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December  2021, 11(4): 513-531. doi: 10.3934/naco.2020053

A primal-dual interior point method for $ P_{\ast }\left( \kappa \right) $-HLCP based on a class of parametric kernel functions

Nadia Hazzam and  Zakia Kebbiche

Laboratoire de Mathématiques Fondamentales et Numériques, Université Ferhat Abbas Sétif1. Sétif. Algérie

Received  June 2020 Revised  October 2020 Published  December 2021 Early access  November 2020

In an attempt to improve theoretical complexity of large-update methods, in this paper, we propose a primal-dual interior-point method for $ P_{\ast}\left( \kappa \right) $-horizontal linear complementarity problem. The method is based on a class of parametric kernel functions. We show that the corresponding algorithm has $ O\left( \left( 1+2\kappa \right) p^{2}n^{\frac{2+p}{2\left( 1+p\right) }}\log \frac{n}{\epsilon }\right) $ iteration complexity for large-update methods and we match the best known iteration bounds with special choice of the parameter $ p $ for $ P_{\ast }\left(\kappa \right) $-horizontal linear complementarity problem that is $ O\left(\left( 1+2\kappa \right) \sqrt{n}\log n\log \frac{n}{\epsilon }\right) $. We illustrate the performance of the proposed kernel function by some comparative numerical results that are derived by applying our algorithm on five kernel functions.

Keywords: Horizontal linear complementarity problem, P*(κ)-matrix, interior-point method, kernel function, complexity bound.
Mathematics Subject Classification: Primary: 90C33; Secondary: 90C51.
Citation: Nadia Hazzam, Zakia Kebbiche. A primal-dual interior point method for $ P_{\ast }\left( \kappa \right) $-HLCP based on a class of parametric kernel functions. Numerical Algebra, Control and Optimization, 2021, 11 (4) : 513-531. doi: 10.3934/naco.2020053
References:
[1]

M. Achache, Complexity analysis of a weighted-full-Newton step interior-point algorithm for $P_{\ast }\left(\kappa \right)$-LCP, RAIRO-Oper. Res., 50 (2016), 131-143.  doi: 10.1051/ro/2015020.

[2]

M. Achache and N. Tabchouche, Complexity analysis and numerical implementation of large-update interior-point methods for SDLCP based on a new parametric barrier kernel function, Optimization, 67 (2018), 1211-1230.  doi: 10.1080/02331934.2018.1462356.

[3]

S. Asadi and H. Mansouri, Polynomial interior-point algorithm for $P_{\ast }\left(\kappa \right)$-horizontal linear complementarity problems, Numer. Algorithms, 63 (2013), 385-398.  doi: 10.1007/s11075-012-9628-0.

[4]

S. Asadi, H. Mansouri and M. Zangiabadi, A class of path-following interior-point methods for $P_{\ast }\left(\kappa \right)$-horizontal linear complementarity problems, J. Oper Res Soc China, 3 (2015), 17-30.  doi: 10.1007/s40305-015-0070-6.

[5]

S. Asadi, M. Zangiabadi and H. Mansouri, An infeasible interior-point algorithm with full-Newton steps for $P_{\ast }\left(\kappa \right)$-horizontal linear complementarity problems based on a kernel function, J. Appl. Math. Comput., 50 (2016), 15-37.  doi: 10.1007/s12190-014-0856-4.

[6]

Y. Q. Bai, M. El Ghami and C. Roos, A new efficient large-update primal-dual interior-point method based on a finite barrier, SIAM J. Optim., 13 (2003), 766-782.  doi: 10.1137/S1052623401398132.

[7]

Y. Q. Bai, M. El Ghami and C. Roos, A comparative study of kernel functions for primal-dual interior-point algorithms in linear optimization, SIAM J. Optim., 15 (2004), 101-128.  doi: 10.1137/S1052623403423114.

[8]

Y. Q. Bai and C. Roos, A primal-dual interior-point method based on a new kernel function with linear growth rate, Proceedings of the 9th Australian Optimization Day, 2002.

[9]

M. Bouafia, D. Benterki and A. Yassine, An efficient primal–dual interior point method for linear programming problems based on a new kernel function with a trigonometric barrier term, J. Optim. Theory Appl., 170 (2016), 528-545.  doi: 10.1007/s10957-016-0895-0.

[10]

Ch. Chennouf, Extension d'une Méthode de Point Intérieur au Problème Complémentaire Linéaire avec $P_{\ast }\left(\kappa \right)-$matrice, Mémoire de Master $2, $ Université Ferhat Abbas Sétif1, Algérie, 2018.

[11]

G. M. Cho and M. K. Kim, A new large-update interior point algorithm for $P_{\ast}\left(\kappa \right)$ LCPs based on kernel functions, Appl. Math. Comput., 182 (2006), 1169-1183.  doi: 10.1016/j.amc.2006.04.060.

[12]

G. M. Cho, A new large–update interior point algorithm for $P_{\ast}\left(\kappa \right)$ linear complementarity problems, J. Comput. Appl. Math., 216 (2008), 256-278.  doi: 10.1016/j.cam.2007.05.007.

[13]

G. M. Cho, Large–update interior point algorithm for $P_{\ast }$-linear complementarity problem, J. Inequalities Appl., 363 (2014), 1-12.  doi: 10.1186/1029-242X-2014-363.

[14] R. W. Cottle, J. S. Pang and R. E. Stone, The Linear Complementarity Problem, Academic Press, San Diego, 1992. 
[15]

M. El Ghami and T. Steihaug, Kernel–function based primal-dual algorithms for $P_{\ast }\left(\kappa \right) $ linear comlementarity problem, RAIRO-Oper. Res., 44 (2010), 185-205.  doi: 10.1051/ro/2010014.

[16]

M. El Ghami and G. Q. Wang, Interior–point methods for $P_{\ast}\left(\kappa \right) $ linear comlementarity problem based on generalized trigonometric barrier function, International Journal of Applied Mathematics, (2017), 11–33. doi: 10.12732/ijam.v30i1.2.

[17]

S. Fathi-Hafshejani and A. Fakharzadeh Jahromi, An interior point method for $P_{\ast}\left(\kappa \right) $-horizontal linear complementarity problem based on a new proximity function, J. Appl. Math. Comput., 62 (2020), 281-300.  doi: 10.1007/s12190-019-01284-9.

[18]

S. Fathi-Hafshejani, H. Mansouri and M. Peyghamic, An interior-point algorithm for $P_{\ast }\left(\kappa \right) $-linear complementarity problem based on a new trigonometric kernel functions, Journal of Mathematical Modeling, 5 (2017), 171-197. 

[19]

P. Ji, M. Zhang and X. Li, A Primal-dual large-update interior-point algorithm for $P_{\ast}(\kappa)$-LCP based on a new class of kernel functions, Acta Mathematicae Applicatae Sinica, English Series, 34 (2018), 119-134.  doi: 10.1007/s10255-018-0729-y.

[20]

M. Kojima, N. Megiddo, T. Noma and A. Yoshise, A unified approach to interior point algorithms for linear complementarity problems: A summary, Operations Research Letters, 10 (1991), 247-254.  doi: 10.1016/0167-6377(91)90010-M.

[21]

Y. Lee, Y. Cho and G. Cho, Kernel function based interior-point methods for horizontal linear complementarity problems, Journal of Inequalities and Applications, (2013), Article number: 215. doi: 10.1186/1029-242X-2013-215.

[22]

G. Lesaja and C. Roos, Unified analysis of kernel–based interior–point methods for $P_{\ast } (\kappa) $-LCP, SIAM Journal on Optimization, 20 (2010), 3014-3039.  doi: 10.1137/090766735.

[23]

J. Peng, C. Roos and T. Terlaky, Self-regular functions and new search directions for linear and semidefinite optimization, Math. Program., Ser., 93 (2002), 129-171.  doi: 10.1007/s101070200296.

[24]

Z. G. Qian and Y. Q. Bai, Primal-dual interior point algorithm with dynamic step-size based on kernel function for linear programming, Journal of Shanghai University (English Edition), 9 (2005), 391-396.  doi: 10.1007/s11741-005-0021-2.

[25]

M. Reza Peyghami, S. Fathi Hafshejani and L. Shirvani, Complexity of interior-point methods for linear optimization based on a new trigonometric kernel function, Journal of Computational and Applied Mathematics, 2 (2014), 74-85.  doi: 10.1016/j.cam.2013.04.039.

[26]

C. Roos, T. Terlaky and J. Ph. Vial, Theory and Algorithms for Linear optimization. An Interior Point Approach, John Wiley and Sons, Chichester, 1997.

[27]

G. Q. Wang and Y. Q. Bai, Polynomial interior-point algorithm for $P_{\ast}\left(\kappa \right)$ horizontal linear complementarity problem, J. Comput. Appl. Math., 233 (2009), 248-263.  doi: 10.1016/j.cam.2009.07.014.

[28]

S. J. Wright, Primal-Dual Interior Point Methods, SIAM, 1997. doi: 10.1137/1.9781611971453.

show all references

References:
[1]

M. Achache, Complexity analysis of a weighted-full-Newton step interior-point algorithm for $P_{\ast }\left(\kappa \right)$-LCP, RAIRO-Oper. Res., 50 (2016), 131-143.  doi: 10.1051/ro/2015020.

[2]

M. Achache and N. Tabchouche, Complexity analysis and numerical implementation of large-update interior-point methods for SDLCP based on a new parametric barrier kernel function, Optimization, 67 (2018), 1211-1230.  doi: 10.1080/02331934.2018.1462356.

[3]

S. Asadi and H. Mansouri, Polynomial interior-point algorithm for $P_{\ast }\left(\kappa \right)$-horizontal linear complementarity problems, Numer. Algorithms, 63 (2013), 385-398.  doi: 10.1007/s11075-012-9628-0.

[4]

S. Asadi, H. Mansouri and M. Zangiabadi, A class of path-following interior-point methods for $P_{\ast }\left(\kappa \right)$-horizontal linear complementarity problems, J. Oper Res Soc China, 3 (2015), 17-30.  doi: 10.1007/s40305-015-0070-6.

[5]

S. Asadi, M. Zangiabadi and H. Mansouri, An infeasible interior-point algorithm with full-Newton steps for $P_{\ast }\left(\kappa \right)$-horizontal linear complementarity problems based on a kernel function, J. Appl. Math. Comput., 50 (2016), 15-37.  doi: 10.1007/s12190-014-0856-4.

[6]

Y. Q. Bai, M. El Ghami and C. Roos, A new efficient large-update primal-dual interior-point method based on a finite barrier, SIAM J. Optim., 13 (2003), 766-782.  doi: 10.1137/S1052623401398132.

[7]

Y. Q. Bai, M. El Ghami and C. Roos, A comparative study of kernel functions for primal-dual interior-point algorithms in linear optimization, SIAM J. Optim., 15 (2004), 101-128.  doi: 10.1137/S1052623403423114.

[8]

Y. Q. Bai and C. Roos, A primal-dual interior-point method based on a new kernel function with linear growth rate, Proceedings of the 9th Australian Optimization Day, 2002.

[9]

M. Bouafia, D. Benterki and A. Yassine, An efficient primal–dual interior point method for linear programming problems based on a new kernel function with a trigonometric barrier term, J. Optim. Theory Appl., 170 (2016), 528-545.  doi: 10.1007/s10957-016-0895-0.

[10]

Ch. Chennouf, Extension d'une Méthode de Point Intérieur au Problème Complémentaire Linéaire avec $P_{\ast }\left(\kappa \right)-$matrice, Mémoire de Master $2, $ Université Ferhat Abbas Sétif1, Algérie, 2018.

[11]

G. M. Cho and M. K. Kim, A new large-update interior point algorithm for $P_{\ast}\left(\kappa \right)$ LCPs based on kernel functions, Appl. Math. Comput., 182 (2006), 1169-1183.  doi: 10.1016/j.amc.2006.04.060.

[12]

G. M. Cho, A new large–update interior point algorithm for $P_{\ast}\left(\kappa \right)$ linear complementarity problems, J. Comput. Appl. Math., 216 (2008), 256-278.  doi: 10.1016/j.cam.2007.05.007.

[13]

G. M. Cho, Large–update interior point algorithm for $P_{\ast }$-linear complementarity problem, J. Inequalities Appl., 363 (2014), 1-12.  doi: 10.1186/1029-242X-2014-363.

[14] R. W. Cottle, J. S. Pang and R. E. Stone, The Linear Complementarity Problem, Academic Press, San Diego, 1992. 
[15]

M. El Ghami and T. Steihaug, Kernel–function based primal-dual algorithms for $P_{\ast }\left(\kappa \right) $ linear comlementarity problem, RAIRO-Oper. Res., 44 (2010), 185-205.  doi: 10.1051/ro/2010014.

[16]

M. El Ghami and G. Q. Wang, Interior–point methods for $P_{\ast}\left(\kappa \right) $ linear comlementarity problem based on generalized trigonometric barrier function, International Journal of Applied Mathematics, (2017), 11–33. doi: 10.12732/ijam.v30i1.2.

[17]

S. Fathi-Hafshejani and A. Fakharzadeh Jahromi, An interior point method for $P_{\ast}\left(\kappa \right) $-horizontal linear complementarity problem based on a new proximity function, J. Appl. Math. Comput., 62 (2020), 281-300.  doi: 10.1007/s12190-019-01284-9.

[18]

S. Fathi-Hafshejani, H. Mansouri and M. Peyghamic, An interior-point algorithm for $P_{\ast }\left(\kappa \right) $-linear complementarity problem based on a new trigonometric kernel functions, Journal of Mathematical Modeling, 5 (2017), 171-197. 

[19]

P. Ji, M. Zhang and X. Li, A Primal-dual large-update interior-point algorithm for $P_{\ast}(\kappa)$-LCP based on a new class of kernel functions, Acta Mathematicae Applicatae Sinica, English Series, 34 (2018), 119-134.  doi: 10.1007/s10255-018-0729-y.

[20]

M. Kojima, N. Megiddo, T. Noma and A. Yoshise, A unified approach to interior point algorithms for linear complementarity problems: A summary, Operations Research Letters, 10 (1991), 247-254.  doi: 10.1016/0167-6377(91)90010-M.

[21]

Y. Lee, Y. Cho and G. Cho, Kernel function based interior-point methods for horizontal linear complementarity problems, Journal of Inequalities and Applications, (2013), Article number: 215. doi: 10.1186/1029-242X-2013-215.

[22]

G. Lesaja and C. Roos, Unified analysis of kernel–based interior–point methods for $P_{\ast } (\kappa) $-LCP, SIAM Journal on Optimization, 20 (2010), 3014-3039.  doi: 10.1137/090766735.

[23]

J. Peng, C. Roos and T. Terlaky, Self-regular functions and new search directions for linear and semidefinite optimization, Math. Program., Ser., 93 (2002), 129-171.  doi: 10.1007/s101070200296.

[24]

Z. G. Qian and Y. Q. Bai, Primal-dual interior point algorithm with dynamic step-size based on kernel function for linear programming, Journal of Shanghai University (English Edition), 9 (2005), 391-396.  doi: 10.1007/s11741-005-0021-2.

[25]

M. Reza Peyghami, S. Fathi Hafshejani and L. Shirvani, Complexity of interior-point methods for linear optimization based on a new trigonometric kernel function, Journal of Computational and Applied Mathematics, 2 (2014), 74-85.  doi: 10.1016/j.cam.2013.04.039.

[26]

C. Roos, T. Terlaky and J. Ph. Vial, Theory and Algorithms for Linear optimization. An Interior Point Approach, John Wiley and Sons, Chichester, 1997.

[27]

G. Q. Wang and Y. Q. Bai, Polynomial interior-point algorithm for $P_{\ast}\left(\kappa \right)$ horizontal linear complementarity problem, J. Comput. Appl. Math., 233 (2009), 248-263.  doi: 10.1016/j.cam.2009.07.014.

[28]

S. J. Wright, Primal-Dual Interior Point Methods, SIAM, 1997. doi: 10.1137/1.9781611971453.

Table 1.  Some kernel functions
${\psi _1}(t) = p\left( {\frac{{{t^2} - 1}}{2}} \right) + \frac{4}{\pi }\left( {{{\tan }^p}\left( {\frac{\pi }{{2t + 2}}} \right) - 1} \right),t > 0,p \ge \sqrt 2 ,$
$\alpha_{1}=\frac{1}{(1+2 \kappa)\left(9 p+4 \pi p^{2}\right)\left(\frac{8}{p}+2\right)^{\frac{p+2}{p+1}}}.$
$\psi_{2}(t)=\frac{t^{2}-1}{2}-\log t+\frac{1}{8} \tan ^{2}\left(\frac{\pi(1-t)}{2+4 t}\right), t>0,[25]$
$\alpha_{2}=\frac{1}{(1+2 \kappa) 5020 \delta^{\frac{4}{3}}}.$
$\psi_{3}(t)=t-1+\frac{t^{1-q}-1}{q-1}, t>0, q \geq 2,[8]$
$\alpha_{3}=\frac{1}{(1+2 \kappa) q(2 \delta+1)^{\frac{1}{q}}(4 \delta+1)}.$
$\psi_{4}(t)=\frac{t^{2}-1}{2}+\frac{q^{\frac{1}{t}-1}}{q \log q}-\frac{q-1}{q}(t-1), q>1,[2]$
$\alpha_{4}=\frac{1}{(1+2 \kappa)(\log q+2)(1+4 \delta)\left(2+\frac{\log (1+4 \delta)}{\log q}\right)}.$
$\psi_{5}(t)=\frac{t^{2}-1}{2}+\frac{4}{p \pi}\left(\tan ^{p}\left(\frac{\pi}{2 t+2}\right)-1\right), t>0, p \geq 2,[9]$
$\alpha_{5}=\frac{1}{(1+2 \kappa)(9+4 p \pi)(8 \delta+2)^{\frac{p+2}{p+1}}}.$
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${\psi _1}(t) = p\left( {\frac{{{t^2} - 1}}{2}} \right) + \frac{4}{\pi }\left( {{{\tan }^p}\left( {\frac{\pi }{{2t + 2}}} \right) - 1} \right),t > 0,p \ge \sqrt 2 ,$
$\alpha_{1}=\frac{1}{(1+2 \kappa)\left(9 p+4 \pi p^{2}\right)\left(\frac{8}{p}+2\right)^{\frac{p+2}{p+1}}}.$
$\psi_{2}(t)=\frac{t^{2}-1}{2}-\log t+\frac{1}{8} \tan ^{2}\left(\frac{\pi(1-t)}{2+4 t}\right), t>0,[25]$
$\alpha_{2}=\frac{1}{(1+2 \kappa) 5020 \delta^{\frac{4}{3}}}.$
$\psi_{3}(t)=t-1+\frac{t^{1-q}-1}{q-1}, t>0, q \geq 2,[8]$
$\alpha_{3}=\frac{1}{(1+2 \kappa) q(2 \delta+1)^{\frac{1}{q}}(4 \delta+1)}.$
$\psi_{4}(t)=\frac{t^{2}-1}{2}+\frac{q^{\frac{1}{t}-1}}{q \log q}-\frac{q-1}{q}(t-1), q>1,[2]$
$\alpha_{4}=\frac{1}{(1+2 \kappa)(\log q+2)(1+4 \delta)\left(2+\frac{\log (1+4 \delta)}{\log q}\right)}.$
$\psi_{5}(t)=\frac{t^{2}-1}{2}+\frac{4}{p \pi}\left(\tan ^{p}\left(\frac{\pi}{2 t+2}\right)-1\right), t>0, p \geq 2,[9]$
$\alpha_{5}=\frac{1}{(1+2 \kappa)(9+4 p \pi)(8 \delta+2)^{\frac{p+2}{p+1}}}.$
Table 2.   
$ \psi _{1}\left( t\right) $ $ \psi _{2}\left( t\right) $ $ \psi _{3}\left( t\right) $ $ \psi _{4}\left( t\right) $ $ \psi _{5}\left( t\right) $
Inn $ 5 $ $ 282 $ $ 10 $ $ 16 $ $ 5 $
Time $ 0.07 $ $ 0.18 $ $ 0.062 $ $ 0.06 $ $ 0.06 $
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$ \psi _{1}\left( t\right) $ $ \psi _{2}\left( t\right) $ $ \psi _{3}\left( t\right) $ $ \psi _{4}\left( t\right) $ $ \psi _{5}\left( t\right) $
Inn $ 5 $ $ 282 $ $ 10 $ $ 16 $ $ 5 $
Time $ 0.07 $ $ 0.18 $ $ 0.062 $ $ 0.06 $ $ 0.06 $
Table 3.   
$ \psi _{1}\left( t\right) $ $ \psi _{2}\left( t\right) $ $ \psi _{3}\left( t\right) $ $ \psi _{4}\left( t\right) $ $ \psi _{5}\left( t\right) $
Inn $ 7 $ $ 167 $ $ 12 $ $ 19 $ $ 10 $
Time $ 0.06 $ $ 0.10 $ $ 0.06 $ $ 0.063 $ $ 0.07 $
Table Options

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$ \psi _{1}\left( t\right) $ $ \psi _{2}\left( t\right) $ $ \psi _{3}\left( t\right) $ $ \psi _{4}\left( t\right) $ $ \psi _{5}\left( t\right) $
Inn $ 7 $ $ 167 $ $ 12 $ $ 19 $ $ 10 $
Time $ 0.06 $ $ 0.10 $ $ 0.06 $ $ 0.063 $ $ 0.07 $
Table 4.   
$ n=3 $ $ n=100 $ $ n=150 $
Inn Time Inn Time Inn Time
$ \psi _{1}\left( t\right) $ $ 4 $ $ 0.06 $ $ 12 $ $ 0.45 $ $ 18 $ $ 3.06 $
$ \psi _{2}\left( t\right) $ $ 179 $ $ 0.11 $ $ 70 $ $ 9.07 $ $ 1447 $ $ 11.6 $
$ \psi _{3}\left( t\right) $ $ 15 $ $ 0.06 $ $ 52 $ $ 0.19 $ $ 77 $ $ 0.91 $
$ \psi _{4}\left( t\right) $ $ 14 $ $ 0.063 $ $ - $ $ - $ $ - $ $ - $
$ \psi _{5}\left( t\right) $ $ 8 $ $ 0.068 $ $ 23 $ $ 0.63 $ $ 28 $ $ 2.30 $
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$ n=3 $ $ n=100 $ $ n=150 $
Inn Time Inn Time Inn Time
$ \psi _{1}\left( t\right) $ $ 4 $ $ 0.06 $ $ 12 $ $ 0.45 $ $ 18 $ $ 3.06 $
$ \psi _{2}\left( t\right) $ $ 179 $ $ 0.11 $ $ 70 $ $ 9.07 $ $ 1447 $ $ 11.6 $
$ \psi _{3}\left( t\right) $ $ 15 $ $ 0.06 $ $ 52 $ $ 0.19 $ $ 77 $ $ 0.91 $
$ \psi _{4}\left( t\right) $ $ 14 $ $ 0.063 $ $ - $ $ - $ $ - $ $ - $
$ \psi _{5}\left( t\right) $ $ 8 $ $ 0.068 $ $ 23 $ $ 0.63 $ $ 28 $ $ 2.30 $
Table 5.   
$ n=3 $ $ n=50 $ $ n=150 $
Inn Time Inn Time Inn Time
$ \psi _{1}\left( t\right) $ $ 3 $ $ 0.07 $ $ 10 $ $ 0.12 $ $ 19 $ $ 1.73 $
$ \psi _{2}\left( t\right) $ $ 259 $ $ 0.15 $ $ 615 $ $ 0.67 $ $ 1301 $ $ 13.96 $
$ \psi _{3}\left( t\right) $ $ 16 $ $ 0.05 $ $ 215 $ $ 0.45 $ $ 71 $ $ 0.64 $
$ \psi _{4}\left( t\right) $ $ 18 $ $ 0.05 $ $ - $ $ - $ $ - $ $ - $
$ \psi _{5}\left( t\right) $ $ 12 $ $ 0.07 $ $ 18 $ $ 0.15 $ $ 39 $ $ 2.47 $
Table Options

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$ n=3 $ $ n=50 $ $ n=150 $
Inn Time Inn Time Inn Time
$ \psi _{1}\left( t\right) $ $ 3 $ $ 0.07 $ $ 10 $ $ 0.12 $ $ 19 $ $ 1.73 $
$ \psi _{2}\left( t\right) $ $ 259 $ $ 0.15 $ $ 615 $ $ 0.67 $ $ 1301 $ $ 13.96 $
$ \psi _{3}\left( t\right) $ $ 16 $ $ 0.05 $ $ 215 $ $ 0.45 $ $ 71 $ $ 0.64 $
$ \psi _{4}\left( t\right) $ $ 18 $ $ 0.05 $ $ - $ $ - $ $ - $ $ - $
$ \psi _{5}\left( t\right) $ $ 12 $ $ 0.07 $ $ 18 $ $ 0.15 $ $ 39 $ $ 2.47 $
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