葉絨絨,暢大為，韓俊佳

(陜西師范大學(xué) 數(shù)學(xué)與信息科學(xué)學(xué)院，陜西 西安 710119)

2-循環(huán)矩陣下MASOR迭代法的收斂性分析

葉絨絨,暢大為，韓俊佳

(陜西師范大學(xué) 數(shù)學(xué)與信息科學(xué)學(xué)院，陜西 西安 710119)

2-循環(huán)矩陣;MASOR迭代矩陣;Jacobbi迭代矩陣;MASOR特征值;Jacobbi特征值

0 引 言

迭代法是求解大型稀疏方程組Ax=b的一種重要方法,在實(shí)際應(yīng)用和計(jì)算中具有重要作用．為此很多學(xué)者對(duì)方程組的解法進(jìn)行了深入研究,在此基礎(chǔ)上提出了各種迭代法,由此而來(lái)的是每個(gè)迭代法的收斂性,收斂范圍及其他一些相關(guān)性質(zhì)的研究．

近年來(lái)關(guān)于迭代方法已經(jīng)進(jìn)行過(guò)很多分析．最初胡家贛在文獻(xiàn)[1]中提出了SOR,AOR迭代方法,并對(duì)其基本性質(zhì)進(jìn)行了研究,之后文獻(xiàn)[2-6]給出了AOR,SOR的一些相關(guān)性質(zhì)及其收斂標(biāo)準(zhǔn),文獻(xiàn)[7-14]分析了MSOR,SSOR方法的收斂性,最優(yōu)參數(shù)估計(jì)及其他一些問(wèn)題,在此基礎(chǔ)上,文獻(xiàn)[15-16]詳細(xì)分析了一類2-循環(huán)矩陣下MASOR的充分條件,文中借助以上相關(guān)結(jié)論及方法研究當(dāng)μ2=mi時(shí),σ1=σ2或者σ1=-σ2情形下MASOR方法可以收斂,并且給出了|mk|=1,|mk|>1,|mk|<1時(shí),MASOR的收斂范圍.

1 預(yù)備知識(shí)

考慮線性方程組

Ax=b

(1)

其中A∈Cn×n,x∈Cn,b為已知,x為所求．

當(dāng)A為p×p塊矩陣時(shí)，即

于是A=D-B-C.其中

設(shè)L=D-1B,U=D-1C,則B=DL,C=DU,于是L與U也必為嚴(yán)格下三角陣與嚴(yán)格上三角陣.

由于A的Jacobbi迭代矩陣為

J=D-1(B+C)=L+U,

取

其中wi≠wj(i≠ j),Ii為與Aii對(duì)應(yīng)的單位矩陣.此時(shí)對(duì)應(yīng)(1)的MASOR迭代矩陣為

φΩ=(I-ΩL)-1(I-Ω+ΩU).

特別地,當(dāng)A為p-循環(huán)矩陣時(shí),即

(2)

或

(3)

其中Aii為非奇異矩陣,1≤i≤p,且p≥2.此時(shí)Jacobbi迭代矩陣可以寫(xiě)成

2 基本結(jié)果

當(dāng)A為式(2)或式(3)的結(jié)果時(shí), 考慮此時(shí) Jacobbi 迭代矩陣特征值和塊 SOR 迭代矩陣特征值之間的關(guān)系.

引理1[8]當(dāng)矩陣A為形式(2),則λ∈σ(ρΩ)當(dāng)且僅當(dāng)存在μ∈σ(J)滿足

特別地,當(dāng)p=2時(shí),

(4)

此時(shí)λ與μ之間的關(guān)系為

(λ+ω1-1)(λ+ω2-1)=λμ2ω1ω2.

引理2[15]設(shè)一元二次方程為λ2-bλ+c=0,則其根的模均小于1當(dāng)且僅當(dāng)

(5)

3 收斂性分析

當(dāng)σ1=-σ2時(shí),

此時(shí),MASOR迭代矩陣收斂.

證明 由于

由題設(shè)可得σ1=1-ω1,σ2=1-ω2,σ1=-σ2,于是要使|λk|<1,根據(jù)文獻(xiàn)[15]可得

當(dāng)σ1=σ2時(shí),

|mk|=1,滿足σ1,σ2∈(0,1)；

證明 由于

由題設(shè)可得σ1=1-ω1,σ2=1-ω2,σ1=-σ2,要使|λk|<1,根據(jù)文獻(xiàn)[15]可得

解式(8)可以得到

σ1,σ2∈(-1,1).

當(dāng)|mk|=1時(shí),解得σ1∈(0,1);

當(dāng)|mk|≥2時(shí),無(wú)解.

由于x∈(-1,1),可得f′(x)>0,即f(x)為增函數(shù)．

4 數(shù)值例子

例1 若線性方程組AX=b的系數(shù)矩陣

此時(shí)

求得

例2 若線性方程組AX=b的系數(shù)矩陣

此時(shí)

求得

[1] 胡家贛.線性代數(shù)方程組的迭代解法[M].北京:科學(xué)出版社,1991:25-29.

HU Jiagan.Iterative solution of linear algebraic equations[M].Beijing:Science Press,1991:25-29.

[2] 高樹(shù)玲,暢大為.相容次序矩陣AOR迭代收斂的充要條件[J].紡織高?；A(chǔ)科學(xué)學(xué)報(bào),2009,22(2):229-231.

GAO Shuling,CHANG Dawei.Sufficient and necessary conditions for compatible order matrix AOR[J].Basic Sciences Journal of Textile Universities,2009,22(2):229-231.

[3] LANZKRON P J,ROSE D J,SZYLD D B.Convergence of nested classical iterative methods for linear systems[J].Numeriche Mathematic,1991,58(1):685-702.

[4] 康鋒艷.奇異P-循環(huán)矩陣AOR迭代的半收斂性與單調(diào)矩陣雙分裂的收斂定理及比較定理[D].西安:陜西師范大學(xué),2011:5-14.

KANG Fengyan.The semiconverence of singularP-cyclic matrix AOR iteration and the convergence theorem and the comparison theorem of the dual division of monotone matrix[D].Xi′an:Shaanxi Normal University,2011:5-14.

[5] JAMES K R,RIHA W.Convergence criteria for successive overrelaxation[J].SIAM Journal on Numerical Analysis,1995,12(2):13-145.

[6] 熊勁松,暢大為.2-循環(huán)系數(shù)矩陣對(duì)稱MSOR法收斂的充分必要條件[J].紡織高?；A(chǔ)科學(xué)學(xué)報(bào),2011,24(4):2-4.

XIONG Jingsong,CHANG Dawei.The necessary and sufficient condition for the convergence of the symmetric MSOR for 2-cyclic coefficient matrices[J].Basic Science Journal of Textile Universities,2011,24(4):2-4.

[7] 熊勁松,暢大為,郭煜.2-循環(huán)系數(shù)矩陣對(duì)稱MSOR法最優(yōu)參數(shù)估計(jì)[J].紡織高校基礎(chǔ)科學(xué)學(xué)報(bào),2012,25(2):169-172.

XIONG Jinsong,CHANG Dawei，GUO Yu.Optimal parameter estimation of symmetric MSOR under the condition of 2-cyclic matrix[J].Basic Sciences Journal of Textile Universities,2012,25(2):169-172.

[8] SONG Yongzhong.Semiconvergence of block SOR method for singular linear systems withp-cyclic matrices[J].Journal of Computational and Applied Mathematics,2001,130(1):217-229.

[9] MARTIS M M.A note on convergence of the MSOR method[J].Linear Algebra Application,1990,141:223-226.

[10] TARVISHI M T,KHOSRO-AGHDAM R.Summetric successive overrelaxation methods for rank deficient linear systems[J].Applied Mathematics and Computation,2006,173(1):404-420.

[11] 龔林國(guó),蔡大用.SSOR法與Jacobbi迭代法在一類P-弱循環(huán)矩陣下特征值之間的關(guān)系[J].高等學(xué)校計(jì)算數(shù)學(xué)學(xué)報(bào),1985,7(1):79-84.

GONG Linguo,CAI Dayong. The relationship between the eigenvalue of SSOR iteration and Jacobbi iteration under the condition ofP-weak cyclic matrix[J].Higher School Journal of Computational Mathematics,1985,7(1):79-84.

[12] SONG Yongzhong.On the convergence of the MSOR[J].Journal of Computational and Applied Mathematics,1997,79(2):295-320.

[13] HADJIDIMOS A,YEYIOS A.The symmetric accelerated overrelaxation(SAOR)method [J].Mathmatics and Computers,1982,24(1):72-76.

[14] DARVISHI M T.On convergence of the symmetric modified successive overrelaxation method for 2-cyclic matrices[J].Applied Mathematics and Computation,2006,183:953-960.

[15] 董瑾.一類2-循環(huán)矩陣MASOR收斂的充分條件[D].西安：陜西師范大學(xué),2014:3-22.

DONG Jin.Sufficient conditions for the convergence of a class of 2-cyclic matrix MASOR[D].Xi′an:Shaanxi Normal University,2014:3-22.

[16] 董瑾,暢大為，楊青青,等.一類2-循環(huán)系數(shù)矩陣對(duì)稱MSOR法收斂的充分條件[J].紡織高?；A(chǔ)科學(xué)學(xué)報(bào)，2014，27(2):222-226.

DONG Jin,CHANG Danwei,YANG Qingqing,et al.The sufficient condition of symmetrical MSOR convergence for one type of 2-cyclic coefficient matrices[J].Basic Sciences Journal of Textile Universities,2014，27(2):222-226.

編輯：武 暉；校對(duì)：師 瑯

The convergence analyse of MASOR iteration method on the basis of 2-cylic matrix

YERongrong,CHANGDawei,HANJunjia

(School of Mathematics and Information Science, Shaanxi Normal University, Xi′an 710119, China)

2-cyclic matrix; MASOR iteration matrix; Jacobbi iteration matrix; the eigenvalue of MASOR; the eigenvalue of Jacobbi

1006-8341(2016)04-0428-07

10.13338/j.issn.1006-8341.2016.04.003

2016-04-08

國(guó)家自然科學(xué)基金資助項(xiàng)目(11226266,11401361)

暢大為(1963—),男,陜西省西安市人，陜西師范大學(xué)副教授,研究方向?yàn)橛?jì)算數(shù)學(xué).E-mail:529729551@qq.com

葉絨絨,暢大為，韓俊佳.2-循環(huán)矩陣下MASOR迭代法的收斂性分析[J].紡織高校基礎(chǔ)科學(xué)學(xué)報(bào),2016,29(4)：428-434.

YE Rongrong,CHANG Dawei,HAN Junjia.The convergence analyse of MASOR iteration method on the basis of 2-cylic matrix[J].Basic Sciences Journal of Textile Universities,2016,29(4):428-434.

O159；TP301.1

A