Figure 2 - uploaded by Paul Castillo
Content may be subject to copyright.
Source publication
A preconditioning technique based on the application of a fixed
but arbitrary number of I + Smax 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. T...
Contexts in source publication
Context 1
... the next numerical experiment we consider an isotropic random permeability field as shown in Figure 2. The permeability tensor of cell i is of the form K i = 10 α i I, where α i is a random number in the interval [ − 6 , 1] . In this experiment the minimum and maximum values of α are 5 . 9674 and 0 . 9936 respectively. In Table 6 we present the iteration count for the preconditioned Gauss-Seidel method to reach a relative residual error of 10 − 6 in the l 2 norm. The initial guess is the null vector. Since this problem is more difficult to solve than the previous example, the ...
Similar publications
In this work, given a positive definite matrix A, we introduce a class of matrices related to A which is obtained by suitably combining projections of its powers onto algebras of matrices simultaneously diagonalized by a unitary transform. After a detailed theoretical study of some spectral properties of the matrices of this class - which suggests...
Citations
... Following this idea and using a different definition of matrix K, another preconditioner, namely I +S max was proposed in [7]. An extension of this preconditioner was recently proposed by the authors in [1]. The preconditioner is based on the application of a fixed but arbitrary number of I + S max steps. ...
... In this work a preconditioner, S, that applies Kotakemori's idea while preserving symmetry is introduced. As in [1], the proposed preconditioner can be applied a fixed but arbitrary number of steps. A block version is also numerically tested and compared against its point version. ...
... Following the idea presented in [1] we define a recursive preconditioner S [k] by applying a fixed but arbitrary number of times the preconditioner S as follows. Let A be a symmetric matrix; we set ...
A preconditioning technique to improve the convergence of the
Gauss-Seidel method applied to symmetric linear systems while preserving
symmetry is proposed. The preconditioner is of the form I + K and can be
applied an arbitrary number of times. It is shown that under certain conditions
the application of the preconditioner a finite number of steps reduces the
matrix to a diagonal. A series of numerical experiments using matrices from
spatial discretizations of partial differential equations demonstrates that both
versions of the preconditioner, point and block version, exhibit lower iteration
counts than its non-symmetric version.
Resumen. Se propone una técnica de precondicionamiento para mejorar la
convergencia del método Gauss-Seidel aplicado a sistemas lineales simétricos
pero preservando simetría. El precondicionador es de la forma I + K y
puede ser aplicado un número arbitrario de veces. Se demuestra que bajo ciertas
condiciones la aplicación del precondicionador un número finito de pasos
reduce la matriz del sistema precondicionado a una diagonal. Una serie de
experimentos con matrices que provienen de la discretización de ecuaciones
en derivadas parciales muestra que ambas versiones del precondicionador, por
punto y por bloque, muestran un menor número de iteraciones en comparación
con la versión que no preserva simetría.
