Basic Matrix Operations Research Articles

Neural networks for critical load factors of unstiffened plates according to the reduced stresses method of EN 1993-1-5

The critical load factor resulting from the complete stress field is the main input parameter for the verification against plate buckling according to the reduced stresses method (RSM) from Eurocode EN 1993-1-5. Currently, the code provides just a simple and often conservative interaction formula with a limited application range. Realistic values can only be obtained by numerical methods, like the FEM, which is tedious for many load cases. To enhance the ease of use of the RSM, simple-to-use but accurate methods are needed, which can be automated and implemented into EN 1993-1-5. The current research deals with this problem by applying artificial neural networks (ANN) to predict the critical load factors of unstiffened plates under longitudinal, shear, and patch loading stresses, including their interaction behaviour. ANNs were used because approximated solutions with polynomials (as actual in the standard) are not suitable due to the complexity of the problem. The developed ANNs are highly accurate (as shown by an independent verification), can be implemented into the Eurocode, and can easily be used, even without in-depth programming skills. Moreover, a simple Python code is provided to demonstrate the ease of use. Still, the code can also be implemented into other software packages (like Excel), as only basic matrix operations are needed. The interaction surfaces are visualized, gaining a deeper understanding of plates under multiaxial loading situations. Finally, recommendations are given for generating ANNs for plate buckling problems.

Matrix numerical method for probability densities of stochastic delay differential equations

Stochastic processes with time delay are invaluable for modeling in science and engineering when finite signal transmission and processing speeds can not be neglected. However, they can seldom be treated with sufficient precision analytically if the corresponding stochastic delay differential equations (SDDEs) are nonlinear. This work presents a numerical algorithm for calculating the probability densities of processes described by nonlinear SDDEs. The algorithm is based on Markovian embedding and solves the problem by basic matrix operations. We validate it for a broad class of parameters using exactly solvable linear SDDEs and a cubic SDDE. Besides, we show how to apply the algorithm to calculate transition rates and first passage times for a Brownian particle diffusing in a time-delayed cusp potential.

Maximally entangled real states and SLOCC invariants: the 3-qutrit case

The absolute values of polynomial SLOCC invariants (which always vanish on separable states) can be seen as entanglement measures. We study the case of real 3-qutrit systems and discover a new set of maximally entangled states (from the point of view of maximizing the hyperdeterminant). We also study the basic fundamental invariants and find real 3-qutrit states that maximize their absolute values. It is notable that the Aharonov state is a simultaneous maximizer for all three fundamental invariants. We also study the evaluation of these invariants on random real 3-qutrit systems and analyze their behavior using histograms and level-set plots. Finally, we show how to evaluate these invariants on any 3-qutrit state using basic matrix operations.

State‐dependent indirect hp‐pseudospectral method for rigid spacecraft attitude optimal control problem

AbstractThe state‐dependent indirect ‐pseudospectral (SDIHP) method for rigid spacecraft attitude control is presented in this paper. The optimal controller is obtained by the SDIHP method through classifying the attitude optimal control problem as a nonlinear quadratic optimal control problem. In the proposed control scheme, successive state‐dependent coefficient (SDC) parameterization is initially applied to the spacecraft dynamics to derive an available linear system to decrease the calculation cost and speed. Then, the indirect ‐adaptive spectral discretization is developed to convert the resultant optimal control problem into a sequence of linear equations, which can improve the global computational efficiency and guarantee the solving precision. Hence, the optimal control can be efficiently solved using basic matrix operations under the necessary collocation points based on the ‐adaptive features. Considering the inertia uncertainties and external disturbances, a state feedback controller is generated by the receding horizon control (RHC) framework to improve the robustness. Simulation results are included to show the effectiveness and performance of the suggested optimal method.

Coding theory for h(x)-Fibonacci polynomials

The amount of information transmitted over the internet network has dramatically increased with the prevailing of internet use. As a result of this increase, the algorithms used in data encryption methods have gained importance. In this paper, h(x)-Fibonacci coding/decoding method for h(x)-Fibonacci polynomials is introduced. The proposed method is fast because it is based on basic matrix operations, and it is suitable for cryptographic applications because it uses the ASCII character encoding system. For this reason, it differs from the classical algebraic methods in literature. Furthermore, the fact that h(x) is a polynomial improves the security of cryptography.

Symbolic model checking quantum circuits in Maude.

This article presents a symbolic approach to model checking quantum circuits using a set of laws from quantum mechanics and basic matrix operations with Dirac notation. We use Maude, a high-level specification/programming language based on rewriting logic, to implement our symbolic approach. As case studies, we use the approach to formally specify several quantum communication protocols in the early work of quantum communication and formally verify their correctness: Superdense Coding, Quantum Teleportation, Quantum Secret Sharing, Entanglement Swapping, Quantum Gate Teleportation, Two Mirror-image Teleportation, and Quantum Network Coding. We demonstrate that our approach/implementation can be a first step toward a general framework to formally specify and verify quantum circuits in Maude. The proposed way to formally specify a quantum circuit makes it possible to describe the quantum circuit in Maude such that the formal specification can be regarded as a series of quantum gate/measurement applications. Once a quantum circuit has been formally specified in the proposed way together with an initial state and a desired property expressed in linear temporal logic (LTL), the proposed model checking technique utilizes a built-in Maude LTL model checker to automatically conduct formal verification that the quantum circuit enjoys the property starting from the initial state.

Nonparametric covariance estimation for mixed longitudinal studies, with applications in midlife women's health

In mixed longitudinal studies, a group of subjects enter the study at different ages (cross-sectional) and are followed for successive years (longitudinal). In the context of such studies, we consider nonparametric covariance estimation with samples of noisy and partially observed functional trajectories. The proposed algorithm is based on a noniterative sequential-aggregation scheme with only basic matrix operations and closed-form solutions in each step. The good performance of the proposed method is supported by both theory and numerical experiments. We also apply the proposed procedure to a study on the working memory of midlife women, based on data from the Study of Women's Health Across the Nation (SWAN).

An improved ellipsoid optimization algorithm in subspace predictive control

PurposeThe purpose of this paper is to derive the output predictor for a stationary normal process with rational spectral density and linear stochastic discrete-time state-space model, respectively, as the output predictor is very important in model predictive control. The derivations are only dependent on matrix operations. Based on the output predictor, one quadratic programming problem is constructed to achieve the goal of subspace predictive control. Then an improved ellipsoid optimization algorithm is proposed to solve the optimal control input and the complexity analysis of this improved ellipsoid optimization algorithm is also given to complete the previous work. Finally, by the example of the helicopter, the efficiency of the proposed control strategy can be easily realized.Design/methodology/approachFirst, a stationary normal process with rational spectral density and one stochastic discrete-time state-space model is described. Second, the output predictors for these two forms are derived, respectively, and the derivation processes are dependent on the Diophantine equation and some basic matrix operations. Third, after inserting these two output predictors into the cost function of predictive control, the control input can be solved by using the improved ellipsoid optimization algorithm and the complexity analysis corresponding to this improved ellipsoid optimization algorithm is also provided.FindingsSubspace predictive control can not only enable automatically tune the parameters in predictive control but also avoids many steps in classical linear Gaussian control. It means that subspace predictive control is independent of any prior knowledge of the controller. An improved ellipsoid optimization algorithm is used to solve the optimal control input and the complexity analysis of this algorithm is also given.Originality/valueTo the best knowledge of the authors, this is the first attempt at deriving the output predictors for stationary normal processes with rational spectral density and one stochastic discrete-time state-space model. Then, the derivation processes are dependent on the Diophantine equation and some basic matrix operations. The complexity analysis corresponding to this improved ellipsoid optimization algorithm is analyzed.

Introduction to Applied Linear Algebra: Vectors, Matrices, and Least Squares [Bookshelf

The book consists of three parts. Part 1 focuses on vectors and their manipulation. Vector algebra, linear functions, linearization, inner products, norms, linear independence, the concept of a basis, and orthogonality are covered. Part 2 is devoted to matrices. Basic matrix operations (including a clear discussion of all four ways to interpret matrix multiplication), linear systems of equations, the QR factorization, and various matrix inverses (left, right, and pseudoinverses) are discussed. Part 3 covers least squares and emphasizes different approaches to finding xt. A notable feature of this book is the excellent end-ofchapter problem sets that span a wide range of interesting applications. A detailed solutions manual is also available to instructors. The book is clear and direct, written by experienced authors. The need for textbooks that show mathematics in action is also clear. Linear algebra has become more important to modern students (arguably, more important than calculus), partly because of the rise of data science and deep learning. Through it all, this wonderful subject retains the beauty of good mathematics.

KLERC: kernel Lagrangian expectile regression calculator

As a generalization to the ordinary least square regression, expectile regression, which can predict conditional expectiles, is fitted by minimizing an asymmetric square loss function on the training data. In literature, the idea of support vector machine was introduced to expectile regression to increase the flexibility of the model, resulting in support vector expectile regression (SVER). This paper reformulates the Lagrangian function of SVER as a differentiable convex function over the nonnegative orthant, which can be minimized by a simple iterative algorithm. The proposed algorithm is easy to implement, without requiring any particular optimization toolbox besides basic matrix operations. Theoretical and experimental analysis show that the algorithm converges r-linearly to the unique minimum point. The proposed method was compared to alternative algorithms on simulated data and real-world data, and we observe that the proposed method is much more computationally efficient while yielding similar prediction accuracy.

Chirality through classical physics

Chirality, or handedness, is a topic that is common in biology and chemistry, yet is rarely discussed in physics courses. We provide a way of introducing the topic in classical physics, and demonstrate the merits of its inclusion—such as a simple way to visually introduce the concept of symmetries in physical law—along with giving some simple proofs using only basic matrix operations, thereby avoiding the full formalism of the three-dimensional point group.

Rigid spacecraft robust adaptive attitude Stabilization Using state-dependent indirect Chebyshev pseudospectral method

In this paper, a robust control method is proposed for rigid spacecraft attitude stabilization under inertia matrix uncertainties and external disturbances. The attitude stabilization problem is firstly reformulated into an optimal control problem, then a state-dependent indirect Chebyshev pseudospectral (SDICP) method is designed to solve the resultant optimal control problem. It is proved that, by incorporating the uncertainties into the performance index and adaptively estimating the parameters, the stabilization problem can be converted into an infinite horizon optimal control problem. Based on successive state-dependent coefficient parameterization, time domain transformation and Chebyshev spectral discretization, the proposed SDICP method turns the resultant optimal control problem into a sequence of systems of linear equations, which can be efficiently solved by basic matrix operations. Simulation results also verify the effectiveness of the proposed robust adaptive attitude stabilization method.

A DEMATEL-Based Disassembly Line Balancing Heuristic

Circular economy has emerged as a response to increasing environmental problems. As opposed to linear economy, circular economy aims at the preservation of energy, material, and labor contents of used products. A critical process in circular economy is product recovery which involves the recovery of materials or components from returned products through various recovery options including recycling, refurbishing, and remanufacturing. All recovery options require some level of disassembly and disassembly operations that are generally carried out in a disassembly line. Like assembly lines, disassembly lines must be balanced in order to ensure the effective operation of the line. Mathematical programming techniques, metaheuristics, and various heuristic procedures were employed in order to solve different types of disassembly line balancing problem (DLBP). However, the use of multi-attribute decision making techniques is limited to few studies. In this study, we propose a DEMATEL-based disassembly line balancing approach which does not require extensive knowledge in operations research and computer programming. A solution can be obtained by carrying out basic matrix operations and following the steps of the approach. Two numerical examples are also provided in order to present the applicability of the proposed approach. The results indicate that the proposed approach presents a satisfactory performance compared to the previously proposed approaches.

Generalized Additive Models for Gigadata: Modeling the U.K. Black Smoke Network Daily Data

We develop scalable methods for fitting penalized regression spline based generalized additive models with of the order of 104 coefficients to up to 108 data. Computational feasibility rests on: (i) a new iteration scheme for estimation of model coefficients and smoothing parameters, avoiding poorly scaling matrix operations; (ii) parallelization of the iteration’s pivoted block Cholesky and basic matrix operations; (iii) the marginal discretization of model covariates to reduce memory footprint, with efficient scalable methods for computing required crossproducts directly from the discrete representation. Marginal discretization enables much finer discretization than joint discretization would permit. We were motivated by the need to model four decades worth of daily particulate data from the U.K. Black Smoke and Sulphur Dioxide Monitoring Network. Although reduced in size recently, over 2000 stations have at some time been part of the network, resulting in some 10 million measurements. Modeling at a daily scale is desirable for accurate trend estimation and mapping, and to provide daily exposure estimates for epidemiological cohort studies. Because of the dataset size, previous work has focused on modeling time or space averaged pollution levels, but this is unsatisfactory from a health perspective, since it is often acute exposure locally and on the time scale of days that is of most importance in driving adverse health outcomes. If computed by conventional means our black smoke model would require a half terabyte of storage just for the model matrix, whereas we are able to compute with it on a desktop workstation. The best previously available reduced memory footprint method would have required three orders of magnitude more computing time than our new method. Supplementary materials for this article are available online.

Processing Succinct Matrices and Vectors

We study the complexity of algorithmic problems for matrices that are represented by multi-terminal decision diagrams (MTDD). These are a variant of ordered decision diagrams, where the terminal nodes are labeled with arbitrary elements of a semiring (instead of 0 and 1). A simple example shows that the product of two MTDD-represented matrices cannot be represented by an MTDD of polynomial size. To overcome this deficiency, we extended MTDDs to + MTDDs by allowing componentwise symbolic addition of variables (of the same dimension) in rules. It is shown that accessing an entry, equality checking, matrix multiplication, and other basic matrix operations can be solved in polynomial time for + MTDD-represented matrices. On the other hand, testing whether the determinant of a MTDD-represented matrix vanishes is P S P A C E-complete, and the same problem is N P-complete for + MTDD-represented diagonal matrices. Computing a specific entry in a product of MTDD-represented matrices is # P-complete.

Multi-core-CPU and GPU-accelerated radiative transfer models based on the discrete ordinate method

The operational processing of remote sensing data usually requires high-performance radiative transfer model (RTM) simulations. To date, multi-core CPUs and also Graphical Processing Units (GPUs) have been used for highly intensive parallel computations. In this paper, we have compared multi-core and GPU implementations of an RTM based on the discrete ordinate solution method. To implement GPUs, the original CPU code has been redesigned using the C-oriented Compute Unified Device Architecture (CUDA) developed by NVIDIA.GPU memory management is a crucial issue regarding the performance. To cope with limitations of GPU registers, we have adapted an RTM based on the matrix operator technique together with the interaction principle for multilayer atmospheric systems. The speed-up of such an implementation depends on the number of discrete ordinates used in the RTM. To reduce the CPU/GPU communication overhead, we have exploited the asynchronous data transfer between host and device. To obtain optimal performance, we have also used overlapping of CPU and GPU computations by distributing the workload between them.With GPUs, we have achieved a 20x–40x speed-up for the multi-stream RTM, and 50x speed-up for the two-stream RTM with respect to the original single-threaded CPU codes. Based on these performance tests, an optimal workload distribution scheme between GPU and CPU is proposed. Additionally, CPU/GPU benchmark tests regarding basic matrix operations are given. Finally, we discuss the performance obtained with the multi-core-CPU and GPU implementations of the RTM.

Matrix-Based Algorithm for Integrating Inheritance Relations of Access Rights for Policy Generation

This paper presents a matrix-based algorithm for integrating inheritance relations of access rights for generating integrated access control policies which unify management of various access control systems. Inheritance relations of access rights are found in subject, resource, and action categories. Our algorithm first integrates inheritance relations in each category, and next, integrates inheritance relations of all categories. It is shown that these operations can be carried out by basic matrix operations. This enables us to implement the integration algorithm very easily.

On solving Diophantine equations by real matrix manipulation

This note presents simple algorithms for obtaining the solutions of the Diophantine equation. Our methods can produce classes of all solutions with lower degree than a specified number. The previous algorithms involve some troublesome computations, e.g., the calculation of both the controllability indexes and the observability indexes or the solution of a pole assignment problem, etc. Our contribution is that our algorithm requires only basic matrix operations such as addition, subtraction, multiplication, and inversion of given real matrices. In addition, by solving simple linear equations, the class of all minimum degree solutions can be given. Therefore the computational efforts are reduced compared with previous algorithms. >

Efficiency Indices for Second‐Order Designs

AbstractDesign efficiencies are studied for inferences regarding the coefficients of second‐order models. Subspaces of parameters are identified wherein each of two designs dominates the other locally, or the two designs are equally efficient. Both Fisher efficiency in estimation and Pitman efficiency in hypothesis testing are considered. Eight small second‐order designs in k=3 regressor variables are screened using a four‐part efficiency index. Of these, the central composite, Box‐Behnken, small composite, and hybrid H310 and H311 B designs are found to be generally superior and roughly comparable. Detailed comparisons among these five designs are then undertaken. This work provides further tools for use in design evaluation, requiring only basic matrix operations independently of experimental data.

PROFIL/BIAS—A fast interval library

The interval data type is currently not supported in common programming languages. Therefore the implementation of algorithms using interval arithmetic requires special programming environments or at least special libraries. In this paper we present the C++ class library PROFIL which provides a user friendly environment for implementing interval algorithms. The main goals in the design of PROFIL were speed and portability. Therefore all interval operations in PROFIL use BIAS (Basic Interval Arithmetic Subroutines) [16]. BIAS defines a concise and portable interface for the basic scalar, vector, and matrix operations. The interface is independent of a specific interval representation or computation but permits machine specific and fast implementations. Based on this general specification we present an implementation in C using a lower/upper bound representation of intervals and directed roundings. By using few assembler instructions for switching the rounding modes and avoiding sign tests and rounding mode switches wherever possible, the computational costs of the interval operations were reduced significantly. This is especially important for RISC machines, where floating point instructions can be executed in few machine cycles. Comparisons with other interval arithmetic packages show an improvement in speed of about one order of magnitude.