CONVERGENCE OF MULTISPLITTING METHOD FOR A SYMMETRIC POSITIVE DEFINITE MATRIX

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1 J. Appl. Math. & Computing Vol No. 1-2 pp CONVERGENCE OF MULTISPLITTING METHOD FOR A SYMMETRIC POSITIVE DEFINITE MATRIX JAE HEON YUN SEYOUNG OH AND EUN HEUI KIM Abstract. We study convergence of symmetric multisplitting method associated with many different multisplittings for solving a linear system whose coefficient matrix is a symmetric positive definite matrix which is not an H-matrix. AMS Mathematics Subject Classification : 65F10 65F15. Key word and phrases : Symmetric multisplitting method multisplitting incomplete factorization symmetric positive definite matrix H-matrix. 1. Introduction In this paper we consider multisplitting method for solving a linear system of the form Ax = b x b R n 1 where A R n n is a symmetric positive definite matrix which is not an H- matrix. Multisplitting method was introduced by O Leary and White [7] and was further studied by many authors [ ]. But the attention was mainly paid to a monotone matrix and an H-matrix. Thus the purpose of this paper is to study the convergence of symmetric multisplitting method for a symmetric positive definite matrix A which is not an H-matrix by using the diagonally dominated matrix [11] of A. This paper is organized as follows. In Section 2 we present some preliminary results which we refer to later. In Section 3 we present convergence results of the symmetric multisplitting method for solving the linear system 1. Received October This work was supported by Chungbuk National University Grant in c 2005 Korean Society for Computational & Applied Mathematics and Korean SIGCAM. 59

2 60 Jae Heon Yun SeYoung Oh and Eun Heui Kim 2. Preliminaries For a vector x R n x 0x>0 denotes that all components of x are nonnegative positive and x denotes the vector whose components are the absolute values of the corresponding components of x. For two vectors x y R n x y x >y means that x y 0x y > 0. These definitions carry immediately over to matrices. For a square matrix A ρa denotes the spectral radius of A. A matrix A =a ij R n n is called an M-matrix if a ij 0 for i j and A 1 0. The comparison matrix A =α ij of a matrix A =a ij is defined by { a ij if i = j α ij = a ij if i j. A matrix A is called an H-matrix if A is an M-matrix. A representation A = M N is called a splitting of A when M is nonsingular. A splitting A = M N is called regular if M 1 0 and N 0 and convergent if ρm 1 N < 1. When A is symmetric a splitting A = M N is called symmetric if M is symmetric. A splitting A = M N is called an H-compatible splitting if A = M N. It was shown in [3] that if A is an H-matrix and A = M N is an H- compatible splitting then M is also an H-matrix and ρm 1 N < 1. Note that M-matrices and strictly or irreducibly diagonally dominant matrices are contained in the class of all H-matrices. Actually an n nh-matrix A =a ij can be equivalently characterized by being generalized strictly diagonally dominant [1] i.e. a ii u i > j i a ij u j i =1 2...n for some vector u =u 1 u 2...u n T > 0. A collection of triples M k N k E k k =1 2...l is called a multisplitting of A if A = M k N k is a splitting of A for k =1 2...l and E k s called l weighting matrices are nonnegative diagonal matrices such that E k = I. k=1 Lemma 2.1 [2]. If A = D B is an H-matrix then A 1 A 1. Let S n denote the set of all pairs of indices of off-diagonal matrix entries that is { } S n = i j i j 1 i n 1 j n.

3 Convergence of multisplitting method 61 The following theorem shows the existence of an incomplete factorization for a symmetric H-matrix A. Theorem 2.2 [5 11]. Let A R n n be a symmetric H-matrix. Then for every symmetric zero pattern set Q S n i.e. i j Q implies j i Q there exist an upper triangular matrix U =u ij a diagonal matrix D whose kth diagonal element is u 1 kk and a symmetric matrix N =n ij with u ij =0if i j Q and n ij =0if i j / Q such that A = U T DU N. Moreover U is an H-matrix. In Theorem 2.2 A = U T DU N is called an incomplete factorization of A corresponding to a symmetric zero pattern set Q. When A is a symmetric H- matrix with positive diagonal elements the diagonal matrix D in the incomplete factorization A = U T DU N given in Theorem 2.2 has also positive diagonal elements and thus U T DU is a symmetric positive definite matrix cf. [4]. When A is a symmetric M-matrix the incomplete factorization A = U T DU N in Theorem 2.2 is a regular splitting of A and U T DU is also a symmetric positive definite matrix. It is easy to show that the incomplete factorization A = U T DU N in Theorem 2.2 is a convergent splitting. Theorem 2.3 [11]. Let A R n n be a symmetric H-matrix. Let A = U T DU N and A = Ũ T DŨ Ñ be the incomplete factorizations of A and A corresponding to a symmetric zero pattern set Q S n respectively. Then each of the following holds: 1 a U DU T T 1 Ũ DŨ b N Ñ. 3. Convergence results Let A R n n be a symmetric positive definite matrix which is not an H-matrix. Then we can choose a nonnegative diagonal matrix Λ such that  = A + Λ is a generalized strictly diagonally dominant matrix i.e. an H- matrix. For example the easiest way of finding such a matrix  is to choose a nonnegative diagonal matrix Λ such that  = A+Λ is a strictly diagonally dominant matrix. From now on the transformed matrix  = A+Λ is called a diagonally dominated matrix of A. Since  is a symmetric positive definite H-matrix Theorem 2.2 implies that  has an incomplete factorization  = U T DU N

4 62 Jae Heon Yun SeYoung Oh and Eun Heui Kim for any symmetric zero pattern set Q S n and the U T DU is symmetric positive definite. Example 3.1 shows how a diagonally dominated matrix  can be obtained from a symmetric positive definite matrix A. In Example 3.1 matrix inverses and eigenvalues have been computed using MATLAB. Example 3.1. Consider a 4 4 matrix A of the form A = A is a symmetric positive definite matrix since all eigenvalues of A are positive. However A is not an H-matrix since A 1 < 0. Let Λ = diag { } which denotes a diagonal matrix whose consecutive diagonal elements are and 0. That is Λ= Then  = A + Λ is a symmetric positive definite H-matrix since  1 > 0. Hence  is a diagonally dominated matrix of A. Lemma 3.2 [11]. Let A be a symmetric positive definite matrix and  be a diagonally dominated matrix of A. Let  = M ˆN be a splitting of  with M being symmetric positive definite and let N = M A. If the splitting  = M ˆN is convergent then the splitting A = M N is convergent. From Lemma 3.2 it can be seen that the problem of constructing a convergent splitting for a symmetric positive definite matrix A can be reduced to that for a diagonally dominated matrix  of A. Let M k N k E k k =1 2...l be a multisplitting of a symmetric positive definite matrix A. Then the parallel symmetric multisplitting method introduced in [9] can be constructed as follows. For m = compute x m+1 = x m + Gr m 2 where G = 1 l E k M 1 k + M T k E k and r m = b Ax m. 3 2 k=1

5 Convergence of multisplitting method 63 The equation 2 can be transformed into x m+1 =I GAx m + Gb. 4 Hence the iteration matrix of parallel symmetric multisplitting method is H = I GA. If G is nonsingular A = G 1 G 1 H is a splitting of A. Now let us construct the multisplitting M k N k E k k =1 2...l of A by using the diagonally dominated matrix  of A. For simplicity of exposition suppose that l = 3. Then A and  are partitioned into 3 3 block matrices of the form A 1 C 12 C 13  1 C 12 C 13 A = C 21 A 2 C 23  = C 21  2 C 23 5 C 31 C 32 A 3 C 31 C 32  3 where A i and Âi are n i n i matrices with 3 n i = n. Since A and  are symmetric C ij = Cji T for i j. Let Âi = B i C i be an H-compatible splitting of  i for each i = Then Âi = B i C i for each i. Since Âi is an H-matrix B i is also an H-matrix. Let B B B M 1 = C 21 B 2 0 C 31 0 B 3 M 2 = i=1 0 B C 32 B 3 M 3 = 0 B B 3 6 ˆN 1 = M 1  ˆN2 = M 2  ˆN3 = M 3  7 N 1 = M 1 A N 2 = M 2 A N 3 = M 3 A. 8 For each k =1 2 3 let d 1k I E k = 0 d 2k I d 3k I 3 where 0 d ik 1 3 d ik = 1 and I i is an n i n i identity matrix for each k=1 i = Then M k N k E k k =1 2 3 is a multisplitting of A and M k ˆN k E k k =1 2 3 is a multisplitting of Â. It can be easily shown that  = M k ˆN k is an H-compatible splitting of  for

6 64 Jae Heon Yun SeYoung Oh and Eun Heui Kim each k. Hence M k ˆN k E k k =1 2 3 is a multisplitting of  with regular splittings  = M k ˆN k. Some simple but tedious matrix manipulations show 3 G = 1 E k M 1 k + M T k E k 2 G = 1 2 = 1 2 where = 1 2 k=1 B B T 1 d 21B T 1 C 12B T 2 d 31B T 1 C 13B T 3 d 21B 1 2 C21B 1 1 B B T 2 d 32B T 2 C 23B T 3 d 31B 1 3 C31B 1 1 d 32B 1 3 C32B 1 2 B B T 3 E k M k 1 + M k T E k 10 3 k=1 B1 1 + B 1 T d 21 B 1 T C 12 B 2 T d 31 B 1 T C 13 B 3 T d 21 B 2 1 C 21 B 1 1 B B 2 T d 32 B 2 T C 23 B 3 T d 31 B 3 1 C 31 B 1 1 d 32 B 3 1 C 32 B 2 1 B B 3 T Ĥ = I G = ĥ 11 ĥ 12 ĥ 13 ĥ 21 ĥ 22 ĥ 23 ĥ 31 ĥ 32 ĥ 33 H = I G  = ĥ 11 = 1 2 ĥ 12 = 1 2 ĥ 13 = 1 2 B 1 1 C 1 + B T 1 C T 1 + ĥ 21 = 1 2 ĥ 22 = 1 B 1 2 ĥ 23 = j=2 d j1b T 1 C 1jB T j C j1 h 11 h12 h13 h 21 h22 h23 1 d 21B T 1 C 12 + B 1 1 C 12 + d 21B T 1 C 12B T 2 C T 2 d 31B T 1 C 13B T 3 C 32 1 d 31B T 1 C 13 + B 1 1 C 13 + d 31B T 1 C 13B T 3 C T 3 d 21B T 1 C 12B T 2 C 23 1 d 21B 1 2 C 21 + B T 2 C 21 + d 21B 1 2 C 21B 1 1 C 1 d 32B T 2 C 23B T 3 C 31 ĥ 31 = h 31 h32 h C 2 + B T 2 C 2 + d 21B 1 2 C 21B 1 1 C 12 + d 32B T 2 C 23B T d 21B 1 2 C 21B 1 1 C 13 1 d 32B T 2 C 23 B 1 2 C 23 d 32B T 2 C 23B T 3 C T 3 1 d 31B 1 3 C 31 + B T 3 C 31 + d 31B 1 3 C 31B 1 1 C 1 3 C 32

7 and ĥ 32 = 1 2 d 32B 1 3 C 32B 1 2 C 21 Convergence of multisplitting method 65 d 31B 1 3 C 31B 1 1 C 12 1 d 32B 1 3 C 32 B T 3 C 32 d 32B 1 3 C 32B 1 2 C 2 ĥ 33 = 1 B 1 3 C 3 + B T 3 C3 T + 2 h 11 = 1 2 h 12 = 1 2 h 13 = 1 2 h 21 = 1 2 h 22 = 1 2 h 23 = 1 2 h 31 = 1 2 h 32 = 1 2 h 33 = j=1 B 1 1 C 1 + B 1 T C 1 T + d 3jB 1 3 C 3jB 1 j C j3 3 j=2 d j1 B 1 T C 1j B j T C j1 1 d 21 B 1 T C 12 + B 1 1 C 12 + d 21 B 1 T C 12 B 2 T C 2 T +d 31 B 1 T C 13 B 3 T C 32 1 d 31 B 1 T C 13 + B 1 1 C 13 + d 31 B 1 T C 13 B 3 T C 3 T +d 21 B 1 T C 12 B 2 T C 23 1 d 21 B 2 1 C 21 + B 2 T C 21 + d 21 B 2 1 C 21 B 1 1 C 1 +d 32 B 2 T C 23 B 3 T C 31 B 2 1 C 2 + B 2 T C 2 + d 21 B 2 1 C 21 B 1 1 C 12 +d 32 B 2 T C 23 B 3 T C 32 d 21 B 2 1 C 21 B 1 1 C d 32 B 2 T C 23 + B 2 1 C 23 +d 32 B 2 T C 23 B 3 T C 3 T 1 d 31 B 3 1 C 31 + B 3 T C 31 + d 31 B 3 1 C 31 B 1 1 C 1 +d 32 B 3 1 C 32 B 2 1 C 21 d 31 B 3 1 C 31 B 1 1 C d 32 B 3 1 C 32 + B 3 T C 32 +d 32 B 3 1 C 32 B 2 1 C 2 B 3 1 C 3 + B 3 T C 3 T + 2 j=1 d 3j B 3 1 C 3j B j 1 C j3. Since B i is an H-matrix from Lemma 2.1 B 1 i B i 1 for each i. Using this inequality and equation 12 it is easy to show that ĥij h ij for all i and

8 66 Jae Heon Yun SeYoung Oh and Eun Heui Kim j. Hence Ĥ H from which one obtains ρĥ ρ H. 13 Theorem 3.3. Let A R n n be a symmetric positive definite matrix and  be a diagonally dominated matrix of A. Assume that  is partitioned into an l l block matrix as in the form 5 with Âk = B k C k being an H-compatible splitting of  k for each k =1 2...l and assume that M k ˆN k E k k =1 2...l is a multisplitting of  which is constructed as in the form 6 7 and 9. Then ρĥ < 1 and G is nonsingular. Proof. Since Âk = B k C k is an H-compatible splitting of an H-matrix Âk Âk = B k C k is a regular splitting of Âk. It follows from 11 and 12 that G 0 and H 0. Since G  = I H we have G = I H  1 and thus 0 I + H + + H m 1 G = I + H + + H m 1 I H  1 14 = I H m  1  1 for any positive integer m. Since it is easy to see from 11 that each row of G has at least one nonzero component 14 implies that lim H m = 0 which m is equivalent to ρ H < 1. From 13 ρĥ < 1. Since G = I Ĥ G is nonsingular. Since  is a symmetric H-matrix  k is also a symmetric H-matrix and thus there exists an incomplete factorization of Âk corresponding to a symmetric zero pattern set see Theorem 2.2. Hence the following theorem can be obtained. Theorem 3.4. Let A R n n be a symmetric positive definite matrix and  be a diagonally dominated matrix of A. Assume that  is partitioned into an l l block matrix as in the form 5 with  k = Uk T D k U k C k being an incomplete factorization of  k corresponding to a symmetric zero pattern set Q k for each k =1 2...l. Also assume that M k ˆN k E k k =1 2...l

9 Convergence of multisplitting method 67 is a multisplitting of  which is constructed as in the form 6 7 and 9 where B k = Uk T D ku k in 6. Then ρĥ < 1 and G is nonsingular. Proof. Let Âk = Ũ T k D k Ũ k C k be an incomplete factorization of Âk corresponding to the symmetric zero pattern set Q k and let B k = Ũ k T D k Ũ k. Then from Theorem 2.3 one obtains B 1 k 1 B k and C k C k. 15 Let G and H be defined as in 11 and 12 except that B k and C k are replaced by B k and C k respectively. From 12 and 15 it can be easily shown that Ĥ H G 0 and H 0. Since Âk = B k C k is a regular splitting of Âk ρ H < 1 is obtained as in the proof of Theorem 3.3. Therefore ρĥ < 1 and thus G is nonsingular. Theorem 3.5. Let A R n n be a symmetric positive definite matrix and  be a diagonally dominated matrix of A. Assume that  is partitioned into an l l block matrix as in the form 5 with  k = B k C k being an H-compatible splitting of  k for each k =1 2...l and assume that M k ˆN k E k k =1 2...l is a multisplitting of  which is constructed as in the form 6 7 and 9. If G is symmetric positive definite then ρh < 1. Proof. Since GA = I H and G = I Ĥ we have two splittings A = G 1 G 1 H and  = G 1 G 1 Ĥ of A and  respectively. Since ρĥ < 1 from Theorem 3.3 the latter is a convergent splitting. Hence Lemma 3.2 implies that the former is a convergent splitting i.e. ρh < 1. If Âk = B k C k is a symmetric splitting of Âk for k =1 2...l the matrix G in 10 has the form G = diag G diag 16 B1 1 B 1 2 B 1 3 B1 1 B 1 2 B 1 3

10 68 Jae Heon Yun SeYoung Oh and Eun Heui Kim where G = B 1 d21 d21 2 C 12 d31 2 C 13 2 C 21 B 2 d32 d31 2 C 31 d32 2 C 32 B 3 2 C Equation 16 implies that G is symmetric positive definite if and only if G is symmetric positive definite. Theorem 3.6. Let A R n n be a symmetric positive definite matrix and  be a diagonally dominated matrix of A. Assume that  is partitioned into an l l block matrix as in the form 5 with  k = U T k D k U k C k being an incomplete factorization of  k corresponding to a symmetric zero pattern set Q k for each k =1 2...l. Also assume that M k ˆN k E k k =1 2...l is a multisplitting of  which is constructed as in the form 6 7 and 9 where B k = Uk T D ku k in 6. If G of the form 17 is symmetric positive definite then ρh < 1. Proof. Since G is symmetric positive definite G is symmetric positive definite. Using Theorem 3.4 and Lemma 3.2 ρh < 1 is obtained as in the proof of Theorem 3.5. If d ik = 0 for i>kin Theorem 3.6 then the assumption that G is symmetric positive definite is not necessary since G = diagb 1 B 2...B l and B k = Uk T D ku k is symmetric positive definite for each k. The following theorem shows that the assumption that G is symmetric positive definite in Theorem 3.5 can be omitted if  k = B k C k is a symmetric H- compatible splitting of  k for each k. Theorem 3.7. Let A R n n be a symmetric positive definite matrix and  be a diagonally dominated matrix of A. Assume that  is partitioned into an l l block matrix as in the form 5 with  k = B k C k being a symmetric H-compatible splitting of  k for each k = l and assume that M k ˆN k E k k =1 2...l

11 Convergence of multisplitting method 69 is a multisplitting of  which is constructed as in the form 6 7 and 9. Then ρh < 1. Proof. Since Âk = B k C k is a symmetric H-compatible splitting of Âk this theorem follows from Theorem 3.5 if we show that G of the form 17 is symmetric positive definite. Notice that Âk = B k C k B k. Since 0 d ik 1  G. It follows that G is an H-matrix since  is an H-matrix. First we show that each B k has positive diagonal elements. Let From Âk = B k C k From Âk = B k C k  k =â ij B k =b ij and C k =c ij. â ii = b ii c ii > 0 and thus b ii >c ii. â ii = b ii c ii > 0 and thus b ii > c ii. Suppose that b ii 0 for some i. Then c ii <b ii 0 and hence c ii > b ii. This is a contradiction to b ii > c ii. Hence b ii must be positive for all i i.e. B k has positive diagonal elements. From 17 G has positive diagonal elements. Since G is a symmetric H-matrix with positive diagonal elements G is symmetric positive definite. We now give some examples of symmetric splittings of Âk. Let Âk = D k L k L T k where D k = diagâk and L k is the strictly lower triangular part of  k.  k = B k C k is called a Jacobi splitting of  k if B k = D k and C k = L k L T k and it is called an SSORω splitting of  k if 1 B k = ω2 ω D k ωl k D 1 k 1 C k = 1 ωd k + ωl k ω2 ω D k ωl T k D 1 k 1 ωd k + ωl T k 18 where 0 < ω < 2. Note that the Jacobi splitting of  k is a symmetric H- compatible splitting of Âk. Hence Theorem 3.7 holds for the Jacobi splitting of  k. Since Âk = D k L k L T k

12 70 Jae Heon Yun SeYoung Oh and Eun Heui Kim Âk = B k C k is an SSORω splitting of Âk where 1 B k = D k ω L k D k 1 D k ω L T k ω2 ω 1 C k = 1 ω D k + ω L k D k 1 1 ω D k + ω L T k ω2 ω. 19 Theorem 3.8. Let A R n n be a symmetric positive definite matrix and  be a diagonally dominated matrix of A. Assume that  is partitioned into an l l block matrix as in the form 5 with  k = B k C k being an SSORω splitting of  k for each k =1 2...l and assume that M k ˆN k E k k =1 2...l is a multisplitting of  which is constructed as in the form 6 7 and 9. If 0 <ω 1 then ρĥ < 1 and G is nonsingular. Proof. Let Âk = B k C k be an SSORω splitting of Âk which is defined as in 19. Since D k ωl k and D k ωl T k are H-matrices D k ωl k D k ωl k = D k ω L k D k ωl T k 1 D k ωl T 1 1. k = D k ω L T k Hence one obtains B 1 1 k D ω2 ω k ωl T k Dk D k ωl k = ω2 ω D k ω L T k Dk D k ω L k 20 = B 1 k. Since 0 <ω 1 C k C k is directly obtained from 18 and 19. Let G and H be defined as in 11 and 12 except that B k and C k are replaced by B k and C k respectively. From 20 and C k C k it is clear that Ĥ H G 0 and H 0. Since Âk = B k C k is a regular splitting of Âk ρ H < 1 is obtained as in the proof of Theorem 3.3. Therefore ρĥ < 1 and thus G is nonsingular.

13 Convergence of multisplitting method 71 Theorem 3.9. Let A R n n be a symmetric positive definite matrix and  be a diagonally dominated matrix of A. Assume that  is partitioned into an l l block matrix as in the form 5 with  k = B k C k being an SSORω splitting of  k for each k =1 2...l and assume that M k ˆN k E k k =1 2...l is a multisplitting of  which is constructed as in the form 6 7 and 9. If 0 <ω 1 and d ik =0for i>k then ρh < 1. Proof. Notice that Âk = B k C k is a symmetric splitting and B k is symmetric positive definite. Since d ik = 0 for i>k G = diagb 1 B 2...B l is symmetric positive definite and hence G is symmetric positive definite. Using Theorem 3.8 and Lemma 3.2 ρh < 1 is obtained as in the proof of Theorem 3.5. References 1. K. Fan Topological proofs of certain theorems on matrices with nonnegative elements Monatshefte für Mathematik A. Frommer and G. Mayer Convergence of relaxed parallel multisplitting methods Linear Algebra Appl A. Frommer and D.B. Szyld H-splittings and two-stage iterative methods Numer. Math T.A. Manteuffel An incomplete factorization technique for positive definite linear systems Math. Comput A. Messaoudi On the stability of the incomplete LUfactorizations and characterizations of H-matrices Numer. Math M. Neumann and R.J. Plemmons Convergence of parallel multisplitting methods for M- matrices Linear Algebra Appl D.P. O Leary and R.E. White Multisplittings of matrices and parallel solution of linear systems SIAM J. Algebraic Discrete Meth D.B. Szyld and M.T. Jones Two-stage and multisplitting methods for the parallel solution of linear systems SIAM J. Matrix Anal. Appl R.E. White Multisplitting with different weighting schemes SIAM J. Matrix Anal. Appl R.E. White Multisplitting of a symmetric positive definite matrix SIAM J. Matrix Anal. Appl

14 72 Jae Heon Yun SeYoung Oh and Eun Heui Kim 11. J.H. Yun and S.W. Kim Convergence of two-stage iterative methods using incomplete factorization J. Comput. Appl. Math Jae Heon Yun received M.Sc. from Kyungpook National University and Ph.D. from Iowa State University. He is currently a professor at Chungbuk National University since His research interests are computational mathematics iterative method and parallel computation. Department of Mathematics Institute for Basic Sciences & College of Natural Sciences Chungbuk National University Cheongju Korea gmjae@cbucc.chungbuk.ac.kr SeYoung Oh received M.Sc. from Seoul National University and Ph.D at University of Minnesota. Since 1992 he has been at Chungnam National University. His research interests include numerical optimization and biological computation. Department of Mathematics Chungnam National University Daejeon Korea soh@cnu.ac.kr Eun Heui Kim received M.Sc. in Applied Mathematics from Chungbuk National University. She is currently a temporary instructor at Chungbuk National University. Department of Mathematics College of Natural Sciences Chungbuk National University Cheongju Korea chaos75@chungbuk.ac.kr