THE Hadamard product of two nonnegative matrices and

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1 IAENG Internatonal Journal of Appled Mathematcs 46:3 IJAM_46_3_5 Some New Bounds for the Hadamard Product of a Nonsngular M-matrx and Its Inverse Zhengge Huang Lgong Wang and Zhong Xu Abstract Some new convergent sequences of the lower bounds of the mnmum egenvalue for the Hadamard product of a nonsngular M-matrx and an nverse M-matrx are gven Numercal examples show that these bounds could reach the true value of the mnmum egenvalue n some cases These bounds n ths paper mprove some exstng ones Index Terms sequences nonsngular M-matrx Hadamard product mnmum egenvalue lower bounds I INTRODUCTION THE Hadamard product of two nonnegatve matrces and the Fan product of two M-matrces are two specal matrces product In mathematcs many problems can be transformed nto computatonal problems of the Hadamard product and the Fan product such as numercal method for solvng volterra ntegral equatons wth a convoluton ernel ] solvng a system of Wener-Hopf Integral Equatons ] and so on The upper bounds for the spectral radus ρa B) of the Hadamard product of two nonnegatve matrces A and B the lower bounds for the mnmum egenvalue τa B ) of the Hadamard product of a M- matrx A and an nverse of M-matrx B are research hotspots n matrx theory researchng In ths paper by constructng sequence of teratons and combnng wth the slls of nequaltes scalng we wll conduct further research n the lower bounds for the mnmum egenvalue τa B ) of the Hadamard product of a M-matrx A and an nverse of M-matrx B In theory we prove that the sequences obtaned n ths paper are more accurate than the exstng ones Some results of the comparson are also consdered To llustrate our results some examples are gven For a postve nteger n N denotes the set N = n throughout The set of all n n real matrces s denoted by R n n and C n n denotes the set of all n n complex matrces Let Z n denote the set of n n real matrces all of whose off-dagonal entres are nonpostve A matrx A = a ) R n n s called an M-matrx 3] f there exst a nonnegatve matrx B and a nonnegatve real number λ such that A = λi B λ ρb) where I s the dentty matrx ρb) s the spectral radus of the matrx B If λ = ρb) then A s a sngular M-matrx; f λ > ρb) then A s a nonsngular M-matrx If C s Manuscrpt receved August 4 05; revsed October 05 Ths wor was supported by the Natonal Natural Scence Foundatons of Chna No773) Zhengge Huang s wth the Department of Appled Mathematcs Northwestern Polytechncal Unversty X an Shaanx 7007 PR Chna e-mal: ZhenggeHuang@malnwpueducn Lgong Wang e-mal: lgwang@nwpueducn) and Zhong Xu e-mal: zhongxu@nwpueducn) are wth the Department of Appled Mathematcs Northwestern Polytechncal Unversty X an Shaanx 7007 PR Chna a prncpal submatrx of A then C s also a nonsngular M-matrx Denote by M n the set of all n n nonsngular M-matrces Let us denote τa) = mnreλ) : λ σa) and σa) denotes the spectrum of A It nown that 4] τa) = ρa ) s a postve real egenvalue of A M n The Hadamard product of two matrces A = a ) n n and B = b ) n n s defned as the matrx A B = a b ) n n If A and B are nonsngular M-matrces then t s proved n 5] that A B s also a nonsngular M-matrx A matrx A s rreducble 6] 7] 8] f there does not exst a permutaton matrx P such that ) P AP T A A = 0 A where A and A are square matrces For α N denote by α the cardnalty of α and α = N α If α β N then Aα β) s the submatrx of A lyng n the rows ndcated by α and the columns ndcated by β In partcular Aα α) s abbrevated to Aα) Assume that Aα) s nonsngular Then A/α = A/Aα) = Aα ) Aα α)aα)] Aα α ) s called the Schur complement of A respect to Aα) 9] Let A = a ) R n n be a strctly row dagonally domnant matrx for any N t = denote R = a d = R a a + a d s = s = max a s ; a + a m = m = max a m ; a r = a a r = max r ; a + a r t = t = max a t ; a + a t u = u = max a u ; a l = max a t a t a + a t l w = a Advance onlne publcaton: 6 August 06)

2 IAENG Internatonal Journal of Appled Mathematcs 46:3 IJAM_46_3_5 w = max w q = mns t a h = max a q a q a + a q h v 0) = a p t) = a + h t) = max v t) = a + a a p t) a v t ) a a p t) ht) a p t) a p t) = max pt) Let A be a nonsngular M-matrx and A = α ) n n be a doubly stochastc matrx Before presentng our results we revew the exstng results that relate to lower bounds of the mnmum egenvalue for the Hadamard product of a nonsngular M-matrx and an nverse M-matrx as follows It was proved n 0] that 0 < τa A ) Fedler and Marham 5] gave a lower bound on τa A ): τa A ) n and conectured that τa A ) n Chen ] Song ] and Yong 3] have ndependently proved ths conecture L et al n 4] and L et al n 5] obtaned the followng results τa A ) mn N τa A ) mn N a s R + s a t R + t respectvely In 6] Cheng et al mproved the results n 4] and 5] showng that τa A ) mn τa A ) mn N a u R + u Recently Chen n 7] mproved the results n 4] and gave the followng result: a α + a α a α a α ) τa A ) mn ) m a α m a α )] In 8] Cheng et al proposed τa A ) ρ J A ) and n 9] L et al presented the followng result: a α + a α a α a α ) ) w a α w a α )] In the sequel Zhao et al 0] mproved the results n 4] 5] 6] and arrved at τa A a p t) R ) max mn N + a s mn N p t) a p t) s + p t) In ] Zhou et al proved the followng result: If A = a ) B = b ) R n n are nonsngular M-matrces and A = α ) n n then b t τb A ) mn N a b In ths paper we exhbt some new lower bounds for τa A ) and τb A ) These bounds mprove the results n 4] 5] 6] 7] 8] 9] 0] ] The rest of ths paper s organzed as follows In Secton II we propose some notatons and lemmas whch are useful n the followng proofs We propose some new lower bounds for τb A ) and τa A ) n Secton III Secton IV s devoted to some numercal experments to show that the advantages and precse of the new convergent sequences of the lower bounds of the mnmum egenvalue for the Hadamard product of a nonsngular M-matrx and an nverse M-matrx Fnally some concludng remars are gven II PRELIMINARIES In ths secton we start wth some notatons and lemmas that nvolve nequaltes for the entres of A and the strctly dagonally domnant matrx They wll be useful n the proofs Let A = a ) R n n For N t = denote a + a t l g 0) = = w a a + a g t ) f t) = a l t) = max g t) = a + a f t) a a f t) lt) a f t) a f t) = max t) f Lemma 7] Let A = a ) n n be a nonsngular M- matrx B = b ) n n Z n and A B Then B s a nonsngular M-matrx and A B O Lemma 7] Let A = a ) n n α N and assume that Aα) s nonsngular Then det A = det Aα) det A/α Lemma 3 0] If A = a ) R n n s a strctly row dagonally domnant nonsngular M-matrx then for all Advance onlne publcaton: 6 August 06)

3 IAENG Internatonal Journal of Appled Mathematcs 46:3 IJAM_46_3_5 N t = a) > q v 0) p ) v ) p ) v ) p t) v t) 0; b) h 0 h t) 0 Usng the same technque as the proof of Lemma n 0] we can obtan the Lemma 4 Lemma 4 If A = a ) R n n s a strctly row dagonally domnant nonsngular M-matrx then for all N t = a) > t g 0) f ) g ) f ) g ) f t) g t) 0; b) l 0 l t) 0 Lemma 5 0] Let A = a ) R n n be a strctly row dagonally domnant nonsngular M-matrx Then for A = b ) n n N t = 0 we have a + α a a v t) α = p t+) α Lemma 6 Let A = a ) R n n be a strctly row dagonally domnant nonsngular M-matrx Then for A = b ) n n we have α z t+) α z t+) α N t = 0 where = mnp t) f t) zt) = max zt) t = Proof Usng the same technques as the proof of Lemma n ] for N t = 0 we have α a + a a g t) α = f t+) α Then t follows from the above nequalty and Lemma 5 that α mnp t+) f t+) α = z t+) α for N t = 0 Remar By Lemma 3 we fnd that 0 h n vew of q = mns t s and by Lemma 3 for N t = we have a + a q h p t) v 0) = a a + a a = m Furthermore by Lemma 4 0 l then we have a + a t l f t) g 0) = a a + a t = w = u a By Theorem 33 n 4] 5] and the above two nequaltes for N t = we nfer that v 0) m s g 0) = w u t whch results n v 0) m s g 0) = w u t N Moreover nasmuch as = max zt) = maxzt) zt) zt) + zt) n maxp t) pt) pt) + pt) n = max pt) = pt) = max zt) = maxzt) zt) zt) + zt) n maxf t) f t) f t) + f t) n t) = maxf = f t) for N the result of Lemma 6 s sharper than the results of Theorem n 4] Lemma n 5] 6] 7] and Lemma n 9] 0] Lemma 7 Let A = a ) R n n be a strctly row dagonally domnant nonsngular M-matrx Then for A = α ) n n N t = we have a a a a α a a ) where = mnp t) f t) N Proof Let B = A Snce A s a nonsngular M-matrx B 0 Denote A s the submatrx of A obtaned by deletng the -th row and the -th column of A Then A s a nonsngular M-matrx and A daga a a ++ a nn ) = A Thus by Lemma and A beng a nonsngular M-matrx we have A daga a a ++ a nn) ) For N by Lemma 3 we have α = det A det A = det A det A det A/A = By Inequalty ) we deduce that Hence det A/A a a a a det A/A α = det A det A a a a N 3) a Combnng Lemma 7 and AB = I results n n = a α e = a α = a α a α a α = a ) a α α a N 4) a Advance onlne publcaton: 6 August 06)

4 IAENG Internatonal Journal of Appled Mathematcs 46:3 IJAM_46_3_5 By Inequaltes 3) and 4) the result s obtaned Remar Accordng to Remar for N t = we have s m t u w = g 0) pt) whch mples that a a a a a a a a m a a s a Moreover t s easy to get a a p t) a a g 0) = a a w a a u a a t a a a a N So the bounds of Lemma 7 are sharper than the ones of Theorem 5 n 4] Lemma 3 n 5] 6] 7] and Lemma 3 n 0] Lemma 8 ] If A s a doubly stochastc matrx then Ae = e A T e = e where e = ) T Lemma 9 7] Let A = a ) R n n be a nonsngular M-matrx Then 0 τa) a for all N Lemma 0 3] Let A = a ) C n n and x x x n be postve real numbers Then all the egenvalues of A le n the regon n z C : z a z a = x ) a x x ) a x Lemma 7] Let A Z n A s a nonsngular M-matrx f and only f all ts leadng prncpal mnors are postve III MAIN RESULTS In ths secton we exhbt some new lower bounds for τb A ) and τa A ) whch mprove the ones n 4] 5] 6] 7] 8] 9] 0] ] Theorem 3 Let A = a ) B = b ) R n n be nonsngular M-matrces and A = α ) n n Then for t = τb A ) mn α b + α b α b α b ) ) b α b α )] = t 5) Proof It s evdent that the result 5) holds wth equalty for n = Below we assume that n Snce A s a nonsngular M-matrx there exsts a postve dagonal matrx D such that D AD s a strctly row dagonally domnant nonsngular M-matrx and τb A ) = τd B A )D) = τb D AD) ) Therefore for convenence and wthout loss of generalty we assume that A s a strctly row dagonally domnant matrx Now let us dstngush two cases: ) Frst we assume that A and B are rreducble matrces Then for N t = wthout loss of generalty we assume that = max zt) zt) + zt) n = maxp t) pt) pt) + pt) n By the defnton of p t) obvously 0 < zt) < for N t = Let τb A ) = λ By Lemma 0 and Lemma 6 there exsts a par ) of postve ntegers wth such that λ α b λ α b ) α b = ) b α ) b α ) α b ) b α b α ) 6) Snce A and B are nonsngular M-matrces B A s a nonsngular M-matrx as well By Lemma 9 we have 0 λ α b for all N It follows from Inequalty 6) that λ α b )λ α b ) ) b α b α ) 7) Thus Inequalty 7) s equvalent to τb A ) α b + α b α b α b ) that s τb A ) mn ) b α b α )] α b + α b α b α b ) ) b α b α )] ) Now assume that one of A and B s reducble By Lemma we now that all the leadng prncpal mnors of A and B are postve If we denote by P = p ) the n n permutaton matrx wth p = p 3 = = p n n = p n = the remanng p beng zero Then for any chosen postve real number ε suffcently small such that all the leadng prncpal mnors of A εp and B εp are postve t follows that A εp and B εp are rreducble nonsngular M-matrx Now we substtute A εp for A B εp for B respectvely n the prevous case and then lettng ε 0 8) Advance onlne publcaton: 6 August 06)

5 IAENG Internatonal Journal of Appled Mathematcs 46:3 IJAM_46_3_5 the result follows by contnuty Ths completes our proof of Theorem 3 Remar 3 Wthout loss of generalty for t = assume that α b b α α b Thus Inequalty 9) can be wrtten as b α α b α b + b α 9) b α 0) From Inequaltes 5) 0) and by Lemma 0 and Lemma 3 n 0] t follows that α b + α b α b α b ) ) b α )] b α α b + α b ) b α = α b + α b α b α b ) + 4 b α b α ) + 4 α b α b + ) b α ) )] b α α b α b = α b + α b α b α b + = α b b b α = b a Ths means that mn α b + α b ) b α b mn N a b t mn N a b b b α ) ] b ) b α b α b ) mn N b α )] α b α )] b p t) a b So the lower bound n Theorem 3 s better than the lower bounds n Theorem n 0] and Theorem 48 n ] Moreover snce mn mn w for N t = we have α b + α b ) b α α b α b ) b α )] α b + α b α b α b ) ) w b α w b α )] Thus the lower bound n Theorem 3 s better than that n Theorem n 9] Theorem 3 The sequence t t = obtaned from Theorem 3 s monotone ncreasng wth an upper bound τb A ) and consequently s convergent Proof By Lemma 3 and Lemma 4 we have p t) p t+) 0 and f t) f t+) 0 t = So by the defntons of p t) sequence p t) snce and f t) = mnf t) = mnf t) pt) pt) so by the defnton of t = we have and f t) t s easy to see that are monotone decreasng and for t = we have mnf t+) = max p t+) = z t+) zt) = max N zt) zt) zt) + zt) n max N zt+) z t+) zt+) for N for + zt+) n = z t+) whch mples the sequence s monotone decreasng sequence Then t s a monotoncally ncreasng sequence Hence the sequence s convergent Let A = a ) R n n be a nonsngular M-matrx By Lemma 9 we now that f A s a doubly stochastc matrx then A T e = e Ae = e that s a = + a = + a So A s a strctly dagonally domnant matrx by row and by column If B = A accordng to Theorem 3 the followng corollary s establshed Corollary 3 Let A = a ) n n be a nonsngular M- matrx and A = α ) n n be doubly stochastc Then for t = τa A ) mn α a + α a α a α a ) ) a α a α )] = Γ t ) Remar 3 Accordng to Remar for N N and t = we have s m u w = g 0) pt) s m u w = g 0) p t) Advance onlne publcaton: 6 August 06)

6 IAENG Internatonal Journal of Appled Mathematcs 46:3 IJAM_46_3_5 whch results n τa A ) mn mn a α + a α a α a α ) τa A ) mn ) a α a α + a α ) m a α m a α )] a α a α ) a α )] and smlar to the proof of Remar 3 and by Lemma 3 n 6] we have a α + a α a α a α ) ) a α mn a α + a α ) w a α w mn N a u R + u a α )] a α a α ) Furthermore by Theorem 33 n 6] we have a u R mn N + a t R mn u N + t a α )] So the lower bound n Corollary 3 s an mprovement on the lower bounds n Theorem 3 n 7] Corollary 3 n 9] and Theorem 3 n 5] 6] In addton by Theorem 33 n 7] we have mn a b + a b ) m a b m mn N a s R + s a b a b ) a b )] Hence the result of Corollary 3 s sharper than that of Theorem 3 n 4] Theorem 33 Let A = a ) B = b ) R n n be nonsngular M-matrces and A = α ) n n Then for t = τb A ) mn α b + α b s α α b α b ) b ) s α b )] = Ω t ) Proof It s not dffcult to verfy that the result holds wth equalty for n = We next assume that n Snce A s a nonsngular M-matrx there exsts a postve dagonal matrx D such that D AD s a strctly row dagonally domnant nonsngular M-matrx and τb A ) = τd B A )D) = τb D AD) ) So for convenence and wthout loss of generalty we assume that A s a strctly row dagonally domnant matrx ) If A and B are rreducble matrces Then for any N we derve a + a d 0 < s = max < Let τb A ) = λ By Lemma 0 and Lemma 6 there exsts a par ) of postve ntegers wth for t = such that λ α b λ α b s s = s α b α b α a ) s ) s ) b α b ) s α b α ) b ) 3) Snce A and B are nonsngular M-matrces B A s a nonsngular M-matrx Havng n mnd that 0 λ α b for all N from Inequalty 3) we see that whch yelds that λ α b )λ α b ) s α b ) s α τb A ) α b + α b α b α b ) s α and therefore τb A ) b ) s α mnα b + α b b s α b ) 4) b )] α b α b ) ) s α b )] 5) If one of A and B s reducble smlar to the proof of Theorem 3 ) the result s obtaned Remar 33 Smlar to the proof of Remar 3 and by Advance onlne publcaton: 6 August 06)

7 IAENG Internatonal Journal of Appled Mathematcs 46:3 IJAM_46_3_5 Lemma 3 n 5] for t = s gven by τb A ) mn α b + α b s α b s mn N α b α b ) b ) s α a b p t) s b )] Therefore the bound of Theorem 33 s sharper than that n 0] Usng the same method as the proof of Theorem 3 the followng theorem can be deduced Theorem 34 The sequence Ω t t = obtaned from Theorem 33 s monotone ncreasng wth an upper bound τb A ) and consequently s convergent Tang B = A and usng Theorem 33 we can get the followng corollary Corollary 3 Let A = a ) n n be a nonsngular M- matrx and let A = α ) n n be doubly stochastc Then for t = τa A ) mn α a + α a s α a ) s α α a α a ) a )] = Υ t Remar 34 Accordng to Remar 3 and by Lemma 4 n 0] for t = we can conclude that the bound of Corollary 3 s sharper than that n Corollary 4 n 0] Remar 35 The sequence Υ t t = obtaned from Corollary 3 s monotone ncreasng wth an upper bound τa A ) and consequently s convergent Let Φ t = max t Ω t Combnng Theorems 3 and 33 Theorem 35 s easly obtaned Theorem 35 Let A = a ) B = b ) R n n be nonsngular M-matrces and A = α ) n n Then for t = 6) τb A ) Φ t 7) Let Ψ t = maxγ t Υ t By Corollares 3 and 3 Theorem 36 s easly derved Theorem 36 Let A = a ) n n be a nonsngular M- matrx and let A = α ) n n be doubly stochastc Then for t = τa A ) Ψ t 8) IV NUMERICAL EXAMPLES Example 4 Consder the followng two nonsngular M- matrces 9]: A = B = Numercal results are gven n Table I and Table II for the total number of teratons T = 5 In fact τb A ) = We defne RES = T a) τb A ) where T a) stands for the result derved by usng sequence a when the number of teraton s T In Table I for gven matrces A and B we lst the lower bounds of τb A ) calculated by Theorem 48 n ] Theorem n 9] Theorem n 9] Theorem n 0] and Theorem 3 n ths paper For the sequences obtaned by Theorem n 0] and Theorem 3 n ths paper the results are dsplayed for every step of teraton In Table II we present the lower bounds of τb A ) calculated by Theorem 3 n 0] and Theorem 33 n ths paper respectvely For the sequences obtaned by Theorem 3 n 0] and Theorem 33 n ths paper the results are dsplayed for every step of teraton Numercal results n Table I and Table II show that: a) Lower bounds obtaned from Theorem 3 and Theorem 33 are greater than the ones n Theorem n 9] Theorem n 0] Theorem 48 n ] and Theorem 3 n 0] b) Sequences obtaned from Theorems 3 and 33 both are monotone ncreasng c) The sequences obtaned from Theorems 3 and 33 are closer to the true value of τb A ) generally when the number of teratons T s ncreasng d) The sequences obtaned from Theorems 3 and 33 are much closer to the true value of τb A ) than those obtaned from Theorems and 3 n 0] respectvely In Fgure and Fgure we dsplay the RES generated by Theorem n 0] Theorem 3 and Theorem 3 n 0] Theorem 33 respectvely aganst number of teratons for T = 00 From these two fgures we note that the four Advance onlne publcaton: 6 August 06)

8 IAENG Internatonal Journal of Appled Mathematcs 46:3 IJAM_46_3_5 TABLE I THE LOWER BOUNDS OF τb A ) Method t τb A ) Method t τb A ) Theorem 48 n ] 0007 Theorem n 9] Theorem n 9] 07 Theorem n 0] 03 Theorem sequences of τb A ) are convergent whle the RES of the sequence obtaned from Theorem 3 are much less than that obtaned from Theorem n 0] For the sequences obtaned from Theorem 33 and Theorem 3 n 0] we can get the same concluson TABLE II THE LOWER BOUNDS OF τb A ) Method t τb A ) Method t τb A ) Theorem 3 n 0] Theorem Example 4 0] Let A = By Ae = e A T e = e we now that A s a strctly dagonally domnant by row and column Base on A Z n t s easy to see that A s a nonsngular M-matrx and A s doubly stochastc Numercal results are gven n Table III and Table IV for the total number of teratons T = 0 In fact τa A ) = In Table III for gven matrx A we lst the lower bounds of τa A ) calculated by Theorem 3 n 4] 5] 6] Corollary 5 n 8] Theorem 3 n 7] and Corollary 3 n 9] In Table IV we show the lower bounds of τa A ) calculated by Theorem 5 n 0] and Theorem 36 n ths paper respectvely For the sequences obtaned by Theorem 5 n 0] and Theorem 36 n ths paper the results are dsplayed for every step of teraton Numercal results n Table III and Table IV show that: a) Lower bounds obtaned from Theorem 36 are greater than those n Theorem 3 n 4] 5] 6] Corollary 5 n 8] Theorem 3 n 7] Corollary 3 n 9] and Theorem 5 n 0] b) Sequence obtaned from Theorem 36 s monotone ncreasng c) Sequence obtaned from Theorem 36 s convergent to the value whch s close to the true value of τa A ) d) The sequence obtaned from Theorem 36 s much closer to the true value of τa A ) than that obtaned from Theorem 5 n 0] and the sequence obtaned from Theorem 36 approxmates effectvely to the true value of τa A ) so we can estmate τa A ) by Theorem 36 In Fgure 3 we present the RES generated by Theorem 5 n 0] and Theorem 36 respectvely aganst number of teratons for T = 0 where RES s defned as Example 4 From ths fgure we fnd that the two sequences of τa A ) are convergent whle the RES of the sequence obtaned from Theorem 36 are much less than that obtaned from Theorem 5 n 0] TABLE III THE LOWER BOUNDS OF τa A ) Method τa A ) Theorem 3 n 4] 059 Theorem 3 n 5] 045 Theorem 3 n 6] 0447 Corollary 5 n 8] 040 Theorem 3 n 7] 0473 Corollary 3 n 9] Example 43 0] Let A = a ) R n n where a = a = = a nn = a = a 3 = = a n n = a n = and a = 0 elsewhere It s easy to see that A s a nonsngular M-matrx and A s doubly stochastc The results obtaned from Theorem 5 n 0] and Theorem 36 for n = 5 and T = 0 are lsted n Table 5 where T s defned n Example 4 In fact τa A ) = In Table V we show the lower bounds of τa A ) calculated by Theorem 5 n 0] and Theorem 36 n ths paper respectvely For the sequences obtaned by Theorem 5 n 0] and Theorem 36 n ths paper the results are dsplayed for every step of teraton Numercal results n Table V show that the lower bound obtaned from Theorem 36 could reach the true value of τa A ) n some cases Moreover applyng Theorem 36 s faster to reach the convergence value than Theorem 5 n 0] From Table 5 t can be seen that applyng Theorem 36 reaches the true value of τa A ) after only one teraton but applyng Theorem 5 n 0] needs 9 teratons for n = 5 when they reach the true value of τa A ) Advance onlne publcaton: 6 August 06)

9 IAENG Internatonal Journal of Appled Mathematcs 46:3 IJAM_46_3_5 TABLE IV THE LOWER BOUNDS OF τa A ) for τb A ) and τa A ) s also a further wor Method t τa A ) Method t τa A ) Theorem 5 n 0] Theorem TABLE V THE LOWER UPPER OF τa A ) N=5 Method t τa A ) Method t τa A ) Theorem 5 n 0] 0905 Theorem V CONCLUSIONS In ths paper base on the constructng sequence of teratons we have establshed new convergent sequences Φ t and Ψ t t = n Theorem 35 and Theorem 36 whch are more accurate than the exstng ones n 4] 5] 6] 7] 8] 9] 0] ] to approxmate τb A ) and τa A ) respectvely Numercal results gven n Secton IV Tables I-IV) show that the results obtaned by the new convergent sequences Φ t and Ψ t t = are more sharper than the those obtaned by the lower bounds n 4] 5] 6] 7] 8] 9] 0] ] Numercal results also present the feasblty and effectveness of the new convergent sequences when they are used to estmate τb A ) and τa A ) Inasmuch as the calculaton n experments are nvolve n controllng accuracy so there an nterestng problem s how to accurately these bounds can be computed whch may mprove the accuracy of the of these sequences At present t s very dffcult for us to gve the error analyss We wll contnue to study ths problem n the future In addton although the effcency convergent sequences for Φ t and Ψ t are gven they are almost useless for the mplementaton of these new sequences for Φ t and Ψ t when the sze n of matrx s large snce the calculatons of the nverse of the matrces s very expensve How to fnd easer calculated convergent sequences for Φ t and Ψ t s stll a tough tas whch should be further studed n the further Fnally we can fnd that the sequences obtaned by Theorem 3 Theorem 33 Theorem 35 and Theorem 36 are sharper than the exstng ones by theory analyzng and numercal examples also verfy that whereas the sequences are less precse sometmes we can see that the result 006 n Table I obtaned by Theorem 3 and the result 305 n Table II obtaned by Theorem 33 they are not close to the true value Hence fndng more accurate sequences ACKNOWLEDGMENTS Ths wor s supported by the Natonal Natural Scence Foundatons of Chna No773) REFERENCES ] C Q Yang and J H Hou Numercal method for solvng volterra ntegral equatons wth a convoluton ernel IAENG Internatonal Journal of Appled Mathematcs vol 4 no 4 pp ] A T P Naafabad and D Z Kucerovsy On solutons of a system of wener-hopf ntegral equatons IAENG Internatonal Journal of Appled Mathematcs vol 44 no pp ] A Berman and R J Plemmons Nonnegatve Matrces n the Mathematcal Scences SIAM Phladelpha 979 4] R A Horn and C R Johnson Topcs n Matrx Analyss Cambrdge Unversty Press 99 5] M Fedler and T L Marham An nequalty for the Hadamard product of an M-matrx and ts nverse Lnear Algebra Appl vol 0 pp ] A J L Improvng AOR teratve methods for rreducble L- matrces Engneerng Letters vol 9 no pp ] Z Xu Q Lu K Y Zhang and X H An Theory and Applcatons of H-matrces Scence Press 03 8] A J L A New Precondtoned AOR Iteratve Method and Comparson Theorems 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Some nequaltes for the mnmum egenvalue of the Hadamard product of an M-matrx and ts nverse J Inequal Appl vol 03 no ] F B Chen New nequaltes for the Hadamard product of an M- matrx and ts nverse J Inequal Appl vol 05 no ] D M Zhou G L Chen G X Wu and X Y Zhang Some nequaltes for the Hadamard product of an M-matrx and ts nverse J Inequal Appl vol 03 no ] Y T L F Wang C Q L and J X Zhao Some new bounds for the mnmum egenvalue of the Hadamard product of an M-matrx and an nverse M-matrx J Inequal Appl vol 03 no ] J X Zhao F Wang and C L Sang Some nequaltes for the mnmum egenvalues of the Hadamard product of an M-matrx and an nverse M-matrx J Inequal Appl vol 05 no 9 05 ] D M Zhou G L Chen G X Wu and X Y Zhang On some new bounds for egenvalues of the Hadamard product and the Fan product of matrces Lnear Algebra Appl vol 438 pp ] X Y Yong On a conecture of Fedler and Maham Lnear Algebra Appl vol 88 pp ] R A Horn and C R Johnson Matrx Analyss Cambrdge Unversty Press 985 Advance onlne publcaton: 6 August 06)

10 RES RES RES IAENG Internatonal Journal of Appled Mathematcs 46:3 IJAM_46_3_ Theorem of 9] * Theorem Number of teratonst) Fg The teratons curves of two sequences obtaned from Theorem n 0] and Theorem 3 respectvely wth T = Theorem 3 n 9] 3 * Theorem Number of teratons) Fg The teratons curves of two sequences obtaned from Theorem 3 n 0] and Theorem 33 respectvely wth T = Theorem 5 n 9] * Theorem Number of teratons Fg 3 The teratons curves of two sequences obtaned from Theorem 5 n 0] and Theorem 36 respectvely wth T = 0 Advance onlne publcaton: 6 August 06)