Elsevier

Applied Mathematics and Computation

Volume 275, 15 February 2016, Pages 156-164
Applied Mathematics and Computation

Preconditioned parallel multisplitting USAOR method for H-matrices linear systems

https://doi.org/10.1016/j.amc.2015.11.068Get rights and content

Abstract

In this paper, the preconditioned multisplitting USAOR method is established for solving the system of linear equations. The convergence and comparison results of the method are given when the coefficient matrices of the linear systems are H-matrices. The method for H-matrices is proved to be more efficient than the multisplitting USAOR method for M-matrices. Finally, a numerical example is given to illustrate the efficiency of our method.

Introduction

Sometimes, one has to solve a nonsingular linear system as Ax=bwhere A=(aij)n×nRn×n is nonsingular, b is n-dimensional vector. The basic iterative method for solving (1) is Mx(k+1)=Nx(k)+b,k=0,1,,where A=MN, and M is nonsingular, so (2) can be written as x(k+1)=Tx(k)+c,k=0,1,,where T=M1N,c=M1b.

O'leary and White [12] presented the matrix multisplitting method in 1985 for parallelly solving the large sparse linear systems on the multiprocessor systems and it was further studied by many authors. See [1], [2], [6], [7], [8], [11], [14], [16]. Neumann and Plemmons [11] developed some more refined convergence results for one of the cases considered in [12]. Elsner [7] established the comparison theorems about the asymptotic convergence rate of this case. Frommer and Mayer [8] discussed two different variants of relaxed multisplitting methods. White [14] studied the convergence properties of the above matrix multisplitting methods for the symmetric positive definite matrix. Bai [1] studied the convergence domain of the matrix multisplitting relaxation methods. Cao and Liu [6] studied the convergence of two different variants of multisplitting relaxation methods with different weighting schemes. Zhang et al. [16] presented parallel multisplitting method for H-matrices. For parallel matrix multisplitting methods, a very good comprehensive survey is [2] and its references are worth reading. Bai et al. [3] discussed weak-convergence of the splitting methods for solving singular linear systems. In this paper, we will study the preconditioned multisplitting USAOR method for H-matrices linear systems and analyze its convergence theoretically.

The multisplitting method is as follows.

If A is a nonsingular n × n matrix, and Mk,Nk,EkRn×n,k=1,2,,K(KN) satisfy:

  • (1)

    A=MkNk,

  • (2)

    Mk is nonsingular,

  • (3)

    Ek is a nonnegative diagonal matrix, and k=1KEk=I,

then (Mk, Nk, Ek) is a multisplitting of A.

The multisplitting method for solving (1) is x(m+1)=k=1KEkMk1Nkx(m)+k=1KEkMk1b,m=0,1,,we denote T=k=1KEkMk1Nk and G=k=1KEkMk1, T is called the iteration matrix.

In paper [16], Zhang et al. presented the local relaxed parallel multisplitting method as follows.

Algorithm 1

(Local Relaxed Parallel Multisplitting Method)

Given an initial vector x0. For m=0,1,2,, repeat (I) and (II), until convergence.

  • (I)

    For k=1,2,,α, solving yk:Mkyk=Nkx(m)+b.

  • (II)

    Computing:x(m+1)=k=1αEkyk.

Let A=ILkUk,k=1,2,,α, where I is an identity matrix, Lk are strictly lower triangular matrices, Uk are general matrices.

Now, we consider one kind of parallel multisplitting unsymmetric accelerated over-relaxation (USAOR) method called local relaxed parallel multisplitting unsymmetric accelerated over-relaxation method (LUSAOR).

Algorithm 1 associated with LUSAOR method can be written as x(m+1)=HLUSAORx(m)+GLUSAORb,m=0,1,,where HLUSAOR=k=1αEkUω2γ2(k)Lω1γ1(k),Uω2γ2(k)=(Iγ2Uk)1[(1ω2)I+(ω2γ2)Uk+ω2Lk],Lω1γ1(k)=(Iγ1Lk)1[(1ω1)I+(ω1γ1)Lk+ω1Uk],GLUSAOR=k=1αEk(Iγ2Uk)1[(ω1+ω2ω1ω2)I+ω2(ω1γ1)Lk+ω1(ω2γ2)Uk](Iγ1Lk)1.

In order to solve (1) fasterly, a nonsingular preconditioner P is introduced. The original systems (1) can be transformed into the preconditioned form PAx=Pb,then, we can define the basic iterative scheme: x(k+1)=Mp1Npx(k)+Mp1Pb,k=0,1,2,,where PA=MpNp is a splitting of PA, Mp is nonsingular.

Suppose that A has unit diagonal elements, and A is an H-matrix. In the literature, various authors have suggested different models of preconditioner P for linear systems (1). See [10], [15].

In this paper, we consider the preconditioner P as follows P=I+S=[1α1a12000β2a211α2a23000β3a32100αn2an2,n1000βn1an1,n21αn1an1,n000βnan,n11]

In this paper, we will analyze the convergence of the preconditioned multisplitting USAOR method theoretically and give some comparison results of spectral radius. Finally, we provide one numerical example.

Section snippets

The preconditioned parallel multisplitting USAOR method

Let A˜=PA=D˜L˜kU˜k,k=1,2,,K, where D˜=diag(1α1a12a21,1β2a21a12α2a23a32,,1βnan,n1an1,n),L˜k(k=1,2,,K) are strictly lower triangular matrices, U˜k(k=1,2,,K) are general matrices.

Then Algorithm 1 associated with preconditioned parallel multisplitting USAOR method (LPUSAOR) can be written as x(m+1)=H˜LPUSAORx(m)+G˜LPUSAORb,m=0,1,,where H˜LPUSAOR=k=1αEkU˜ω2γ2(k)L˜ω1γ1(k),U˜ω2γ2(k)=(D˜γ2U˜k)1[(1ω2)D˜+(ω2γ2)U˜k+ω2L˜k],L˜ω1γ1(k)=(D˜γ1L˜k)1[(1ω1)D˜+(ω1γ1)L˜k+ω1U˜k],G˜LUSAOR=k=1αEk(

Preliminaries

We need the following results.

Lemma 1

[9]. A is an H-matrix if and only if there is a positive vector r such that 〈Ar > 0, where r=(r1,r2,,rn)T.

Lemma 2

[5]. LetA=MN be an M-splitting of A, thenρ(M1N)<1 if and only if A is an M-matrix.

Lemma 3

[5]. Let A and B be two n × n matrices with 0 ≤ BA, then ρ(B) ≤ ρ(A).

Lemma 4

[9]. If A is an H-matrix, then|A1|A1.

Lemma 5

[13]. Let A be a Z-matrix, then the following statements are equivalent:

  • (1)

    A is an M-matrix.

  • (2)

    There is a positive vector xRn such that Ax > 0.

  • (3)

    A10.

Lemma 6

[11].

Convergence result

In this section, we will present the convergence result of the preconditioned multisplitting USAOR method.

Theorem 1

LetA be an H-matrix with unit diagonal elements. If

  • (1)

    for α1, βn0|α1|1|a12|(2A11),0|βn|1|an,n1|(2A11),

  • (2)

    forαi,βi(i=2,3,,n1)0|βi||ai,i1|+|αi||ai,i+1|1(2A11),

    then PA is an H-matrix.

Proof

Let (PA)ij={aijαiai,i+1ai+1,ji=1aijβiai,i1ai1jαiai,i+1ai+1,ji=2,,n1aijβiai,i1ai1,ji=n

Since A is an H-matrix with unit diagonal elements, then 〈A〉 is an M-matrix and let A=IP

Comparison results of spectral radius

Let A=M^kN^k=1ω1(|I|γ1|Lk|)1ω1[(1ω1)|I|+(ω1γ1)|Lk|+ω1|Uk|]=M^^kN^^k=1ω2(|I|γ2|Uk|)1ω2[(1ω2)|I|+(ω2γ2)|Uk|+ω2|Lk|],

where M^k=1ω1(|I|γ1|Lk|),N^k=1ω1[(1ω1)|I|+(ω1γ1)|Lk|+ω1|Uk|],M^^k=1ω2(|I|γ2|Uk|),N^^k=1ω2[(1ω2)|I|+(ω2γ2)|Uk|+ω2|Lk|],then the iteration matrix of parallel multisplitting USAOR method for 〈A〉 is as follows H^LUSAOR=k=1αEkM^^k1N^^kM^k1N^k.

Let PA=M˜kN˜k=1ω1(|D˜|γ1|L˜k|)1ω1[(1ω1)|D˜|+(ω1γ1)|L˜k|+ω1|U˜k|]=M˜˜kN˜˜k=1ω2(|D˜|γ2|U˜k|)1ω2[(1ω2)|D˜|+(ω2γ2)|U˜k|

Numerical example

Consider the linear system Ax=b,where A=(1141411414114141),bT=(1,14,,1n2).

By simple computation, we know that A is an M-matrix. Obviously, A=A, then we have HLUSAOR=H^LUSAOR.

We take P=[1α1a12000β2a211α2a23000β3a32100αn2an2,n1000βn1an1,n21αn1an1,n000βnan,n11],and r=A1e, where e=(1,1,...,1)T, αi,βi(i=1,2,...,n) satisfy the conditions of Theorem 1.

Let S1={1,2,...,m1},S2={m1+1,...,n}, and determine (IL1,U1,E1) and (IL2,U2,E2) are the splittings of matrix A

Acknowledgments

The authors would like to thank the referees for their valuable comments and suggestions, which greatly improved the original version of this paper.

This work was supported by the National Natural Science Foundation of China (Grant no. 11001144) and the Natural Science Foundation of Shandong Province (no. ZR2012AL09).

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