TY - JOUR
T1 - On the Minimum Property of the Pseudo $x$-Condition Number for a Linear Operator
AU - Jiao-Xun Kuang
JO - Journal of Computational Mathematics
VL - 4
SP - 316
EP - 324
PY - 1987
DA - 1987/05
SN - 5
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/jcm/9555.html
KW - 
AB - <p style="text-align: justify;">It is well known that the $x$-condition number of a linear operator is a measure of ill condition with respect to its generalized inverses and a relative error bound with respect to the generalized inverses of operator $T$ with a small perturbation operator E, namely,$$\frac{\|(T+E)^+-T^+\|}{\|T^+\|}\leq \frac{x(T)\frac{\|E\|}{\|T\|}}{1-x(T)\frac{\|E\|}{\|T\|}},$$ where $x{T}=\|T\|\dot\|T^+\|$. The problem is whether there exists a positive number $μ(T)$ independent of $E$ but dependent on $T$ such that the above relative error bound holds and $μ(T)＜x(T)$.<br/>In this paper, an answer is given to this problem.</p>