Modified parallel multisplitting iterative methods for non-hermitian positive definite systems

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1 Adv Coput ath DOI 0.007/s odfed parallel ultsplttng teratve ethods for non-hertan postve defnte systes Chuan-Long Wang Guo-Yan eng Xue-Rong Yong Receved: Septeber 20 / Accepted: 4 Noveber 20 Sprnger Scence+Busness eda, LLC 20 Abstract In ths paper we present three odfed parallel ultsplttng teratve ethods for solvng non-hertan postve defnte systes Ax = b. The frst s a drect generalzaton of the standard parallel ultsplttng teratve ethod for solvng ths class of systes. The other two are the teratve ethods obtaned by optzng the weghtng atrces based on the sparsty of the coeffcent atrx A. In our ultsplttng there s only one that s requred to be convergent n a standard ethod all the splttngs ust be convergent, whch not only decreases the dffculty of constructng the ultsplttng of the coeffcent atrx A, but also releases the constrants to the weghtng atrces unlke the standard ethods, they are not necessarly be known or gven n advance. We then prove the convergence and derve the convergent rates of the algorths by akng use of the standard quadratc optzaton technque. Fnally, our nuercal coputatons ndcate that the ethods derved are feasble and effcent. Councated by Charles A. cchell. Ths work s supported by NSF of Chna 0784 and NSF of Shanx Provnce C.-L. Wang B Departent of atheatcs, Tayuan Noral Unversty, Tayuan 03002, Shanx Provnce, People s Republc of Chna e-al: clwang28@26.co G.-Y. eng Departent of Coputer Scence, Xnzhou Noral Unversty, Xnzhou , Shanx Provnce, People s Republc of Chna X.-R. Yong Departent of atheatcal Scences, Unversty of Puerto Rco at ayaguez, ayaguez, PR , USA

2 C.-L. Wang et al. Keywords Optal weghtng atrces Parallel ultsplttng teratve ethod Non-Hertan atrx Convergence atheatcs Subject Classfcatons F0 5A06 Introducton and prelnares In solvng a large sparse lnear syste of equatons Ax = b,. where A = a j C n n s nonsngular, and b C n, O Leary and Whte [9] see to be the frst to ntroduce the parallel algorths by ultsplttng A and derve the convergence propertes. Ther forulas can be wrtten as A = N, =, 2,,, x k = N x k + b, k =, 2,. x k = = E x k, where E, called the weghtng atrces, are nonnegatve dagonal atrces that satsfy the equaton E = I. The trples, N, E = s sad to be a ultsplttng splttngs of A. Subsequently, any authors studed the ethods = for the cases that A s an -atrx [6], an H-atrx [ 5, 8] and an Hertan postve defnte atrx e.g., [9 2, 5, 2, 22], respectvely. However, we notced that all the above parallel ultsplttng teratve ethods are based on the followng constrants to A: all splttngs A = N, =, 2,,, ust be convergent; the weghtng atrces, E k =, 2,...,; k =, 2, e.g. [2,, 4] ust be gven n advance. When the coeffcent atrx A of the syste. s non-hertan postve defnte, the frst author of ths paper and Ba [8] presented soe suffcent condtons that guarantee the convergence of the sngle = teratve ethods and we would lke to address that Ba et al. [6, 7] also dscussed two alternatve ethods, called HSS and PSS ethods, whch converge uncondtonally to the unque soluton of the syste of equatons.. But a coon drawback of those ethods s that, f A s Hertan or skew- Hertan, the correspondng equatons ust be solved at each teraton step. The research nto a skew-hertan syste of lnear equatons s also conducted n [3, 9, 20]. But untl now we have not seen an artcle that dscusses the convergence of a parallel ultsplttng teratve algorth for the non-hertan postve defnte systes of lnear equatons. Ths otvated us to coe up wth the algorths proposed here.

3 odfed parallel ultsplttng teratve ethods The a of ths paper s to nvestgate the convergent parallel ultsplttng teratve algorths for the non-hertan postve defnte systes of lnear equatons. By akng use of the standard quadratc optzaton technque we choose the optal weghtng atrces at each step. And n coputaton, we need only one of the splttngs A = N, =, 2,,, to be convergent and all the others can be constructed arbtrarly. Thus, we not only decrease the dffculty of constructng the ultsplttng of A, but also release the constrants to the weghtng atrces they are not necessarly be known or gven n advance. The proposed three parallel ultsplttng teratve ethods are as follows. a drect generalzaton of the tradtonal parallel ultsplttng teratve ethods for solvng non-hertan postve-defnte systes; a ethod based on cobnng a specal ultsplttng that has conjugate property and the sparsty of the atrx A; a parallel ultsplttng teratve ethod wth optal weghtng atrces. It s convenent to ntroduce soe essental notatons and prelnares. As usual, we use C n n to denote the n n coplex atrx set and C n the n- densonal coplex vector space. X represents the conjugate transpose of a atrx or a vector X and x, y stands for the angle between the vectors x and y.weuse 2 to denote the Eucldean nor. If A s an n n Hertan postve defnte or se-defnte atrx then t s wrtten as A 0 or 0. A atrx A C n n s called postve defnte, f for all nonzero x C n, Rex Ax >0 s always true. For a coplex or real atrx A we let HA = 2 A + A, SA = 2 A A..2 The wdth l of a sparse atrx A s defned as l = ax{ j, a j = 0}..3 For a large sparse atrx A, we always assue that l n.also,[ n ] represents the nteger part of the nuber n. For the resdual vectors rk = Ax k b, k =, 2,,ftheysatsfy r k 2 N k 2 r 0 2,.4 then the convergent rate q s defned as q = ln N 2..5 In the followng Secton 2 we descrbe three dfferent parallel ultsplttng teratve algorths and then n Secton 3 we provde the proofs of ther convergence. Fnally, We apply our algorths to two concrete exaples and then llustrate the advantages of the algorths.

4 C.-L. Wang et al. 2 Parallel algorths In ths secton we present three parallel ultsplttng teratve algorths. The convergence theores for the algorths wll be establshed n the next secton. The frst s a drect generaton of the standard parallel ultsplttng teratve algorth. The next two are the parallel ultsplttng teratve algorths obtaned by optzng the weghtng atrces. I Let the ultsplttng of A be gven by [9, 0] A = B C, =, 2,,, 2. where B that s =, 2,, are Hertan dagonal block atrces, B = dag B,, B,2,, B,, 2.2 where B,, B,2,, B, are Hertan, and the weghtng atrces E satsfy E = I, E = dag α I,α 2 I 2,,α I 0, =, 2,,. = 2.3 Algorth 2. Gve an ntal pont x 0 and a tolerance ɛ>0,fork =, 2, untl the process converges, do Step. Solve n parallel the equatons x k = N x k + b, =, 2,,. Step 2. Copute x k by the forula x k = E x k. Step 3. = If Ax k b 2 <ɛ, stop; Otherwse, k k + and go back to Step. II Let the ultsplttng of A be gven by A = N, =, 2,,, 2.4 where s a Hertan postve defnte atrx, and let = P, = 2,,, 2.5 P = 0,, P [ n ] n,, 0,

5 odfed parallel ultsplttng teratve ethods P [ n ] n = [ n ] n, 2.6 where, the nonzero entres can be fro row [ n ]+to row + [ n ] l and where the zero entres are fro row + [ n ] l to row + [ n ]. Slarly, the weghtng atrces satsfy = E k = I, E k = α k I, =, 2,,, k =, 2,, we now descrbe our second algorth. 2.7 Algorth 2.2 Gve an ntal pont x 0 and a tolerance ɛ>0, let the resdual vector r 0 = Ax 0 b. Fork =, 2, untl the algorth converges, do Step. Copute n parallel Step 2. x k = x k r k, =, 2,,, r k = Ax k b, =, 2,,. Copute α n parallel for = 2,, r k α k r k = r k r k r k rk r k, 2.8 Step 3. Step 4. Copute x k and r k by the followng forulas x k = α k x k = x k + α k x k x k, = r k = = =2 α r k. If r k 2 <ɛ, stop; Otherwse, k k + andgobacktostep. III Let the ultsplttng of A be gven by A = F G, =, 2,,, 2.9

6 C.-L. Wang et al. where F s a Hertan postve defnte atrx, and E k = α k I, =, 2,,, k =, 2,, 2.0 now we release the constrant = I and descrbe our thrd algorth below. Algorth 2.3 Gve an ntal pont x 0 and a tolerance ɛ>0, letr 0 = Ax 0 b. Fork = 0,, 2, untl the process converges, do Step. Copute n parallel F x k = G x k + b, =, 2,,. Step 2. Copute x k = α k x k, = = E k Step 3. r k = Ax k b, where α k = α k,αk 2,,αk s the soluton to the followng quadratc prograng n α 2 r r, 2. r = A = α x k b. 2.2 If r k 2 <ɛ, stop; Otherwse, k k + andgobacktostep. In fact, f we defne Xk = x k,, xk, then the soluton of 2. and 2.2sgvenby α k = Xk T A T AXk Xk T A T b. 2.3 Reark We would lke to address the releases of the constrants that are requred n the algorths appeared n the prevous artcles. Algorth 2. s a drect generalzaton of syetrc splttng dscussed n [9]. The advantage of the algorth s that the weghtng atrces do not need to be scaled. In Algorth 2.2, the splttng has conjugate property. By akng use of the sparsty of the coeffcent atrx A, we obtaned ore general weghtng atrces E k = α k I, =, 2,, the atrces do not need to be nonnegatve or statonary. In Algorth 2.3, The splttngs do not need to be convergent for all. Instead, we need only one convergent splttng of A n coputaton. Also, the weghtng atrces E k = α k I, =, 2,, do not

7 odfed parallel ultsplttng teratve ethods need to be nonnegatve or statonary and we reove the constrant E = I = n coputaton. 3 Analyss of convergence In ths secton we dscuss the convergence of the ultsplttng algorths addressed n the prevous secton. Our dea s by cobnng the sparsty of the atrx A and the property of the soluton to a quadratc prograng. Theore 3. Let A = N, det = 0 be a splttng and T = N. If HA 0.Then 2 HA + SA HA SA f and only f 2 T 2 < Proof Note that provng 2 T 2 2 < s equvalent to showng that 2 N N 2 I, whch s also equvalent to clang that Fro N = A, we have N N. 3.2 A A = A A + A A. By cobnng ths dentty wth forula 3.2 we obtan ther another equvalent condton: and t s also equvalent to A A A + A, A A + AA = A + A. 3.3 To prove the theore we rewrte A n ters of HA 2 A = HA + SA and SA. Snce = HA 2 I + ŜA HA 2 = HA 2 I + ŜA I ŜA I ŜA HA 2 = HA 2 I Ŝ 2 A I ŜA HA 2 = HA 2 I Ŝ 2 A HA 2 HA 2 I Ŝ 2 A ŜAHA 2, where ŜA = HA 2 SAHA 2, we have that A + A = 2HA 2 I Ŝ 2 A HA 2.

8 C.-L. Wang et al. Cobnng ths relaton wth 3.3 yelds 2HA 2 I Ŝ 2 A HA 2, whch s equvalent to 2 HA + SA HA SA. Observng the assuptons n the theore yelds 3.. Theore 3.2 Let B, C, E = be the ultsplttng gven by the forulas and B = dagb,, B,2,, B, be of the sae fors as B =, 2,,.If B B 2 HA + SA HA SA, HA 0, E = I, =, 2,,, E 0, = then the sequence {x k } generated by Algorth 2. converges to the soluton of.. Proof Fro the proofs gven n [9, 7] t s trvally seen that the ultsplttng B, C, E = s convergent f and only f the splttng A = E B E B E B C = = = s convergent. Thus, t suffces to prove E B B. 3.4 Fro 2.2and2.3 we have that = E B = = dag α B,,α 2 B,2,,α B, dag α B,α 2 B 22,,α B. Consequently, E B dag α B, α 2 B 22,, α B = = = = B, whch s equvalent to 3.4. Now, fro Theore 3. we see that the splttng A = E B E B E B C s convergent. Ths proves = = = the theore.

9 odfed parallel ultsplttng teratve ethods Lea 3.3 If the wdths of A and B are l and d, respectvely, then the wdth of AB s at ost l + d. Proof Let A = a j, B = b j, C = c j = AB.Then c j = n a k b kj = a l b lj + +a +l b +lj. k= If l j > d, thenb lj = =b +lj = 0,soc j = 0. If + l < j d, thenb lj = =b +lj = 0,soc j = 0. Hence, f j > l + d,thenc j = 0. Lea 3.4 Conjugate property Let the wdth of the atrx A be l, and let l, << n, [ n ]= n. Assue P = 0,,P [ n ] n,, 0, where P [ n ] n s -th block atrx wth nonzero rows beng fro [ n ]+ to + [ n ] l. Then Proof Snce P AP j = P AP j = 0, = j ,, P [, 0,, 0 n ] n A 0,, P j [,, 0 n ] n and the nonzero rows of AP j are fro j [ n ] l + to j + [ n ], the nonzero coluns of P are at ost fro [ n ]+ to + [ n ] l. Ths ples P AP j = 0 for all, j such that = j. Reark If [ n ] < n,wecanselectp = 0,, 0, P k n,wherek = n [ n ],thenp k n has [ n ] l nonzero rows. Hence, for the case that [ n ] = n, Lea 3.4 s stll true. Theore 3.5 Let the ultsplttng, N, E = be gven by forulas Assue that P =, 2,, satsfy P A AP j = 0. If 2 HA + SA HA SA and HA 0. Then {x k } generated by Algorth 2.2 converges to the soluton of.. Furtherore, f 2 r k, 2 r k θ, then the convergent rate s gven by q = ln 2 N ln cos θ, 3.6 Proof Let r k = Ax k b. Then whch ples r k x k = r k A = x k r k, =, 2,,, 3.7 r k = I A r k, =, 2,,. 3.8

10 C.-L. Wang et al. On the other hand, snce r = = α r k = r k + =2 α r k we consder the followng quadratc prograng r r = r k + α r k r k r k + = + = + r k =2 r k =2 rk + 2 α r k rk + 2, j=2 fro 3.8, t follows α α j r k r k =2 r k =2 r k α r k =2 α r k r k, =2 r k rk α r k r k j r k r k rk r k = A r k = AP r k, α r k r k r k, 3.9 and r k r k r k j Hence, 3.9 s reduced to r r = r k + r k = r k P A AP j r k = 0. rk + 2 α 2 =2 r k r k =2 α r k r k r k rk r k. 3.0 Now, by dfferentatng r r wth respect to α, =, 2,,, we obtan forula 2.8. That s, the crtcal pont,α k, that nzes the quadratc for r r: α k = r k r k r k r k r k rk r k, = 2, 3,,. 3.

11 odfed parallel ultsplttng teratve ethods Fg. The orthogonalty between 2 r k and 2 r k r k Next, we dscuss the convergence and convergent rate of {x k }.Forr k and r k,3. yelds r k r k r k = 0, whch ples that the vectors 2 r k and 2 r k r k are orthogonal vectors shown n Fg.. Fro the assupton 2 r k, 2 r k θ, we see that Hence, r k rk cos 2 θ r k rk r k 2 cos θ 2 r k 2 = cos θ 2 N 2 rk cos θ 2 N 2 2 r k 2 2. cos k θ 2 N 2 k 2 r 0 2 Fro Theore 3., we have that 2 N 2 2 <. Thus, l 2 r k 2 = 0, 3.3 k whch ples l k rk = 0. Furtherore, by the defnton of the convergent rate see.4 and.5, we obtan 3.6. Theore 3.6 Let the ultsplttng F, G, E = be gven by forulas , andletf 2 HA + SA HA SA. Then the sequence {x k } that s generated by Algorth 2.3 converges to the soluton of.. Furtherore, f F 2 r k, F 2 r k θ, then the convergent rate s q = ln F 2 G F 2 + ln cos θ

12 C.-L. Wang et al. Proof Let r k = Ax k b. Then cobnaton of 2. and2.2 ples 3.2. Thus, proceedng a slar dervaton results n the followng property l k rk = and 3.4 Ther dervatons are otted, whch proves the theore. 4 Nuercal results Exaple [6] Consder the generalzed convecton-dffuson equatons n a two-densonal case: 2 u x 2 u u du + q expx + y x + q expx + y y 2 y2 x dy = f 4. wth the hoogeneous Drchlet boundary condton. We use the standard Rtz-Galerkn Fnte Eleent ethod and apply the conforng lnear trangular eleents to approxate ts contnuous solutons u = x y x y n the doan =[0, ] [0, ], where the step-szes along both x and y drectons are selected to be the sae h = 28. After dscretzaton the atrx A of ths equaton s gven by A B 2 C 2 A 22 B 23 A = , C p p 2 A p p B p p C pp A pp where A, =,, p are n-by-n nonsyetrc atrces and B T + = C +., and np = Let q =. Gven x 0 = 0,, 0 T and a tolerance ε = 0 5, the teraton fals n coputaton for the teratve nuber that s up to 30,000. Now, let HA = D L L T, where D = dagha,, HA pp, and L the block strctly lower trangle atrx of HA. SnceA s a sparse atrx wth the block wdth l =, we choose = D = dagha,, HA pp. the splttngs, N, = 2, 3 are constructed by 2.5 and2.6, where the nonzero part of P s the correspondng part of the followng atrx [ p 3 ] p 0 B HA 2 HA 22 C 2 0 B C HApp p p 2 0 B p p C pp 0 p p

13 odfed parallel ultsplttng teratve ethods Our nuercal results of Algorth 2.2, when p = 64, are gven n Table. When we use the followng ultsplttng teratve ethod = D L T D D L T, N = A; 2 the block SOR ethod paraeter ω =.8; 3 The block Jacob ethod. 4 Algorth 2.3 The coputaton generates the followng results see Table 2. If we replace the splttng wth the new splttng = HA, N = SA, and unchange the others, our algorth generates uch better results, llustratng n the followng Table 3. Fro the above nuercal results we see that the ultsplttng parallel algorths for non-hertan lnear systes are convergent. In addton, the ultsplttng parallel algorth has less teratve nuber than the sngle splttng teraton. The rate of convergence of the ultsplttng parallel algorth depends on the secton of the an splttng. Table Coparsons of Algorth 2.2 and the sngle splttng teratve ethods when p = 64 ethods N 2 N 2 3 N 3 Algorth 2.2 IT Table 2 Coparsons of Algorth 2.3 and the sngle splttng teratve ethods ethods Block Jacob Block SOR Splttng Algorth 2.3 IT Table 3 Coparsons of Algorth 2.3 and the sngle splttng teratve ethods ethods Block Jacob Block SOR The new splttng Algorth 2.3 IT

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