خوارزميتان متوازيتان فعالتان لحل جمل المعادلات الخطية خماسية الأقطار المتناظرة
Abstract
في هذه المقالة، نصف خوارزميتين متوازيتين لإيجاد حل جمل المعادلات الخطية خماسية الأقطار المتناظرة المربعة من المرتبة. تتطلب الخوارزميتين معالجاً وكل معالج يمتلك ذاكرة موضعية. تتضمن الخوارزمية الأولى كتابة المصفوفة خماسية الأقطار على شكل جداء مصفوفتين كل منهما مصفوفة ثلاثية الأقطار. اقترحنا لحل جمل المعادلات الخطية ثلاثية الأقطار الناتجة خوارزمية متوازية. أما الخوارزمية الثانية فتتضمن تحليل المصفوفة خماسية الأقطار وفق شكل ما بحيث يمكن تنفيذ جمل المعادلات الناتجة وفق خوارزمية متوازية. أجرينا العديد من تجارب المحاكاة العددية لتوضيح فعالية، وسرعة، ودقة الخوارزميتين المقترحتين لحل جمل المعادلات الخطية خماسية الأقطار المتناظرة المدروسة. تبين من التجارب العددية أنّ الخوارزميتين فعّالتين وأن إحداهما أسرع من الأخرى بمرتين لحل نفس مسائل الاختبار.
In this paper, we described tow parallel algorithms for finding the solution of symmetric pentadiagonal linear systems of equations of order. The proposed algorithms require 2 processors; each of both possesses local memory.
The first algorithm includes writing the pentadiagonal matrix in the form of product of tow tridiagonal matrices. We suggested a parallel algorithm for solving tridiagonal linear systems of equations. The second algorithm consists of decomposition of the pentadiagonal matrix in a form such that we can carry out the resulting linear systems of equations by using parallel algorithm. We carried out many numerical experiments to illustrate the efficiency, speeding up and accuracy for solving symmetric pentadiagonal linear systems of equations. The numerical experiments showed that the proposed algorithms were efficient and one of both was much faster in factor of 2 than the other one for solving the same test problems.
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