On the theory of deflation and singular symmetric positive semi-definite matrices

On the theory of deflation and singular symmetric positive semi-definite matrices

Tang, J.M.,
Vuik, C.

Electrical Engineering, Mathematics and Computer Science

2005

In this report we give new insights into the properties of invertible and singular deflated and preconditioned linear systems where the coefficient matrices are also symmetric and positive (semi-) definite. First we prove that the invertible de ated matrix has always a more favorable effeective condition number compared to the original matrix. So, in theory, the solution of the deflated linear system converges faster in iterative methods than the original one. Thereafter, some results are presented considering the singular systems originally from the Poisson equation with Neumann boundary conditions. In practice these linear systems are forced to be invertible leading to a worse (eective) condition number. We show that applying the deflation technique remedies this problem of a worse condition number. Moreover, we derive some useful equalities between the de ated variants of the singular and invertible matrices. Then we prove that the de ated singular matrix has always a more favorable effective condition number compared by the original matrix.

singularity
deflation
conjugate gradient method
preconditioning, Poisson equation
spectral analysis
symmetric positive semi-definite matrices

http://resolver.tudelft.nl/uuid:e85b1287-1460-4d93-aaa6-9f678603516c

Delft University of Technology, Faculty of Electrical Engineering, Mathematics and Computer Science, Delft Institute of Applied Mathematics

1389-6520

Reports of the Department of Applied Mathematical Analysis, 05-06

Institutional Repository

report

(c) 2005 Department of Applied Mathematical Analysis