Arithmetic Algorithms Research Articles

Efficient Ternary Logic Circuits Optimized by Ternary Arithmetic Algorithms

Efficient Ternary Logic Circuits Optimized by Ternary Arithmetic Algorithms

Lévy arithmetic optimization for energy Management of Solar Wind Microgrid with multiple diesel generators for off-grid communities

This paper presents an improved optimization algorithm for the energy management of a renewable energy solar/wind microgrid with multiple diesel generators applied to off-grid remote communities. The main objective aims to solve the economic emission dispatch problem with a price penalty factor to minimize the energy cost and the emission level. An enhanced metaheuristic optimization algorithm, Lévy arithmetic algorithm, is applied to improve the searchability for optimal solution compared to the conventional arithmetic algorithm. The Lévy arithmetic method is used for the management of the microgrid and compared to other metaheuristic optimization algorithms for the same application. Comparative analysis demonstrates good cost savings using the Lévy arithmetic algorithm, compared to other optimization algorithms such as the arithmetic algorithm, crow search algorithm, hybrid modified grey wolf algorithm, interior search algorithm, cuckoo search algorithm, particle swarm algorithm, colony algorithm, and genetic algorithm.

A comprehensive exploration of approximate DNN models with a novel floating-point simulation framework

This paper introduces TorchAxf11https://github.com/knuscent/TorchAxf., a framework for fast simulation of diverse approximate deep neural network (DNN) models, including spiking neural networks (SNNs). The proposed framework utilizes various approximate adders and multipliers, supports industrial standard reduced precision floating-point formats, such as bfloat16, and accommodates user-customized precision representations. Leveraging GPU acceleration on the PyTorch framework, TorchAxf accelerates approximate DNN training and inference. In addition, it allows seamless integration of arbitrary approximate arithmetic algorithms with C/C++ behavioral models to emulate approximate DNN hardware accelerators.We utilize the proposed TorchAxf framework to assess twelve popular DNN models under approximate multiply-and-accumulate (MAC) operations. Through comprehensive experiments, we determine the suitable degree of floating-point arithmetic approximation for these DNN models without significant accuracy loss and offer the optimal reduced precision formats for each DNN model. Additionally, we demonstrate that approximate-aware re-training can rectify errors and enhance pre-trained DNN models under reduced precision formats. Furthermore, TorchAxf, operating on GPU, remarkably reduces simulation time for complex DNN models using approximate arithmetic by up to 131.38× compared to the baseline optimized CPU implementation. Finally, we compare the proposed framework with state-of-the-art frameworks to highlight its superiority.

The Design of FIR Filter Based on improved DA Algorithm and its FPGA implementation: REVIEW

The Design of FIR Filter Based on improved DA Algorithm and its FPGA implementation: REVIEW

Quantum Circuit Optimization of Arithmetic Circuits using ZX Calculus

Quantum computing is an emerging technology in which quantum mechanical properties are suitably utilized to perform certain compute-intensive operations faster than classical computers. Quantum algorithms are designed as a combination of quantum circuits that each require a large number of quantum gates, which is a challenge considering the limited number of qubit resources available in quantum computing systems. Our work proposes a technique to optimize quantum arithmetic algorithms by reducing the hardware resources and the number of qubits based on ZX calculus. We have utilized ZX calculus rewrite rules for the optimization of fault-tolerant quantum multiplier circuits where we are able to achieve a significant reduction in the number of ancilla bits and T-gates as compared to the originally required numbers to achieve fault-tolerance. Our work is the first step in the series of arithmetic circuit optimization using graphicalrewrite tools and it pavesthe way for advancing the optimization of various complex quantum circuits and establishing the potential for new applications of the same.

Optimal controller design for reactor core power stabilization in a pressurized water reactor: Applications of gold rush algorithm.

Nuclear energy (NE) is seen as a reliable choice for ensuring the security of the world's energy supply, and it has only lately begun to be advocated as a strategy for reducing climate change in order to meet low-carbon energy transition goals. To achieve flexible operation across a wide operating range when it participates in peak regulation in the power systems, the pressurised water reactor (PWR) NE systems must overcome the nonlinearity problem induced by the substantial variation. In light of this viewpoint, the objective of this work is to evaluate the reactor core (main component) of the NE system via different recent optimization techniques. The PWR, which is the most common form, is the reactor under investigation. For controlling the movement of control rods that correspond with reactivity for power regulation the PWR, PID controller is employed. This study presents a dynamic model of the PWR, which includes the reactor core, the upper and lower plenums, and the piping that connects the reactor core to the steam alternator is analyzed and investigated. The PWR dynamic model is controlled by a PID controller optimized by the gold rush optimizer (GRO) built on the integration of the time-weighted square error performance indicator. Additionally, to exhibit the efficacy of the presented GRO, the dragonfly approach, Arithmetic algorithm, and planet optimization algorithm are used to adjust the PID controller parameters. Furthermore, a comparison among the optimized PID gains with the applied algorithms shows great accuracy, efficacy, and effectiveness of the proposed GRO. MATLAB\ Simulink program is used to model and simulate the system components and the applied algorithms. The simulation findings demonstrate that the suggested optimized PID control strategy has superior efficiency and resilience in terms of less overshoot and settling time.

Modified Arithmetic Algorithms for Optimal Power System Operation Considering Uncertainties from Intermittent Renewable Energy and Electric Vehicles

Modified Arithmetic Algorithms for Optimal Power System Operation Considering Uncertainties from Intermittent Renewable Energy and Electric Vehicles

The Archimedean Origin of Modern Positional Number Systems

A symbolic analysis of Archimedes’s periodical number system is developed, from which a natural link emerges with the modern positional number systems with zero. After the publication of Fibonacci’s Liber Abaci, the decimal Indo-Arabic positional system was the basis of the algorithmic and algebraic trend of modern mathematics, but even if zero plays a crucial role in algebra and mathematical analysis, zeroless positional systems show the same capability of producing efficient arithmetical algorithms based on operation tables over digits. The crucial role of digits is assessed, by considering a representation of numbers based on strings in lexicographic order. A new algorithm for the determination of decimal periods is presented by remarking on the cruciality of this topic in number theory. Periods of ordinal numbers and enumerations of recursive enumerability are shortly recalled. Concluding remarks are formulated about the deep relationship between numbers and information, which shed new light on a red line passing through the whole history of mathematics.

A New Hybrid Diffie-Hellman and Caesar Cipher Algorithm for Cryptography

Caesar cipher algorithm is one of the ancient algorithms. However, due to advances in technology the algorithm is now much simple to breach. This is because; each letter in the message is replaced by the same letter as specified. To raise the safety, some changes can be made. So, here we are going to use another arithmetic algorithm Diffie - Hellman's mode of changing the key to find the secret key and use simple calculations to verify data encryption. After obtaining a private shared key using the Diffie-Hellman method, depending on the operating mode with 26 keys to get the value of 26 or less, then the current character is taken with the key when the additional value of the new character. In any letter in the position of an 'x', the key first increases with an 'x' and is adjusted to get the encrypted letter. Therefore, 2 messages are repeated 2 times and the third letter has 3. This increases safety. This technology can be

Lévy Arithmetic Algorithm: An enhanced metaheuristic algorithm and its application to engineering optimization

Lévy Arithmetic Algorithm is an upgraded metaheuristic optimization approach proposed to enhance the Arithmetic Optimization Algorithm using the Lévy random step. Arithmetic Optimization Algorithm addresses diverse optimization problems by employing arithmetic operators. However, its linear search capability might hinder attaining optimal solutions, which leads to stagnation. In this study, the Lévy Arithmetic Algorithm is introduced by combining the Arithmetic Optimization Algorithm and the Lévy random step to enhance search capabilities and minimize computational demands for improved outcomes. The evaluation encompassed ten CEC2019 benchmark functions, four established real-world engineering problems, and the Economic Load Dispatch of microgrids with renewable energy integration. Using evaluation metrics such as standard deviation and mean, along with hypothesis tests, the study conducts a thorough comparison between the performance of the proposed algorithm and that of the conventional Arithmetic Optimization Algorithm. The results show that the Lévy Arithmetic Algorithm achieves optimization with a minimized number of evaluations in terms of standard deviation and mean, compared to the Arithmetic Optimization Algorithm. Additionally, the proposed method was compared with various well-known and recent metaheuristics algorithms, and the Lévy Arithmetic Algorithm consistently demonstrates superior performance, especially in contrast to the Arithmetic Optimization Algorithm.

How to use a CAS for Hardware Design Automation

With the rise of new technologies, the demand for efficient cryptographic hardware is rising. This work is focusing on finite field based cryptography. To explore the vast design space obtained from possible finite field parameters, automated generation of hardware submodules using designer-specified algorithms for finite field arithmetic is needed. This work presents a hardware design automation framework exploiting symbolic computation capabilities of GAP to generate the expressions, needed for hardware implementations, on-the-fly, and compiles them into synthesizable datapaths, test-vectors, and testbences.

Elements of Algorithmic Thinking in the Teaching of School Geometry through the Application of Geometric Problems

Algorithmic thinking and the creation of algorithms have traditionally been associated with mathematics. It is based on the general perception of an algorithm as a logically unambiguous and precise prescription for performing a certain set of operations, through which we reach a result in real time in a finite number of steps. There are well-known examples from history, such as the division algorithm used by ancient Babylonian mathematicians, Eratosthenes algorithm for finding prime numbers, Euclid’s algorithm for finding the greatest common divisor of two numbers, and cryptographic algorithm for coding and breaking, invented by Arabic mathematicians in the 9th century. Although the usage of algorithms and the development of algorithmic thinking currently fall within the domain of computer science, algorithms still play a role in mathematics and its teaching today. Contemporary mathematics, and especially its teaching in schools of all grades, prefers specific algorithms in arithmetic, algebra, and calculus. For example, operations with numbers, modifications of algebraic expressions, and derivation of functions. Teaching geometry in schools involves solving a variety of problems, many of which are presented as word problems. Algorithmization of school geometric tasks is therefore hardly visible and possible at first glance. However, there are ways to solve examples of a certain kind and to establish a characteristic and common algorithmic procedure for them. Algorithmic thinking in geometry and the application of algorithms in the teaching of thematic parts of school geometry are specific issue that we deal with in this study. We will focus on a detailed analysis of the possibilities of developing algorithmic thinking in school geometry and the algorithmization of geometric tasks.

Integer Arithmetic Algorithm for Fundamental Frequency Identification of Oceanic Currents.

Underwater sensor networks play a crucial role in collecting valuable data to monitor offshore aquaculture infrastructures. The number of deployed devices not only impacts the bandwidth for a highly constrained communication environment, but also the cost of the sensor network. On the other hand, industrial and literature current meters work as raw data loggers, and most of the calculations to determine the fundamental frequencies are performed offline on a desktop computer or in the cloud. Belonging to the edge computing research area, this paper presents an algorithm to extract the fundamental frequencies of water currents in an underwater sensor network deployed in offshore aquaculture infrastructures. The target sensor node is based on a commercial ultra-low-power microcontroller. The proposed fundamental frequency identification algorithm only requires the use of an integer arithmetic unit. Our approach exploits the mathematical properties of the finite impulse response (FIR) filtering in the integer domain. The design and implementation of the presented algorithm are discussed in detail in terms of FIR tuning/coefficient selection, memory usage and variable domain for its mathematical formulation aimed at reducing the computational effort required. The approach is validated using a shallow water current model and real-world raw data from an offshore aquaculture infrastructure. The extracted frequencies have a maximum error below a 4%.

AritPIM: High-Throughput In-Memory Arithmetic

Digital processing-in-memory (PIM) architectures are rapidly emerging to overcome the memory-wall bottleneck by integrating logic within memory elements. Such architectures provide vast computational power within the memory itself in the form of parallel bitwise logic operations. We develop novel <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">algorithmic</i> techniques for PIM that, combined with new perspectives on computer arithmetic, extend this bitwise parallelism to the four fundamental arithmetic operations (addition, subtraction, multiplication, and division), for both fixed-point and floating-point numbers, and using both bit-serial and bit-parallel approaches. We propose a state-of-the-art suite of arithmetic algorithms, demonstrating the first algorithm in the literature of digital PIM for a majority of cases – including cases previously considered impossible for digital PIM, such as floating-point addition. Through a case study on memristive PIM, we compare the proposed algorithms to an NVIDIA RTX 3070 GPU and demonstrate significant throughput and energy improvements.

D0L-System Inference from a Single Sequence with a Genetic Algorithm

In this paper, we proposed a new method for image-based grammatical inference of deterministic, context-free L-systems (D0L systems) from a single sequence. This approach is characterized by first parsing an input image into a sequence of symbols and then, using a genetic algorithm, attempting to infer a grammar that can generate this sequence. This technique has been tested using our test suite and compared to similar algorithms, showing promising results, including solving the problem for systems with more rules than in existing approaches. The tests show that it performs better than similar heuristic methods and can handle the same cases as arithmetic algorithms.

Design Space Exploration for Efficient Quantum Most-Significant Digit-First Arithmetic

Quantum computing has been considered as an emerging approach in addressing problems which are not easily solvable using classical computers. In parallel to the physical implementation of quantum processors, quantum algorithms have been actively developed for real-life applications to show quantum advantages, many of which benefit from quantum arithmetic algorithms and their efficient implementations. As one of the most important operations, quantum addition has been adopted in Shor's algorithm and quantum linear algebra algorithms. Although various least-significant digit-first quantum adders have been introduced in previous work, interest in investigating the efficient implementation of most-significant digit-first addition is growing. In this work, we propose a novel design method for most-significant digit-first addition with several quantum circuit optimisations to reduce the number of quantum bits ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i.e.</i> qubits), quantum gates, and circuit depth. An open-source library of different arithmetic operators based on our proposed method is presented, where all circuits are implemented on IBM Qiskit SDK. Extensive experiments demonstrate that our proposed design, together with the optimisation techniques, reduces T-depth by up-to 4.0×, T-count by 3.5×, and qubit consumption by 1.2×.

Optimized Distributive Arithmetic-based Hardware Accelerator for Dual Tree Complex Wavelet Transform Computation

Hardware architectures for fast computation of complex wavelet transforms for image processing require optimized design approaches. The Dual Tree Complex Wavelet Transform (DTCWT) is twice as complex as the Discrete Wavelet Transform (DWT) and was designed while considering the distributive arithmetic (DA) algorithm, which is customized for the design of a 10-tap filter architecture. Redundancy in the filter coefficients was considered in optimizing the DA partial products, reducing the area resources by 97.65%. The reduced architecture was modeled in Verilog HDL and implemented on a Xilinx FPGA. The operating frequency is 312 MHz, and the power dissipation is less than 1 W. The proposed model is suitable for high-speed computation of DTCWT sub-bands on an FPGA platform.

Design and Implementation of an Area and Power-efficient Reconfigurable Hearing Aid using Interpolated Sub-band Distribution Technique

A hearing aid is a compensatory device that helps in overcoming various hearing disabilities. Since different people possess different hearing problems and the requirement gets changed over time, it is necessary to design a reconfigurable hearing aid which is generic in nature such that it supports various hearing disabilities without modifying the hardware components. The objective of the paper is to implement a reconfigurable digital hearing aid which is hardware efficient and it is auto-adaptable to disabilities ranging from mild to severe intensities. Since multipliers and LUTs are power hungry elements, we have proposed a design which is multiplier less and LUT-less DA Architecture. The prototype filter is a FIR filter which is designed using LUT-less Distributed Arithmetic Algorithm that saves 64% of logic elements &amp; memory and 76% of power utilization. As it is a multiplier-less as well as LUT- less architecture, hence this can be claimed to be area and power efficient design. The delays and matching errors are within the standard limits which are accepted globally. The input audio spectrum is divided into three regions and for each region there are four different filter banks are proposed using interpolated sub-bands distribution. Xilinx System Generator is used to implement the proposed design. The proposed design requires least manual configuration for selection of filter banks for audiogram matching which also minimizes the trial-anderror method to establish the best match with the person’s audiogram.

Drinking water quality assessment based on index values incorporating WHO guidelines and Bangladesh standards

Drinking water quality is one of the most severe concerns around the globe. This study aimed to assess drinking water quality for a densely populated region, the Jashore University of Science and Technology (JUST), and the surrounding villages. Fifty samples, including some filtered and boiled groundwater samples, were collected. The widely used arithmetic algorithm, the water quality index (WQI), was used to categorize the groundwater quality as per World Health Organization (WHO) guidelines and Bangladesh (BD) standards. Here, the WQI for all samples was calculated based on the concentrations of seventeen parameters (pH, TDS, TH, Ca2+, Na+, Mg2+, Cl−, SO42−, HNO3−, K, PO43−, EC, As, Zn, Mn, Cr, and Fe). The result showed that the water quality varies laterally as well as on the sampling conditions. The calculated WQI values ranged from 16.66 to 253.50 and the average value was 91.9 for the BD standards, whereas the WHO guidelines showed WQI values between 16.74 and 605.0 with an average value of 146.90. Based on the WQI values, water samples were divided into excellent, good, poor, very poor, and unfit categories which varied from BD standards to WHO guidelines for drinking purposes. 14%, 54%, 26%, and 6% of samples fell into excellent, good, poor, and inferior categories respectively as BD standards, while these values were 38%, 22%, 14%, and 14% as WHO guidelines. The result showed that deep tube-wells (>170 ft) and shallow tube-wells (<170 ft) provide mostly good and bad quality drinking water, respectively.

Lookahead analysis in exact real arithmetic with logical methods

Program extraction from proofs can be used to obtain verified algorithms in exact real arithmetic for e.g., the signed-digit code representation. In Minlog this has been done in the past with the use of certain coinductive predicates. In a next step we want to analyze the lookahead of these extracted programs. Doing it by hand is quite cumbersome, so instead we change our definition. Instead of a coinductive predicate we use an inductive predicate for the representation of reals that already incorporates the lookahead. In this way the lookahead becomes part of the specification which is carried through all the proofs. In the end we extract programs for computations on a signed-digit representation and we can just read off the lookahead from the specification.