The direct-projection methods

Abstract

In this paper, the direct-projection method given by Benzi and Meyer (1995) is derived by slightly different way - Basic Solutions of corresponding homogeneous system. The idea of the method differs from the idea of the Gaussian Elimination (or LU decomposition). The method works for every nonsingular coefficient matrix in the absence of rounding error. The Gaussian Elimination is explained by the method. The corresponding numerical algorithms of the method are also given. The method is applied to find the preconditioner for least squares problems and solvers of singular systems. The numerical experiments illustrate that the method has better numerical stability than the Gaussian Elimination.

Keywords:

• Direct-Projection Method
• Basic Solutions
• Gaussian Elimination
• Least Squares Problem
• Preconditioner
• Singular System
• Rank Determination
• Markov Chain

AMS Subjective Classification:

• 65F10

C.R. Categories:

• G.1.3

^∗The work of the first author was supported by NNSFLNC/962108 [email protected].

^†The work of the second author was supported by FUNPAR, Brazil. Deceased.

^‡The work of the third author was supported by grant 301035193-8 of CNPq, Brazil. [email protected] [email protected].

^∗The work of the first author was supported by NNSFLNC/962108 [email protected].

^†The work of the second author was supported by FUNPAR, Brazil. Deceased.

^‡The work of the third author was supported by grant 301035193-8 of CNPq, Brazil. [email protected] [email protected].

Notes

^∗The work of the first author was supported by NNSFLNC/962108 [email protected].

^†The work of the second author was supported by FUNPAR, Brazil. Deceased.

^‡The work of the third author was supported by grant 301035193-8 of CNPq, Brazil. [email protected] [email protected].