Note on error bounds for linear complementarity problems of Nekrasov matrices

Abstract

García-Esnaola and Peña (Numer. Algor. 67, 655–667, 2014) presented an error bound involving a parameter for linear complementarity problems of Nekrasov matrices. This bound is not effective in some cases because it tends to infinity when the involved parameter tends to zero. In this paper, the optimal value of this error bound is determined completely by using the monotonicity of functions of this parameter. Numerical examples are given to verify the corresponding results.

Appendices

Appendix A: 32 cases

Appendix B: The values of $\inf \limits _{\varepsilon \in (0,\tilde {w}_0) } f(\varepsilon )$ and some related notations

In the table above, some related notations are listed as follows:

$$\begin{array}{ll} \rho_{bnd}^{(1)}:=\frac{\frac{h_{n}(M)}{m_{nn}}+\tilde{s}_{i_{n-1}}}{\tilde{s}_{i_{n-1}}m_{nn}},&\rho_{bnd}^{(2)}:=\frac{h_{n}(M)+w_{i_{n-1}} }{m_{nn}w_{i_{n-1}}}, \\ \rho_{bnd}^{(3)}:= \frac{\frac{h_{n}(M)}{m_{nn}}+\tilde{s}_{i_{n-1}}}{ \min\limits_{1\leq i\leq n-1} \{\bar{s}_{i}- \tilde{s}_{i_{n-1}} |m_{in}|\}},&\rho_{bnd}^{(4)}:=\frac{ \frac{h_{n}(M)}{m_{nn}}+\tilde{s}_{i_{n-1}}}{w_{i_{n-1}}},\\ \rho_{bnd}^{(5)}:=\frac{1}{m_{nn}-h_{n}(M)},&\rho_{bnd}^{(6)}:=\frac{w_{i_{1}}}{\tilde{s}_{i_{n-1}} m_{nn}},\\ \rho_{bnd}^{(7)}:=\frac{w_{i_{1}}}{\frac{h_{n}(M)}{m_{nn}}+\tilde{s}_{i_{n-1}}},&\rho_{bnd}^{(8)}:=\frac{w_{i_{1}}}{w_{i_{n-1}}},\\\rho_{bnd}^{(9)}:=\frac{w_{i_{1}}}{\min\limits_{1\leq i\leq n-1}\{\bar{s}_{i}-\tilde{s}_{i_{n-1}} |m_{in}|\}},&\rho_{bnd}^{(10)}:=\frac{w_{i_{1}}}{\min\limits_{1\leq i\leq n-1}\frac{\bar{s}_{i}-\frac{h_{n}(M)}{m_{nn}}}{1+|m_{in}|}+\frac{h_{n}(M)}{m_{nn}}},\\ \rho_{bnd}^{(11)}:=\frac{w_{i_{1}}}{\min\limits_{1\leq i\leq n-1} \{\bar{s}_{i}-(w_{i_{n-1}}-\frac{h_{n}(M)}{m_{nn}}) |m_{in}|\}},&\rho_{bnd}^{(12)}=\frac{ w_{i_{1}}}{\min\limits_{1\leq i\leq n-1} \{\bar{s}_{i}-(w_{i_{1}}-\frac{h_{n}(M)}{m_{nn}}) |m_{in}|\}},\\ \rho_{bnd}^{(13)}=\frac{w_{i_{1}}}{m_{nn}w_{i_{n-1}}-h_{n}(M)},&\rho_{bnd}^{(14)}=\frac{w_{i_{1}}}{m_{nn}w_{i_{1}}-h_{n}(M)},\\ \rho_{bnd}^{(15)}=\frac{w_{i_{n-1}}-h_{n}(M)}{\min\limits_{1\leq i\leq n-1 } \{m_{nn}\bar{s}_{i}-w_{i_{n-1}}|m_{in}|\}},&\rho_{bnd}^{(16)}=\frac{m_{nn}w_{i_{1}}}{w_{i_{n-1}}+h_{n}(M)} \end{array} $$

and

$$\rho_{bnd}^{(17)}=\frac{m_{nn}w_{i_{1}}}{\min\limits_{1\leq i\leq n-1 } \{{m_{nn}\bar{s}_{i}-w_{i_{n-1}}}|m_{in}|\}}.$$

Remark here that for a given Nekrasov matrix M, each $\rho _{bnd}^{(i)}$ for i = 1, 2,…, 17 depends only on the entries of M, and thus is computable.