forces, or more complex combinations) and their individual components. Subsequently, the impact of the driving constraints on the uniqueness of the joint reactions is discussed. Next, the uniqueness analysis using these three methods is extended to perform a newly proposed body-wise analysis instead of the usual constraint-wise analysis. Two examples of spatial systems (one with nonholonomic constraints) are considered to illustrate the approach. Moreover, the computational efficiency of selected methods is analyzed.

Time-shift selection for reservoir computing using a rank-revealing QR algorithm.

Reservoir computing, a recurrent neural network paradigm in which only the output layer is trained, has demonstrated remarkable performance on tasks such as prediction and control of nonlinear systems. Recently, it was demonstrated that adding time-shifts to the signals generated by a reservoir can provide large improvements in performance accuracy. In this work, we present a technique to choose the time-shifts by maximizing the rank of the reservoir matrix using a rank-revealing QR algorithm. This technique, which is not task dependent, does not require a model of the system and, therefore, is directly applicable to analog hardware reservoir computers. We demonstrate our time-shift selection technique on two types of reservoir computer: an optoelectronic reservoir computer and the traditional recurrent network with a t a n h activation function. We find that our technique provides improved accuracy over random time-shift selection in essentially all cases.

THE EIGENVALUES OF SYMMETRIC TRIDIAGONAL MATRIX METHOD WITH ACCELERATION MBY THE QR SHIFT

This research investigates the application of the QR - method for computing all the eigenvalues of the real symmetric tridiagonal matrix. The Householder method will be used for reduction of the real symmetric matrix to symmetric tridiagonal form, and then the so called QR - method with acceleration shift applies a sequence of orthogonal transformations to the symmetric tridiagonal matrix which converges to a similar matrix that is tridiagonal. This tridiagonal matrix possesses an eigenvalues similar to the eigenvalues of the symmetric tridiagonal matrix. Particular attention is paid to the shift technique that accelerates the rate of convergence. Computer algorithms for implementing the Householder's method and QR – method are presented. Computer Matlab programs for performing the Householder algorithm and the QR algorithm (with acceleration shift) are listed in the Appendix.

A flexible block classical Gram–Schmidt skeleton with reorthogonalization

AbstractWe investigate a variant of the reorthogonalized block classical Gram–Schmidt method for computing the QR factorization of a full column rank matrix. Our aim is to bound the loss of orthogonality even when the first local QR algorithm is only conditionally stable. In particular, this allows the use of modified Gram–Schmidt instead of Householder transformations as the first local QR algorithm. Numerical experiments confirm the stable behavior of the new variant. We also examine the use of non‐QR local factorization and show by example that the resulting variants, although less stable, may also be applied to ill‐conditioned problems.

On the Numerical Treatment of the Temporal Discontinuity Arising from a Time-Varying Point Mass Attachment on a Waveguide

A vibrating pylon, modeled as a waveguide, with an attached point mass that is time-varying poses a numerically challenging problem regarding the most efficient way for eigenvalue extraction. The reason is three-fold, starting with a heavy mass attachment that modifies the original eigenvalue problem for the stand-alone pylon, plus the fact that the point attachment results in a Dirac delta function in the mixed-type boundary conditions, and finally the eigenvalue problem becomes time-dependent and must be solved for a sequence of time steps until the time interval of interests is covered. An additional complication is that the eigenvalues are now complex quantities. Following the formulation of the eigenvalue problem as a system of first-order, time-dependent matrix differential equations, two eigenvalue extraction methods are implemented and critically examined, namely the Laguerre and the QR algorithms. The aim of the analysis is to identify the most efficient technique for interpreting time signals registered at a given pylon as a means for detecting damage, a procedure which finds application in structural health monitoring of civil engineering infrastructure.

QR algorithm with two‐sided Rayleigh quotient shifts

AbstractWe introduce the two‐sided Rayleigh quotient shift to the QR algorithm for non‐Hermitian matrices to achieve a cubic local convergence rate. For the singly shifted case, the two‐sided Rayleigh quotient iteration is incorporated into the QR iteration. A modified version of the method and its truncated version are developed to improve the efficiency. Based on the observation that the Francis double‐shift QR iteration is related to a 2D Grassmann–Rayleigh quotient iteration, A doubly shifted QR algorithm with the two‐sided 2D Grassmann–Rayleigh quotient double‐shift is proposed. A modified version of the method and its truncated version are also developed. Numerical examples are presented to show the convergence behavior of the proposed algorithms. Numerical examples also show that the truncated versions of the modified methods outperform their counterparts including the standard Rayleigh quotient single‐shift and the Francis double‐shift.

Prediction interval estimation of dynamic thermal rating considering weather uncertainty

Uncertainty and randomness of weather bring challenges to the accurate estimation of dynamic thermal rating (DTR). To reduce the impact of weather factors on the DTR predictions and to further enhance transmission line ampacity. An ensemble empirical mode decomposition quantile regression temporal convolutional network (EQRTCN)-based interval prediction model is proposed for DTR. Different from the existing DTR techniques, the model overcomes the effects of weather non-smoothness and uncertainty on prediction intervals with low quality by ensemble empirical mode decomposition (EEMD) and quantile regression theory. In this work, a k-means clustering algorithm is first used for the time boundary condition division by considering the coupling between line currents and weather parameters. Then, based on different time boundary conditions, the DTR interval prediction model is proposed by using EEMD, quantile regression theory and temporal convolutional networks. Finally, the proposed improved method for ampacity calculation is incorporated in the EQRTCN to calculate the prediction interval of DTR. The model simulation results demonstrated that the proposed DTR prediction framework effectively reduced the uncertainty introduced by meteorological parameters in interval predictions and provided safe guidance for power dispatch. • Propose a novel EEMD-guided TCN model based on the QR method. • Design a novel time boundary condition division method. • Propose an improved method for line ampacity calculation.

A practical approach to computing Lyapunov exponents of renewal and delay equations.

We propose a method for computing the Lyapunov exponents of renewal equations (delay equations of Volterra type) and of coupled systems of renewal and delay differential equations. The method consists of the reformulation of the delay equation as an abstract differential equation, the reduction of the latter to a system of ordinary differential equations via pseudospectral collocation and the application of the standard discrete QR method. The effectiveness of the method is shown experimentally and a MATLAB implementation is provided.