Max-norm bounds for the inverse of S-Nekrasov matrices

Applied Mathematics and Computation 218 (2012) 9498–9503

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Max-norm bounds for the inverse of S-Nekrasov matrices Ljiljana Cvetkovic´ a,⇑, Vladimir Kostic´ a, Ksenija Doroslovacˇki b a b

Department of Mathematics and Informatics, Faculty of Science, University of Novi Sad, Serbia Faculty of Technical Sciences, University of Novi Sad, Serbia

a r t i c l e

i n f o

a b s t r a c t Max norm bounds of the inverse of an H-matrix appear in a wide range of applications. Motivated by this fact, we will start with some preliminary estimations already obtained for S-SDD matrices, and present new, applicable and practical bounds for a wider class of S-Nekrasov matrices. Then, we will comment how the scaling technique proves that new bounds improve the known ones for Nekrasov matrices. Finally, we will illustrate new results by numerical examples. Ó 2012 Elsevier Inc. All rights reserved.

Keywords: Max-norm Nekrasov matrices S-Nekrasov matrices H-matrices

1. Introduction We start with some preliminaries. First, denote: the complex (real) n dimensional vector space by Cn ðRn Þ, collection of all n  n matrices with complex (real) entries by Cn;n ðRn;n Þ, and set of indices by N :¼ f1; 2; . . . ; ng. P Given a matrix A ¼ ½aij  2 Cn;n and a nonempty subset S # N, for i 2 N, denote r Si ðAÞ :¼ nj2Snfig jaij j, and, in addition, define S the values hi ðAÞ and zi ðAÞ; i 2 N, recursively: S

h1 ðAÞ :¼ rS1 ðAÞ;

z1 ðAÞ :¼ 1;

S

hi ðAÞ :¼

zi ðAÞ :¼

iP 1 ja j ij j¼1 jajj j

iP 1 ja j ij j¼1 jajj j

n P

S

hj ðAÞ þ

zj ðAÞ þ 1;

jaij j;

ð1Þ

j¼iþ1;j2S

i ¼ 2; 3; . . . n:

ð2Þ N

To simplify notations, when S ¼ N, we will write ri ðAÞ :¼ rNi ðAÞ and hi ðAÞ :¼ hi ðAÞ, for all i 2 N. S S It is easy to see that, for every S # N; r i ðAÞ ¼ r Si ðAÞ þ rSi ðAÞ and hi ðAÞ ¼ hi ðAÞ þ hi ðAÞ, where S :¼ N n S. Furthermore, given a matrix A, by A ¼ D  L  U we denote the standard splitting of A into its diagonal (D), strictly lower ðLÞ and strictly upper ðUÞ triangular parts. In this paper we will estimate max-norm of inverses of matrices that belong to a well-known class of H-matrices, defined as follows. A matrix A ¼ ½aij  2 Cn;n is called an H-matrix if its comparison matrix hAi ¼ ½mij  defined by

hAi ¼ ½mij  2 Cn;n ;

mij ¼



jaii j;

i¼j

jaij j; i – j;

is an M-matrix, i.e. hAi1 P 0. ⇑ Corresponding author. E-mail address: [email protected] (L. Cvetkovic´). 0096-3003/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2012.03.040

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Among many, one of the well-known properties of H-matrices (see [1]) is that for any nonsingular H-matrix A ¼ ½aij  2 Cn;n , jA1 j 6 hAi1 holds. The most important subclass of H-matrices is the class of strictly diagonally dominant (SDD) matrices, i.e. matrices with the following property:

jaii j > r i ðAÞ; for all i 2 N: Besides this one, three more subclasses of H-matrices will be important in the following. First, the class of Nekrasov matrices, see [5,7], defined by

jaii j > hi ðAÞ; for all i 2 N: Second, the class of S-SDD matrices, defined by:

jaii j > r Si ðAÞ for all i 2 S and 

jaii j  r Si ðAÞ

  jajj j  r Sj ðAÞ > rSi ðAÞr Sj ðAÞ for all i 2 S;

j 2 S;

where S is an arbitrary nonempty proper subset of N. And, the third class, called S-Nekrasov matrices, see [4], defined by: S

jaii j > hi ðAÞ for all i 2 S and    S S S S jaii j  hi ðAÞ jajj j  hj ðAÞ > hi ðAÞhj ðAÞ for all i 2 S;

j 2 S:

Again, S is an arbitrary nonempty proper subset of N. To conclude the introduction, let us recall some of the known estimates:  Varah bound for SDD matrices [11]:

kA1 k1 6

1 ; minðjaii j  ri ðAÞÞ

ð3Þ

i2N

 the first bound from [3], for Nekrasov matrices:

kA1 k1 6

maxi2N zi ðAÞ ; mini2N ðjaii j  hi ðAÞÞ

ð4Þ

 the second bound from [3], for Nekrasov matrices:

kA1 k1 6

maxi2N zjai ðAÞ ii j 1  maxi2N hjai ðAÞj

ð5Þ

;

ii

 Kolotilina bound for S-SDD matrices, [6]:

n o kA1 k1 6 max max qSij ðAÞ; qSji ðAÞ ;

ð6Þ

i2S;j2S

where

qSij ðAÞ :¼

jaii j  rSi ðAÞ þ rSj ðAÞ ðjaii j  r Si ðAÞÞðjajj j  rSj ðAÞÞ  r Si ðAÞr Sj ðAÞ

:

ð7Þ

A known fact, see, for example, [2,10], about the classes of matrices that we have introduced is that they stand in the following position (with arbitrary S  N):

SDD  S  SDD  S-Nekrasov  H-matrices; SDD  Nekrasov  S  Nekrasov  H-matrices; while S-SDD and Nekrasov matrices are not subset of each other, in general. Obviously, a bound for a wider class of matrices can be applied to all of its subclasses, too. Since in the literature, up to the authors’ knowledge, there are no bounds for S-Nekrasov matrices, in the following section we will present two such bounds, together with some examples. Afterwards, we will explain why these new bounds improve the known ones for Nekrasov class.

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2. Bounds for S-Nekrasov matrices In order to obtain new bounds, we start with the following lemma, proved by Robert in [9]. Lemma 1. Given any matrix A ¼ ½aij  2 Cn;n ; n P 2, with aii – 0 for all i 2 N, then

h i hi ðAÞ ¼ jaii j ðjDj  jLjÞ1 jUje ;

ð8Þ

i

where e 2 Cn is the vector with all components equal to 1. An immediate corollary of this Lemma is the following characterization of Nekrasov matrices, given by Szulc in [10]: Lemma 2. A matrix A ¼ ½aij  2 Cn;n ; n P 2 is a Nekrasov matrix if and only if

ðjDj  jLjÞ1 jUje < e;

ð9Þ 1

i.e. if and only if E  ðjDj  jLjÞ jUj is an SDD matrix, where E is the identity matrix. However, we are able to generalize Lemma 1, by introducing an arbitrary nonempty subset S of N into the story. Lemma 3. Given any matrix A ¼ ½aij  2 Cn;n ; n P 2, with aii – 0 for all i 2 N, and an arbitrary nonempty subset S of N, then

h i S hi ðAÞ ¼ jaii j ðjDj  jLjÞ1 jUjeS ;

ð10Þ

i

where eS is defined as:

eSi ¼



1; i 2 S; 0;

i 2 S:

Proof. Define vector x component-wise

h i xi :¼ jaii j ðjDj  jLjÞ1 jUjeS ;

i 2 N;

i

and observe that:

jDj1 x ¼ ðjDj  jLjÞ1 jUjeS : Therefore,

x ¼ jLjjDj1 x þ jUjeS ; or, equivalently,

xi ¼

i1 P jaij j j¼1 jajj j

n P

xj þ

for all i 2 N:

jaij j;

ð11Þ

j¼iþ1;j2S S

By mathematical induction, we will show that, for each i 2 N; hi ðAÞ ¼ xi . Obviously, for i ¼ 1, we have

x1 ¼

P

S

ja1j j ¼ rS1 ðAÞ ¼ h1 ðAÞ:

j2Snf1g

S

For a given 2 6 k 6 n, suppose that hi ðAÞ ¼ xi , for all i 6 k  1. Then, from (11), we have

xk ¼

k1 P jakj j j¼1

jajj j

xj þ

n P

jakj j ¼

j¼kþ1;j2S

k1 P jakj j j¼1

jajj j

n P

S

hj ðAÞ þ

j¼kþ1;j2S

S

jakj j ¼ hk ðAÞ;

which concludes the proof. h The first main result of this paper is the following theorem: Theorem 1. Suppose that A ¼ ½aij  2 Cn;n is an S-Nekrasov matrix. Then,

kA1 k1 6 maxzi ðAÞ  max max i2N

i2S;j2S

where zi ðAÞ is given by (2), and

n

o

vSij ðAÞ; vSji ðAÞ ;

ð12Þ

L. Cvetkovic´ et al. / Applied Mathematics and Computation 218 (2012) 9498–9503 S

vSij ðAÞ :¼

S

jaii j  hi ðAÞ þ hj ðAÞ S

9501

S

S

S

ð13Þ

:

ðjaii j  hi ðAÞÞðjajj j  hj ðAÞÞ  hi ðAÞhj ðAÞ

Proof. Given an S-Nekrasov matrix A, there exists diagonal matrix W ¼ diagðw1 ; w2 ; . . . ; wn Þ, where



wi ¼

c > 0; i 2 S; 1;

ð14Þ

i 2 S;

such that AW is a Nekrasov matrix, see [4]. From Lemma 2, we have that

E  ðjDjW  jLjWÞ1 jUjW ¼ W 1 ðE  ðjDj  jLjÞ1 jUjÞW is an SDD matrix, which is equivalent to the fact that ðE  ðjDj  jLjÞ1 jUjÞ is an S-SDD matrix (more details can be found in [4]). Obviously, multiplying by diagonal matrix jDj, and denoting

B :¼ jDj  jDjðjDj  jLjÞ1 jUj;

ð15Þ

we have that B is also an S-SDD matrix, so Kolotilina bound (6) can be applied:

n o kB1 k1 6 max max qSij ðBÞ; qSji ðBÞ : i2S;j2S

Knowing that all diagonal entries of matrix ðjDj  jLjÞ1 jUj are less than 1 (which follows from Lemma 2), we have:

h i jbii j ¼ jaii j  jaii j ðjDj  jLjÞ1 jUj ; ii

rSi ðBÞ ¼

P

h

i

jaii j ðjDj  jLjÞ1 jUj ; ij

j2Snfig

and, therefore, for i 2 S:

jbii j  r Si ðBÞ ¼ jaii j 

h i i P h S jaii j ðjDj  jLjÞ1 jUj ¼ jaii j  jaii j ðjDj  jLjÞ1 jUjeS ¼ jaii j  hi ðAÞ: ij

j2S

i

h i S ¼ jaii j  jaii j ðjDj  jLjÞ1 jUjeS ¼ jaii j  hi ðAÞ: i

Similarly, we obtain: S

jbjj j  r Sj ðBÞ ¼ jajj j  hj ðAÞ; S

rSi ðBÞ ¼ hi ðAÞ;

i 2 S;

j 2 S; S

and r Sj ðBÞ ¼ hj ðAÞ;

j 2 S:

Hence,

qSij ðBÞ ¼ vSij ðAÞ and qSji ðBÞ ¼ vSji ðAÞ; for all i 2 S; j 2 S: Now, in order to estimate kA1 k1 , it remains to find a link between max-norms of matrices B1 and A1 . Since

B ¼ jDjðjDj  jLjÞ1 hAi; it holds that

hAi ¼ ðE  jLjjDj1 ÞB; and

kA1 k1 6 khAi1 k1 6 kB1 k1 kðE  jLjjDj1 Þ1 k1 : Finally, we estimate kðE  jLjjDj1 Þ1 k1 . Since E  jLjjDj1 is an M matrix, we have

kðE  jLjjDj1 Þ1 k1 ¼ kðE  jLjjDj1 Þ1 ek1 : On the other hand, it is easy to see that zi ðAÞ, given by (2), satisfies zðAÞ ¼ ðE  jLjjDj1 Þ1 e, and, therefore,

kðE  jLjjDj1 Þ1 k1 ¼ kzðAÞk1 ¼ maxzi ðAÞ; i2N

which completes the proof. h While the previous theorem gives the bound that corresponds to (4), the following one corresponds to (5).

L. Cvetkovic´ et al. / Applied Mathematics and Computation 218 (2012) 9498–9503

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Theorem 2. Suppose that A ¼ ½aij  2 Cn;n is an S-Nekrasov matrix. Then,

kA1 k1 6 max i2N

zi ðAÞ e Sij ðAÞ; v e Sji ðAÞg;  max maxf v jaii j i2S;j2S

ð16Þ

where zi ðAÞ is given by (2), and S

ve Sij ðAÞ :¼

S

jaii jjajj j  jajj jhi ðAÞ þ jaii jhj ðAÞ S

S

S

ð17Þ

:

S

ðjaii j  hi ðAÞÞðjajj j  hj ðAÞÞ  hi ðAÞhj ðAÞ

Proof. The proof of this theorem is analogous to the previous one. The main difference is that, instead of the matrix B in (15), we use

e :¼ E  ðjDj  jLjÞ1 jUj: B In this way, for all i 2 S; j 2 S, we obtain: S

e ¼ 1  hi ðAÞ ; je b ii j  r Si ð BÞ jaii j

S

e ¼ hi ðAÞ ; r Si ð BÞ jaii j

S

e ¼1 je b jj j  r Sj ð BÞ

hj ðAÞ ; jajj j

S

e ¼ r Sj ð BÞ

hj ðAÞ ; jajj j

e ¼v e ¼v e Sij ðAÞ and qSji ð BÞ e Sji ðAÞ: qSij ð BÞ Since

e hAi ¼ ðjDj  jLjÞ B; the proof is completed by the fact that

jDj1 zðAÞ ¼ ðjDj  jLjÞ1 e; which implies

kðjDj  jLjÞ1 k1 ¼ kjDj1 zðAÞk1 ¼ max i2N

zi ðAÞ : jaii j



In [8] it was shown, using the scaling technique, that bound (6) is less or equal to the bound (3), for every SDD matrix. In the same way, using arguments from [4], one can obtain that, for every Nekrasov matrix, Theorems 1 and 2 give bounds which are less or equal to the bounds (4) and (5), respectively. To conclude the paper, we give some illustrative examples. 3. Numerical examples We consider the following six matrices:

2

50

0

30 10

3

6 10 40 10 20 7 7 6 A1 ¼ 6 7; 4 10 20 50 20 5 0

90 2

3 6 0 6 A3 ¼ 6 4 1

1 20 0

1 2 2

9

0 1

1

3

2

8 6 9 6 A4 ¼ 6 4 6

7

45

0:5 0:5 0:5 16

5

4

15

4:9 0:9 0:9 3

6 1 10 1 2 7 7 6 A5 ¼ 6 7; 4 1 2 4 2 5 2 1 3

15 15 15

15 15 15

4

4 1 2

60

3

6 75 105 45 0 7 7 6 A2 ¼ 6 7; 4 60 60 120 15 5

70

2 2 7 7 7; 10 2 5 0

2

2

21

3

5 7 7 7; 3 5 6

9:1 4:2 2:1

3

6 0:7 9:1 4:2 2:1 7 7 6 A6 ¼ 6 7: 4 0:7 0:7 4:9 2:1 5 0:7 0:7

0:7

2:8

L. Cvetkovic´ et al. / Applied Mathematics and Computation 218 (2012) 9498–9503

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By simple analysis, one can see that matrix A1 is S-Nekrasov for S ¼ f2; 3g. On the other hand, there is no S such that A1 is an S-SDD matrix. It is not a Nekrasov matrix, either, thus, only bounds (12) and (16) can be applied. Matrix A2 is a Nekrasov matrix, implying that it is S-Nekrasov, too, for arbitrary S. Again, there is no S such that A2 is an S-SDD matrix. Thus, only bounds (4), (5), (12) and (16) can be applied. Matrix A3 is an S-SDD matrix for S ¼ f1g, implying that it is S-Nekrasov, as well. Since it is not a Nekrasov matrix, only bounds (6), (12) and (16) can be applied. Both matrices A4 and A5 are Nekrasov matrices. They are also S-SDD matrices for S ¼ f1g and S ¼ f2g, respectively. Thus, only Varah bound (3) could not be applied. Finally, matrix A6 is an SDD matrix, implying that it is Nekrasov, S-SDD and S-Nekrasov, for any choice of nonempty S. Therefore, all the bounds can be applied. For these six matrices, the following table summarizes the mentioned bounds. In the last three rows of the table, when appropriate, set S for which the corresponding matrix is S-SDD (i.e. S-Nekrasov) is given. Moreover, among all the possibilities for such a set, the one which gives the best bound is chosen. In addition, for each matrix, the best bound is bolded. Matrix

A1

A2

A3

A4

A5

A6

k  k1 (3) (4) (5) (6)

1.4800 – – – –

0.6843 – 1.5333 2.4667 –

(12)

2.6320 S ¼ f2; 3g 1.7680 S ¼ f2; 3g

1.1176 S ¼ f2; 4g 1.6864 S ¼ f2; 4g

0.4479 – – – 1.6667 S ¼ f1g 1.0291 S ¼ f1g 0.9096 S ¼ f1g

0.4474 – 0.5702 1.557 0.4928 S ¼ f1g 0.5658 S ¼ f2; 3g 0.9076 S ¼ f1; 3g

1.0093 – 5.0000 2.8571 6.0000 S ¼ f2g 2.6227 S ¼ f2; 4g 2.0155 S ¼ f1; 4g

0.8759 1.4286 0.9676 1.8076 1.1429 S ¼ f1; 3g 0.9109 S ¼ f2; 3g 1.7672 S ¼ f1; 3g

(16)

Acknowledgments This work is partly supported by the Ministry of Education and Science of Serbia (174019), and by the Provincial Secretariat of Science and Technological Development of Vojvodina (2002). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]

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