- Email: [email protected]
Erratum to: “Convergence of the preconditioned AOR method for irreducible L-matrices” [Appl. Math. Comput. 201 (2008) 56–64]
Applied Mathematics and Computation 212 (2009) 551–552
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Erratum
Erratum to: ‘‘Convergence of the preconditioned AOR method for irreducible L-matrices” [Appl. Math. Comput. 201 (2008) 56–64] q Shi-Liang Wu *, Ting-Zhu Huang School of Applied Mathematics, University of Electronic Science and Technology of China, Chengdu, Sichuan 610054, PR China
Recently, in [1], Yun et al. presented the following results. Lemma 1. Let A 2 Rnn be an L-matrix. Suppose that there exists a nonempty set a N ¼ f1; 2; . . . ; n 1g such that ai;iþ1 – 0 for all i 2 a and ai;iþ1 aiþ1;i < 1 for all i 2 N. Let Lrw and Lrw be defined by (3) and (8). If 0 6 r 6 w 6 1ðw – 0Þðr – 1Þ, A is irreducible even for ai;iþ1 set to 0 for every i 2 a, then Lrw and Lrw are nonnegative and irreducible. Theorem 1. Let A 2 Rnn be an L-matrix. Suppose that there exists a nonempty set a N ¼ f1; 2; . . . ; n 1g such that ai;iþ1 – 0 for all i 2 a and ai;iþ1 aiþ1;i < 1 for all i 2 N. Let Lrw and Lrw be defined by (3) and (8). If 0 6 r 6 w 6 1ðw – 0Þðr – 1Þ, A is irreducible even for ai;iþ1 set to 0 for every i 2 a, then (1) qðLrw Þ < qðLrw Þ, if qðLrw Þ < 1; (2) qðLrw Þ ¼ qðLrw Þ, if qðLrw Þ ¼ 1; (3) qðLrw Þ > qðLrw Þ, if qðLrw Þ > 1. However, in fact, Lemma 1 and Theorem 1 do not hold. To this end, the following Example 1 is given to show that Lwr may be reducible under Lemma 1 and Theorem 1. Example 1. Consider the following matrix A
3 1 0:5 0 0 0 7 6 1 0:2 0:4 0 7 6 0 7 6 A¼6 0 1 0:2 0 7 7: 6 0 7 6 0 0:3 1 0:2 5 4 0 0:1 0 0 0 1 2
It is clear that A is an irreducible L-matrix and a ¼ f2g. Obviously, a23 ¼ 0:2 – 0, a12 a21 ¼ a23 a32 ¼ a45 a54 ¼ 0 and a34 a43 ¼ 0:06 < 1. If a23 sets to 0, i.e., a23 ¼ 0, then A is reduced to
3 1 0:5 0 0 0 7 6 1 0 0:4 0 7 6 0 7 6 A¼6 0 1 0:2 0 7 7: 6 0 7 6 0 0:3 1 0:2 5 4 0 0:1 0 0 0 1 2
DOI of original article: 10.1016/j.amc.2007.11.045 q
This research was supported by 973 Programs (2008CB317110), NSFC (10771030), the Scientific and Technological Key Project of the Chinese Education Ministry (107098), the Specialized Research Fund for the Doctoral Program of Chinese Universities (20070614001), Sichuan Province Project for Applied Basic Research (2008JY0052). * Corresponding author. E-mail address: [email protected] (S.-L. Wu). 0096-3003/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2009.02.046
552
S.-L. Wu, T.-Z. Huang / Applied Mathematics and Computation 212 (2009) 551–552
It is not difficult to get that A is still an irreducible. Then
2
1 6 6 0 6 A¼6 6 0 6 4 0:02 0:1
0
0
0:2
1
0
0:4
0
0:94
0
0 0:3
1
0
0
0
0
3
7 7 7 0:04 7 7: 7 0 5 1 0
Further, we obtain that A is reducible. Assume that 0 6 r 6 w 6 1ðw – 0Þðr – 1Þ, then we get that
2 Lwr
6 6 6 ¼6 6 6 4
1w
0
0
0
1w
0
0 wð1rÞ 50 wð1rÞ 10
0 0
1w 3rð1wÞ 100
0
þ 0
3ðwrÞ 10
w 5 2w 5
0
3
0
0
2w 47
wr 1 w þ 250
3wr 2350
7 7 7 7; 7 7 5
wr 50
1w
which is clearly nonnegative, but reducible. This is a contradiction with Lemma 1 and Theorem 1. Subsequently, the following correct results corresponding to Lemma 1 and Theorem 1 are presented, and their proofs which are almost the same as those of Lemma 1 and Theorem 1 are omitted here. Lemma 2. Let A 2 Rnn be an L-matrix. Let N ¼ f1; 2; . . . ; n 1g and a ¼ fi 2 Njai;iþ1 – 0g. Suppose that a is nonempty and ai;iþ1 aiþ1;i < 1 for all i 2 N. Let Lrw and Lrw be defined by (3) and (8). If 0 6 r 6 w 6 1ðw – 0Þðr – 1Þ and A is irreducible even for ai;iþ1 set to 0 for every i 2 a, then Lrw and Lrw are nonnegative and irreducible. Theorem 2. Let A 2 Rnn be an L-matrix. Let N ¼ f1; 2; . . . ; n 1g and a ¼ fi 2 Njai;iþ1 – 0g. Suppose that a is nonempty and ai;iþ1 aiþ1;i < 1 for all i 2 N. Let Lrw and Lrw be defined by (3) and ð8Þ. If 0 6 r 6 w 6 1ðw – 0Þðr – 1Þ and A is irreducible even for ai;iþ1 set to 0 for every i 2 a, then (1) qðLrw Þ < qðLrw Þ, if qðLrw Þ < 1; (2) qðLrw Þ ¼ qðLrw Þ, if qðLrw Þ ¼ 1; (3) qðLrw Þ > qðLrw Þ, if qðLrw Þ > 1. Acknowledgement The authors would like to thank the referee very much for his extremely helpful suggestions for improving the presentation of this manuscript. Reference [1] J.H. Yun, S.W. Kim, Convergence of the preconditioned AOR method for irreducible L-matrices, Appl. Math. Comput. 201 (2008) 56–64.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Erratum
Erratum to: ‘‘Convergence of the preconditioned AOR method for irreducible L-matrices” [Appl. Math. Comput. 201 (2008) 56–64] q Shi-Liang Wu *, Ting-Zhu Huang School of Applied Mathematics, University of Electronic Science and Technology of China, Chengdu, Sichuan 610054, PR China
Recently, in [1], Yun et al. presented the following results. Lemma 1. Let A 2 Rnn be an L-matrix. Suppose that there exists a nonempty set a N ¼ f1; 2; . . . ; n 1g such that ai;iþ1 – 0 for all i 2 a and ai;iþ1 aiþ1;i < 1 for all i 2 N. Let Lrw and Lrw be defined by (3) and (8). If 0 6 r 6 w 6 1ðw – 0Þðr – 1Þ, A is irreducible even for ai;iþ1 set to 0 for every i 2 a, then Lrw and Lrw are nonnegative and irreducible. Theorem 1. Let A 2 Rnn be an L-matrix. Suppose that there exists a nonempty set a N ¼ f1; 2; . . . ; n 1g such that ai;iþ1 – 0 for all i 2 a and ai;iþ1 aiþ1;i < 1 for all i 2 N. Let Lrw and Lrw be defined by (3) and (8). If 0 6 r 6 w 6 1ðw – 0Þðr – 1Þ, A is irreducible even for ai;iþ1 set to 0 for every i 2 a, then (1) qðLrw Þ < qðLrw Þ, if qðLrw Þ < 1; (2) qðLrw Þ ¼ qðLrw Þ, if qðLrw Þ ¼ 1; (3) qðLrw Þ > qðLrw Þ, if qðLrw Þ > 1. However, in fact, Lemma 1 and Theorem 1 do not hold. To this end, the following Example 1 is given to show that Lwr may be reducible under Lemma 1 and Theorem 1. Example 1. Consider the following matrix A
3 1 0:5 0 0 0 7 6 1 0:2 0:4 0 7 6 0 7 6 A¼6 0 1 0:2 0 7 7: 6 0 7 6 0 0:3 1 0:2 5 4 0 0:1 0 0 0 1 2
It is clear that A is an irreducible L-matrix and a ¼ f2g. Obviously, a23 ¼ 0:2 – 0, a12 a21 ¼ a23 a32 ¼ a45 a54 ¼ 0 and a34 a43 ¼ 0:06 < 1. If a23 sets to 0, i.e., a23 ¼ 0, then A is reduced to
3 1 0:5 0 0 0 7 6 1 0 0:4 0 7 6 0 7 6 A¼6 0 1 0:2 0 7 7: 6 0 7 6 0 0:3 1 0:2 5 4 0 0:1 0 0 0 1 2
DOI of original article: 10.1016/j.amc.2007.11.045 q
This research was supported by 973 Programs (2008CB317110), NSFC (10771030), the Scientific and Technological Key Project of the Chinese Education Ministry (107098), the Specialized Research Fund for the Doctoral Program of Chinese Universities (20070614001), Sichuan Province Project for Applied Basic Research (2008JY0052). * Corresponding author. E-mail address: [email protected] (S.-L. Wu). 0096-3003/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2009.02.046
552
S.-L. Wu, T.-Z. Huang / Applied Mathematics and Computation 212 (2009) 551–552
It is not difficult to get that A is still an irreducible. Then
2
1 6 6 0 6 A¼6 6 0 6 4 0:02 0:1
0
0
0:2
1
0
0:4
0
0:94
0
0 0:3
1
0
0
0
0
3
7 7 7 0:04 7 7: 7 0 5 1 0
Further, we obtain that A is reducible. Assume that 0 6 r 6 w 6 1ðw – 0Þðr – 1Þ, then we get that
2 Lwr
6 6 6 ¼6 6 6 4
1w
0
0
0
1w
0
0 wð1rÞ 50 wð1rÞ 10
0 0
1w 3rð1wÞ 100
0
þ 0
3ðwrÞ 10
w 5 2w 5
0
3
0
0
2w 47
wr 1 w þ 250
3wr 2350
7 7 7 7; 7 7 5
wr 50
1w
which is clearly nonnegative, but reducible. This is a contradiction with Lemma 1 and Theorem 1. Subsequently, the following correct results corresponding to Lemma 1 and Theorem 1 are presented, and their proofs which are almost the same as those of Lemma 1 and Theorem 1 are omitted here. Lemma 2. Let A 2 Rnn be an L-matrix. Let N ¼ f1; 2; . . . ; n 1g and a ¼ fi 2 Njai;iþ1 – 0g. Suppose that a is nonempty and ai;iþ1 aiþ1;i < 1 for all i 2 N. Let Lrw and Lrw be defined by (3) and (8). If 0 6 r 6 w 6 1ðw – 0Þðr – 1Þ and A is irreducible even for ai;iþ1 set to 0 for every i 2 a, then Lrw and Lrw are nonnegative and irreducible. Theorem 2. Let A 2 Rnn be an L-matrix. Let N ¼ f1; 2; . . . ; n 1g and a ¼ fi 2 Njai;iþ1 – 0g. Suppose that a is nonempty and ai;iþ1 aiþ1;i < 1 for all i 2 N. Let Lrw and Lrw be defined by (3) and ð8Þ. If 0 6 r 6 w 6 1ðw – 0Þðr – 1Þ and A is irreducible even for ai;iþ1 set to 0 for every i 2 a, then (1) qðLrw Þ < qðLrw Þ, if qðLrw Þ < 1; (2) qðLrw Þ ¼ qðLrw Þ, if qðLrw Þ ¼ 1; (3) qðLrw Þ > qðLrw Þ, if qðLrw Þ > 1. Acknowledgement The authors would like to thank the referee very much for his extremely helpful suggestions for improving the presentation of this manuscript. Reference [1] J.H. Yun, S.W. Kim, Convergence of the preconditioned AOR method for irreducible L-matrices, Appl. Math. Comput. 201 (2008) 56–64.