Newton Tabanlı Kök Bulma Yöntemleri İçin Simülatör Tasarımı

Abstract

Mühendislik problemlerinin birçoğunda denklemlerin köklerinin hesaplanması gerekmektedir. Bunun için birçok yöntemler geliştirilmiştir. Ancak, özellikle gerçek zamanlı uygulamalarda köklerin en az işlemle, en kısa sürede, yüksek hassasiyetle bulunması istenen başlıca özelliktir. Gerçekleştirilen çalışmada, Newton tabanlı 42 yöntemi barındıran grafiksel arayüz programı geliştirilmiştir. Kullanıcı dostu ve eğitim amaçlı da kullanılabilecek simülatörde tanımlanan/girilen denklemlerin, belirtilen aralıkta ve istenen hassasiyette kökleri hesaplanabilmekte; köke yakınsama adımları (iterasyonları) hem sayısal hem de grafiksel (animasyonlu veya animasyonsuz) olarak görülebilmekte, yöntemlerle ilgili konu anlatımları sunulmaktadır. Ayrıca yöntemlerin performans analizleri (iterasyon sayısı, bulunan kök, hesaplama süresi)  de karşılaştırmalı olarak yapılabilmektedir. Böylece simülatör ile kullanıcılar farklı yöntemlerle kök bulma işlemlerini karşılaştırmalı olarak gerçekleştirebilmekte; öğrenciler bu alandaki yöntemleri görsel olarak öğrenip uygulayabilmekte; tasarımcılar sistemleri için performans açısından en uygun yöntemi kolaylıkla, etkin ve verimli bir şekilde seçebilmektedirler.

Keywords

• Kök bulma
• Newton yöntemleri
• Simülatör

Design the Simulator for Root-finding based on Newton's Methods

Abstract

In most of the engineering problems, the calculation of the roots of the equations is required. Many methods have been developed for this purpose. However, especially to obtain the roots in real-time applications with a minimum operation in minimum time, and having high accuracy are main properties. In the performed study, a graphical user interface program that contains 42 Newton-based methods is developed. In the user friendly simulator which can be used also for educational purposes, the roots of the defined/entered equations can be calculated within the specified range with a desired precision; the convergence steps (iterations) to the root can be seen as both numerical and graphical (animated or non-animated), the descriptions of subjects related to the methods are presented. Also, the performance analyzes (the iteration number, the obtained root, the computation time) of the methods can be performed comparatively. Thus, the users can perform the root-finding operations with different methods comparatively by the simulator; the students can learn and apply the methods visually in this field; the designers can choose the most appropriate method in terms of performance easily, effectively and efficiently for their systems.

Keywords

• Root-finding
• Newton's methods
• Simulator

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