Abstract
Evaluation the performance of the algorithms and the method that is used to implement it play a major role in the assessment of the performance of many applications and it help the researchers to decide which algorithm to use and which method to implement it, it also give indicate of the performance of the hardware that the algorithm is tested over. In this paper we evaluate the performance of solving linear equation application over supercomputer which was implemented and using Message Passing interface (MPI) library. The sequential and multithreaded algorithm for solving linear equations has been experimented too and the results has been recorded, the speedup and efficiency of the algorithm has been calculated and the results showed that the parallel algorithm outperforms other methods with the large size matrix of 8192 * 8192 over the number of processors of 64. For large input size, the results also showed that there is a noticeable decrease in running time as the number of processors increase. But in case of multithreaded the results showed that as the matrix size increase the time required for running the algorithm is rapidly increasing although the number of threads increased. This indicates that the parallel performance over for large matrix input size is better and outperforms other methods.
Highlights
From the beginning of computer invention until today the computers processing has been developed very rapidly a new generation of computer systems sets higher standards with regards to performance, size, price, and usability
The continued development of computer hardware and improvements in the computer price/performance ratio coupled with improvements in the usability and functionality parallel computers have raised the expectations of computers to the point where they appear to believe that any problem and any size model can be solved in short time, regardless of the size or complexity of the problem
Te number of steps that is required for solving linear equation system with N * N matrix and a vector S of N x 1 is N -1 Step, through the iteration of the algorithm and in any i iteration any non zero value lies below the diagonal in column i are changed by replacing with every j row, where as i + 1 ≤ j < n, replaced by the sum of row j and – aj, I /ai,i multiplied by row I (Dumas, 2002)
Summary
From the beginning of computer invention until today the computers processing has been developed very rapidly a new generation of computer systems sets higher standards with regards to performance, size, price, and usability. The most important high performance computer categories are as follows: SIMD machines: Single instruction machines that manipulate many data items in parallel Such machines have large number of processors, ranging from 1,024 to 16,384. Many researches has been done on parallel processing (Pasetto, and Akhriev, 2011, Maria, et al, 2015, Atif, and Rauf, 2009, Rajalakshmi, 2009, Delic, and Juric, 2013) many researches invistigate matrix and linear equation solving and numerical analysis over large number of processors using parallel processing (Scholl, Stumm and Wehn, 2013, Rajalakshmi, 2009, Saeed, et al, 2015), these researches focus on measuring the performance evolution of parallel processing and differentiate it from sequential analysis and parallelization of the sequential methods many researches discussed parallel algorithms based on Cholesky factorization, Gaussian elimination, LU decomposition, Gauss-Jordan, and many methods gave solution for dense linear systems, Erich Kaltofen, and Victor Pan proposed Processor Efficient Parallel Solution of Linear Systems over an Abstract Field, The algorithms utilize within an O (log n) factor as many processors as are needed to multiply two n × n matrices. This paper evaluate the performance of matrix multiplication and an application to it in solve linear equation on super computer, Gaussian elimination algorithm which is been used for solving a system of linear equations in parallel super computer
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