Exploring Solvents and Sensitivity of Systems of Linear Equations Arising from Real World Phenomena Via Optimal Successive Over-relaxation Method
Authors: Kigodi Odeli , Chacha Chacha , Mtunya Adeline 
Published: November 2024

This study places a significant emphasis on assessing the efficiency of numerical methods, specifically in the context of solving linear equations of the form &lt;i&gt;Ax&lt;/i&gt; = &lt;i&gt;b&lt;/i&gt;, where &lt;I&gt;A&lt;/I&gt; is a square matrix, &lt;i&gt;x&lt;/i&gt; is a solvent vector, and &lt;i&gt;b&lt;/i&gt; is a column vector representing real-world phenomena. The investigation compares the effectiveness of the Refined Successive Over Relaxation (RSOR) method to the standard Successive Over-relaxation (SOR) method. The core evaluation criteria encompass computational time (in seconds), convergence behavior, and the number of iterations necessary to approximate the solvents of five distinct real-world phenomena: Model Problem 1 (MP1) involving an Electrical Circuit, Model Problem 2 (MP2) focusing on Beam Deflection, Model Problem 3 (MP3) addressing Damped Vibrations of a Stretching Spring, Model Problem 4 (MP4) dealing with Linear Springs and Masses, and Model Problem 5 (MP5) focusing on Temperature Distribution on Heated Plate. The RSOR method generally outperforms the SOR method, particularly with a constant relaxation parameter (&lt;i&gt;ω&lt;/i&gt;) in the range 1.0 &lt; &lt;i&gt;ω&lt;/i&gt; &lt; 1.2. The RSOR method is favored for its robustness and efficiency with less need for fine-tuning &lt;i&gt;ω&lt;/i&gt;, whereas the SOR method can achieve superior performance if the optimal &lt;i&gt;ω&lt;/i&gt; is found, although this often requires time-consuming trial and error. Despite the potential for better performance with an optimal &lt;i&gt;ω&lt;/i&gt;, the RSOR method’s consistent results make it the more practical choice in many cases. The study also explores the stability of the systems of linear equations arising from these phenomena by calculating their condition numbers (&lt;I&gt;K&lt;/I&gt;(&lt;I&gt;A&lt;/I&gt;)). More interestingly, the results reveal that all systems MP1 to MP4 exhibit instability when subjected to even modest perturbations, shedding light on potential challenges in their solvents. This research not only underscores the advantages of the RSOR method but also emphasizes the importance of understanding the stability of numerical solvents in the context of real-world problems. Additionally, the results for MP5 demonstrates that tiny changes to the original matrix’s coefficients have no effect on the desired solvent because the perturbed matrix’s condition number is the same as the original matrix’s, making the problem well-structured. The problem becomes ill-conditioned if there is an increase or decrement to the matrix’s coefficients that is bigger than 10&lt;sup&gt;−5&lt;/sup&gt;. In summary, Sparse systems are sensitive to perturbations, resulting in instability. If the tolerance |&lt;i&gt;k&lt;/i&gt;(&lt;I&gt;A&lt;/I&gt;&lt;sup&gt;0&lt;/sup&gt;&lt;sub&gt;&lt;i&gt;i&lt;/i&gt;&lt;/sub&gt;) − &lt;i&gt;k&lt;/i&gt;(&lt;i&gt;A&lt;sub&gt;i&lt;/sub&gt;&lt;/i&gt;)| &gt; 10&lt;sup&gt;−5&lt;/sup&gt; for all positive integers &lt;i&gt;i&lt;/i&gt;, then the problem becomes poorly structured.