University of Tehran
Journal of Algorithms and Computation
2476-2776
53
1
2021
06
01
A New Numerical Solution for System of Linear Equations
23
39
81267
10.22059/jac.2021.81267
EN
Iman
Shojaei
Align Technology Inc, San Jose, CA 95134, USA
Hossein
Rahami
School of Engineering Science, College of Engineering, University of Tehran, Tehran, Iran
0000-0001-7540-8412
Journal Article
2021
05
11
In this paper we have developed a numerical method for solving system of linear equations through taking advantages of properties of repetitive tridiagonal matrices. <br />A system of linear equations is usually obtained in the final step of many science and engineering problems such as problems involving partial differential equations. <br />In the proposed algorithm, the problem is first solved for repetitive tridiagonal matrices (i.e., system of linear equations) and a closed-from relationship is obtained.<br /> This relationship is then used for solving a general matrix through converting the matrix into a repetitive tridiagonal matrix and a remaining matrix that is moved to the right-hand side of the equation. <br />Therefore, the problem is converted into a repetitive tridiagonal matrix problem where we have a vector of unknowns on the right-hand side (in addition to the left-hand side) of the equation. <br />The problem is solved iteratively by first using an initial guess to define the vector on the right-hand side of the equation and then solving the problem using the closed-from relationship for repetitive tridiagonal matrices. The new obtained solution is then substituted in the right-hand side of the equation and the tridiagonal problem is solved again. <br />This process is carried out iteratively until convergence is achieved. Computational complexity of the method is investigated and efficiency of the method is shown through several examples. <br />As indicated in the examples, one of the advantages of the proposed method is its high rate of convergence in problems where the given matrix includes large off-diagonal entries. <br />In such problems, methods like Jacobi, Gauss-Seidel, and Successive Over-Relaxation will either have a low rate of convergence or be unable to converge.
https://jac.ut.ac.ir/article_81267_23db04a9319ce7c9f6bbf2eb577623b4.pdf