Study on parallel processing method of matrix multiplication--A method to calculate the N-th power of massive matrix

Parallelizing Strassen's method for matrix multiplication on distributed-memory MIMD architectures

The presence of rounding errors is frequently inevitable when performing arithmetic operations in computers due to the use of floating-point number system whose elements have finite precision. Consequently, error-free transformation algorithms are proposed as solutions, and a particular instance of these is the error-free transformation algorithm for real matrix multiplication. Based on the previously mentioned algorithm, this study proposes two error-error free transformation algorithms for complex matrix multiplication. By conducting the numerical experiment, one finds that the combination of both proposed algorithms with an accurate summation algorithm generates the most accurate results, but it also suffers from a notable increase in the computing time ratio. On the other hand, the second proposed algorithm, which is simply another form of the first one with the change of order in summation, combined with the pure floating-point addition produces results that are nearly as accurate as those generated by the former combination. Since these results are achieved with significantly smaller computing time ratio, then the study concludes that the last combination is the best method for complex matrix multiplication in terms of accuracy and computing time efficiency.

The intersection of climate change and human activities exacerbates the prominent contradiction between environmental resources and economic development. The synergistic development of water resources, economy and society, and ecological environment is particularly urgent. Therefore, accurately identifying its influencing factors and related pathways holds significant practical importance in resolving the contradiction between resources and economic development. This study constructs a system of influencing factors for the synergistic development of “water resources, economic society and ecological environment” from three dimensions, reveals the influencing relationship between indicators through the Decision Making Trial and Evaluation Laboratory (DEMATEL), determines the hierarchical structure of the influencing factors and the related paths through the Interpretive Structural Model (ISM), and classifies the influencing factors through the Matrix Multiplication Method of Cross Influence Matrix Multiplication (MICMAC). The results show that water resource factors are important cause factors and economic, social and ecological factors are essential effect factors. The factors influencing the synergistic development of “water resources, economic society and ecological environment” can be divided into four levels, of which the total water resources, precipitation and NDVI are at the bottom and are the most fundamental factors. The percentage of groundwater supply, total water supply, water resource utilization rate and energy consumption of 10,000 yuan of GDP are the deep core elements. Total grain output, water resources per capita, and water use per capita are the direct influencing factors. Finally, taking the Central Plains Urban Agglomeration as an example, selecting the identified important influencing factors and analyzing the level of synergistic development of its coupled system found that the degree of coupling coordination increased from 0.324 in 2011 to 0.978 in 2020. The water resources system showed a fluctuating change, and it is still the key to synergistic development. Analyzing the relationship between subsystems and clarifying the key factors for synergistic development of coupled systems are important for policy formulation, improving environmental governance and promoting high-quality economic development.

We present a novel heterogeneous parallel matrix multiplication algorithm that utilizes both central processing units (CPUs) and graphics processing units (GPUs) for large-scale matrices. Based on Strassen’s method, we represent matrix multiplication work as a set of matrix addition and multiplication tasks among their sub-matrices. Then, we distribute the tasks to CPUs and GPUs while considering the characteristics of the tasks and computing resources to minimize the data communication overhead and fully utilize the available computing power. To handle a large matrix efficiently with limited GPU memory, we also propose a block-based work decomposition method. We then further improve the performance of our method by exploiting the concurrent execution abilities of a heterogeneous parallel computing system. We implemented our method on five different heterogeneous systems and applied it to matrices of various sizes. Our method generally shows higher performance than the prior GPU-based matrix multiplication methods. Moreover, compared with the state-of-the-art GPU matrix multiplication library (i.e., CUBLAS), our method achieved up to 1.97 times higher performance using the same GPUs and CPU cores. In some cases, our method using a low-performance GPU (e.g., GTX 1060, 3 GB) achieved performance comparable to that of CUBLAS using a high-performance GPU (e.g., RTX 2080, 8 GB). Also, our method continually improves performance as we use more computing resources like additional CPU cores and GPUs. We could achieve such high performance because our approach fully utilized the capacities of the given heterogeneous parallel computing systems while employing the Strassen’s method, which has a lower asymptotic complexity. These results demonstrate the efficiency and robustness of our algorithm.

The paper presents exact analytical solutions of one-dimensional periodic heat conduction through an inhomogeneous slab for a certain class of thermal conductivity profiles (including linear and exponential). The exact analytical solutions for some of these profiles have been compared with those obtained by considering the slab to be made up of a number of homogeneous layers with different thermal conductivities varying from layer to layer and using the layered structure (or matrix multiplication) method. The numerical results arrived at by the layered-structure method converge rapidly (with increasing number of layers considered) to the values obtained from the exact analytical solutions. This gives confidence in the application of the layered-structure method to periodic heat conduction through inhomogeneous slabs. The numerical results have been presented in the form of elements of a 2 × 2 matrix, relating the sinusoidal steady-state temperature and heat flux on the two sides of the slab.

Matrix and vector multiplications are widely used in signal processing in operations such as FIR and IIR filtering, feature extraction and classification. Frequently, signal processing must be done in real time requiring the use of special purpose VLSI hardware. Regular structures such as systolic arrays are well suited for matrix and vector operations and are also amenable to VLSI implementation. This paper describes efficient systolic arrays for matrix and vector multiplication, at both word and bit levels, and outlines three applications relevant for signal processing: a convolver, an IIR filter and a linear classifier.

In this work, we obtain a new formula for Fibonacci’s family m-step sequences. We use our formula to find the nth term with less time complexity than the matrix multiplication method. Then, we extend our results for all linear homogeneous recurrence m-step relations with constant coefficients by using the last few terms of its corresponding Fibonacci’s family m-step sequence. As a computational number theory application, we develop a method to estimate the square roots.

This paper proposes a cellular method of matrix multiplication. The method reduces the multiplicative and additive complexities of well-known matrix multiplication algorithms by 12.5%. The computational complexities of cellular analogs of such algorithms are estimated. A fast cellular analog is presented whose multiplicative and additive complexities are equal to ?0.382n3 multiplications and ?1.147n3 additions, respectively, where n is the order of the matrices being multiplied.

This paper examines the performance issues associated with computing devices performing arithmetic operations on large matrices. One of the optimal methods for matrix multiplication is to calculate a large matrix by dividing it into blocks using the block-based method. This is achieved by multiplying matrices of different sizes using the parallel block-based method on a computer’s graphics processing unit using Compute Unified Device Architecture technology, as well as on a central processor using the Open Multi-Processing parallel library for devices without a graphics processing unit. The study examines the time-consuming task of multiplying matrices of sizes 64x64, 128x128, 512x512, 1024x1024, and 2048x2048 using a simple sequential naive method and a parallel block-based method using these parallel processing technologies. The performance of the block parallel method implemented on a graphics processing unit using Compute Unified Device Architecture technology and on a central processing unit using Open Multi-Processing technology is also compared with the existing CUDA basic linear algebra subprograms libraries for NVIDIA graphics processing units and Intel Math Kernel Library for Intel processors. The proposed approach allows the user to fully control the programming model, customize the algorithm, change the block size, and perform computations as quickly as existing libraries.

The paper presents a study of medium and coarse grain numerical computations in a new cluster-based shared memory parallel architecture oriented into implementation in "Systems on Chip" (SoC) technology. The assumed architecture is based on dynamic processor clusters, organized around shared memory modules. Fast shared data transfers between processors from different clusters are performed through communication on the fly, which is a synergy of processor switching between clusters and intra-cluster data reads on the fly. Dynamic processor clusters are implemented inside SoC modules additionally connected by a global inter-cluster network. The paper discusses speedup and parallelization efficiency of parallel matrix multiplication estimated by symbolic execution of program graphs. Simulation results are presented for algorithms with two kinds of data decomposition: recursive division of matrices into quadrants and division of matrices into stripes. In the quadrant-based method, elementary square sub-matrix multiplications are performed using the serial Strassen method and the communication on the fly is applied. The experiments reveal much higher efficiency of the proposed quadrant-based matrix multiplication method than that of the "stripe" method, considered very efficient in conventional parallel shared memory systems

Parallel direct method for linear equations

Matrix multiplication (MM) is a fundamental operation in various scientific and engineering computations, as well as in artificial intelligence algorithms. Efficient implementation of MM is crucial for speeding up numerous applications. Photonics presents an opportunity for efficient acceleration of dense matrix computation, owing to its intrinsic advantages, such as huge parallelism, low latency, and low power consumption. However, most optical matrix computing architectures have been limited to realizing single-channel vector-matrix multiplication or using complex configurations to expand the number of channels, which does not fully exploit the parallelism of optics. In this study, we propose a novel, to the best of our knowledge, scheme for the implementation of large-scale two-dimensional optical MM with truly massive parallelism based on a specially designed Dammann grating. We demonstrate a sequence of MMs of 50 pairs of randomly generated 4 × 8 and 8 × 4 matrices in our proof-of-principle experiment. The results indicate that the mean relative error is approximately 0.048, thereby demonstrating optical robustness and high accuracy.

For many scientific applications, dense matrix multiplication is one of the most important and computation intensive linear algebra operations. An efficient matrix multiplication on high performance and parallel computers requires optimizations on how matrices are decomposed and exchanged between computational nodes to reduce communication and synchronization overhead, as well as to efficiently exploit the memory hierarchy within a node to improve both spatial and temporal data locality. In this paper, we presented our studies of performance, cache behavior, and energy efficiency of multiple parallel matrix multiplication algorithms on a multicore desktop computer and a medium-size shared memory machine, both being considered as referenced sizes of nodes to create a medium- and largescale computational clusters for high performance computing used in industry and national laboratories. Our results highlight both the performance and energy efficiencies, and also provide implications on the memory and resources pressures of those algorithms. We hope this could help users choose the appropriate implementations according to their specific data sets when composing larger-scale scientific applications that use parallel matrix multiplication kernels on a node.

Parallel processing is a key way to get good speed on difficult computer jobs thanks to how quickly computer technology is improving. Multistage Interconnection Networks (MINs) are very important in this situation because they make it easy for multiple working elements in parallel computer systems to share data and work together. The focus of this study is on how parallel methods can be used in MINs, with a focus on how these networks make it possible for nodes to communicate and process data quickly. We look at the layout and design of MINs and talk about how well they can handle apps that use a lot of data by spreading out work. We look at some of the most important parallel algorithms, like matrix multiplication, sorting, and Fast Fourier Transform (FFT), to show how they use the unique features of MINs to make computations faster and more flexible. The study also looks into the problems that come up when you try to use parallel methods, such as delay, timing issues, and load balance. It also talks about ways to deal with these problems, like route algorithms, timing methods, and fault-tolerant systems. The study shows how important parallel processing is for making high-performance computer systems work better. This helps make progress in areas like science models, big data analytics, and artificial intelligence. Through looking at how parallel algorithms and MINs work together, this study shows how parallel processing can speed up complicated calculations and make systems more efficient, leading to more advanced and scalable computing solutions.

In this paper, we have studied a multi - LED matrix optical sensing technology for high speed transmission in a visible light communication system based on an optical camera CCD sensor. In order to improve transmission capacity in visible light communication, data must be transmitted from different LEDs at the same time and received. However, a system environment with multiple LEDs may cause interference between adjacent LEDs. Generally, since a visible light communication system generally uses a luminance modulation method, when a plurality of LEDs transmit data at the same time, it is difficult to accurately detect each LED due to light scattering interference of an adjacent LED. To solve this problem, we proposed a spatial filtering technique to group multiple LEDs and to reduce the interference of grouped LED matrix. We also propose a pixel image processing technique for detecting each LED. This technology accurately recognizes each matrix of LEDs in a multi-matrix LED environment and improves system performance by increasing the throughput of the MISO-OCVLC (Optical Camera Visible Light Communication) system.