Eigenvalue localization refinements for the Schur complement

Applied Mathematics and Computation 218 (2012) 8341–8346

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

Eigenvalue localization refinements for the Schur complement Ljiljana Cvetkovic´ a,⇑, Maja Nedovic´ 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 The theory of Schur complement plays an important role in many fields, as well as the theory of H-matrices. In this paper, we obtain some bounds for the eigenvalues of Schur complement by the entries of the original matrix instead of those of the Schur complement, for some special H-matrices. Also, we show how this result for a wider class can be applied for estimating bounds for the eigenvalues of Schur complement of an SDD matrix. Ó 2012 Elsevier Inc. All rights reserved.

Keywords: H-matrices Schur complement Diagonal scaling Eigenvalue distribution

0. Introduction The starting point for the considerations that follow is the fact that a nonsingular matrix A is an H-matrix if and only if there exists a diagonal nonsingular matrix W such that AW is a strictly diagonally dominant (SDD) matrix. In other words, see [5], the class of H-matrices is diagonally derived from the class of SDD matrices. Some special subclasses of H-matrices could be characterized by the form of the corresponding scaling matrix W. These characterizations will be recalled in the first section, for they will be used to obtain some new results on the eigenvalue distribution of the Schur complement. Throughout the paper we will use the following notations:

N :¼ f1; 2; . . . ; ng for the set of indices; S for any nonempty proper subset of N; S :¼ N n S for the complement of S; X jaik j for ith deleted row sum and r i ðAÞ :¼ k2N;k–i

r Si ðAÞ

:¼

X

jaik j for part of ith deleted row sum; corresponding to S:

k2S;k–i

Obviously, for arbitrary subset S and each index i 2 N,

ri ðAÞ ¼ r Si ðAÞ þ r Si ðAÞ: It is important to emphasize that all the time we are dealing with nonsingular H-matrices, calling them shortly H-matrices. To be precise, we recall the definition of SDD matrices and H-matrices, as well as some more preliminaries.

⇑ Corresponding author. E-mail address: [email protected] (L. Cvetkovic´). 0096-3003/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2012.01.058

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1. Preliminaries Definition 1.1. A matrix A ¼ ½aij  2 Cn;n is called an SDD matrix if, for each i 2 N, it holds that

jaii j > r i ðAÞ: Theorem 1.2. If a matrix A 2 Cn;n is an SDD matrix, then it is nonsingular, moreover it is an H-matrix. That SDD matrices are nonsingular is an old and recurring result in matrix theory. In [12] it is pointed out that this nonsingularity result and the well-known Geršgorin’s Theorem (1931) that gives the eigenvalue localization are actually equivalent. Theorem 1.3 (Geršgorin Theorem). For any matrix A 2 Cn;n and any k 2 rðAÞ, there is a positive integer k in N such that

jk  ak;k j 6 r k ðAÞ: The set

Ci ðAÞ ¼ fz 2 Cjjz  ai;i j 6 ri ðAÞg is called the ith Geršgorin disk. The union of these disks contains all the eigenvalues and it is called the Geršgorin set,

rðAÞ # CðAÞ ¼

[

Ci ðAÞ:

i2N

Definition 1.4. A matrix A ¼ ½aij  2 Cn;n is called an H-matrix if its comparison matrix hAi ¼ ½mij  defined by

mii ¼ jaii j;

mij ¼ jaij j; 1

is an M-matrix, i.e., hAi

i; j ¼ 1; 2; . . . ; n;

i–j

P 0.

Theorem 1.5. A matrix A is an H-matrix if and only if there exists a diagonal nonsingular matrix W such that AW is an SDD matrix. Moreover, we can always assume that W has only positive diagonal entries. The following subclass of H-matrices has been investigated in [6,7]. Definition 1.6. A matrix A ¼ ½aij  2 Cn;n is called a Dashnic–Zusmanovich matrix if there exists an index i 2 N such that

jaii j  ðjajj j  rj ðAÞ þ jaji jÞ > ri ðAÞ  jaji j;

for all j – i;

j 2 N:

Theorem 1.7. If a matrix A 2 Cn;n is a Dashnic–Zusmanovich matrix, then it is nonsingular, moreover it is an H-matrix. Class of R  SDD matrices, a generalization of Dashnic–Zusmanovich class, was defined in the present form in [2,12]. Here we will recall one of several equivalent definitions of the R  SDD class, for more details see [5]. Definition 1.8. Given any matrix A ¼ ½ai;j  2 Cn;n ; n P 2, and given any nonempty proper subset S of N, then A is an S-strictly diagonally dominant (S-SDD) matrix if

jaii j > r Si ðAÞ for all i 2 S and    jaii j  r Si ðAÞ jajj j  r Sj ðAÞ > r Si ðAÞr Sj ðAÞ for all i 2 S; j 2 S: Definition 1.9. If there exists a nonempty proper subset S of N, such that A ¼ ½ai;j  2 Cn;n ; n P 2 is an S-SDD matrix, then we will say that A belongs to class of R  SDD matrices. Theorem 1.10. If a matrix A 2 Cn;n is an R  SDD matrix, then it is nonsingular, moreover it is an H-matrix. According to Theorem 1.5, a matrix A 2 Cn;n is an H-matrix if and only if there exists a nonsingular diagonal matrix W such that AW is an SDD matrix. But, such a matrix W could be found in a very few special cases. Up to now, we are aware of two such cases: Dashnic Zusmanovich matrices and R  SDD matrices. Namely, Dashnic Zusmanovich class can be characterized as a subclass of H-matrices for which the corresponding scaling matrix W belongs to the set F , defined as the set of diagonal matrices, whose diagonal entries are equal to 1, all except one, which is an arbitrary positive number, i.e.

F¼

8 > < > :

W ¼ diagðw1 ; w2 ; . . . ; wn Þ :

9 wi ¼ c > 0 for one i 2 N; > = wj ¼ 1

for j – i:

> ;

ð1Þ

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From the other hand, the R  SDD class can be characterized as a subclass of H-matrices for which the corresponding scaling matrix W belongs to the set W, defined as the set of all diagonal matrices whose diagonal entries are either 1 or c, where c is an arbitrary positive number, i.e.

W¼

[

WS;

ð2Þ

SN

WS ¼

8 > < > :

wi ¼ c > 0 for i 2 S; W ¼ diagðw1 ; w2 ; . . . ; wn Þ : wi ¼ 1

otherwise:

9 > = > ;

If A is an S  SDD matrix, we choose scaling parameter c from the interval ðc1 ðAÞ; c2 ðAÞÞ, with

0 6 c1 ðAÞ :¼ max i2S

rSi ðAÞ ; jaii j  r Si ðAÞ

c2 ðAÞ :¼ min j2S

jajj j  rSj ðAÞ r Sj ðAÞ

;

where, if rSj ðAÞ ¼ 0 for all j 2 S, then c2 ðAÞ is defined to be þ1. According to the definition, if A is an S  SDD matrix, then the interval ðc1 ðAÞ; c2 ðAÞÞ is not empty, and for each W from W S , with arbitrary c from this ‘good’ interval, matrix AW is an SDD matrix. 2. Eigenvalue distribution of the Schur complement for some H-matrices 2.1. Vertical eigenvalue bands The Schur complement of A with respect to a proper subset of N; a, is denoted by A=a and defined to be

AðaÞ  Aða; aÞðAðaÞÞ1 Aða; aÞ; where Aða; bÞ stands for the submatrix of A 2 Cn;n lying in the rows indexed by a and the columns indexed by b, while Aða; aÞ is abbreviated to AðaÞ. Troughout the paper we assume that AðaÞ is a nonsingular matrix. As stated in [8], investigating the distribution for the eigenvalues of the Schur complement is of great significance. If the eigenvalues of the Schur complement can be estimated by the elements of the original matrix, it is easy to know whether a linear system could be transformed into a smaller one which can be solved by iteration. This kind of iteration, which has many advantages, is called the Schur-based iteration, as it converts the original system into two smaller ones by the Schur complement. Fore more details about properties of the Schur complement, with respect to the original matrix, see [1,3,4,9,10,13]. In [11], the following theorem concerning the eigenvalues of the Schur complement is proved. Theorem 2.1. Let a matrix A 2 Cn;n be an SDD matrix with real diagonal entries, and let a be a proper subset of the index set. Then, A=a and AðaÞ have the same number of eigenvalues whose real parts are greater (less) than wðAÞ (resp. wðAÞ), where

wðAÞ ¼ min½jajj j  r j ðAÞ þ min j2a

i2a

jaii j  r i ðAÞ X jajk j: jaii j k2a

Using this result together with the scaling approach, we can prove the following statement. Theorem 2.2. Given arbitrary nonempty proper subset S # N, let A 2 Cn;n be an S-SDD matrix with real diagonal entries, and let a be a proper subset of the index set. Then, A=a and AðaÞ have the same number of eigenvalues whose real parts are greater (less) than wðW 1 AWÞ (resp. wðW 1 AWÞ), where wðAÞ is defined as above and W is a corresponding scaling matrix: W ¼ diagðw1 ; w2 ; . . . ; wn Þ, with wi ¼ c 2 ðc1 ðAÞ; c2 ðAÞÞ, for i 2 S, and wi ¼ 1, otherwise. Proof. Since A is an S  SDD matrix with real diagonal entries, and W is the corresponding scaling matrix, we know that W 1 AW is an SDD matrix (also with real diagonal entries). Then, if a is a proper subset of the index set, we have

ðW 1 AWÞ=a ¼ W 1 ðaÞA=aWðaÞ; which is always similar to A=a. Moreover, if a ¼ S or a ¼ S this matrix is exactly A=a. Obviously, matrices ðW 1 AWÞðaÞ and AðaÞ are similar (for any choice of a), so they have the same eigenvalues. Now, we apply Theorem 2.1 to SDD matrix W 1 AW, and obtain that A=a and AðaÞ have the same number of eigenvalues whose real parts are greater (less) than wðW 1 AWÞ (resp. wðW 1 AWÞ). h In previous section, we stated that the scaling matrix for the fixed S  SDD matrix A is not unique, i.e., the scaling parameter c can be chosen from the ‘good’ interval ðc1 ðAÞ; c2 ðAÞÞ. In other words, we can transform the given S  SDD matrix to many different SDD matrices, by choosing different values for c, as long as c belongs to the ‘good’ interval. Therefore, we

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can in this way obtain many different values for wðW 1 AWÞ for the given S  SDD matrix A, and all of them have the same separating property. Geometrical explanation of this is that, if the principal submatrix AðaÞ has no eigenvalues whose real parts are between the vertical lines x ¼ wðW 1 AWÞ and x ¼ wðW 1 AWÞ, then the Schur complement A=a has no eigenvalues whose real parts are in the band, either. By taking the maximum value for wðW 1 AWÞ, while c belongs to ðc1 ðAÞ; c2 ðAÞÞ, we obtain the widest (in the sense of Theorem 2.2) band. Now, we go one step back, and apply this result on SDD matrices. Note that, if a matrix A is SDD, then it is also S  SDD for any subset S of the index set N. If we fix any subset S of N and choose c from the ‘good’ interval, we can scale the given SDD matrix to some other SDD matrix and we can get w1 ¼ wðW 1 AWÞ ‘better’ than w ¼ wðAÞ with the same separating property. This means that A=a and AðaÞ have the same number of eigenvalues whose real parts are greater than (resp. smaller than) w1 (resp. w1 ). Note that, if a matrix A is both S  SDD and T  SDD, than we can get two different values for w, i.e. wðW 1 S AW S Þ and wðW 1 T AW T Þ. Moreover, different values of c from the ‘good’ intervals can also produce different vertical bands for eigenvalues. 2.2. Geršgorin-like disks for Schur complement In [8], the following result concerning the dominant degree and the eigenvalue localization for the Schur complement, is proven. Let N r ðAÞ denote the set of indices for which the corresponding row of the matrix A is strictly diagonally dominant:

Nr ðAÞ ¼ fi 2 N : jaii j > r i ðAÞg: Theorem 2.3. Let A 2 Cn;n ; a ¼ fi1 ; i2 ; . . . ; ik g # N r ðAÞ; a ¼ fj1 ; j2 ; . . . ; jl g. Then, for every eigenvalue k of A=a, there exists 1 6 t 6 l such that

jk  ajt jt j 6 rjt ðAÞ  wjt 6 r jt ðAÞ: Here, wjt ¼

Pk

u¼1 jajt iu j

jaiu iu jr iu ðAÞ . jaiu iu j

This means that the eigenvalues of A=a are contained in the union of those Geršgorin disks for the matrix A whose indices are in a. We can improve this result, using our scaling approach, in the same fashion as it is done when forming the weighted Geršgorin disks, as a starting point for the minimal Geršgorin set. Let us note that Theorem 2.3 holds when a ¼ fi1 ; i2 ; . . . ; ik g # Nr ðAÞ, i.e., only if we choose a-indices among indices of SDD-rows. Therefore, in order to apply this result on a scaled matrix, we must provide that these rows stay SDD after diagonal scaling. We will present here our result concerning the class of S  SDD matrices. Moreover, we can go one step back, to the class of SDD matrices and apply this result for any ‘good’ scaling matrix (preserving SDD property in a-rows), finding in that way the best possible Geršgorin disks (in the sense of scaling). Theorem 2.4. Let A 2 Cn;n be an S-SDD matrix, let a # N, and let W be a corresponding scaling matrix for A. Then,

rðA=aÞ ¼ rððW 1 AWÞ=aÞ #

[

Cj ðW 1 AWÞ:

j2a

Proof. Let W be any corresponding scaling matrix for A, with c belonging to the ‘good’ interval. We have

ðW 1 AWÞ=a ¼ W 1 ðaÞA=aWðaÞ; which is always similar to A=a. Therefore, it holds that

rðA=aÞ ¼ rððW 1 AWÞ=aÞ: As ðW 1 AWÞ is an SDD matrix, we can apply Theorem 2.3 to the matrix ðW 1 AWÞ, which proves our statement. h The benefits from this are the following. First of all, scaling allows us to deal with wider class of matrices. Second, as we will see from the examples, scaled disks give a tighter eigenvalue inclusion area than Geršgorin disks for the original matrix. Compared to the disks obtained in Theorem 2.3, for some matrices our scaled radius is smaller. We can take the intersection of our scaled Geršgorin set and the set obtained in Theorem 2.3 and in that way, for each disk choose the smaller radius. 3. Numerical examples Example 1 The given matrix, A, is S  SDD for S ¼ f1; 2; 3g. We take a ¼ S, and determine the ‘good’ interval for c. It is easy to see that c ¼ 0:002 belongs to this ‘good’ interval. For the chosen c, we scale the matrix A, and, as in Theorem 2.2, obtain the vertical band (wðW 1 AWÞ ¼ 7:98531) given in the Fig. 1.

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10

5

0

5

10 10

5

0

5

10

Fig. 1. Vertical band for A=a.

30 20 10 0 10 20 30 30

20

10

0

10

20

30

Fig. 2. Geršgorin-like disks for B=a.

2 6 6 6 6 6 6 A¼6 6 6 6 6 4

1125000

1000

0

1125000

1000

1000

1000

2000

0

2000

1000

0

0

1000

1000

1

1

1

0

1

0

3

1 7 7 7 1225000 1 1 1 2 7 7 7 1000 25 1 1 1 7: 7 1000 0 25 0 1 7 7 7 1000 1 1 25 1 5 1000 1 2 1 12 1000

1

Example 2. Matrix B is S  SDD for S ¼ f1; 2; 3g. For a ¼ S, we determine the ‘good’ interval for c, and choose c ¼ c1 ðBÞ ¼ 1=90. As in Theorem 2.4, in Fig. 2 we obtain the Geršgorin-like set that contains all the eigenvalues of the Schur complement B=a.

2 6 6 6 6 6 6 B¼6 6 6 6 6 4

600

50

0

650

50 50 0 50 0

50

1

1

1

0

1

0

3

1 7 7 7 50 550 1 1 1 2 7 7 7 100 50 12 1 1 1 7: 7 100 50 0 21 0 1 7 7 7 0 50 1 1 25 1 5 50 50 1 2 1 10 50

1

Example 3 Matrix C is SDD, so, it is S  SDD for any choice of S # N. If we choose S ¼ a ¼ f1; 2; 3g, scale the given matrix C to W 1 CW, with c ¼ c1 ðCÞ ¼ 3=11123, and then determine Geršgorin-like disks (see Fig. 3) as in Theorem 2.4, our scaled radius for any of the four disks is smaller than the corresponding radius obtained in Theorem 2.3.

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40 20 0 20 40 40

20

0

20

40

Fig. 3. Geršgorin-like disks for C=a.

2 6 6 6 6 6 6 C¼6 6 6 6 6 4

11125 1 0 21125 1

1

7

1

18

0

25

0

35

0

3 0 1 7 7 7 31225 1 1 1 2 7 7 7 1 12 1 0 1 7: 7 0 0 21 0 1 7 7 7 1 1 1 31 1 5 1 1 0 1 41 1 1

1 1

1 0

1 1

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