Matrix Multiplication Application Research Articles

Applications of Matrix Multiplication

In this paper we present some interesting applications of the matrix’s multiplication, that include the Leslie matrix and population change, which we calculate this kind of changes from year to other year by matrix multiplication. Another important part of the paper is Analysis of Traffic Flow, we represent the flow of traffic through a network of one-way streets. Another much important part is the production costs, this is fantastic usage of matrix multiplication, in which, A company manufactures three products. Its production expenses are divided into three categories, here in this paper we well describe this beautiful issue. By matrix multiplication, we can encode and decode messages. To encode a message, we choose an invertible matrix and multiply the uncoded row matrices (on the right) by to obtain coded row matrices, this idea will be clarify by some useful examples. Also we used to study certain relationships between objects by matrix multiplication. We will clarify all of these applications by useful examples. In this paper, we present six different applications of matrix multiplication.

Experimentally-Validated Crossbar Model for Defect-Aware Training of Neural Networks

Nanoscale memristors in a crossbar configuration have demonstrated their ability to accelerate modern computing workloads in various applications, including machine learning, image processing, and data analytics. Modeling the crossbar behavior is critical to software-hardware co-design, but most previous works focused on single or several memristor devices. So, challenges still exist in large-scale crossbar implementations due to non-idealities that only emerge at the system level. In this brief, we build a crossbar model based on experimentally characterized device statistics in large crossbar arrays. We identify different types of imperfections, including statistical device relaxation, fluctuation, peripheral circuits, etc. The experimentally-validated model is then used to co-optimize analog matrix multiplication and neural network applications. Specifically, we propose and implement defect-aware training and verify that the neural network trained with our algorithm can provide better accuracy and reliability when deployed to physical crossbars. Finally, we achieve an experimental accuracy of 93.4% on the MNIST dataset in a physical crossbar by training based on our crossbar model to compensate for statistical stochasticity, 8.4% higher than the vanilla model. More importantly, the accuracy remains larger than 90.0% after two days, while the accuracy with the vanilla model drops to 73.2% because of conductance relaxation. The method is also scaleable to more practical networks and demonstrates a 92.3% CIFAR-10 accuracy with the VGG-16 model on the simulated crossbar model.

Design and Analysis of Approximate 4–2 Compressors for High-Accuracy Multipliers

Approximate multipliers are applicable in error-resilient applications with relaxed precision constraints, including image processing, multimedia, and data recognition. Such multipliers that sacrifice some accuracy can gain a corresponding increase in electrical performance. This article presents an analysis of the architectures of previously proposed compressors to investigate their performance and accuracy. In this article, we propose five high-accuracy approximate 4–2 compressors with better delay, area, power, and better performance–accuracy tradeoff. Pro1–Pro4 rely on the critical path optimization, while Pro5 derives from the modified sorting technique. This article implements <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$8 \times 8$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$16 \times 16$ </tex-math></inline-formula> multipliers by employing the proposed approximate compressors in TSMC 28 nm. The experimental results indicate that our designs have about 18% delay, 43%–52% area-delay product (ADP) reduction compared to the exact multiplier, and 20%–55% ADP optimization compared to compressors with the same accuracy. This article further verifies the efficacy of the proposed compressors through image blending and matrix multiplication applications.

JAMPI: Efficient Matrix Multiplication in Spark Using Barrier Execution Mode

The new barrier mode in Apache Spark allows for embedding distributed deep learning training as a Spark stage to simplify the distributed training workflow. In Spark, a task in a stage does not depend on any other tasks in the same stage, and hence it can be scheduled independently. However, several algorithms require more sophisticated inter-task communications, similar to the MPI paradigm. By combining distributed message passing (using asynchronous network IO), OpenJDK’s new auto-vectorization and Spark’s barrier execution mode, we can add non-map/reduce-based algorithms, such as Cannon’s distributed matrix multiplication to Spark. We document an efficient distributed matrix multiplication using Cannon’s algorithm, which significantly improves on the performance of the existing MLlib implementation. Used within a barrier task, the algorithm described herein results in an up to 24% performance increase on a 10,000 × 10,000 square matrix with a significantly lower memory footprint. Applications of efficient matrix multiplication include, among others, accelerating the training and implementation of deep convolutional neural network-based workloads, and thus such efficient algorithms can play a ground-breaking role in the faster and more efficient execution of even the most complicated machine learning tasks.

RANC: Reconfigurable Architecture for Neuromorphic Computing

Neuromorphic architectures have been introduced as platforms for energy efficient spiking neural network execution. The massive parallelism offered by these architectures has also triggered interest from non-machine learning application domains. In order to lift the barriers to entry for hardware designers and application developers we present RANC: a Reconfigurable Architecture for Neuromorphic Computing, an open-source highly flexible ecosystem that enables rapid experimentation with neuromorphic architectures in both software via C++ simulation and hardware via FPGA emulation. We present the utility of the RANC ecosystem by showing its ability to recreate behavior of the IBM's TrueNorth and validate with direct comparison to IBM's Compass simulation environment and published literature. RANC allows optimizing architectures based on application insights as well as prototyping future neuromorphic architectures that can support new classes of applications entirely. We demonstrate the highly parameterized and configurable nature of RANC by studying the impact of architectural changes on improving application mapping efficiency with quantitative analysis based on Alveo U250 FPGA. We present post routing resource usage and throughput analysis across implementations of Synthetic Aperture Radar classification and Vector Matrix Multiplication applications, and demonstrate a neuromorphic architecture that scales to emulating 259K distinct neurons and 73.3M distinct synapses.

Creating Ciphertext and Decipher using Graph Labeling Techniques

In this paper, we design plaintext through product edge labeling for a Cyclegraph. The edge labels thus obtained is further processed through Caesar cipher method. The ciphertext thus obtained is subjected to Bifid Cipherfor which we apply matrix multiplication. The decryption is performed usingInverse matrix multiplication and subsequent application of Bifid cipher and Caesar Cipher yields our plaintext. This methodology is considered highly efficient and secure as it combines Cryptographic technique together with Graph Labeling thereby making it difficult for any Adversary to hack the plaintext. We investigate the above technique using Non- singular square matrices.

Square Matrix Multiplication Using CUDA on GP-GU

This paper focuses on matrix multiplication algorithm, particularly square parallel matrix multiplication using Computer Unified Device Architecture (CUDA) programming model with C programming language. Matrix multiplication is under the list of time-consuming problems that require s huge computational resources to improve its speedup. As many studies have shown, it is not easy to achieve high performance speedup in sequential matrix multiplication algorithm using larger input. The emphasis of this study is to propose a parallel algorithm to calculate the product of two square matrices with improved speedup performance compared to the sequential and OpenMP algorithms. In this research, biruni (super machine workstation) in the School of Computer Sciences, USM, Malaysia with General Purpose Graphics Processing Unit (GP-GU) was used to parallelize the matrix product algorithm. A comparison between parallel OpenMp versions and sequential algorithm with the proposed CUDA based algorithm of this research was carried out to evaluate the speedup performance of the proposed parallel CUDA based algorithm. The overall results show that CUDA based parallel matrix multiplication is approximately 400 times faster than sequential matrix multiplication and 4 times faster than OpenMp matrix multiplication algorithms, respectively. Therefore, the proposed parallel algorithm can help the researchers working with matrix multiplication application problems. It can also help mathematicians to easily calculate the product of any two matrices and obtain the result in a shorter time.

NMR-MPar: A Fault-Tolerance Approach for Multi-Core and Many-Core Processors

Multi-core and many-core processors are a promising solution to achieve high performance by maintaining a lower power consumption. However, the degree of miniaturization makes them more sensitive to soft-errors. To improve the system reliability, this work proposes a fault-tolerance approach based on redundancy and partitioning principles called N-Modular Redundancy and M-Partitions (NMR-MPar). By combining both principles, this approach allows multi-/many-core processors to perform critical functions in mixed-criticality systems. Benefiting from the capabilities of these devices, NMR-MPar creates different partitions that perform independent functions. For critical functions, it is proposed that N partitions with the same configuration participate of an N-modular redundancy system. In order to validate the approach, a case study is implemented on the KALRAY Multi-Purpose Processing Array (MPPA)-256 many-core processor running two parallel benchmark applications. The traveling salesman problem and matrix multiplication applications were selected to test different device’s resources. The effectiveness of NMR-MPar is assessed by software-implemented fault-injection. For evaluation purposes, it is considered that the system is intended to be used in avionics. Results show the improvement of the application reliability by two orders of magnitude when implementing NMR-MPar on the system. Finally, this work opens the possibility to use massive parallelism for dependable applications in embedded systems.

Parallel Data Partitioning Algorithms for Optimization of Data-Parallel Applications on Modern Extreme-Scale Multicore Platforms for Performance and Energy

Data partitioning algorithms aiming to minimize the execution time and the energy of computations in self-adaptable data-parallel applications on modern extreme-scale multicore platforms must address two critical challenges. First, they must take into account the new complexities inherent in these platforms such as severe resource contention and non-uniform memory access. Second, they must have low practical runtime and memory costs. The sequential data partitioning algorithms addressing the first challenge have a theoretical time complexity of O( $m * m * p * p$ ) where $m$ is the number of points in the discrete speed/energy function and $p$ is the number of available processors. They, however, exhibit high practical runtime cost and excessive memory footprint, therefore, rendering them impracticable for employment in self-adaptable applications executing on extreme-scale multicore platforms. We present, in this paper, the parallel data partitioning algorithms that address both the challenges. They take as input the functional models of performance and energy consumption against problem size and output workload distributions, which are globally optimal solutions. They have a low time complexity of O( $m * m * p$ ) thereby providing a linear speedup of O( $p$ ) and low memory complexity of O( $n$ ) where $n$ is the workload size expressed as a multiple of granularity. They employ dynamic programming approach, which also facilitates the easier integration of performance and energy models of communications. We experimentally study the practical cost of application of our algorithms in two data-parallel applications, matrix multiplication and fast Fourier transform, on a cluster in Grid’5000 platform. We demonstrate that their practical runtime and memory costs are low making them ideal for employment in self-adaptable applications. We also show that the parallel algorithms exhibit tremendous speedups over the sequential algorithms. Finally, using theoretical analysis for a forecast exascale platform, we demonstrate that the parallel algorithms have negligible execution times compared to the matrix multiplication application executing on the platform.

Performance and Power Efficient Massive Parallel Computational Model for HPC Heterogeneous Exascale Systems

The emerging high-performance computing Exascale supercomputing system, which is anticipated to be available in 2020, will unravel many scientific mysteries. This extraordinary processing framework will accomplish a thousand-folds increment in figuring power contrasted with the current Petascale framework. The prospective framework will help development communities and researchers in exploring from conventional homogeneous to the heterogeneous frameworks that will be joined into energy efficient GPU devices along with traditional CPUs. For accomplishing ExaFlops execution through the Ultrascale framework, the present innovations are confronting several challenges. Huge parallelism is one of these challenges, which requires a novel low power consuming parallel programming approach for attaining massive performance. This paper introduced a new parallel programming model that achieves massive parallelism by combining coarse-grained and fine-grained parallelism over inter-node and intra-node computation respectively. The proposed framework is tri-hybrid of MPI, OpenMP, and compute unified device architecture (MOC) that compute input data over heterogeneous framework. We implemented the proposed model in linear algebraic dense matrix multiplication application, and compared the quantified metrics with well-known basic linear algebra subroutine libraries such as CUDA basic linear algebra subroutines library and KAUST basic linear algebra subprograms. MOC outperformed to all implemented methods and achieved massive performance by consuming less power. The proposed MOC approach can be considered an initial and leading model to deal emerging Exascale computing systems.

Task Partitioning and Load Balancing Strategy for Matrix Applications on Distributed System

In this paper, we present a load-balancing strategy (Adaptive Load Balancing strategy) for data parallel applications to balance the work load effectively on a distributed system. We study its impact on computation-hungry matrix multiplication application. The ALB strategy enhances the performance with features such as intelligent node selection, pre-task assignment, adaptive task sizing and buffer allocation, and load balancing. The ALB strategy exhibits reduced nodes idle time and inter process communication time, and improved speed up as compared to Run Time task Scheduling strategy.

Principles of linear algebra with Maple

Preface. Conventions and Notations. 1 An Introduction To Maple. 1.1 The Commands . 1.2 Programming. 2 Linear Systems of Equations and Matrices. 2.1 Linear Systems of Equations. 2.2 Augmented Matrix of a Linear System and Row Operations. 2.3 Some Matrix Arithmetic. 3 Gauss-Jordan Elimination and Reduced Row Echelon Form. 3.1 Gauss-Jordan Elimination and rref. 3.2 Elementary Matrices. 3.3 Sensitivity of Solutions to Error in the Linear System. 4 Applications of Linear Systems and Matrices. 4.1 Applications of Linear Systems to Geometry. 4.2 Applications of Linear Systems to Curve Fitting. 4.3 Applications of Linear Systems to Economics. 4.4 Applications of Matrix Multiplication to Geometry. 4.5 An Application of Matrix Multiplication to Economics. 5 Determinants, Inverses and Cramer s Rule. 5.1 Determinants and Inverses from the Adjoint Formula. 5.2 Determinants by Expanding Along Any Row or Column . 5.3 Determinants Found by Triangularizing Matrices. 5.4 LU Factorization. 5.5 Inverses from rref. 5.6 Cramer s Rule. 6 Basic Linear Algebra Topics. 6.1 Vectors. 6.2 Dot Product. 6.3 Cross Product. 6.4 Vector Projection. 7 A Few Advanced Linear Algebra Topics. 7.1 Rotations in Space. 7.2 Rolling a Circle Along a Curve. 7.3 The TNB Frame. 8 Independence, Basis and Dimension for Subspaces of Rn. 8.1 Subspaces of Rn. 8.2 Independent and Dependent Sets of Vectors in Rn. 8.3 Basis and Dimension for Subspaces of Rn. 8.4 Vector Projection onto a Subspace of Rn. 8.5 The Gram-Schmidt Orthonormalization Process. 9 Linear Maps from Rn to Rm. 9.1 Basics About Linear Maps. 9.2 The Kernel and Image Subspaces of a Linear Map. 9.3 Composites of Two Linear maps and Inverses. 9.4 Change of Bases for the Matrix Representation of a Linear Map. 10 The Geometry of Linear and Affine Maps. 10.1 The Effect of a Linear Map on Area and Arclength in Two Dimensions. 10.2 The Decomposition of Linear Maps into Rotations, Reflections and Rescalings in R2. 10.3 The Effect of Linear Maps on Volume, Area and Arclength in R3. 10.4 Rotations, Reflections and Rescalings in Three Dimensions. 10.5 Affine Maps. 11 Least Squares Fits and Pseudoinverses. 11.1 Pseudoinverse to a Non-Square Matrix and Almost Solving an Overdetermined Linear System. 11.2 Fits and Pseudoinverses. 11.3 Least Squares Fits and Pseudoinverses. 12 Eigenvalues and Eigenvectors. 12.1 What Are Eigenvalues and Eigenvectors, and Why Do We Need Them? 12.2 Summary of Definitions and Methods for Computing Eigenvalues and Eigenvectors as well as the Exponential of a Matrix. 12.3 Applications of the Diagonalizability of Square Matrices. 12.4 Solving a Square First Order Linear. System of Differential Equations ... 12.5 Basic Facts About Eigenvalues and Eigenvectors, and Diagonalizability. 12.6 The Geometry of the Ellipse Using Eigenvalues and Eigenvectors. 12.7 A Maple Eigen-Procedure. Bibliography. Indices. Keyword Index. Index of Maple Commands and Packages.

Comment on "A novel data distribution technique for host-client type parallel applications

A paper by Nick Comino and V. Lakshmi Narasimhan is related to existing literature on divisible load scheduling theory. This includes the use of a Gantt-chart to illustrate timing relationships, basic model assumptions common to the divisible load scheduling literature, allowing time for the reporting back to the originator node of solutions the inclusion of fixed costs (charges) in the model, a proof that such load distribution can be optimized the product form solution for optimal load allocation of the first equation of page and the matrix multiplication application.

Automated tuning of parallel I/O systems: an approach to portable I/O performance for scientific applications

Parallel I/O systems typically consist of individual processors, communication networks, and a large number of disks. Managing and utilizing these resources to meet performance, portability, and usability goals of high performance scientific applications has become a significant challenge. For scientists, the problem is exacerbated by the need to retune the I/O portion of their code for each supercomputer platform where they obtain access. We believe that a parallel I/O system that automatically selects efficient I/O plans for user applications is a solution to this problem. The authors present such an approach for scientific applications performing collective I/O requests on multidimensional arrays. Under our approach, an optimization engine in a parallel I/O system selects high quality I/O plans without human intervention, based on a description of the application I/O requests and the system configuration. To validate our hypothesis, we have built an optimizer that uses rule based and randomized search based algorithms to tune parameter settings in Panda, a parallel I/O library for multidimensional arrays. Our performance results obtained from an IBM SP using an out-of-core matrix multiplication application show that the Panda optimizer is able to select high quality I/O plans and deliver high performance under a variety of system configurations with a small total optimization overhead.

Developing applications for heterogeneous computing environments using simulation: A case study

The objective of this paper is to show the profitability of the development of applications for heterogeneous networked computing environments using simulation tools. Firstly, an overview of the problem is given, discussing the advantages of development on top of a high-performance simulator rather than directly on the real hardware. Then PS, an existing simulator of PVM applications, is described and the development of a distributed matrix multiplication application in a workstation cluster is discussed as a case study. In the proposed example, the performance of two well-known algorithms is analyzed and optimized by the simulated execution of prototypes, i.e., of partially-implemented program designs. The problem of workload sharing in the presence of computing resource heterogeneity is tackled, and the performance figures obtained by simulation are compared to the actual ones, measured by executing the implemented program in a real computing environment.