An extension of the I + S_max preconditioner for
the Gauss-Seidel method

ISNARDO ARENAS^*, PAUL CASTILLO, XUERONG YONG
University of Puerto Rico, Department of Mathematical Sciences, Mayagüez, Puerto Rico, 00681, US.

Abstract. A preconditioning technique based on the application of a fixed but arbitrary number of I +S_max steps is proposed. A reduction of the spectral radius of the Gauss-Seidel iteration matrix is theoretically analyzed for diagonally dominant Z-matrices. In particular, it is shown that after a finite number of steps this matrix reduces to null matrix. To illustrate the performance of the proposed technique numerical experiments on a wide variety of matrices are presented. Point and block versions of the preconditioner are numerically studied.
Keywords: Preconditioning, Gauss-Seidel method, regular splitting, point and block preconditioners.
MSC2010: XX00YY, AA99BB

Una extensión del precondicionador I + S_max para el
método de Gauss-Seidel

Resumen. Se propone una técnica de precondicionamiento para el método de Gauss-Seidel basada en la aplicación de una cantidad de pasos arbitrarios pero fijos del precondicionador I+S_max. Se analiza de manera teórica la reducción del radio espectral de la matriz de iteración del método de Gauss-Seidel para Z-matrices diagonalmente dominantes. En particular, se demuestra que después de un número finito de pasos esta matriz se reduce a una matriz nula. Para ilustrar la eficacia de la técnica propuesta se presentan experimentos numéricos para una amplia variedad de matrices. Se estudian numéricamente versiones puntuales y de bloques del precondicionador.
Palabras claves: Precondicionamiento, método Gauss-Seidel, descomposiciones regulares, precondicionadores de punto y bloque.

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^*Corresponding author: E-mail: isnardo.arenas@upr.edu.
Received: 03 December 2012, Accepted: 23 March 2013.