Kernel function based interior-point methods for horizontal linear complementarity problems

Journal of Inequalities and Applications

Table 3 The third derivative of the kernel functions

j	$ψ_{j}^{(3)} (t)$
1	$- e^{p (g_{1} (t) - e)} g_{1} (t) t^{- 3 r - 3} (p^{2} r^{2} g_{1}^{2} (t) + 3 p r^{2} g_{1} (t) + r^{2} + (r + 1) t^{r} h_{1} (t))$ , where $h_{1} (t) = 3 r (p g_{1} (t) + 1) + (r + 2) t^{r}$
2	$- e^{p (g_{2} (t) - 1)} g_{2} (t) t^{- 3 r - 3} (p^{2} r^{2} g_{2}^{2} (t) + 3 p r^{2} g_{2} (t) + r^{2} + (r + 1) t^{r} h_{2} (t))$ , where $h_{2} (t) = 3 r (p g_{2} (t) + 1) + (r + 2) t^{r}$