Improving AOR Method for a Class of Two-by-Two Linear Systems

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1 Alied Mathematics doi:4236/am22226 Published Online February 2 (htt://scirporg/journal/am) Imroving AOR Method for a Class of To-by-To Linear Systems Abstract Cuixia Li Shiliang Wu 2 College of Mathematics Chengdu University of Infromation Technology Chengdu China 2 School of Mathematics Statistics Anyang Normal University Anyang China lixiatk@26com ushiliang999@26com slu@aynueducn Received October 8 2; revised December 2; acceted December 5 2 In this aer the reconditioned accelerated overrelaxation (AOR) method for solving a class of to-by-to linear systems is resented A ne reconditioner is roosed according to the idea of [] by Wu Huang The sectral radii of the iteration matrix of the reconditioned the original methods are comared The comarison results sho that the convergence rate of the reconditioned AOR methods is indeed better than that of the original AOR methods henever the original AOR methods are convergent under certain conditions Finally a numerical examle is resented to confirm our results Keyords: Preconditioner AOR Method Convergence Comarison Introduction Sometimes e have to solve the folloing linear systems Hx f () here I B D H C I B 2 is non-singular ith ij ij n n ij ij n n B b B b 2 C c D d Systems such as () are imortant aear in many different alications of scientific comuting For examle () are usually faced hen the folloing generalized linear-squares roblem is considered T Axb W Axb min n x here W is the variance-covariance matrix One can see [2-5] for details As is knon the linear systems () can be solved by direct methods or iterative methods Direct methods are idely emloyed hen the order of the coefficient matrix H is not too large are often regarded as robust methods The memory the comutational requirements for solving the large linear systems may seriously challenge the most efficient direct methods available today The alternative is to use iterative methods established for solving the large linear systems Naturally it is necessary that e make the use of iterative methods instead of direct methods to solve the large sarse linear systems Meanhile iterative methods are easier to imlement efficiently on high erformance comuters than direct methods As is knon there exist three ell-knon classical iterative methods ie Jacobi Gauss-Seidel successive overrelaxation (SSOR) method hich ere fully covered in the excellent books by Varge [6] Young [7] To make the convergence rate of SSOR method better accelerated overrelaxation (AOR) method as roosed in [8] by Hadjidimos To solve the linear systems () ith the AOR iterative method based on the structure of the matrix H the matrix H is slit as follos B D H I C B (2) 2 The AOR iterative method for solving () is established as follos i i r x T x g i (3) here T r is iteration matrix is of the folloing form Coyright 2 SciRes

2 C X LI ET AL 237 I B D rc I C B 2 Tr I I B D r CrCBI B2 rcd I g f rc I Obviously if r then the AOR method reduces to the SOR method The sectral radii of the iteration matrix is decisive for the convergence stability of the method the smaller it is the faster the method converges hen the sectral radii is smaller than To accelerate the convergence rate of the iterative method solving the linear systems () reconditioned methods are often used That is Coyright 2 SciRes PHx Pf here the reconditioner P is a non-singular matrix If the matrix PH is exressed as I B D PH C I B 2 then the reconditioned AOR method can be defined by here i i r x T x g i (4) I B D T r r C rcb IB rcd 2 I g Pf rc I In this aer according to the idea of [] by Wu Huang a ne reconditioner is roosed to imrove the convergence rate of the AOR method Be similar to the ork of [] [9] e comare the sectral radii of the iteration matrix of the reconditioned the original methods The comarison results sho that the convergence rate of the reconditioned AOR methods is indeed suerior to that of the original AOR methods henever the original AOR methods are convergent (to see the next section) For convenience e shall no briefly exlain some of nn the terminology lemmas Let C c ij be an n n real matrix By diag C e denote the n n diagonal matix coinciding in its diagonal ith c ii For nn A a ij B b ij e rite A B if aij holds for all i j 2 n Calling A is nonnegative if A aij ; i j 2 n e say that A B if only if A B These definitions carry immediately over to vectors by identifying them ith n matrices denotes the sectral radius of a matrix nn Lemma [6] Let A be a nonnegative irreducible n n matrix Then ) A has a ositive real eigenvalue equal to its sectral radius A ; 2) for A there corresonds an eigenvector x ; 3) A is a simle eigenvalue of A 4) A increases hen any entry of A increases Lemma 2 [] Let A be a nonnegative matrix Then ) If x Ax for some nonnegative vector x x then A 2) If Ax x for some ositive vector x then A Moreover if A is irreducible if x Ax x for some nonnegative vector x then ( A) x is a ositive vector The outline of this aer is as follos In Section 2 the sectral radii of the iteration matrix of the original the reconditioned methods are comared In Section 3 a numerical examle is resented to illustrated our results 2 Preconditioned AOR Methods Comarisons No let us consider the reconditioned linear systems Hx f (2) here H I S H f I S f ith S S According to [] here S is taken as as follos b2 b23 S b b Naturally e assume that there at least exists a nonzero number in the elements of S By simle comutations e obtain I B SI B I SD H C I B2

3 238 ith BSI B b b2b2 b2b22 b b2b2 b2 b23b3 b22 b23b32 b2 b23b 3 b b b2b b2 b b b Be similar to (2) H can be exressed as B SI B I SD H I C B2 Then the reconditioned AOR method for (2) is defined as follos: i i x T r x g i (22) here T r I BSI BrC rc B S I B I S DI B2 rc I S D I g f rc I The folloing theorem is given by comaring the sectral radii of the iteration matrix T r the original iteration matrix T r Theorem 2 Let the coefficient matrix H be irreducible B ith diag B B2 C D r Then ) T r Tr if T r ; 2) T r Tr if T r Proof By simle comutations e obtain I B D Tr r C I B2 (23) r CB CD Clearly if the matrix H satisfies B B2 C D ith r then T r CB CD Since the matrix H is irreducible by observing the structure of (23) it is not difficult to get that the matrix T r is irreducible Similarly the matrix T r is nonnegative irreducible ith diag B By Lemma there is a ositive vector x such that T x x r C X LI ET AL Where T r Obviously is imossible otherise the matrix H becomes singular So e ill mainly discuss to cases: Case : Since T r xx T r xtr x e get S I B SD T r xtr x x rcs I B rcsd S I B D x rcs S Tr Ix rcs S x rcs Since S S then e get S S x x rcs rcs If then T r xtr x but not equal to zero vector By Lemma 2 e get T r Tr That is ) holds Similarly 2) holds ith hich comletes the roof It is ell knon that hen r AOR iteration is reduced to SOR iteration The folloing corollary is easily obtained Corollary 2 Let the coefficient matrix H be irreducible B ith diag B B2 C D Then T T if ; ) 2) T T T if Next e consider the folloing reconditioners Let the matrix S in (2) be defined by S S There exist the folloing three forms for S that is ) If n then c c22 S cn cn n 2) If n then T c c22 S c c n n n n n Coyright 2 SciRes

4 C X LI ET AL 239 3) If n then c c22 S c c n Naturally e assume that there at least exists a nonzero number in the elements of S For the sake of simlicity e assume that n H can be exressed as I B D H SI BC I B2 SD ith SI B C cb c2 cb2 c cb c2 c22b2 c22b22 c2 c22b 2 dn dn2 dn here dn cnb cn nbn d c c b c b d c c b c b n2 n2 n2 n n n2 n n n n n n The matrix H is slit as follos B D H I SI B C B2 SD Then the reconditioned AOR method for (2) is of the folloing form: here i i r x T x g i Tr I J rk B D J SI BC B2 SD K SI BCI B SI BCD I g f rsi B C I Similarly the folloing theorem corollary are given by comaring the sectral radii of the iteration matrix T r the original iteration matrix T r Theorem 22 Let the coefficient matrix H be irreducible B ith diag B B2 C D r Then ) Tr Tr if T r ; 2) Tr Tr if T r Corollary 22 Let the coefficient matrix H be irreducible B ith diag B B2 C D Then T T if ; ) 2) T T T if 3 A Numerical Examle No let us consider the folloing examle to illustrate the results Examle 3 I B D H C I B 2 here ith T 2 ij 2 ij n n B b B b C cij D d n ij n bii i 2 5 i j 4 4 j i i j 2 i j 4 4i ji i 2 j 2 2 bii i 2 n 4 2 i j 4 4 j i i n j 2 n 2 i j 4 4i j i i 2 n j 2 n Coyright 2 SciRes

5 24 C X LI ET AL Table The sectral radii of the AOR reconditioned AOR iteration matrix n r T r T r T r Table 2 The sectral radii of the SOR reconditioned SOR iteration matrix n r T r T r T r c ij 4i j i 4 i n j 2 d ij 4 ji 4 i j 2 n Tables 2 dislay the sectral radii of the corresonding iteration matrix ith different arameters r These calculations are erformed using Matlab 7 Obviously from Table it easy to knon that T r Tr T r Tr That is these are in concord ith Theorem 2 22 T T From Table 2 it is easy to kno that T T hen T That is these are in concord ith Corollary Acknoledgements This research as suorted by NSFC ( ) 5 References [] S-L Wu T-Z Huang A Modified AOR-Tye Iterative Method for L-Matrix Linear Systems Australian & Ne Zeal Industrial Alied Mathematics Journal Vol [2] J-Y Yuan Iterative Methods for Generalized Least Squares Problems PhD Thesis IMPA Rio de Janeiro Brazil 993 [3] J-Y Yuan Numerical Methods for Generalized Least Squares Problems Journal of Comutational Alied Mathematic Vol 66 No doi:6/ (95)67- [4] J-Y Yuan A N Iusem SOR-Tye Methods for Generalized Least Squares Problems Acta Mathematicae Alicatae Sinica Vol doi:7/bf [5] J-Y Yuan X-Q Jin Convergence of the Generalized AOR Method Alied Mathematics Comutation Vol 99 No doi:6/s96-33(97)75-8 [6] R S Varga Matrix Iterative Analysis Sringer Series in Comutational Mathematics: 27 Sringer-Verlag Berlin 2 [7] D M Young Iterative Solution of Large Linear Systems Academic Press Ne York 97 [8] A Hadjidimos Accelerated over Relaxtion Method Mathematics of Comutation Vol 32 No doi:9/s [9] X-X Zhou Y-Z Song L Wang Q-S Liu Preconditioned GAOR Methods for Solving Weighted Linear Least Squares Problems Journal of Comutational Alied Mathematics Vol 224 No doi:6/jcam28434 [] A Berman R J Plemmons Nonnegative Matrices in the Mathematics Sciences SI Philadelhia 994 Coyright 2 SciRes