بعض الطرائق المباشرة و التكرارية المطورة لحل جمل المعادلات الخطية كثيرة الأصفار
Abstract
In this paper, efficient direct and iterative methods are described for solving a large random sparse non-symmetric linear system. Such systems of linear equations of huge order arise in several applications such as physics, mechanics, signal processing and other applications of real life problems. For this reason, we try to develop direct and iterative methods for solving such systems of linear equations. The suggested direct method is based on the sparse LU-decomposition method (DSLU). The developed iterative methods include a Semi-iterative Method (SM), a Splitting-based Iterative Method (SIM) and a preconditioned GMRES method. We consider two types of preconditioners based on Incomplete LU-decomposition (ILU). We test and compare the numerical implementations of these methods on four numerical examples to demonstrate their efficiency. Results show that the proposed ILU preconditioners in GMRES reduce largely number of iterations and give very accurate solutions.
في هذه المقالة، نصف طرائق مباشرة وتكرارية فعالة لحل جمل معادلات خطية، غير متناظرة ، كيفية، كثيرة الأصفار ذات مراتب عليا. تظهر هذه الجمل من المعادلات الخطية ذات المراتب العليا في تطبيقات عديدة كالفيزياء و الميكانيك والمعالجة الرقمية وتطبيقات أخرى من مسائل الحياة الحقيقية. لهذه الأسباب نحاول تطوير طرائق مباشرة وطرائق تكرارية لحل هذا النوع من جمل المعادلات. تعتمد الطريقة المباشرة المقترحة على طريقة تحليل LU كثيرة الأصفار (DSLU). تتضمن الطرائق التكرارية المطورة: طريقة نصف تكرارية (SM) و طريقة تكرارية تعتمد على التجزئة (SIM) و طريقةGMRES المسرعة. ندرس نوعين من المسرعات التي تعتمد على تحليل LU غير التام (ILU). نختبر و نقارن التنفيذات العددية لهذه الطرائق على أربعة أمثلة عددية لتوضيح فعاليتها. تبين النتائج أن المسرعات ILU المحددة في GMRES تخفض عدد التكرارات بشكل كبير و تعطي حلولا دقيقة جدا.
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