Analysis of Matrix Multiplication Computational Methods

Transcription

1 European Journal of Scientific Research ISSN X / X Vol.121 No.3, 2014, pp Analysis of Matrix Multiplication Computational Methods Khaled Matrouk Corrospondent Author, Department of Computer Engineering, Faculty of Engineering khaled.matrouk@ahu.edu.jo Tel: (ext. 8503), Fax: Abdullah Al- Hasanat Department of Computer Engineering, Faculty of Engineering Haitham Alasha'ary Department of Computer Engineering, Faculty of Engineering Ziad Al-Qadi Prof, Department of Computer Engineering, Faculty of Engineering Hasan Al-Shalabi Prof, Department of Computer Engineering, Faculty of Engineering Abstract Matrix multiplication is a basic concept that is used in engineering applications such as digital image processing, digital signal processing and graph problem solving. Multiplication of huge matrices requires a lot of computation time as its complexity is O(n 3 ). Because most engineering applications require higher computational throughputs with minimum time, many sequential and parallel algorithms are developed. In this paper, methods of matrix multiplication are chosen, implemented, and analyzed. A performance analysis is evaluated, and some recommendations are given when using openmp and MPI methods of parallel computing. Keywords: OpenMP, MPI, Processing Time, Speedup, Efficiency 1. Introduction With the advent of parallel hardware and software technologies users are faced with the challenge to choose a programming paradigm best suited for the underlying computer architecture (Alqadi and Abu-Jazzar, 2005a; Alqadi and Abu-Jazzar, 2005b; Alqadi et al, 2008). With the current trend in parallel computer architectures towards clusters of shared memory symmetric multi-processors (SMP) parallel programming techniques have evolved to support parallelism beyond a single level (Choi et al, 1994).

2 Analysis of Matrix Multiplication Computational Methods 259 Parallel programming within one SMP node can take advantage of the globally shared address space. Compilers for shared memory architectures usually support multi-threaded execution of a program. Loop level parallelism can be exploited by using compiler directives such as those defined in the OpenMP standard (Dongarra et al, 1994; Alpatov et al, 1997). OpenMP provides a fork-and-join execution model in which a program begins execution as a single thread. This thread executes sequentially until a parallelization directive for a parallel region is found (Alpatov et al, 1997; Anderson et al, 1987). At this time, the thread creates a team of threads and becomes the master thread of the new team (Chtchelkanova et al, 1995; Barnett et al, 1994; Choi et al, 1992). All threads execute the statements until the end of the parallel region. Work-sharing directives are provided to divide the execution of the enclosed code region among the threads. All threads need to synchronize at the end of parallel constructs. The advantage of OpenMP (web ref.) is that an existing code can be easily parallelized by placing OpenMP directives around time consuming loops which do not contain data dependences, leaving the source code unchanged. The disadvantage is that it is not easy for the user to optimize workflow and memory access. On an SMP cluster the message passing programming paradigm can be employed within and across several nodes. MPI (web ref.) is a widely accepted standard for writing message passing programs (web ref.; Rabenseifner, 2003). MPI provides the user with a programming model where processes communicate with other processes by calling library routines to send and receive messages. The advantage of the MPI programming model is that the user has complete control over data distribution and process synchronization, permitting the optimization data locality and workflow distribution. The disadvantage is that existing sequential applications require a fair amount of restructuring for a parallelization based on MPI Serial Matrix Multiplication Matrix multiplication involves two matrices A and B such that the number of columns of A and the number of rows of B are equal. When carried in sequential, it takes a time O(n 3 ). The algorithm for ordinary matrix multiplication is: for i=1 to n for j=1 to n c(i,j)=0 for k=1 to n c(i,j)=c(i,j)+a(i,k)*b(k,j) end end end 1.2. Parallel Matrix Multiplication Using OpenMp Master thread forks the outer loop between the slave threads, thus each of these threads implements matrix multiplication using a part of rows from the first matrix, when the threads multiplication are done the master thread joins the total result of matrix multiplication Parallel Matrix Multiplication Using MPI The procedures of implementing the sequential algorithm in parallel using MPI can be divided into the following steps: Split the first matrix row wise to split to the different processors, this is performed by the master processor. Broadcast the second matrix to all processors. Each processor performs multiplication of the partial of the first matrix and the second matrix. Each processor sends back the partial product to the master processor.

3 260 Khaled Matrouk, Abdullah Al- Hasanat, Haitham Alasha'ary Ziad Al-Qadi and Hasan Al-Shalabi Implementation Master (processor 0) reads data Master sends size of data to slaves Slaves allocate memory Master broadcasts second matrix to all other processors Master sends respective parts of first matrix to all other processors Every processor performs its local multiplication o All slave processors send back their result. 2. Methods and Tools One station with Pentium i5 processor with 2.5 GHz and 4 GB memory is used to implement serial matrix multiplication. Visual Studio 2008 with openmp library is used as an environment for building, executing and testing matrix multiplication program. The program is tested using Pentium i5 processer with 2.5 GHz and 4 GB memory. A distributed processing system with different number of processors is used, each processor is a 4 core processor with 2.5 MHz and 4 GB memory, the processors are connected though Visual Studio 2008 with MPI environment. 3. Experimental Part Different sets of 2 matrices are chosen (different in sizes and data types ) and each pair of matrices is multiplied serially and in parallel using both openmp and MPI environments, and the average multiplication time is taken Experiment 1 Sequential matrix multiplication program is tested using different size matrices. Different size matrices with different data types (integer, float, double, and complex) are chosen, 100 types of matrices with different data types and different sizes are multiplied. Table 1 shows the average results obtained in this experiment. Table 1: Experiment 1 Results 3.2. Experiment 2 Matrices size Multiplication time in seconds 10* * * * * * * * * Matrix multiplication program is tested using small size matrices. Different size matrices with different data types (integer, float, double, and complex) are chosen, 200 types of matrices with different data types and different sizes are multiplied using openmp environment. Table 2 shows the average results obtained in this experiment.

4 Analysis of Matrix Multiplication Computational Methods 261 Table 2: Experiment 2 Results # of threads 10,10 20,20 40,40 100, ,200 (time in seconds) (time in seconds) (time in seconds) (time in seconds) (time in seconds) Experiment 3 Matrix multiplication program is tested using big size matrices. Different size matrices with different data types (integer, float, double, and complex) are chosen, 200 types of matrices with different data types and different sizes are multiplied using openmp environment with 8 threads. Table 3 shows the average results obtained in this experiment. Table 3: Experiment 3 Results Matrices size Multiplication time (in seconds) 1000, , , , , Experiment 4 Matrix multiplication program is tested using big size matrices. Different size matrices with different data types (integer, float, double, and complex) are chosen, 200 types of matrices with different data types and different sizes are multiplied using MPI environment with different number of processors. Table 4 shows the average results obtained in this experiment. Table 4: Experiment 4 Results Number of processors Multiplication time in second 1000x1000 matrices Multiplication time in second 5000x5000 matrices Multiplication time in second 10000x10000 matrices Results Discussion From the results obtained in the previous section we can categorize the matrices into three groups: Small size matrices with size less than 1000*1000 Mid size matrices with 1000*1000 size 5000*5000

5 262 Khaled Matrouk, Abdullah Al- Hasanat, Haitham Alasha'ary Ziad Al-Qadi and Hasan Al-Shalabi Huge size matrices with size 5000*5000 The following recommendation can be declared: For small size matrices, it is preferable to use sequential matrix multiplication. For mid size matrices, it is preferable to use parallel matrix multiplication using openmp. For huge size matrices, it is preferable to use parallel matrix multiplication using MPI. Also it is recommended to use hybrid parallel systems (MPI with openmp) to multiply matrices with huge sizes. From the results obtained in Table 2 we can see that the speedup of using openmp is limited to the number of actual available physical cores in the computer system as it is shown in Table 5 and Fig. 1. Speedup (times) = Time of execution with 1 thread/parallel time Table 5: Speedup results of Using OpenMP Matrix size #of threads = 1 #of threads = 8 Speedup 300, , , , , , , , , , , , Figure 1: Maximum performance of using openmp From the results obtained in Tables 1 and 2 we can see that the matrix multiplication time increases rapidly when the matrix size increases as shown in Figs 2 and 3.

6 Analysis of Matrix Multiplication Computational Methods 263 Figure 2: Comparing between 8 and 1 threads results time in seconds One Thread 8 threads n matrix(nxn) Figure 3: Relationship between the speedup and the matrix size max # of cores 3.5 speedup :times matrix size(nxn) From the results obtained in Table 4 we can calculate the speedup of using MPI and the system efficiency: Efficiency = speedup/number of processors The calculation results are shown in Table 6:

7 264 Khaled Matrouk, Abdullah Al- Hasanat, Haitham Alasha'ary Ziad Al-Qadi and Hasan Al-Shalabi Table 6: Speedup and efficiency of using MPI Number of processors Speedup of Multiplication 1000x100 matrices Efficiency Speedup of Multiplication 5000x5000 matrices Efficiency Speedup of Multiplication 10000x10000 matrices Efficiency , From Table 6 we can see that increasing the number of processors in an MPI environments leads to enhancing the speedup of matrix multiplication but it also leads to poor system efficiency as shown in Figs 4, 5 and 6. Figure 4: Multiplication time for 10000x10000 matrices Running times for parallel matrix multiplication of two 10000x10000 matrices Time in seconds Number of processors

8 Analysis of Matrix Multiplication Computational Methods 265 Figure 5: Speedup of multiplication for 10000x10000 matrices 45 Speedup of matrix multiplication 10000* Speedup Number of processors Figure 6: System efficiency of matrices multiplication System efficiency 1000* * *10000 Efficiency Number of processors 5. Conclusions Based on the results obtained and shown above, the following conclusions can be drawn: Sequential matrix multiplication is preferable for matrices with small sizes. OpenMP is a good method to use as an environment for parallel matrix multiplication with mid sizes, and here the speedup is limited to the number of available physical cores.

9 266 Khaled Matrouk, Abdullah Al- Hasanat, Haitham Alasha'ary Ziad Al-Qadi and Hasan Al-Shalabi MPI is a good method to use as an environment for parallel matrix multiplication with huge sizes, here we can increase the speedup but negatively affects the system efficiency. To avoid the problems in the two previous conclusions we can recommend a hybrid parallel system References [1] Alqadi Z., and Abu-Jazzar A., CNRS-NSF Workshop on Environments and Program Methods Used for Optimizing Matrix Tools for Parallel Scientific Computing, Saint Hilaire Multiplication, Journal of Engineering 15(1), pp , du Touvet, France, Sept. 7-8, Elsevier Sci. Publishers. [2] Alqadi Z., and Abu-Jazzar A., "Analysis of Program Methods Used for Optimizing Matrix Multiplication", Journal of Engineering 15(1), pp [3] Alqadi Z., Aqel M., and El Emary I. M. M., "Performance Analysis and Evaluation of Parallel Matrix Multiplication Algorithms",World Applied Sciences Journal 5(2). [4] Dongarra, J. J., R.A. Van de Geijn, and D.W. Walker, "Scalability Issues Affecting the Design of a Linear Algebra Library, Parallel Linear Algebra Package Design", Distributed Computing 22( 3), Proceedings of SC 97, pp [5] Alpatov, P., G. Baker, C. Edwards, J. Gunnels, and P. Geng, "Parallel Matrix Distributions: Parallel Linear Algebra Package", Tech. Report TR-95-40, Proceedings of the SIAM Parallel Processing Computer Sciences Conference, The University of Texas, Austin. [6] Choi, J., J. J. Dongarra and D.W. Walker, "A High-Performance Matrix Multiplication Algorithm Pumma: Parallel Universal Matrix Multiplication on a Distributed Memory Parallel Computer Using Algorithms on Distributed Memory Concurrent Overlapped Communication", IBM J. Res. Develop., Computers, Concurrency: Practice and Experience 6(7), pp [7] Chtchelkanova, A., J. Gunnels, and G. Morrow, "IEEE Implementation of BLAS: General Techniques for Level 3 BLAS", Proceedings of the 1986 International Conference on Parallel Processing, pp , TR-95-40, Department of Computer Sciences, University of Texas. [8] Barnett, M., S. Gupta, D. Payne, and L. Shuler, "Using MPI: Communication Library (InterCom), Scalable High Portable Programming with the Message-Passing Performance, Computing Conference, pp [9] Anderson E., Z. Bai, C. Bischof, and J. Demmel, "Solving Problems on Concurrent Processors", Proceedings of Matrix Algorithms Supercomputing '90, IEEE 1, pp [10] Choi J., J. J. Dongarra, R. Pozo and D.W. Walker, "Scalapack: A Scalable Linear Algebra Library for Distributed Memory Concurrent Computers", Proceedings of the Fourth Symposium on the Frontiers of Massively Parallel Computation. IEEE Comput. Soc. Press, pp [11] MPI 1.1 Standard, [12] OpenMP Fortran Application Program Interface, [13] Rabenseifner, R., Hybrid Parallel Programming: Performance Problems and Chances, Proceedings of the 45th Cray User Group Conference, Ohio, May 12-16, 2003.