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Volume 5, Issue 4
On the Minimum Property of the Pseudo $x$-Condition Number for a Linear Operator

Jiao-Xun Kuang

J. Comp. Math., 5 (1987), pp. 316-324.

Published online: 1987-05

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  • Abstract

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 $μ(T)＜x(T)$.
In this paper, an answer is given to this problem.

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@Article{JCM-5-316, author = {Jiao-Xun Kuang}, title = {On the Minimum Property of the Pseudo $x$-Condition Number for a Linear Operator}, journal = {Journal of Computational Mathematics}, year = {1987}, volume = {5}, number = {4}, pages = {316--324}, abstract = {

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 $μ(T)＜x(T)$.
In this paper, an answer is given to this problem.

}, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/9555.html} }
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 -

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 $μ(T)＜x(T)$.
In this paper, an answer is given to this problem.

Jiao-Xun Kuang. (1987). On the Minimum Property of the Pseudo $x$-Condition Number for a Linear Operator. Journal of Computational Mathematics. 5 (4). 316-324. doi:
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