Iterative refinement of the Gauss-Jordan method, in ill conditioned systems.
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Abstract
In this paper, an iterative algorithm is constructed to improve the solution of a system of linear equations, of the form Ax = b, when it is solved using the Gauss-Jordan Method and by using finite arithmetic. Understanding the functioning of the algorithm, showing its scope and analyzing how it is deduced, is achieved through the concept of matrix norm, together with some of its properties. The concept of the condition number of a matrix is introduced, and are found bounds for it by using the matrix norms. Finally, it is explained the iterative algorithm of the Refinement, showing the power of this one, when it is solved a system of linear equations ill conditioned.
Keywords
Matrix norm, condition numbers, Refinement Method.
References
[1] V. Arunachalam y A. Calvache, “Approximation of the Bivariate Renewal Function”, Communications in Statistics -Simulation and Computation, vol.44. , no. 1, pp. 154-167, 2015.
[2] V. Arunachalam y S. Dharmajara, “Fluid Queue Driven by Finite Markov Processes”, Revista Ciencia en Desarrollo, vol. 5, no. 2, pp. 79-86, 2015.
[3] R. L. Burden y J. D. Faires, “Numerical Analysis”, BROOKS/COLE, ed. 9, United States, 2011.
[4] A. Calvache, “The Transient and Asymptotic Moments for the Random Mission Time of a System”, Revista Ciencia en Desarrollo, vol. 7, no. 2, pp.109-124, 2016.
[5] J.I. Díaz. “John von Neumann: de la matemática pura a la matemática aplicada”, Boletín de la Sociedad Española de Matemática Aplicada, vol. 32, pp. 149-169, 2005.
[6] R. A. Horn y C. R. Johnson, “Matrix Analysis”, Cambridge University Press, ed. 2, New York, 2013.
[7] IEEE Computer Society, “IEEE Standard for Floating Point Arithmetic”, IEEE Std, 2008.
[8] J. Neumann y H. H. Goldstine, “Numerical Inverting of Matrices of High Order”, Bull. American Mathematical Society, no. 53, pp. 1021-1099, 1947.
[9] A. Pyzzara, B. Bylina y J. Bylina, “The Influence of a Matrix Condition Number on Iterative Methods Convergence”, Conference on Computer Science and Information Systems, pp.459-464, 2017.
[10] A. M. Turing, “Rounding-off Errors in Matrix Processes”, Quartely J. Mech. Appl. Math, no.1, pp. 287-308, 1948.