Análisis de algunas técnicas iterativas para sistemas de ecuaciones lineales y su estudio de la convergencia a través del número de condicionamiento

Abstract

At present, numerical analysis provides us with powerful tools to determine the solution of various problems whose mathematical model can be represented by a system of linear equations, these tools correspond to a number of direct and iterative methods, among which are Carl's method. Gustav Jakob Jacobi and the Doolittle and Crout method, which we analyze and compare in this document. To do this we will initially explore the concepts of conditioning the problem to determine how stable is the system from which the model was obtained, until we reach the decomposition of LU arrays proposed in the Doolittle and Crout method. As a result of the analysis and comparison in this document, depending on what is sought when solving a system of equations, either very large or small enough for our computer, we can choose an approximation that will bring a short-term result with an error. Due to the starting point as proposed in the Jacobi method, or it is possible to reach a direct result by implementing fewer iterations as proposed in the Doolittle and Crout metho