JAM Journal of Applied Mathematics 1687-0042 1110-757X Hindawi Publishing Corporation 10.1155/2014/708128 708128 Research Article A New Upper Bound on the Infinity Norm of the Inverse of Nekrasov Matrices Gao Lei Li Chaoqian Li Yaotang Wu Shi-Liang School of Mathematics and Statistics Yunnan University Kunming, Yunnan 650091 China ynu.edu.cn 2014 1162014 2014 01 05 2014 24 05 2014 12 6 2014 2014 Copyright © 2014 Lei Gao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

A new upper bound which involves a parameter for the infinity norm of the inverse of Nekrasov matrices is given. And we determine the optimal value of the parameter such that the bound improves the results of Kolotilina, 2013. Numerical examples are given to illustrate the corresponding results.

1. Introduction

The class of Nekrasov matrices is a subclass of H -matrices. Estimating the infinity norm of the inverse of Nekrasov matrices can be used to prove the convergence of matrix splitting and matrix multisplitting iteration methods for solving large sparse systems of linear equations; see . Here, we call a matrix A = ( a i j ) C n , n an H -matrix if its comparison matrix A = [ m i j ] defined by (1) A = [ m i j ] C n , n , m i j = { | a i i | , i = j - | a i j | , i j , is an M -matrix; that is, A - 1 0 [1, 5, 6], and a matrix A = [ a i j ] C n , n is called a Nekrasov matrix if for each i N , (2) | a i i | > h i ( A ) , where h 1 ( A ) = j 1 | a 1 j | and h i ( A ) = j = 1 i - 1 ( | a i j | / | a j j | ) h j ( A ) + j = i + 1 n | a i j | , i = 2,3 , , n [2, 6].

In 1975, Varah  provided the following upper bound for strictly diagonally dominant (SDD) matrices as one most important subclass of Nekrasov matrices, consequently, H -matrices [2, 6, 8]. Here a matrix A = [ a i j ] C n , n is called SDD if for each i N = { 1,2 , , n } , (3) | a i i | > r i ( A ) , where r i ( A ) = j i | a i j | .

Theorem 1 (see [<xref ref-type="bibr" rid="B14">7</xref>]).

Let A = [ a i j ] C n , n be SDD. Then (4) A - 1 1 min i N ( | a i i | - r i ( A ) ) .

We call the bound in Theorem 1 the Varah’s bound. As Cvetković et al.  said, Varah’s bound works only for SDD matrices and even then it is not always good enough. To obtain new upper bounds for the infinity norm of the inverse of a wider class of matrices which sometimes works better in the SDD case, Cvetković et al.  give the following bound of Nekrasov matrices.

Theorem 2 (see [<xref ref-type="bibr" rid="B5">2</xref>, Theorem  2]).

Let A = [ a i j ] C n , n be a Nekrasov matrix. Then (5) A - 1 max i N ( z i ( A ) / | a i i | ) 1 - max i N ( h i ( A ) / | a i i | ) , (6) A - 1 max i N z i ( A ) min i N | a i i | - h i ( A ) , where z 1 ( A ) = 1 and z i ( A ) = j = 1 i - 1 ( | a i j | / | a j j | ) z j ( A ) + 1 , i = 2,3 , n .

In [9, Theorems  2.2 and  2.3], Kolotilina gave an improvement of these upper bounds in Theorem 2 (see Theorems 3 and 4).

Theorem 3 (see [<xref ref-type="bibr" rid="B10">9</xref>, Theorem  2.2]).

Let A = [ a i j ] C n , n be a Nekrasov matrix. Then (7) A - 1 max i N z i ( A ) | a i i | - h i ( A ) .

Theorem 4 (see [<xref ref-type="bibr" rid="B10">9</xref>, Theorem  2.3]).

Let A = [ a i j ] C n , n be a Nekrasov matrix. Then (8) max i N z i ( A ) | a i i | - h i ( A ) max i N ( z i ( A ) / | a i i | ) 1 - max i N ( h i ( A ) / | a i i | ) , max i N z i ( A ) | a i i | - h i ( A ) max i N z i ( A ) min i N | a i i | - h i ( A ) .

In this paper, we also focus on the estimation problem of the infinity norm of the inverse of Nekrasov matrices and give an improvement of the bound in Theorem 3 (Theorem  2.2 in ). Numerical example is given to illustrate the corresponding results.

2. Bounds for the Infinity Norm of the Inverse of Nekrasov Matrices

In order to obtain a new bound, we start with the following lemmas and notations. Given a matrix A = [ a i j ] , by A = D - L - U we denote the standard splitting of A into its diagonal ( D ) , strictly lower ( - L ) , and strictly upper ( - U ) triangular parts. And by [ A ] i j denote the ( i , j ) -entry of A ; that is, [ A ] i j = a i j . Furthermore, we denote | A | = [ | a i j | ] .

Lemma 5 (see [<xref ref-type="bibr" rid="B2">10</xref>]).

Let A = [ a i j ] C n , n be a nonsingular H -matrix. Then (9) | A - 1 | A - 1 .

Lemma 6 (see [<xref ref-type="bibr" rid="B12">11</xref>]).

Given any matrix A = [ a i j ] C n , n , n 2 , with a i i 0 for all i N , then (10) h i ( A ) = | a i i | [ ( | D | - | L | ) - 1 | U | e ] i , where e C n , n is the vector with all components equal to 1.

Lemma 7 (see [<xref ref-type="bibr" rid="B13">12</xref>]).

A matrix A = [ a i j ] C n , n , n 2 , is a Nekrasov matrix if and only if (11) ( | D | - | L | ) - 1 | U | e < e , that is, if and only if E - ( | D | - | L | ) - 1 | U | is an SDD matrix, where E is the identity matrix.

Let C = E - ( | D | - | L | ) - 1 | U | = [ c i j ] . Then from Lemma 7, C is SDD when A is a Nekrasov matrix. Note that c 11 = 1 , c k 1 = 0 , k = 2,3 , , n , and c 1 k = - | a 1 k | / | a 11 | , k = 2,3 , , n , which leads to the following lemma.

Lemma 8.

Let A = [ a i j ] C n , n be a Nekrasov matrix and (12) C ( μ ) = C D ( μ ) = [ E - ( | D | - | L | ) - 1 | U | ] D ( μ ) , where D ( μ ) = diag ( μ , 1 , , 1 ) and μ > r 1 ( A ) / | a 11 | . Then C ( μ ) is SDD.

Proof.

It is not difficult from (12) to see that [ C ( μ ) ] k 1 = μ c k 1 for all k N and [ C ( μ ) ] k j = c k j for all k N and j 1 . Hence (13) [ C ( μ ) ] 11 = μ , r 1 ( C ( μ ) ) = r 1 ( C ) = r 1 ( A ) | a i i | and for i = 2 , , n , (14) [ C ( μ ) ] i i = c i i , r i ( C ( μ ) ) = r i ( C ) . From the fact that C is SDD and μ > r 1 ( A ) / | a 11 | , we have that C ( μ ) is SDD. The proof is completed.

The main result of this paper is the following theorem.

Theorem 9.

Let A = [ a i j ] C n , n be a Nekrasov matrix. Then for μ > r 1 ( A ) / | a 11 | , (15) A - 1 max { μ , 1 } max { max i 1 ( z i ( A ) / ( | a i i | - h i ( A ) ) ) 1 μ | a 11 | - h 1 ( A ) , f f f f f f i i f f f f f f max i 1 z i ( A ) | a i i | - h i ( A ) } .

Proof.

Let C ( μ ) = C D ( μ ) = ( E - ( | D | - | L | ) - 1 | U | ) D ( μ ) , where D ( μ ) = diag ( μ , 1 , , 1 ) . From (12), we have (16) C ( μ ) = ( | D | - | L | ) - 1 A D ( μ ) , which implies that (17) A = ( | D | - | L | ) C ( μ ) D ( μ ) - 1 = ( | D | - | L | ) Δ · Δ - 1 C ( μ ) D ( μ ) - 1 , where (18) Δ = diag ( δ 1 , δ 2 , , δ n ) , δ i > 0 , i = 1,2 , , n . Furthermore, since a Nekrasov matrix is an H -matrix, we have, from Lemma 5, (19) A - 1 A - 1 D ( μ ) · C ( μ ) - 1 Δ · [ ( | D | - | L | ) Δ ] - 1 .

First, we estimate [ ( | D | - | L | ) Δ ] - 1 . Since ( | D | - | L | ) Δ is an M -matrix and there exists a positive diagonal matrix Δ such that ( | D | - | L | ) Δ e = e , see , we get (20) [ ( | D | - | L | ) Δ ] - 1 = [ ( | D | - | L | ) Δ ] - 1 e = 1 .

Secondly, we estimate C ( μ ) - 1 Δ . From Lemma 8, C ( μ ) is SDD. Obviously, multiplying the left-hand side of C ( μ ) by diagonal matrix Δ - 1 does not change SDD property, so Δ - 1 C ( μ ) is also SDD. Thus, Varah’s bound (4) can be applied as follows: (21) C ( μ ) - 1 Δ max i N 1 [ Δ - 1 C ( μ ) e ] i = max i N δ i [ C ( μ ) e ] i = max { δ 1 [ C ( μ ) e ] 1 , max i 1 δ i [ C ( μ ) e ] i } = max { δ 1 | a 11 | μ | a 11 | - h 1 ( A ) , max i 1 δ i | a i i | | a i i | - h i ( A ) } . In addition, since z ( A ) = [ z 1 ( A ) , , z n ( A ) ] T = | D | ( | D | - | L | ) - 1 e and ( | D | - | L | ) Δ e = e , see [9, 13], we have (22) δ i | a i i | = z i ( A ) , i = 1,2 , , n . Substituting (22) into (21), we get that (23) C ( μ ) - 1 Δ max { z 1 ( A ) μ | a 11 | - h 1 ( A ) , max i 1 z i ( A ) | a i i | - h i ( A ) } . Finally, from (20), (23), z 1 ( A ) = 1 , and the fact that D ( μ ) = max { μ , 1 } , we have (24) A - 1 max { μ , 1 } × max { z 1 ( A ) μ | a 11 | - h 1 ( A ) , max i 1 z i ( A ) | a i i | - h i ( A ) } = max { μ , 1 } × max { 1 μ | a 11 | - h 1 ( A ) , max i 1 z i ( A ) | a i i | - h i ( A ) } . The conclusions follow.

Example 10.

Consider the Nekrasov matrix A 1 in [2, 9], where (25) A 1 = [ - 7 1 - 0.2 2 7 88 2 - 3 2 0.5 13 - 2 0.5 3.0 1 6 ] . By computation, h 1 ( A ) = 3.2000 , h 2 ( A ) = 8.2000 , h 3 ( A ) = 2.9609 , h 4 ( A ) = 0.7359 , z 1 ( A ) = 1 , z 2 ( A ) = 2 , z 3 ( A ) = 1.2971 , and z 4 ( A ) = 1.2394 . By the bound of Theorem 3 (the bound of Theorem  2.2 in ), we have (26) A 1 - 1 0.2632 . By Theorem 9, we have (27) A 1 - 1 0.3226 ( Taking μ = 0.90 ) , A 1 - 1 0.2786 ( Taking μ = 0.97 ) , A 1 - 1 0.2549 ( Taking μ = 1.04 ) , A 1 - 1 0.2590 ( Taking μ = 1.10 ) , A 1 - 1 0.2708 ( Taking μ = 1.15 ) . In fact, A 1 - 1 = 0.1921 .

Remark 11.

Example 10 shows that by choosing the value of μ , the bound in Theorem 9 is better than that in Theorem 3 in some cases. We further observe the bound in Theorem 9 by Figure 1 and find that there is an interval such that for any μ in this interval, the bound in Theorem 9 for the matrix A 1 is always smaller than that in Theorem 3. An interesting problem arises: whether there is an interval of μ such that the bound in Theorem 9 for any Nekrasov matrix is smaller than that in Theorem 3. In the following section, we will study this problem.

The bounds in Theorems 9 and 3.

3. The Choice of <inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M124"> <mml:mrow> <mml:mi>μ</mml:mi></mml:mrow> </mml:math></inline-formula>

In this section, we determine the value of μ such that the bound for A - 1 in Theorem 9 is less than or equal to that in . First, we consider the Nekrasov matrix A = [ a i j ] C n , n with (28) 1 | a 11 | - h 1 ( A ) > max i 1 z i ( A ) | a i i | - h i ( A ) , and give the following lemma.

Lemma 12.

Let a , b , and c be positive real numbers, and 0 < a ( b - c ) < 1 . Then (29) 1 < 1 + a c a b < 1 a ( b - c ) .

Proof.

We only need to prove that ( 1 + a c ) / a b - 1 > 0 and 1 / a ( b - c ) - ( 1 + a c ) / a b > 0 . In fact, (30) 1 + a c a b - 1 = 1 - a ( b - c ) a b > 0 , 1 a ( b - c ) - 1 + a c a b = c ( 1 - a ( b - c ) ) a b ( b - c ) > 0 . The proof is completed.

Lemma 13.

Let A = [ a i j ] C n , n be a Nekrasov matrix with (31) 1 | a 11 | - h 1 ( A ) > max i 1 z i ( A ) | a i i | - h i ( A ) . Then (32) 1 < 1 + max i 1 ( z i ( A ) / ( | a i i | - h i ( A ) ) ) · h 1 ( A ) | a 11 | · max i 1 ( ( A ) / ( | a i i | - h i ( A ) ) ) < 1 / ( | a 11 | - h 1 ( A ) ) max i 1 ( z i ( A ) / ( | a i i | - h i ( A ) ) ) .

Proof.

Let a = max i 1 ( z i ( A ) / ( | a i i | - h i ( A ) ) ) , b = | a 11 | , and c = h 1 ( A ) . From (28), we get 0 < a ( b - c ) < 1 . Then from Lemma 12, the first and second inequalities in (32) hold.

We now give an interval of μ such that the bound in Theorem 9 is less than that in Theorem 3.

Lemma 14.

Let A = [ a i j ] C n , n be a Nekrasov matrix with (33) 1 | a 11 | - h 1 ( A ) > max i 1 z i ( A ) | a i i | - h i ( A ) . Then for each μ ( 1 , ( 1 / ( | a 11 | - h 1 ( A ) ) ) / max i 1 ( z i ( A ) / ( | a i i | - h i ( A ) ) ) ) , (34) A - 1 max { μ , 1 } × max { 1 μ | a 11 | - h 1 ( A ) , max i 1 z i ( A ) | a i i | - h i ( A ) } < max i N z i ( A ) | a i i | - h i ( A ) .

Proof.

From Lemma 13, we have (35) μ ( 1 , 1 + max i 1 ( z i ( A ) / ( | a i i | - h i ( A ) ) ) · h 1 ( A ) | a 11 | · max i 1 ( z i ( A ) / ( | a i i | - h i ( A ) ) ) ] [ 1 + max i 1 ( z i ( A ) / ( | a i i | - h i ( A ) ) ) · h 1 ( A ) | a 11 | · max i 1 ( z i ( A ) / ( | a i i | - h i ( A ) ) ) , f f f f 1 / ( | a 11 | - h 1 ( A ) ) max i 1 ( z i ( A ) / ( | a i i | - h i ( A ) ) ) ) and max { μ , 1 } = μ .

( I ) For μ ( 1 , ( 1 + max i 1 ( z i ( A ) / ( | a i i | - h i ( A ) ) ) · h 1 ( A ) ) / ( | a 11 | · max i 1 ( z i ( A ) / ( | a i i | - h i ( A ) ) ) ) ] , then (36) μ | a 11 | - h 1 ( A ) 1 max i 1 ( z i ( A ) / ( | a i i | - h i ( A ) ) ) ; that is, (37) 1 μ | a 11 | - h 1 ( A ) max i 1 z i ( A ) | a i i | - h i ( A ) . Therefore, (38) max { μ , 1 } max { 1 μ | a 11 | - h 1 ( A ) , max i 1 z i ( A ) | a i i | - h i ( A ) } = μ μ | a 11 | - h 1 ( A ) . Consider the function f ( x ) = x / ( x | a 11 | - h 1 ( A ) ) , x [ 1 , ( 1 + max i 1 ( z i ( A ) / ( | a i i | - h i ( A ) ) ) · h 1 ( A ) ) / ( | a 11 | · max i 1 ( z i ( A ) / ( | a i i | - h i ( A ) ) ) ) ] . It is easy to prove that f ( x ) is a monotonically decreasing function of x . Hence, for any μ ( 1 , ( 1 + max i 1 ( z i ( A ) / ( | a i i | - h i ( A ) ) ) · h 1 ( A ) ) / ( | a 11 | · max i 1 ( z i ( A ) / ( | a i i | - h i ( A ) ) ) ) ] , (39) f ( μ ) < f ( 1 ) ; that is, (40) μ μ | a 11 | - h 1 ( A ) < 1 | a 11 | - h 1 ( A ) = max i N z i ( A ) | a i i | - h i ( A ) . Hence, (41) max { μ , 1 } max { 1 μ | a 11 | - h 1 ( A ) , max i 1 z i ( A ) | a i i | - h i ( A ) } < max i N z i ( A ) | a i i | - h i ( A ) .

( II ) For [ ( 1 + max i 1 ( z i ( A ) / ( | a i i | - h i ( A ) ) ) · h 1 ( A ) ) / ( | a 11 | · max i 1 ( z i ( A ) / ( | a i i | - h i ( A ) ) ) ) , ( 1 / ( | a 11 | - h 1 ( A ) ) ) / max i 1 ( z i ( A ) / ( | a i i | - h i ( A ) ) ) ) , then (42) μ | a 11 | - h 1 ( A ) 1 max i 1 ( z i ( A ) / ( | a i i | - h i ( A ) ) ) ; that is, (43) 1 μ | a 11 | - h 1 ( A ) max i 1 z i ( A ) | a i i | - h i ( A ) . Therefore, (44) max { μ , 1 } max { 1 μ | a 11 | - h 1 ( A ) , max i 1 z i ( A ) | a i i | - h i ( A ) } = μ max i 1 z i ( A ) | a i i | - h i ( A ) . Consider the function (45) g ( x ) = x max i 1 z i ( A ) | a i i | - h i ( A ) , x [ 1 + max i 1 ( z i ( A ) / ( | a i i | - h i ( A ) ) ) · h 1 ( A ) | a 11 | · max i 1 ( z i ( A ) / ( | a i i | - h i ( A ) ) ) , f f i 1 / ( | a 11 | - h 1 ( A ) ) max i 1 ( z i ( A ) / ( | a i i | - h i ( A ) ) ) ] . Obviously, g ( x ) is a monotonically increasing function of x . Hence, for any μ [ ( 1 + max i 1 ( z i ( A ) / ( | a i i | - h i ( A ) ) ) · h 1 ( A ) ) / ( | a 11 | · max i 1 ( z i ( A ) / ( | a i i | - h i ( A ) ) ) ) , ( 1 / ( | a 11 | - h 1 ( A ) ) ) / max i 1 ( z i ( A ) / ( | a i i | - h i ( A ) ) ) ) , (46) g ( μ ) < g ( ( 1 / ( | a 11 | - h 1 ( A ) ) ) max i 1 ( z i ( A ) / ( | a i i | - h i ( A ) ) ) ) ; that is, (47) μ max i 1 z i ( A ) | a i i | - h i ( A ) < 1 | a 11 | - h 1 ( A ) = max i N z i ( A ) | a i i | - h i ( A ) . Hence, (48) max { μ , 1 } max { 1 μ | a 11 | - h 1 ( A ) , max i 1 z i ( A ) | a i i | - h i ( A ) } < max i N z i ( A ) | a i i | - h i ( A ) . The conclusion follows from ( I ) and ( II ) .

Lemma 14 provides an interval of μ such that the bound in Theorem 9 is better than the bound in Theorem 3 (the bound in ). Moreover, we can determine the optimal value of μ by the following theorem.

Theorem 15.

Let A = [ a i j ] C n , n be a Nekrasov matrix with (49) 1 | a 11 | - h 1 ( A ) > max i 1 z i ( A ) | a i i | - h i ( A ) . Then (50) min { ( 1 , 1 / ( | a 11 | - h 1 ( A ) ) max i 1 ( z i ( A ) / ( | a i i | - h i ( A ) ) ) ) max { μ , 1 } fffifffff × max { 1 μ | a 11 | - h 1 ( A ) , max i 1 z i ( A ) | a i i | - h i ( A ) } : fffifffff μ ( 1 , 1 / ( | a 11 | - h 1 ( A ) ) max i 1 ( z i ( A ) / ( | a i i | - h i ( A ) ) ) ) } . = z 1 ( A ) | a 11 | + max i 1 z i ( A ) | a i i | - h i ( A ) · h 1 ( A ) | a 11 | . Furthermore, (51) A - 1 max { μ , 1 } ( z 1 ( A ) | a 11 | + max i 1 z i ( A ) | a i i | - h i ( A ) · h 1 ( A ) | a 11 | ) < max i N z i ( A ) | a i i | - h i ( A ) .

Proof.

From the proof of Lemma 14, we have that (52) f ( x ) = x x | a 11 | - h 1 ( A ) , x [ 1 , 1 + max i 1 ( z i ( A ) / ( | a i i | - h i ( A ) ) ) · h 1 ( A ) | a 11 | · max i 1 ( z i ( A ) / ( | a i i | - h i ( A ) ) ) ] is decreasing and that (53) g ( x ) = x max i 1 z i ( A ) | a i i | - h i ( A ) , x [ 1 + max i 1 ( z i ( A ) / ( | a i i | - h i ( A ) ) ) · h 1 ( A ) | a 11 | · max i 1 ( z i ( A ) / ( | a i i | - h i ( A ) ) ) , f f 1 / ( | a 11 | - h 1 ( A ) ) max i 1 ( z i ( A ) / ( | a i i | - h i ( A ) ) ) ] is increasing. Therefore, the minimum of f ( x ) and g ( x ) is (54) f ( 1 + max i 1 ( z i ( A ) / ( | a i i | - h i ( A ) ) ) · h 1 ( A ) | a 11 | · max i 1 ( z i ( A ) / ( | a i i | - h i ( A ) ) ) ) = g ( 1 + max i 1 ( z i ( A ) / ( | a i i | - h i ( A ) ) ) · h 1 ( A ) | a 11 | · max i 1 ( z i ( A ) / ( | a i i | - h i ( A ) ) ) ) = z 1 ( A ) | a 11 | + max i 1 z i ( A ) | a i i | - h i ( A ) · h 1 ( A ) | a 11 | , which implies that (50) holds. Again by Lemma 14, (51) follows easily.

Remark 16.

Theorem 15 provides a method to determine the optimal value of μ for a Nekrasov matrix A = [ a i j ] C n , n with (55) 1 | a 11 | - h 1 ( A ) > max i 1 z i ( A ) | a i i | - h i ( A ) . Also consider the matrix A 1 in Example 10. By computation, we get (56) 1 | a 11 | - h 1 ( A ) = 0.2632 > 0.2354 = max i 1 z i ( A ) | a i i | - h i ( A ) . Hence, by Theorem 15, we can obtain that the bound in Theorem 9 reaches its minimum (57) z 1 ( A ) | a 11 | + max i 1 z i ( A ) | a i i | - h i ( A ) · h 1 ( A ) | a 11 | = 0.2505 at μ = ( 1 + max i 1 ( z i ( A ) / ( | a i i | - h i ( A ) ) ) · h 1 ( A ) ) / ( | a 11 | · max i 1 ( z i ( A ) / ( | a i i | - h i ( A ) ) ) ) = 1.0639 (also see Figure 1).

Next, we study the bound in Theorem 9 for the Nekrasov matrix A = [ a i j ] C n , n with (58) 1 | a 11 | - h 1 ( A ) max i 1 z i ( A ) | a i i | - h i ( A ) .

Theorem 17.

Let A = [ a i j ] C n , n be a Nekrasov matrix with (59) 1 | a 11 | - h 1 ( A ) max i 1 z i ( A ) | a i i | - h i ( A ) . Then we can take μ [ ( 1 + max i 1 ( z i ( A ) / ( | a i i | - h i ( A ) ) ) · h 1 ( A ) ) / ( | a 11 | · max i 1 ( z i ( A ) / ( | a i i | - h i ( A ) ) ) ) , 1 ] such that (60) A - 1 max { μ , 1 } × max { 1 μ | a 11 | - h 1 ( A ) , max i 1 z i ( A ) | a i i | - h i ( A ) } = max i N z i ( A ) | a i i | - h i ( A ) .

Proof.

From (59), we get ( 1 + max i 1 ( z i ( A ) / ( | a i i | - h i ( A ) ) ) · h 1 ( A ) ) / ( | a 11 | · max i 1 ( z i ( A ) / ( | a i i | - h i ( A ) ) ) ) 1 . Then, for μ > r 1 ( A ) / | a 11 | = h 1 ( A ) / | a 11 | , we have (61) μ ( h 1 ( A ) | a 11 | , 1 + max i 1 ( z i ( A ) / ( | a i i | - h i ( A ) ) ) · h 1 ( A ) | a 11 | · max i 1 ( z i ( A ) / ( | a i i | - h i ( A ) ) ) ) [ 1 + max i 1 ( z i ( A ) / ( | a i i | - h i ( A ) ) ) · h 1 ( A ) | a 11 | · max i 1 ( z i ( A ) / ( | a i i | - h i ( A ) ) ) , 1 ] ( 1 , + ) .

( I ) For μ ( h 1 ( A ) / | a 11 | , ( 1 + max i 1 ( z i ( A ) / ( | a i i | - h i ( A ) ) ) · h 1 ( A ) ) / ( | a 11 | · max i 1 ( z i ( A ) / ( | a i i | - h i ( A ) ) ) ) ) , then max { μ , 1 } = 1 and (62) μ < 1 + max i 1 ( z i ( A ) / ( | a i i | - h i ( A ) ) ) · h 1 ( A ) | a 11 | · max i 1 ( z i ( A ) / ( | a i i | - h i ( A ) ) ) ; that is, (63) 1 μ | a 11 | - h 1 ( A ) > max i 1 z i ( A ) | a i i | - h i ( A ) . Therefore, (64) max { μ , 1 } max { 1 μ | a 11 | - h 1 ( A ) , max i 1 z i ( A ) | a i i | - h i ( A ) } = 1 μ | a 11 | - h 1 ( A ) > max i 1 z i ( A ) | a i i | - h i ( A ) = max i N z i ( A ) | a i i | - h i ( A ) . ( II ) For μ [ ( 1 + max i 1 ( z i ( A ) / ( | a i i | - h i ( A ) ) ) · h 1 ( A ) ) / ( | a 11 | · max i 1 ( z i ( A ) / ( | a i i | - h i ( A ) ) ) ) , 1 ] , then max { μ , 1 } = 1 and (65) μ 1 + max i 1 ( z i ( A ) / ( | a i i | - h i ( A ) ) ) · h 1 ( A ) | a 11 | · max i 1 ( z i ( A ) / ( | a i i | - h i ( A ) ) ) ; that is, (66) 1 μ | a 11 | - h 1 ( A ) max i 1 z i ( A ) | a i i | - h i ( A ) . Therefore, (67) max { μ , 1 } max { 1 μ | a 11 | - h 1 ( A ) , max i 1 z i ( A ) | a i i | - h i ( A ) } = max i 1 z i ( A ) | a i i | - h i ( A ) = max i N z i ( A ) | a i i | - h i ( A ) . ( III ) For μ ( 1 , + ) , then max { μ , 1 } = μ and (68) 1 μ | a 11 | - h 1 ( A ) max i 1 z i ( A ) | a i i | - h i ( A ) . Therefore, (69) max { μ , 1 } max { 1 μ | a 11 | - h 1 ( A ) , max i 1 z i ( A ) | a i i | - h i ( A ) } = μ · max i 1 z i ( A ) | a i i | - h i ( A ) > max i 1 z i ( A ) | a i i | - h i ( A ) = max i N z i ( A ) | a i i | - h i ( A ) . The conclusion follows from (I), (II), and (III).

Remark 18.

Theorems 15 and 17 provide the value of μ ; that is, (70) μ = 1 + max i 1 ( z i ( A ) / ( | a i i | - h i ( A ) ) ) · h 1 ( A ) | a 11 | · max i 1 ( z i ( A ) / ( | a i i | - h i ( A ) ) ) such that the bound in Theorem 9 is not worse than that in Theorem 3 for a Nekrasov matrix A = [ a i j ] C n , n . In particular, for the Nekrasov matrix A with 1 / ( | a 11 | - h 1 ( A ) ) > max i 1 ( z i ( A ) / ( | a i i - h i ( A ) | ) ) , the bound in Theorem 9 is better than that in Theorem 3.

Example 19.

Consider the following five Nekrasov matrices in [2, 9]: (71) A 2 = [ 8 1 - 0.2 3.3 7 13 2 - 3 - 1.3 6.7 13 - 2 0.5 3 1 6 ] , A 3 = [ 21 - 9.1 - 4.2 - 2.1 - 0.7 9.1 - 4.2 - 2.1 - 0.7 - 0.7 4.9 - 2.1 - 0.7 - 0.7 - 0.7 2.8 ] , A 4 = [ 5 1 0.2 2 1 21 1 - 3 2 0.5 6.4 - 2 0.5 - 1 1 9 ] , A 5 = [ 6 - 3 - 2 - 1 11 - 8 - 7 - 3 10 ] , A 6 = [ 8 - 0.5 - 0.5 - 0.5 - 9 16 - 5 - 5 - 6 - 4 15 - 3 - 4.9 - 0.9 - 0.9 6 ] . Obviously, A 2 , A 3 , and A 4 are SDD. And it is not difficult to verify that A 4 satisfies the conditions in Theorem 15 and A 2 , A 3 , A 5 , and A 6 satisfy the conditions in Theorem 17. We compute by Matlab 7.0 the upper bounds for the infinity norm of the inverse of A i , i = 2 , , 6 , which are shown in Table 1. It is easy to see from Table 1 that this example illustrates Theorems 15 and 17.

The upper bounds for | | A i - 1 | | , i = 2 , , 6 .

Matrix A 2 A 3 A 4 A 5 A 6
Exact | | A - 1 | | 0.2390 0.8759 0.2707 1.1519 0.4474
Varah (4) 1 1.4286 0.5556
Cvetković et al. (5) 0.8848 1.8076 0.6200 1.4909 1.1557
Cvetković et al. (6) 0.6885 0.9676 0.7937 2.4848 0.5702
Kolotilina (7) 0.5365 0.9676 0.5556 1.4138 0.4928
Theorem 9 0.5365 0.9676 0.5038 1.4138 0.4928
4. Conclusions

In this paper, we give an improvement on the infinity norm bound for the inverse of a Nekrasov matrix in . In particular, for the Nekrasov matrix A = [ a i j ] C n , n with (72) 1 | a 11 | - h 1 ( A ) > max i 1 z i ( A ) | a i i | - h i ( A ) , we prove that new bound is better than that in . However, for the Nekrasov matrix A with (73) 1 | a 11 | - h 1 ( A ) max i 1 z i ( A ) | a i i | - h i ( A ) , we only obtain that new bound is equal to that in . For this case, we try to found some better bounds in future. On the other hand, our bound only considers one parameter μ , that is, D ( μ ) = diag ( μ , 1 , , 1 ) , which poses an interesting problem: whether we further improve this bound by introducing more parameters. In future, we will research this problem.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research is supported by the National Natural Science Foundation of China (11361074 and 11326242), the Science Foundation of the Education Department of Yunnan Province of China (2013FD002), and Science and Technology Innovation Fund projects of Yunnan University (ynuy201366).

Bai Z.-Z. Wang D.-R. Generalized matrix multisplitting relaxation methods and their convergence Numerical Mathematics. A Journal of Chinese Universities. (English Series) 1993 2 1 87 100 MR1390722 ZBL0849.65016 Cvetković L. Dai P.-F. Doroslovački K. Li Y.-T. Infinity norm bounds for the inverse of Nekrasov matrices Applied Mathematics and Computation 2013 219 10 5020 5024 10.1016/j.amc.2012.11.056 MR3009462 ZBL1283.15014 Hu J. G. Estimates of | | B - 1 A | | and their applications Mathematica Numerica Sinica. Jisuan Shuxue 1982 4 3 272 282 MR760891 ZBL0538.65015 Hu J. G. Scaling transformation and convergence of splittings of a matrix Mathematica Numerica Sinica 1983 5 1 72 78 MR721894 Cvetković L. H-matrix theory vs. eigenvalue localization Numerical Algorithms 2006 42 3-4 229 245 10.1007/s11075-006-9029-3 MR2279446 ZBL1107.15012 Cvetković L. Kostić V. Doroslovački K. Max-norm bounds for the inverse of S-Nekrasov matrices Applied Mathematics and Computation 2012 218 18 9498 9503 10.1016/j.amc.2012.03.040 MR2923046 Varah J. M. A lower bound for the smallest singular value of a matrix Linear Algebra and Its Applications 1975 11 3 5 MR0371929 10.1016/0024-3795(75)90112-3 ZBL0312.65028 Li W. On Nekrasov matrices Linear Algebra and Its Applications 1998 281 1–3 87 96 10.1016/S0024-3795(98)10031-9 MR1645339 ZBL0937.15019 Kolotilina L. Y. On bounding inverse to Nekrasov matrices in the infinity norm Zapiski Nauchnykh Seminarov POMI 2013 419 111 120 Berman A. Plemmons R. J. Nonnegative Matrices in the Mathematical Sciences 1979 New York, NY, USA Academic Press xviii+316 MR544666 Robert F. Blocs-H-matrices et convergence des méthodes itératives classiques par blocs Linear Algebra and Its Applications 1969 2 223 265 MR0250463 10.1016/0024-3795(69)90029-9 Szulc T. Some remarks on a theorem of Gudkov Linear Algebra and its Applications 1995 225 221 235 10.1016/0024-3795(95)00343-P MR1341080 ZBL0833.15020 Gudkov V. V. On a certain test for non-singularity of matrices Latv. Mat. Ezhegodnik (1965) 1966 Riga, Latvia Zinatne 385 390 MR0193102