求解對稱不定線性系統(tǒng)的吉爾-默里強(qiáng)迫正定方法

程 軍

(曲靖師范學(xué)院教師教育學(xué)院,云南曲靖655011)

考慮如下2×2塊狀線性系統(tǒng)

其中,A∈Rn×n是對稱不定矩陣,B∈Rm×n(m≤n)滿秩,即秩(B)=m,令BT表示B的轉(zhuǎn)置.向量x,f∈Rn,y,g∈Rm.在此假設(shè)條件下易知線性方程組(1)的解是存在且唯一的,并且方程組的系數(shù)矩陣是非奇異的.具有形如方程組(1)的線性系統(tǒng)有許多實(shí)際應(yīng)用背景,如計算流體力學(xué)[1-2]、電磁計算[3]、Stokes方程和二階橢圓形的混合有限元方法,以及帶約束的優(yōu)化問題等[4-13].線性系統(tǒng)(1)中的A矩陣為對稱正定或?qū)ΨQ半正定的,有許多不同的迭代方法來求解這類問題[8-9],但是當(dāng)(1,1)塊矩陣A是不定矩陣的研究工作相對來說則少很多.本文針對系數(shù)矩陣(1,1)塊矩陣A是不定矩陣,運(yùn)用吉爾-默里強(qiáng)迫正定分裂方法[14]使分解成一個對稱正定矩陣和一個對角矩陣,構(gòu)造一個新的迭代方法,并給出該算法的收斂條件.

1 吉爾 -默里強(qiáng)迫正定分解算法

2 吉爾-默里強(qiáng)迫正定迭代方法的收斂性分析

3 數(shù)值算例

表1 吉爾-默里強(qiáng)迫正定迭代方法的迭代數(shù)及運(yùn)行時間Table 1 Number of iterations and running time of Gill-Murry forced positive definite splitting methods

表1列出了迭代矩陣G的譜半徑的值以及迭代格式(5)收斂所需要的時間.由結(jié)果可知迭代格式(5)收斂,故此算法是有效的.

[1]Cliffe K A,Garratt T J,Spence A.Eigenvalues of block matrices arising from problems in fluid mechanics[J].SIAM J Matrix Analy Appl,1994,15:1310-1318.

[2]Glowinski R.Finite element methods for incompressible viscous flow[C]//Handbook of Num Anal.Amsterdam:North-Holland,2003.

[3]Arbenz P,Geus R.Multilevel preconditioned iterative eigensolvers for Maxwell eigenvalue problems[J].Appl Num Math,2005,54:107-121.

[4]Zhou Y Y,Zhang G F.A generalization of parameterized inexact Uzawa methods for generalized saddle point problems[J].Appl Math Comput,2009,215:599-607.

[5]Ling X F,Hu X.On the iterative algorithm for large sparse saddle point problems[J].Appl Math Comput,2006,178:372-379.

[6]Jiang M Q,Cao Y.On local Hermitian and skew-Hermitian splitting iteration methods for generatized saddle point problems[J].J Comput Appl Math,2009,231:973-982.

[7]Bai Z Z,Wang Z Q.Restrictive preconditioners for conjugate gradient methods for symmetric positive definite linear systems[J].J Comput Appl Math,2006,187:202-226.

[8]Cao Z H.Fast Uzawa algorithm for generalized saddle point problems[J].Appl Num Math,2003,46:157-171.

[9]Chen F,Jiang Y L.A generalization of the inexact parameterized Uzawa methods for saddle point problems[J].Appl Math Comput,2008,206:765-771.

[10]Cao Z H.Constraint Schur complement preconditioners for nonsymmetric saddle point problems[J].Appl Num Math,2009,59:151-169.

[11]Bai Z Z.Structured preconditioners for nonsingular matrices of block two-by-two structures[J].Math Comput,2006,75:791-815.

[12]Bai Z Z,Parlett B N,Wang Z Q.On generalized successive overrelaxation methods for augmented linear systems[J].Num Math,2005,102:1-38.

[13]Cui M R.Analysis of iterative algorithms of Uzawa type for saddle point problems[J].Appl Num Math,2004,56:133-146.

[14]徐成賢,陳志平,李乃成.近代優(yōu)化方法[M].北京:科學(xué)出版社,2002:62-67.

[15]Cao Z H.Positive stable block triangular precondetioners for symmetric saddle point problems[J].Appl Num Math,2007,57:899-910.

[16]Bai Z Z,Pan J Y,Golub G H.Preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite linear systems[J].Num Math,2004,98:1-32.

[17]Bai Z Z,Pan J Y,Ng M K.New preconditioners for saddle point problems[J].Appl Math Comput,2006,172:762-771.

[18]Bai Z Z,Golub G H,Pan J Y.Preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite linear systems[J].Num Math,2004,98:1-32.

[19]程云鵬.矩陣?yán)碚揫M].西安:西北工業(yè)大學(xué)出版社,2005:266-271.