Commenced in January 2007
Frequency: Monthly
Edition: International
Paper Count: 31903
On the Algorithmic Iterative Solutions of Conjugate Gradient, Gauss-Seidel and Jacobi Methods for Solving Systems of Linear Equations

Authors: H. D. Ibrahim, H. C. Chinwenyi, H. N. Ude

Abstract:

In this paper, efforts were made to examine and compare the algorithmic iterative solutions of conjugate gradient method as against other methods such as Gauss-Seidel and Jacobi approaches for solving systems of linear equations of the form Ax = b, where A is a real n x n symmetric and positive definite matrix. We performed algorithmic iterative steps and obtained analytical solutions of a typical 3 x 3 symmetric and positive definite matrix using the three methods described in this paper (Gauss-Seidel, Jacobi and Conjugate Gradient methods) respectively. From the results obtained, we discovered that the Conjugate Gradient method converges faster to exact solutions in fewer iterative steps than the two other methods which took much iteration, much time and kept tending to the exact solutions.

Keywords: conjugate gradient, linear equations, symmetric and positive definite matrix, Gauss-Seidel, Jacobi, algorithm

Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 234

References:


[1] Shewchuk, J. R. (1994). An Introduction to the Conjugate Gradient Method Without the Agonizing Pain, Edition 11/4, School of Computer Science, Carnegie Mellon University Pittsburgh, PA 15213
[2] https://encyclopediaofmath.org/wiki/conjugate_gradients_method_of
[3] Antia, H. M. (2012). Numerical methods for scientists and engineers. Hindustan Book Agency.
[4] D. S. Watkins, D. S. (2010). Fundamentals of matrix computations, 3rd ed. WILEY.
[5] Hestenes, M. R. and Eduard Stiefel, E. (1952). Methods of Conjugate Gradients for Solving Linear Systems. Journal of Research of the National Bureau of Standards, 49 (6).