Jordan Elimination Research Articles

Depth-Optimized Quantum Circuit of Gauss–Jordan Elimination

Quantum computers have the capacity to solve certain complex problems more efficiently than classical computers. To fully leverage these quantum advantages, adapting classical arithmetic for quantum systems in a circuit level is essential. In this paper, we introduce a depth-optimized quantum circuit of Gauss–Jordan elimination for matrices in binary. This quantum circuit is a crucial module for accelerating Information Set Decoding (ISD) using Grover’s algorithm. ISD is a cryptographic technique used in analyzing code-based cryptographic algorithms. When combined with Grover’s search, it achieves a square root reduction in complexity. The proposed method emphasizes the potential for parallelization in the quantum circuit implementation of Gauss–Jordan elimination. We allocate additional ancilla qubits to enable parallel operations within the target matrix and further reuse these ancilla qubits to minimize overhead from our additional allocation. The proposed quantum circuit for Gauss–Jordan elimination achieves the lowest Toffoli depth compared to the-state-of-art previous works.

Jordan Eliminations in Force Analysis When Changing Beam Structural Design

Purpose: The aim of the work is to study changes in forces in cross sections of a statically indeterminate beam when changing the structural design.Methodology: Jordan elimination is used to solve the system of equations. Jordan elimination used in the force analysis is considered on the example of the formation of a statically indeterminate beam by connecting two previously calculated statically indeterminate beams into one.Research findings: The data obtained for two separate statically indeterminate beams and Jordan elimination allow to calculate the forces and displacements for a statically indeterminate beam obtained by connecting two separate beams without the need to solve a new system of resolving equations.

NUMERICAL ANALYSIS OF RICE GROWTH PHASES IN KARAWANG USING THE GAUSS-JORDAN ELIMINATION METHOD

Indonesia's growth cannot be separated from Indonesia's economic growth. The agricultural sector has a massive and significant impact on Indonesia's economic growth. One of the agricultural products that’s important is rice. Rice is a seasonal crop that is very necessary for human survival. Karawang, one of the districts in West Java Province, which is known as Lumbung Padi City, is the location of the main topic in making this journal. In fact, based on BPS data, the area harvested for paddy fields in Karawang from 2010-2015 was found not to have increased regularly from year to year. This is interesting to discuss because the rice harvest area is expected to increase regularly to support Indonesia's economic growth. Of course, this is because the rice harvest area will affect the amount of rice produced in the rice fields. By ignoring other internal and external factors of rice growth, this journal will only focus on rice harvest area based on rice growth phases (vegetative 1, vegetative 2, and generative) and accumulated harvest area using the Gauss Jordan elimination method. Calculations are carried out to determine the possibility of farmers to predict the maximum amount of rice production and avoid losses in purchasing materials to support rice growth

Improving the Efficiency of Quantum Circuits for Information Set Decoding

Code-based cryptosystems are a promising option for Post-Quantum Cryptography, as neither classical nor quantum algorithms provide polynomial time solvers for their underlying hard problem. Indeed, to provide sound alternatives to lattice-based cryptosystems, U.S. National Institute of Standards and Technology (NIST) advanced all round 3 code-based cryptosystems to round 4 of its Post-Quantum standardization initiative. We present a complete implementation of a quantum circuit based on the Information Set Decoding (ISD) strategy, the best known one against code-based cryptosystems, providing quantitative measures for the security margin achieved with respect to the quantum-accelerated key recovery on AES, targeting both the current state-of-the-art approach and the NIST estimates. Our work improves the state-of-the-art, reducing the circuit depth by 2 19 to 2 30 for all the parameters of the NIST selected cryptosystems, mainly due to an improved quantum Gauss–Jordan elimination circuit with respect to previous proposals. We show how our Prange’s-based quantum ISD circuit reduces the security margin with respect to its classical counterpart. Finally, we address the concern brought forward in the latest NIST report on the parameters choice for the McEliece cryptosystem, showing that its parameter choice yields a computational effort slightly below the required target level.

Assessment of robustness of hinged-bar systems

Typically in structural design, foreseeable loads are assumed in a dimensioning exercise. Structures can, however, be exposed to largely unforeseeable events such as intense environmental phenomena, accidents, malicious acts, and planning or execution errors. This circumstance determines the interest in the problem of structural robustness, which has been the subject of many recent works.
 This paper focuses on methods for assessing the robustness of hinged bar systems, considering truss structures as an example. They are the simplest in terms of computation, but make it possible to fully illustrate the proposed approach.
 First, the differences between progressive collapse (description of the process) and the disproportionate propagation of local failures (description of the state) are analyzed. The generalizing nature of the concept of robustness and its differences from the concept of invulnerability are pointed out.
 The paper considers the problem of measuring robustness. The known quantitative estimates of robustness are analyzed focusing on estimates that are invariant with respect to the stress state, as more general ones. The paper considers estimates that use such properties of the stiffness matrix as the condition number, or based on a comparison of the determinants of the original and changed stiffness matrices. It is pointed out that the degree of static indeterminacy can serve only as a necessary, but insufficient measure of robustness.
 The paper considers a well-known method of robustness assessment using a redundancy matrix determined by the forces that must be applied to assemble the system from elements with the length different from the design one. This method is opposed to the use of a projection matrix, the main diagonal elements of which indicate the degree of participation of the bars in ensuring robustness. The main properties of the idempotent projection matrix are considered. The paper illustrates the possibility of recalculating the projection matrix for the changed system with the help of the Jordan elimination step. A simple example demonstrates assembling and changing the projection matrix.
 In addition to the failure of the bar, the case of its damage (partial failure) is also considered, it is shown how it affects the change in the projector and the redistribution of internal forces.

Fast asymmetric encryption and decryption of SimpleMatrix scheme for Internet of Things

Asymmetric cryptography plays an essential role in many areas, including cloud computing, big data, blockchain, and the Internet of Things (IoT). However, most of them are based on the difficulty of factorizing large numbers or discrete logarithm problems, which are not secure to quantum computer attacks. SimpleMatrix is a new multivariate encryption scheme based on simple matrix multiplications, which can resist quantum computer attacks. Because of the low speed and demands of large finite fields, SimpleMatrix is limited in applications that use small finite fields. As a result, it is critical to improve the efficiency of SimpleMatrix to make its applications broader. In this paper, we speed up the encryption and decryption of SimpleMatrix by building efficient small finite field arithmetics based on Field-Programmable Gate Arrays (FPGAs) technology. We propose a fast architecture for encryption and decryption of SimpleMatrix based on table look-up based composite field multiplications and inversions and fast Gauss–Jordan elimination for solving systems of linear equations in a composite field. We test and verify the hardware architecture of SimpleMatrix on an FPGA, and the experimental results confirm our estimates and comparisons show that our design is much faster than other implementations. Thus, the hardware architecture can be used in FPGA-based systems of cloud computing, IoT, etc., for accelerating encryption and decryption.

Fine‐grain task‐parallel algorithms for matrix factorizations and inversion on many‐threaded CPUs

AbstractWe extend a two‐level task partitioning previously applied to the inversion of dense matrices via Gauss–Jordan elimination to the more challenging QR factorization as well as the initial orthogonal reduction to band form found in the singular value decomposition. Our new task‐parallel algorithms leverage the tasking mechanism currently available in OpenMP to exploit “nested” task parallelism, with a first outer level that operates on matrix panels and a second inner level that processes the matrix either by ‐panels or by tiles, in order to expose a large number of independent tasks. We present a detailed performance analysis, including execution traces, which shows that the two‐level refinement into fine grain tasks allows for an improved load balancing and delivers high performance on current general‐purpose many‐core processors (CPUs) from Intel and AMD.

A solver of single-agent stochastic puzzle: A case study with Minesweeper

People have enjoyed solving puzzles for decades because of the challenge and the satisfaction derived from solving problems. However, although previous researches focused on the puzzle’s complexity, strategy, and solving automatically, few studies worked on sorting out puzzles from a solvability way related to the stochastic elements among the solving process. The contribution of the study is twofold. Firstly, a single-agent stochastic puzzle definition is established via Minesweeper testbed, a well-known puzzle synonymous with Microsoft Windows. Secondly, this study proposes an artificial intelligence (AI) solver based on the obtained information on the board, called the ‘PAFG’ strategy, which stands for the primary reasoning, the advanced reasoning, the first action strategy, and the guessing strategy. The first two strategies take advantage of knowledge-based rules and linear system transformation (Gauss–Jordan elimination algorithms) to determine the probability of making a move independently. The last two strategies explore the beginning and ways to determine hidden puzzle states to enhance the winning rate of the AI solver. The experimental simulation of various configurations and the AI solver with PAFG strategy yielded a high-level winning rate of 96.4%, 86.3%, and 45.6% for the 9×9|10, 16×16|40, and 16×30|99 Minesweeper board configuration, which is comparable to the state of the art study. Thus, such an AI solver could contribute to classifying single-agent stochastic puzzles and establishing the boundary of the puzzle-solving and game-playing paradigm.

A multi-target on-line ranging method based on matrix sparsification and a division-free Gauss–Jordan solver

To improve the on-line laser-ranging performance of light detection and ranging (LiDAR), the run time and resource consumption of the commonly used Gauss–Newton method need to be considered under the condition of high-frequency laser echo scattered from multiple targets. Limited by insufficient hardware processing units and the requirement of increased measurement speed, a multi-target laser ranging based on matrix sparsification and a division-free Gauss–Jordan solver implemented on a field programmable gate array is proposed. A Jacobian matrix sparsification method is used to reduce the number of multiplication operations in the coefficient matrix and constant vector calculations, thereby reducing hardware resources and time costs. In order to increase the hardware implementation efficiency, a division-free Gauss–Jordan elimination method is used to quickly solve linear equations. The Vivado-based simulation results show that the amount of hardware resources, such as look-up tables, flip-flops and digital signal processor (DSP) are reduced by 34.5%, 35.7% and 27.3%, respectively, and the time cost is reduced by 3 μs (750 clock cycle @250 MHz) in the case of five iterations, satisfying the real-time requirements for a measurement rate of 45.45 kHz at most. Experimental results show that the proposed method maintains comparable performance of point extraction ability, ranging precision and accuracy, which are 100.0%, 1.78 cm and 0.62 cm@35.64 dB for two-target laser ranging and 98.03% 1.89 cm and 0.60 cm@35.91 dB for three-target laser ranging. For the field of multi-target real-time ranging full-waveform LiDAR, it is a promising method due to its low time cost and resource consumption while ensuring ranging performance.

Jordan eliminations in bar system analysis with changes in design model

The article discusses the use of the Jordan elimination method for solving the system of resolvent equations when analyzing the bar systems. The use of the Jordan elimination method makes it possible to determine the forces in cross-sections of bars and displacements of the system units in the case of changes in the design model of the system without the solution of new resolvent equations at each change. Changes in the design model indicate the introduction or removal of the support or internal connections, changes in the stiffness parameters of elements of statically indeterminate systems, and others. The Jordan elimination method is used for a simple statically indeterminate beam, which is a special case of the bar system.

Gauss–Jordan elimination-based image tampering detection and self-recovery

This paper proposes a novel Gauss–Jordan elimination-based image tampering detection and self-recovery scheme, aiming at dealing with the problem of malicious tampering on digital images. To deal with the copy–move tampering which is challenging because the tampered region may contain the watermark information, we propose the Improved Check Bits Generation algorithm during watermark generation, to generate the check bits for tampering detection. Meanwhile, the recovery bits are reconstructed according to the fundamental of Gauss–Jordan Elimination, for purpose of image contents self-recovery. To improve the accuracy of detection and the quality of recovered images, we propose the Morphological Processing-Based Enhancement method and the Edge Extension preprocessing respectively during and after the tampering detection Finally, the Gauss–JordanElimination-Based Self-Recovery method is proposed to recover the damaged content mathematically on basis of the detected results. By employing the unchanged recovery bits which are embedded in the non-tampered region, the failure in recovery caused by the damaged recovery bits can be completely avoided. A large number of experiments have been conducted to show the very good performance of the proposed scheme. The precision, recall, and F1 score are calculated for evaluation of tampering detection, while the PSNR values are calculated for evaluation of image recovery. The comparisons with the state-of-the-art methods show that the proposed scheme shows the superiorities in terms of imperceptibility, security and recovery capability. The experimental result indicates the average PSNR of recovered image is 44.415dB.

Methods of Gauss–Jordan elimination to compute core inverse and dual core inverse

In this paper, two formulas, which were studied by Wang and Liu (2015) and Ma and Li (2019), respectively, for the core inverse, are simplified. Then two methods for computing the core inverse and dual core inverse are investigated through Gauss–Jordan elimination on the two appropriate block partitioned matrices. The corresponding algorithms are also summarized. The computational complexities of the these two algorithms are analysed in detail. In the end, some numerical examples are presented to demonstrate the efficiency of the two algorithms.

Analysis of acoustic sensor placement for PD location in power transformer

Partial discharge PD is an abnormal activity that occurs in high-voltage components, such as power cables, switchgear, machines, and power transformers. Such activity needs to be diagnosed for the equipment to last longer as PD could harm the insulation and potentially lead to asset destruction from time to time. Moving one or more externally mounted acoustic sensors to different locations on the transformer tank is commonly used in order to detect and locate PD signal occurring in the power transformer. However, this procedure may lead to less accuracy in PD identification. Therefore, this research paper presents an analysis of acoustic sensor placement based on time of arrival TOA technique for PD location in a power transformer. The detection and location can be determined by permanently installing the acoustic sensor to provide valuable data in an early stage of occurrence for online condition PD monitoring. Several methods are available for the detection of PD signal, whereby one of the best choices is via acoustic emission AE . PD creates an ultrasonic signal used for PD detection. This paper proposes the possible placement of AE sensors to be mounted on the power transformer wall based on ideal and static PD signals. The sensors were placed in order to capture the PD signal without any disturbance signal from inside or outside the tank. The time for the signal for the first approach for each sensor is recorded to estimate the PD location using the TOA technique. A comparison between the least square method LSM and Gauss--Jordan elimination GJE for the TOA technique was analyzed to differentiate the resulting performance. This research utilized three different PD sources to apply the performance analysis on PD locations, while five cases were proposed to represent the five different placements of four sensors for the analysis. This research ultimately suggests that sensors be placed and randomly mounted on the four sides of the transformer tank, with one sensor allocated to one side. Among all five cases, Case 1 and Case 5 yielded a displacement error DE less than others, while between these two cases, Case 5 gave the lowest DE. The findings were recorded based on LSM and GJE methods used to differentiate the resulting performance.

Solving fully fuzzy linear systems by Gauss Jordan Elimination Method

In this paper, we discuss fully fuzzy linear systems with triangular fuzzy numbers. A Gauss Jordan Elimination Method is proposed for solving fully fuzzy linear systems (FFLS). We used elementary row operations augmented matrices of crisp linear system of equation to arrive the row reduced form. The method in detail is discussed and illustrated by a numerical example.

Using Gauss - Jordan elimination method with The Application of Android for Solving Linear Equations

Problems involving mathematical models appear in many scientific disciplines. Complex mathematical models sometimes cannot be solved by analytic methods using standard algebraic formulas. Computers play a major role in the development of the field of numerical methods because the calculation uses numerical methods in the form of arithmetic operations, the number of arithmetic operations is very large and repetitive, so manual calculations are often tedious and errors occur. This study aims to develop software solutions for linear equations by implementing the Gauss-Jordan elimination(GJ-elimination) method, building software for linear equations carried out through five stages, namely: (1) System Modeling (2) Simplification of Models, (3) Numerical Methods and algorithms, (4) programming languages using The Android Studio and (5) Simulation programs. Overall regarding content, proper software that can be used by students and lecturers in implementing numerical methods because there are ways to use the application and steps to solve linear equation problems using the GJ-elimination method.

A parameter method for linear algebra and optimization with uncertainties

ABSTRACTSystems of linear equations are studied with uncertainties in the coefficients, often representing imprecise data. Such imprecisions are seen as orders of magnitude. The orders of magnitude are not functionally modelled in terms of or , but within nonstandard analysis as neutrices and external numbers, convex external sets of nonstandard real numbers.In this setting, linear systems with external coefficients are solved by a parameter method. This method assigns a parameter to each imprecision and specifies its range. Then we obtain an ordinary linear system and solve by common methods. At the end we substitute each parameter by its range. We present general solution formulas, for an arbitrary number of equations and variables. We illustrate with a concrete system that the parameter method gives more precise results, under more general conditions, than dealing with propagation of errors in direct Gauss–Jordan elimination or the Cramer rule.We apply the results to linear optimization with uncertainties. In a general setting, nearly optimal solutions are obtained and the shape of the domains of validity of the nearly optimal solutions.

A Teaching Model for Undergraduate Students

Mathematics, along with the need for logic and thinking, is becoming more important in many fields. Therefore, many universities in Korea have opened and operated a college basic math course to improve basic math skills for freshmen in science and engineering. The new generation of digital generation is creative, familiar with cooperation and active. They are already rapidly changing and ready for new education, and education needs a new paradigm to evolve. Flipped Learning is being suggested which is well known as a teaching method which lets students learn the contents they will learn in advance through the advance online video and have a discussion through the team interaction in the main class for them to solve the assignment through the cooperation in a self-initiated way. In this study, we have taken a merit of flipped learning, made a model of Gauss Jordan elimination method in matrix that students cannot easily understand in the lecture. Here, we will introduce a teaching model that combines flipping learning and existing lecture methods.

Progressive Multicore RLNC Decoding With Online DAG Scheduling

A complete generation of packets coded with Random Linear Network Coding (RLNC) can be quickly decoded on a multicore system by scheduling the involved matrix block operations in parallel with an offline (pre-recorded) directed acyclic graph (DAG). The waiting for a complete generation of packets can be avoided with progressive RLNC decoding that commences the decoding (and can decode some packets) before all packets in a generation have been received. This article develops and evaluates a novel progressive RLNC decoding strategy based on the principle of DAG scheduling of parallel matrix block operations. The novel strategy involves helper matrices for conducting the Gauss Jordan elimination based on rows of blocks of matrix elements. The matrix block computations are dynamically scheduled by an online DAG which permits branching, e.g., to skip unnecessary matrix block operations. The throughput and delay of the novel progressive RLNC decoding strategy are evaluated with experiments on two heterogeneous multicore processor boards. The novel progressive RLNC decoding achieves throughput levels on par with state-of-the-art non-progressive (full-generation) RLNC decoding and achieves three times higher throughput than the fastest (highest-throughput) known progressive RLNC decoder for small generation sizes and short data packets. Also, our progressive RLNC decoding greatly reduces receiver delays for moderate to large generation sizes; the delay reductions are particularly pronounced when a low-delay RLNC version is employed (e.g., reduction to one tenth of the non-progressive decoding delay for a generation size of 256 packets).

A Novel Aesthetic QR Code Algorithm Based on Hybrid Basis Vector Matrices

Recently, more and more research has focused on the beautification technology of QR (Quick Response) codes. In this paper, a novel algorithm based on the XOR (exclusive OR) mechanism of hybrid basis vector matrices and a background image synthetic strategy is proposed. The hybrid basis vector matrices include the reverse basis vector matrix (RBVM) and positive basis vector matrix (PBVM). Firstly, the RBVM and PBVM are obtained by the Gauss–Jordan elimination method, according to the characteristics of the RS code. Secondly, the modification of the parity area of the QR code can be applied with the XOR operation of the RBVM, and the XOR operation of the PBVM is used to change the data area of the QR code. So, the QR code can be modified to be very close to the background image without impacting the error-correction ability. Finally, in order to further decrease the difference between the QR code and the background image, a new synthesis strategy is adopted in order to obtain a better aesthetic effect. The experimental results show that it obtains a better visual effect without the sacrificing recognition rate.

Mathematical Model for Traffic Flow

Every year countless hours are lost in traffic jams. When the density of traffic is sufficiently high small disturbances in vehicle’s accelerations can cause phantom traffic jams. We can relate the traffic flow to mathematics and physics like that of liquids and gases. This paper presents mathematical model for phantom jams and Gauss Jordan elimination for traffic flow.