Arenas-Navarro, Isnardo

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    A new preconditioner for solving linear systems with ill-conditioned Z-matrices
    (2011-06) Arenas-Navarro, Isnardo; Yong, Xuerong; College of Arts and Sciences - Sciences; Castillo, Paul; Acar, Robert; Department of Mathematics; Cedeño, José
    The class of Z-matrices is the set of matrices whose off-diagonal entries are non positive. Solving a linear system of the form Ax = b is possible if the coefficient matrix A has certain characteristics such as being of full rank. Linear systems in which the coefficient matrix A is a Z-matrix can be found in many processes used in different applied fields from engineering to economics. For instance, they can be found when approximating the solution of a partial differential equation (PDE) by finite difference methods. Usually, the resulting linear systems are very large and using direct methods is impractical. Then it is necessary to use iterative methods. The success of iterative methods depends on the condition number of the system. The condition number is the maximum ratio of the relative error in x divided by the relative error in b. This is a measurement that relates the behavior of the system given small changes on its right ii hand side (RHS). If the condition number in a linear system is small then it is said that the system is well-conditioned, otherwise it is ill-conditioned. In many cases, large linear systems with Z-matrices as coefficient matrices have large condition numbers, which can cause iterative methods to fail. To alleviate this problem, a technique known as preconditioning is used. Basically, preconditioning is any form of modification of an original linear system that produces an equivalent system that is faster to solve than the original system. The Gauss-Seidel is one of the most reliable and oldest iterative methods for solving linear systems, but it tends to converge slowly for ill-conditioned systems. In 2002, Hisashi Kotakemori [9] proposed a preconditioner for the Gauss-Seidel of the form P = I + Smax for the particular case where the coefficient matrix is a diagonally dominant Z-matrix, with unit diagonal elements. Then, the problem Ax = b is changed to an equivalent one, which is P Ax = P b. This thesis will investigate the properties of the preconditioner P = I + Smax. As it will be shown, using this preconditioner preserves the convergence characteristics of the problem and keeps P A as a Z-matrix. Then, based on these properties, a new preconditioner will be proposed based on P, which can be used for diagonally dominant Z-matrices with positive diagonal elements, not only for unit diagonal elements. In addition, this new preconditioner can be used iteratively to improve the convergence characteristics of the problem.