@Article{JICS-2-77,
author = {},
title = {Preconditioned Conjugate Gradient is M-Error-Reducing},
journal = {Journal of Information and Computing Science},
year = {2024},
volume = {2},
number = {1},
pages = {77--80},
abstract = { The  Preconditioned  Conjugate Gradient (PCG) method has  proven  to be  extremely  powerful for 
solving symmetric positive definite linear systems. This method can also be applied to nonsymmetric linear 
systems  when  combined  with  the  NR/NE  techniques.  It  has  been  shown  in  [1]  that  the  CGNR  algorithm, 
which is a nonsymmetric variant of the Conjugate Gradient (CG) method, is error-reducing with respect to 
the Euclidean norm. However, in practice the simple CGNR algorithm is seldom used because of the squared 
condition number of the iteration matrix. Preconditioning is frequently needed to overcome this difficulty. In 
the present paper we give a much richer result concerning the error-reducing property of the CG procedure. 
Assume  that  the  preconditioner  M  is  also  symmetric  positive  definite.  It  is  shown  that  the  PCG  method  is 
error-reducing with respect to the M-norm. 
},
issn = {1746-7659},
doi = {https://doi.org/},
url = {http://global-sci.org/intro/article_detail/jics/22824.html}
}