Modified Gauss Seidel type methods and Jacobi type methods for Z-matrices

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1 Linear Algebra and its Applications 7 (2) Modified Gauss Seidel type methods and Jacobi type methods for Z-matrices Wen Li a,, Weiwei Sun b a Department of Mathematics, South China Normal University, Guangzhou 56, People s Republic of China b Department of Mathematics, City University of Hong Kong, Hong Kong, People s Republic of China Received 24 September 998; accepted 2 April 2 Submitted by M. Neumann Abstract In this paper, we present the convergence analysis for some modified Gauss Seidel and Jacobi type iterative methods and provide a comparison of spectral radius among the Gauss Seidel iterative method and these modified methods. Some recent results are improved. 2 Elsevier Science Inc. All rights reserved. AMS classification: 65F Keywords: Z-matrix; Modified Gauss Seidel type method; Modified Jacobi type method; Convergence; Spectral radius. Introduction Consider the following linear system Ax = b, (.) where A is an n n square matrix, and x and b are n-dimensional vectors. For any splitting, A = M N with the non-singular matrix M, the basic iterative methods for solving Eq. (.) is This work was supported in part by the City University of Hong Kong Research Grant and Natural Science Foundation of Guangdong Province, China. Corresponding author. addresses: liwen@scnu.edu.cn (W. Li), maweiw@math.cityu.edu.hk (W. Sun) //$ - see front matter 2 Elsevier Science Inc. All rights reserved. PII:S24-795()4-

2 228 W. Li, W. Sun / Linear Algebra and its Applications 7 (2) x (k+) = M Nx (k) + M b, k =,, 2,... For simplicity, we assume that A has unit diagonal entries and let A = I L U, where L and U are strictly lower and strictly upper triangular matrices, respectively. Then the iteration matrices of the classical Jacobi and classical Gauss Seidel methods are J = L + U and T = (I L) U, respectively. We consider a preconditioned system of (.) PAx = Pb. The corresponding basic iterative method is given in general by x (k+) = M p N px (k) + Mp Pb, k =,, 2,..., where PA = M p N p is a splitting of PA. A simple modified Gauss Seidel (MGS) method was first proposed by Gunawardena et al. [] with P = I + S and a 2 a 2 S = a n,n The main theorem in [] is given as follows. Theorem. []. Let A=(a ij ) R n n with <a i,i+ a i+,i <, i,j=,..., n, and A = I L U, where L and U are strictly lower and strictly upper triangular matrices, respectively. Let T GS = (I L) U and T = (I L SL) (U S + SU).Then (a) ρ(t)<ρ(t GS ) if ρ(t GS )<, (b) ρ(t)= ρ(t GS ) if ρ(t GS ) =, (c) ρ(t)>ρ(t GS ) if ρ(t GS )>. Recently, Kohno et al. [2] extended Gunawardena et al. s work to a more general case and presented a new MGS method by using the preconditioned matrix P = I + S α, where α a 2 α 2 a 2 S α = α n a n,n

3 W. Li, W. Sun / Linear Algebra and its Applications 7 (2) They showed that if A is a non-singular diagonally dominant Z-matrix with some conditions, then there exists an α > such that (I + S α )A is a strictly diagonally dominant Z-matrix for all α i [,α ] and presented some numerical investigation for the choice of the optimal parameter. It is shown in [2] by numerical investigation that their method is superior to other methods. Now we give one main result of [2] as follows. Theorem.2 [2, Theorem ]. Let A = (a ij ) R n n be a non-singular diagonally dominant Z-matrix with unit diagonal entries and n j= a nj >. Assume that nj= a i+,j > if n j= a ij = for some i<n.let A = I L U, where L and U are strictly lower and strictly upper triangular matrices, respectively. Then (I + S α )A is a strictly diagonally dominant Z-matrix, and ρ(t α )< for α i ( i<n),where T α = (I L S α L) (U S a + S α U). Theorem.2 provides a convergence result that the general MGS method is convergent when A is a non-singular diagonally dominant M-matrix. Throughout this paper, we always assume that a i,i+ /=, i=,...,n and a ii =, i =,...,n, A= I L m U m, where L m and U m are strictly lower triangular non-negative and general non-negative matrices, respectively. Now we consider more general MGS methods, which say to be an MGS type method, its iterative form is given by x (k+) = T α x (k) + b, k =,, 2,..., (.2) where x () is an initial approximation, and T α = (I L m S α L m ) (U m S α + S α U m ), b = (I L m S α L m ) (I + S α )b. Notice that, if we take some specific values of α and U m, then the MGS type method reduces to the well-known methods as classical Jacobi, classical Gauss Seidel and MGS method given in [,2]. By T and J we denote T = T α= and J = L m + U m, which are iteration matrices of Gauss Seidel type method and Jacobi type method, respectively. Our work in the presentation is to provide convergence analysis. The main results are summarized below: We prove that if A is a non-singular M-matrix, the MGS type method converges for all parameters in [, ] and the convergence rates for the MGS type methods are better than those of the corresponding GS type methods without the condition that A is diagonally dominant as given in Theorem.2 and the condition <a i,i+ a i+,i < as given in Theorem.; see Theorems. and.2. We present the convergence of some modified Jacobi type iteration and Stein Rosenberg type theorem for the general MGS and modified Jacobi (MJ) type methods; see Theorems 4. and 4.2 and Corollary 4..

4 2 W. Li, W. Sun / Linear Algebra and its Applications 7 (2) Preliminaries For an n n matrix A, the directed graph Γ(A) of A is defined to be the pair (V, E), where V ={,...,n} is a set of vertices and E ={(i, j): a ij /=, i, j =,...,n} is a set of arcs. A path from i to i p is an ordered tuple of vertices (i,i 2,...,i p ) such that for each k, (i k,i k+ ) E. Apath(i,i 2,...,i p ) is said to be a cycle provided that i,...,i p are pairwise distinct and i = i p. We say a directed graph to be strongly connected if for any two vertices i, j there is a path from i to j. AmatrixA is said to be irreducible if Γ(A) is strongly connected. Let A R n n,αand β be subsets of V. We denote by A[α β] the submatrix of A whose rows are indexed by α and columns by β and A[α] =A[α β]. By α we denote α = V \α. AmatrixB is non-negative, semi-positive and positive if each entry of B is nonnegative, non-negative but at least a positive entry and positive, respectively. We denote them by B, B>andB. A matrix A = (a ij ) is called a Z-matrix if for any i/= j, a ij, and an M-matrix if A = si B, B ands ρ(b), where ρ(b) denotes the spectral radius of B. A = M N is said to be a splitting of A if M is non-singular. A splitting is said to be convergent if the iteration matrix M N is convergent. A splitting A = M N is said to be regular if M andn, M-splitting if M is a non-singular M-matrix and N, and weak non-negative if M N, respectively. In general, M-splitting regular splitting weak nonnegativesplitting. Some basic properties are given below, which will be used in the proof of our main theorems. Theorem 2. []. Let A be a Z-matrix. Then the following statements are equivalent: (a) A is a non-singular M-matrix. (b) There is a positive vector x such that Ax. (c) All principal submatrices of A are non-singular M-matrices. (d) All principal minors are positive. Theorem 2.2 []. Let A be a Z-matrix. Then the following statements are equivalent: (a) A is an M-matrix. (b) All principal submatrices of A are M-matrices. (c) All principal minors are non-negative. Theorem 2. []. Let A be a Z-matrix. (a) If there is a positive vector x such that Ax, then A is an M-matrix with property c. (b) If A is a singular irreducible M-matrix, then there is a positive vector x such that Ax =.

5 W. Li, W. Sun / Linear Algebra and its Applications 7 (2) Convergence theorem The following lemmas are useful to prove our main theorem. Lemma.. Let A be a diagonally dominant Z-matrix. Then A is a non-singular M-matrix if and only if A is non-singular. Proof. We only need to prove sufficiency. If A is a diagonally dominant Z-matrix, Ae, where e = (,...,) T.ThenAis an M-matrix from Theorem 2.. Lemma.2 [4,7]. Let A = M N be an M-splitting of A. Thenρ(M N) < (= ) if and only if A is a non-singular (singular) M-matrix. Lemma.. Let A be a Z-matrix. Then A is a non-singular M-matrix if and only if for any α i [, ], i=,...,n,(i+ S α )A is a non-singular M-matrix. Proof. If A is a non-singular M-matrix, for any α i [, ], i=,...,n, let à = (I + S α )A = (â ij ).Then { aij α â ij = i a i,i+ a i+,j, i<n, a nj, i = n. Since A is a Z-matrix, â ij fori/= j, i.e., à is a Z-matrix. Since A is a non-singular M-matrix, by Theorem 2. there exists a positive vector x such that Ax. Also we have Ãx = (I + S α )Ax. By Theorem 2. à is a non-singular M-matrix. If à is a non-singular M-matrix, à T is also a non-singular M-matrix. By Theorem 2. there is a positive vector x such that à T x, i.e., A T (I + Sα T )x. Let y = (I + Sα T)x. Theny andat y, which imply that A T is a non-singular M-matrix from Theorem 2., and so is A. Lemma.4. Let A = (a ij ) be an n n matrix (n 2) with a i,i+ /=,i=,..., n. Then A can be written by the following block form: A A 2 A k A 22 A 2k A =......, (.) A kk where A ii is irreducible,i=,...,k. Proof. We use the induction on n to prove this theorem. It is obvious that this is true for n = 2. We assume that it is true for any matrices with dimension <n.let i = max j n {j there is a path from j to in Γ(A)} and δ ={,...,i }. First we show that A[δ] is irreducible. In fact, since a i,i+ /=, i=,...,n, (, 2,...,i ) is a path in Γ(A). By the definition of i, there is a path from i to. Let this path be

6 22 W. Li, W. Sun / Linear Algebra and its Applications 7 (2) (i,i 2,...,i t, ). Then i p i by the definition of i,p=,...,t.hence, for any i, j δ, (i,i +,...,i,i 2,...,i t,, 2,...,j)is a path from i to j in Γ(A[δ]),and thus A[δ] is irreducible. If i = n, A has the form (.) with k =. If i <n,then δ /=. We need to show that A[δ δ] =. Otherwise, if A[δ δ] /=, then there exist i δ and j δ such that a ij /=, i.e., (i, j) Γ(A). Since A[δ] is irreducible and,j δ, there is a path from j toinγ(a). Since (i, j) Γ(A), there is a path from i to in Γ(A). Noting i δ, then i>i, which contradicts the definition of i. Hence A[δ δ] =. Now let A = A[δ]. Since A[δ ] also satisfies the condition of the theorem and is of dimension <n. By the inductive assumption, A[δ ] can be written as the upper triangular block form A[δ ]=(A ij ) with A ii is irreducible, i = 2,...,k. Thus, A has the upper triangular block form (.), which proves the lemma. Further properties for matrix splitting are given in the following two lemmas. The proof of the first one can be obtained by noting Theorem 6. in [5] and the second one follows Lemma. in [6] and Theorem 2.. in []. Lemma.5. Let A be a non-singular M-matrix and A = M N = M 2 N 2 be two convergent splittings, one weak non-negative and one regular. If N N 2, then ρ(m2 N 2) ρ(m N ). Lemma.6. Let A be irreducible, A= M N be an M-splitting. Then there is a positive vector x such that M Nx = ρ(m N)x. Our main result in this section is as follows. Theorem.. Let A = (a ij ) R n n and A = I L m U m, where U m is a nonnegative matrix and L m is a strictly lower triangular non-negative matrix. Then: (a) For any α i [, ], i=,...,n, ρ(t α )< if ρ(t) <. In this case, we have ρ(t α ) ρ(t) ρ(j) <. Moreover, if A is irreducible, then ρ(t α )< ρ(t) for α i (, ], i=,...,n. (b) For any α i [, ), i =,...,n, ρ(t α ) = if ρ(t) =. Proof. Let à = (I + S α )A, M α = (I + S α )(I L m ), N α = (I + S α )U m, E α = I (I + S α )L m, F α = U m S α + S α U m, where E α = (e ij ), L m = (l ij ) and U m = (u ij ). Clearly, we have (.2) à = M α N α = E α F α. (.)

7 W. Li, W. Sun / Linear Algebra and its Applications 7 (2) (a) Since L m is a strictly lower triangular non-negative matrix, I L m is a nonsingular M-matrix and therefore, A = (I L m ) U m is an M-splitting. Since ρ(t) <, it follows from Lemma.2 that A is a non-singular M-matrix. Hence ρ(j) <. By Lemma., à is also a non-singular M-matrix and E α is a lower triangular Z-matrix with { + αi a e ii = i,i+ l i+,i i<n, (.4) i = n. Since l i+,i = a i+,i u i+,i a i+,i, by (.4) e ii α i a i,i+ a i+,i, i =,...,n. Since all 2 2 principal minors of A are positive, e ii α i a i,i+ a i+,i > forα i [, ]. This implies that E α is a non-singular M-matrix and F α is a non-negative matrix. Then à = E α F α is an M-splitting of the non-singular M- matrix Ã. By Lemma.2, ρ(t α )<. Notice that Mα N α = (I L) U and ρ(mα N α) = ρ((i L m ) U m ) = ρ(t) <. So à = M α N α = E α F α are two convergent splittings, à = M α N α is weak non-negative and à = E α F α is regular. It is easy to see that N α F α. By Lemma.5, ρ(eα F α) ρ(mα N α)<, i.e., ρ(t α ) ρ(t). We obtain ρ(t) ρ(j) by noting Lemma 2.() in [4]. The first part of (a) is proved. For the second part of (a), let A be irreducible. By Lemma.6 there is a positive vector x such that Tx = ρ(t)x and by the same proof as [, p. ], we have T α x ρ(t)x = (ρ(t ) )(I L m S α L m ) S α x. Since all diagonal entries of (I L m S α L m ) are positive and the first n rowsofs α are non-zero, the first n rowsof(i L m S α L m ) S α are semi-positive. Let y = ( ρ(t ))(I L m S α L m ) S α x. Then y>andthefirstn entries of y are positive. Clearly, we have T α x = ρ(t)x y. By Lemma.4 of [6], there exists a permutation matrix P, such that [ ] PT α P T H =, H 2 where H 2 is an irreducible square matrix whose dimension is equal to the number of non-zero columns in F α. It is readily to see (F α ) i,i+2 = a i,i+2 + α i a i,i+ a i+,i+2 > forα i >, i=,...,n 2, i.e., F α has at least n 2 non-zero columns. Hence, the dimension of H 2 is equal to or larger than n 2. When n 4, the dimension of H 2 is at least 2. Since T α x = ρ(t)x y we have PH 2 P T P x < ρ(t )P x, which deduces ρ(h 2 )<ρ(t)from Perron Frobenius Theorem (see []), and therefore, ρ(t α )<ρ(t).whenn, we consider the following two cases, respectively. Case. If the last row of L m is non-zero, then there is i<nsuch that l ni /=. We see that ((I L m S α L m ) ) ni and (S α ) i,i+ are non-zero. Since (I L m S α L m ) and S α are semi-positive, ((I L m S α L m ) S α ) n,i+ is positive. This implies that each row of (I L m S α L m ) S α is semi-positive, and hence y. Then we obtain ρ(t)x T α x, and thus, ρ(t α )<ρ(t)from Perron Frobenious Theorem.

8 24 W. Li, W. Sun / Linear Algebra and its Applications 7 (2) Case 2. If the last row of L m is and n = 2, then L m =. It is easy to deduce that the first column of T α is non-zero, and the first entry of y is positive. It is easy to see ρ(t α )<ρ(t).ifthelastrowoflis and n =, since A is irreducible, there is k, k 2, such that u k /=. Hence, it is easy to see that the kth and third columns of F α are non-zero, i.e., the dimension of H 2 is at least 2. Similarly, we have ρ(t α ) < ρ(t ). (b) Let ρ(t) =. Since A = (I L m ) U m is an M-splitting, by Lemma.2, A is a singular M-matrix. First we need to prove that E α is non-singular. Clearly, E α is a lower triangular Z- matrix with the diagonal entries as in (.4). It is noted that for any α i [, ), e ii α i a i,i+ a i+,i >, i=,...,n sinceais a singular M-matrix. Then E α is a lower triangular Z-matrix with positive diagonal entries, and therefore, E α is a non-singular M-matrix. Secondly, we prove that à is a singular M-matrix. If A is irreducible, A is an irreducible singular M-matrix. It follows from Theorem 2. that there is a positive vector x such that Ax =, and Ãx = (I + S α )Ax =. Since à is a Z-matrix, à is a singular M-matrix by Theorem 2.. If A is reducible, by Lemma.4, A has the upper triangular block form (.) with k 2. Let I + S α = (S ij ) k k with the same block form as (.). Then I + S α is also an upper triangular block matrix and its diagonal block S ii has the same non-zero structure as I + S α. Hence, à = (à ij ) has the same upper triangular block structure as (.) with diagonal blocks à ii = S ii A ii,i=,...,k.sincea is a singular M-matrix, all diagonal blocks A ii,i=,...,k, are irreducible M-matrices. If A ii is non-singular, by Lemma. à ii is a non-singular M-matrix. If A ii is singular, à ii is a singular M-matrix by the above proof. By the definition of à and Lemma.4, à is an upper triangular block Z-matrix whose diagonal blocks are à ii,i=,...,k. Hence à is a singular M-matrix. We can conclude that à = E α F α is an M-splitting of a singular M-matrix. It follows from Lemma.2 that ρ(t α ) =. From Theorem. we can easily obtain the following theorem. Theorem.2. Let A be a non-singular M-matrix and A = I L m U m.then MGS type method converges and the convergence rates for the MGS type methods are better than those of the corresponding GS type methods for α i [, ], i =,...,n. Remark.. (a) Theorem.(b) is not true for some α with α i [, ] since E α may be singular. For example, taking α = and [ ] [ ] A =, E α =. (b) Under the assumption in Theorem., one still cannot obtain ρ(t) from ρ(t α ). For example, let α = and

9 We have W. Li, W. Sun / Linear Algebra and its Applications 7 (2) [ ] [ ] [ ] 2 2 A =, L =, I + S α =. E α = [ ], F α = [ ] and T α = Eα F α =. It is easy to obtain ρ(t α ) = < andρ(t) = ρ(t GS ) = 2 >. In the following section, we shall consider this problem. Remark.2. U m in Theorem. is assumed to be a general non-negative matrix rather than an upper triangular non-negative matrix as given in [,2]. Then, Jacobi iteration is a special case with U m = I A and L m =. The corresponding results are given in the following corollaries. Corollary.. Let A = (a ij ) R n n be a Z-matrix and J = I A be a Jacobi matrix. Let J α = J S α + S α J. Then: (a) For any α i [, ], i =,...,n, ρ(j α )< if A is a non-singular M-matrix. In this case, we have ρ(j α ) ρ(j) <. Moreover, if A is irreducible, then ρ(j α )<ρ(j)for α i (, ], i=,...,n. (b) For any α i [, ],i=,...,n,ρ(j α ) = if A is a singular M-matrix. Corollary.2. Let A be a non-singular M-matrix. Then MJ iterative methods converge for α i [, ], i=,...,n. Remark.. When U m is a classical upper triangular non-negative matrix, the conditions in Theorem. become that A is Z-matrix and ρ(t), i.e., A is an M-matrix, which are still weaker than the conditions in Theorem.2. Notice that the inequality ρ(t α ) ρ(t GS ) is not true even for diagonally dominant M-matrix case. For example, let [ ] A =. 2 Then ρ(t GS ) = 2 [.Let ] [ ] [ ] α L m =, U 4 m =, and S 4 α =. We have ρ(t α ) = [ α 2 (7 2α + 4α )] 2.

10 26 W. Li, W. Sun / Linear Algebra and its Applications 7 (2) Hence, if we take α< 2, then ρ(t α)>ρ(t GS ). But if we take α =, then ρ(t α )<ρ(t GS ). The inequality ρ(t α= ) ρ(t GS ) is not true in general. For example, A = We consider the classical splitting, i.e., L m = andu m = L + U. In this case, ρ(t GS ) =.96 and ρ(t α= ) =.9784 >ρ(t GS ). Corollary.. Let A be a non-singular M-matrix,A= I L m U m = I L m U m with L m L m, then ρ(t α) ρ(t α ) for all α [, ]. Proof. Since A α = (I + S α )A = (I L m S α L m ) (U m S α + S α U m ) = (I L m S αl m ) (U m S α + S α U m ) M α N α = M α N α, for all α i [, ], it is easy to show that the above two splittings are regular and N α N α. Then by Lemma.5, we have ρ(mα N α) ρ(m α N α ), which proves that ρ(t α ) ρ(t α). By Corollary. and Theorems. and.2 we can obtain the following inequalities: Corollary.4. If A is a non-singular M-matrix, then ρ(t α ) ρ(t α ) ρ(t) ρ(j) <. Proof. In fact, it need only show ρ(t α ) ρ(t α ) from Theorems. and.2. Let L m = L. Then T α = T α. The result follows from Corollary.. Remark.4. Notice that if A is a generalized diagonally dominant and non-singular Z-matrix, then A is a non-singular M-matrix. Therefore, the MGS, MGS type, GS type and Jacobi type methods are all convergent by Corollary Stein Rosenberg type theorem In Section, we have given a counterexample for which ρ(t) > although the conditions in Theorem. are satisfied and ρ(t α ). In order to present some necessary and sufficientconditions, some additional restrictions on A are necessary.

11 W. Li, W. Sun / Linear Algebra and its Applications 7 (2) Lemma 4.. Let A = (a ij ) R n n with a i,i+ a i+,i <, i,j=,...,n, be irreducible. If ρ(t) >, then for any α i [, ], ρ(t α ) ρ(t) >. Proof. Clearly, A = (I L) U is an M-splitting. By Lemma.6, there is x such that Tx = ρ(t)x. Since T = Mα N α (as defined in (.2)), we have Mα Ãx = (I T)x =[ ρ(t)]x. Then Ãx =[ ρ(t)]m α x. This implies that Eα Ãx = [ ρ(t)]eα M αx, i.e., (I T α )x =[ ρ(t)]eα M αx. By the definition of E α and M α we have Eα M α >Iand therefore, T α x>ρ(t)x. (4.) By (4.) and Theorem 2.2 of [], we obtain ρ(t α ) ρ(t) >. The following theorem is called Stein Rosenberg type theorem. Theorem 4.. Let A = (a ij ) R n n with a i,i+ a i+,i <, i,j =,...,n. Then: (a) For any α i [, ], ρ(t α )< if and only if ρ(t) <. In this case, we have ρ(t α ) ρ(t) <. (b) For any α i [, ],ρ(t α ) = if and only if ρ(t) =. (c) For any α i [, ], ρ(t α )> if and only if ρ(t) >. In this case, we have ρ(t α ) ρ(t) >. Proof. By Theorem., we only need to prove the necessity of (a) and (b). Since E α is a lower triangular Z-matrix with diagonal entries given in (.4), E α is a nonsingular M-matrix for α i [, ] and à = E α F α is an M-splitting. (a) Let ρ(t α )<. Since à = E α F α is an M-splitting, by Lemma.2, à is a non-singular M-matrix. It follows from Lemma., that, A is also a non-singular M-matrix. Then A = (I L m ) U m is an M-splitting of a non-singular M-matrix. By Lemma.2 we obtain ρ(t) <. (b) Let ρ(t α ) =. Since à = E α F α is an M-splitting, by Lemma.2, à is a singular M-matrix. We consider the following two cases: Case. Assume that A is irreducible. Then J = L m + U m is also irreducible. Hence, there exists a positive vector y such that y T J = ρ(j)y T from Theorem 2.(b) of []. Since à is a singular M-matrix, there is a non-zero non-negative vector x such that Ãx = from Theorem 2., i.e., (I + S α )Ax =. This implies that (I J)x = Ax = and we have y T x = y T Jx = ρ(j)y T x.sincey T x>, we have ρ(j) =. Then A is a singular M-matrix and, ρ(t) = from Lemma.2. Case 2. Assume that A is reducible. By Lemma.4, A has the upper triangular block form (.). From our proof of Theorem.(b) we obtain à ii = S ii A ii, i =,...,k,where A ii is irreducible, and S ii has the same non-zero structure as I + S α. Then each S ii,i=,...,k, is non-singular and all à ii,i=, 2,...,k,areM-matrices. It follows from Case and Lemma. that diagonal blocks A ii,i=,...,k,

12 28 W. Li, W. Sun / Linear Algebra and its Applications 7 (2) are all M-matrices. Since A is an upper triangular block form with diagonal blocks A ii,i=,...,k, A(= I J)is a singular M-matrix, which deduces that ρ(j) =. So we have ρ(t) =. (c) From the parts (a) and (b), we see that ρ(t α )> ρ(t) >. When A is irreducible, (c) follows from Lemma 4. immediately. When A is reducible, we define A ɛ = (a(ɛ) ij ) as a perturbed matrix of A, which is given by { aij, if a ij /=, a(ɛ) ij = i, j n, ɛ, if a ij =, where ɛ is a positive real number. Let A ɛ = I L m (ɛ) U m (ɛ) and J ɛ = L m (ɛ) + U m (ɛ). Then both A ɛ and J ɛ are irreducible matrices. Let ɛ be small enough such that ɛa i,i+ <, i =,...,n. Then the conditions in this theorem are satisfied for A ɛ.sincej ɛ is irreducible, there is x suchthatj ɛ x = ρ(j ɛ )x from Theorem 2. of []. Since J ɛ = L m (ɛ) + U m (ɛ) L m + U m = J,wehaveJx J ɛ x = ρ(j ɛ )x and ρ(j ɛ ) ρ(j) >. By Lemma.2, ρ(t ɛ )>, where T ɛ = (I L m (ɛ)) U m (ɛ) and moreover by Lemma 4., we have ρ(t ɛ ) ρ(t ɛ) where T ɛ = (I L m (ɛ) S α L m (ɛ)) [(I + S α )U m (ɛ) S α ]. Part (c) can be obtained by letting ɛ. Remark 4.. Notice that, in Theorem 4., the conditions a i,i+ a i+,i >, i =,...,n, are not necessary, i.e., A may be reducible. Remark 4.2. The equalities in Theorem 4. may hold in some case. For example, [ ] A =. It is easy to see that for any α [, ], ρ(t α ) = ρ(t) =. In order to obtain the strict inequalities, the irreducibility of A is necessary. We present the result in the following theorem and the proof is similar to those for Theorems 4. and.(a). Theorem 4.2. Let A be irreducible and A = (a ij ) R n n with a i,i+ a i+,i <, i, j =,...,n.then: (a) For each α i (, ], ρ(t α )< if and only if ρ(t) <. In this case, we have ρ(t α )<ρ(t)<. (b) For each α i [, ],ρ(t α ) = if and only if ρ(t) =. (c) For each α i (, ], ρ(t α )> if and only if ρ(t) >. In this case, we have ρ(t α )>ρ(t)>. Since U m is a general non-negative matrix, it is easy to extend the results to some Jacobi type iteration methods. Corollary 4.. Let A = (a ij ) R n n with a i,i+ a i+,i <, i,j =,...,n. Then, for any α i [, ], we have

13 W. Li, W. Sun / Linear Algebra and its Applications 7 (2) (a) ρ(t α )< if and only if ρ(j) <. In this case, we have ρ(t α ) ρ(t) ρ(j) <. (b) ρ(t α ) = if and only if ρ(j) =. (c) ρ(t α )> if and only if ρ(j) >. In this case, we have ρ(t α ) ρ(t) ρ(j) >. Remark 4.. We have given some inequalities of spectral radii of iteration matrices. However, the spectral radius of the MGS type method also depends upon the choice of the parameters α i,i=,...,n. For example, let A =. 2 We have à = (I + S)A = 2, T = 2, 2 2 and ρ(t)= 2. If we choose α i =,i=, 2, I + S α =, à = (I + S α )A = and 2 T α = 6 2 = Then ρ(t α ) = > 2 = ρ(t). If we choose α = 2 and α 2 =, 2 2 I + S α =, à = (I + S α )A = 2 2 and T α = =

14 24 W. Li, W. Sun / Linear Algebra and its Applications 7 (2) Then ρ(t α ) = ρ(t)= 2. A natural and open problem is whether the convergence rate of MGS type methods is a monotonic function of the parameter α. Acknowledgements The authors would like to thank the referee for his helpful comments and Prof. M. Neumann for his kind help. References [] A.N. Gunawardena, S.K. Jain, L. Snyder, Modified iterative methods for consistent linear systems, Linear Algebra Appl (99) 2 4. [2] T. Kohno, H. Kotakemori, H. Niki, M. Usui, Improving modified iterative methods for Z-matrices, Linear Algebra Appl. 267 (997) 2. [] A. Berman, R.J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, SIAM, Philadelphia, PA, 994. [4] W. Li, Z.Y. You, The multi-parameters overrelaxation method, J. Comput. Math. 6 (4) (998) [5] Z.I. Woznicki, Nonnegative splitting theory, Japan J. Indust. Appl. Math. (994) [6] H. Schneider, Theorems on M-splittings of a singular M-matrix which depend on graph structure, Linear Algebra Appl. 58 (984) [7] R.S. Varga, Matrix Iterative Analysis, Prentice-Hall, Englewood Cliffs, NJ, 962. [8] W. Li, M. Zhang, On upper triangular block weak regular splittings of a singular M-matrix, Linear Algebra Appl. 2 (996)