Minimization of l 2 -Norm of the KSOR Operator

Transcription

1 ournal of Matheatcs and Statstcs 8 (): 6-70, 0 ISSN Scence Publcatons do:0.38/jssp Publshed Onlne 8 () 0 ( Mnzaton of l -Nor of the KSOR Operator Youssef, I.K. and A.I. Alzak Departent of Matheatcs, Faculty of Scence, An Shas Unversty, Caro, 566, Egypt Receved , Revsed 0--9; Accepted 0--9 ABSRAC We consder the proble of nzng the l -nor of the KSOR operator when solvng a lnear systes of the for AX = b where, A = I +B ( = -B, s the acob teraton atrx), B s skew syetrc atrx. Based on the egenvalue functonal relatons gven for the KSOR ethod, we fnd optal values of the relaxaton paraeter whch nze the l -nor of the KSOR operators. Use the Sngular Value Decoposton (SVD) technques to fnd an easy coputable atrx untary equvalent to the teraton atrx KSOR. he optu value of the relaxaton paraeter n the KSOR ethod s accurately approxated through the nzaton of the l -nor of an assocated atrx ( ) whch has the sae spectru as the teraton atrx. Nuercal exaple llustratng and confrng the theoretcal relatons are consdered. Usng SVD s an easy and effectve approach n provng the egenvalue functonal relatons and n deternng the approprate value of the relaxaton paraeter. All calculatons are perfored wth the help of the coputer algebra syste Matheatca 8.0. Keywords: KSOR Iteratve Method, l -Nor, Sngular Value Decoposton (SVD). INRODUCION We consder lnear systes of the for Equaton : a jx j = b j,a 0, =,, L, () j=.. acob Method Equaton 3: [ n+ ] [ n] [ n] x = b ajx j ajx j a j= j= + (3) Wth a j = - a j for j and the syste adts a unque soluton. hs syste of equatons can be wrtten as Equaton : A X b,ax, b R, A R = () Such lnear systes arse n any dfferent applcatons for exaple n the fnte dfference treatent of the Korteweg de Vres equaton, Buckley (977). Also, s-lar lnear systes appears n the treatent of coupled haronc equatons, Ehrlch (97). In the teratve treatent of lnear systes, we use the splttng, A = D-L-U, where D = d I s the dag-onal part of A, for soe non-zero constant d, -L s the strctly lower-trangular part of A and -U s the strctly uppertrangular part of A, Woznck (00). he acob Method n atrx for s Equaton : X = X + D b = D (L + U) () [n+ ] [n] j s the acob teraton atrx, t s clear that n ths case s a skew syetrc atrx... he SOR Method s Equaton 5: [ n+ ] [ n ] [n ] [n] [n] x = x + b + ajx j ajx j ax a j= j= updated coponents old coponents =,, L,; n = 0,, L Correspondng Author: Youssef, I.K., Departent of Matheatcs, Faculty of Scence, An Shas Unversty, Caro, 566, Egypt (5) Scence Publcatons 6

2 Youssef, I.K. and A.I. Alzak /ournal of Matheatcs and Statstcs 8 (): 6-70, 0 where, (0,) s a relaxaton paraeter, = gves the well-known Gauss-Sedel ethod. he SOR Method n atrx for s Equaton 6: X = X + (D L) b [n+ ] [n] SOR Scence Publcatons ( = + SOR (D L) D U) (6) where, SOR s the SOR teraton atrx. he choce of the relaxaton paraeter s very portant for the convergence rate of the SOR ethod. For certan classes of atrces (-cyclc consstently ordered) wth property A, n the sense of Young (003), for such systes there s a functonal egenvalue relaton of the for Equaton 7: / ( λ + ) = µλ (7) where, λ s an egenvalue of the SOR and µ s a correspondng egenvalue of the. Most work on the choce of s to nze ρ( SOR ) whch s only an asyptotc crtera of the convergence rate of lnear statonary teratve ethod, Hadjdos and Neuann (998). In real coputatons, we have to consder average convergence rate Mlleo et al. (006). he deternaton of the optal value of the relaxaton paraeter opt can be obtaned wth the help of the egenvalue functonal relaton (7). Young (003), deterned opt when has only real egenvalues and ρ( ). In ths case we have: opt = + ( ρ ) where the optalty s understood n the sense of the nzaton of ρ( SOR ). Maleev (006) deterned opt when has only pure agnary egenvalues and ρ ( ). In ths case we have: opt = + + ( ρ ) Golub and Plls (990) ntroduced a sple proof for the egenvalue functonal relaton (7) by the use of the Sngular Value Decoposton (SVD) approach for real syetrc atrces. Yn and Yuan (00) consdered the skew syetrc case as well as the syetrc case. Mlleo et al. (006) consdered the nzaton of l - nors of the SOR and MSOR operators for the skew syetrc case he KSOR Method s In a recent work Youssef (0), ntroduced the KSOR ethod Equaton 8 and 9: [ n+ ] [ n] x = x + a b a x a x a x [n+ ] [n] [n+ ] j j j j 3 j= j= Assued updated old updated =,, L,, R [,0] [ n+ ] x = ( + ) [ n ] [n+ ] [n] x + b ajx j ajx j a j= j= + =,, L,, R [,0] (8) (9) he KSOR Method n atrx notaton s Equaton 0: ( ( X = X + + D L) b [n+ ] [n] KSOR KSOR = + D L) [D + U] (0) where, KSOR s the KSOR teraton atrx (operator). As t was n the SOR the rate of convergence of the KSOR ethod depends on the choce of the relaxaton paraeter. For certan classes of atrces (-cyclc consstently ordered wth property A), Youssef (0) establshed the egenvalue functonal relaton Equaton : / β + β = µ β () where, β I 's are the egenvalues of the KSOR and µ 's are the egenvalues of the acob teraton atrx. he egenvalue functonal relaton () can be proved by the use of the SVD technque... Sngular Value Decoposton Sngular Value Decoposton (SVD) of a atrx B R l s a factorzaton: ( q) l B = U ΣV, Σ = dag s,s, L,s R,q = { l,} where, s s s q 0, U and V are orthogonal atrces such that:

3 Youssef, I.K. and A.I. Alzak /ournal of Matheatcs and Statstcs 8 (): 6-70, 0 U U = IP, V V = Iq We consder n ths study the case studed by Yn and Yuan (00) also by Mlleo et al. (006) n whch the coeffcent atrx take the for Equaton : IP F A = F Iq () where, F R p q wth p + q = and p q. In ths case the acob teraton atrx becoes Equaton 3: 0 F = F 0 (3) It s clear that s skew syetrc and accordngly adts pure agnary egenvalues and the KSOR teraton atrx KSOR becoes Equaton : heore Let Abe the atrx gven by (), then Equaton 5: { µ } = σ = ( σ ) = σ( FF ) = { S } =,...,q (5) where, µ are the egenvalues of, S are the squares of the sngular values of F and (σ( )) s the set of squares of the egenvalues of. Proof Usng the SVD to decopose the corner block atrx F, we obtan Equaton 6: F = UΣ V (6) where, p p atrx U and q q atrx V are orthogonal,.e.: U U = I p,v V = Iq I P F + + = ( + ) + ( + ) KSOR F I q F F () and s the p q dagonal atrx (of sngular values) defned n (7). he egenvalues of the atrx: FF = UΣΣ U are {s,s,...,s } q Usually, researchers work on obtanng the optu value of the relaxaton paraeter n the sense of nzng the spectral radus of the teraton atrx or an equvalent quantty. We use the SVD approach n provng the egenvalue functonal relaton for the KSOR ethod. Also, we use the sae arguent of Golub and Plls (990) to defne a atrx ( ) whch has the sae spectru as the teraton atrx KSOR. Our objectve s to fnd the optal value of the relaxaton paraeter whch nzes the l -nor of the KSOR operator and llustrate the theoretcal results through applcatons to a nuercal exaple.. MAERIALS AND MEHODS We use the SVD n provng the relaton between the egenvalues of the skew syetrc acob teraton atrx and the sngular values of a block sub-atrx F, theore (). We wll prove the relaton between the egenvalue functonal relaton between the egenvalues of and KSOR by usng SVD, theore (). We wll fnd the spectal radus of (( KSOR ) KSOR ), theore (3). We wll fnd the optal value of the relaxaton paraeter to nze the l -nors of the KSOR theore (). Scence Publcatons 63 FF U = UΣΣ U U = UΣΣ Accordngly, the egenvectors of the atrx FF equal to the coluns of orthogonal atrx U. Slarly, F F = V V has ts egenvectors equal to the coluns of orthogonal atrx V. he nuber of nonzero sngular values S of F s equal to the rank of F. Substtutng the sngular value decoposton (6) nto the corner eleents F, F of (3), we obtan (8) Equaton 7 and 8: s 0 L L 0 0 s 0 0 q q M O M sq M Σ = 0 L L L sq 0 0 L (p q) q M L M 0 0 L 0 F 0 UΣV = = F 0 VΣ U 0 (7) (8)

4 Youssef, I.K. and A.I. Alzak /ournal of Matheatcs and Statstcs 8 (): 6-70, 0 Now, we wll fnd a relaton between the sngular values S (dagonal of ) and the egenvalues µ of where =,,..,q. For µ 0 an egenvalues of, we have Equaton 9: Coparng the block entres of n (8) and (), we obtan the equaltes: H F = XY = UΣ V x x x x ff,,,t y = µ y = µ = y y (9) H F = YX = VΣ U So that, the nuber of non-zero egenvalues of equals t that s coe n pars ±µ. o account for zero egenvalues, we wrte Equaton 0: z z 0 (0) 0,,,r z = = = z 0 We construct the n n non-sngular atrx W whose coluns are the orthogonal egenvectors of (9) and (0): Scence Publcatons X X Z W = n = p + q = t + r Y Y Z Note that the t coluns of p t atrx X and q t atrx Y are the t respectve egenvectors of (9), the r coluns of p r atrx Z and q r atrx Z coe fro the r null vectors of (0). Ordnarly, we would scale the coluns of W to produce an orthogonal atrx as a techncal convenence, however, we assue that the coluns of W are scaled so that Equaton : H WW = I () Let the atrx I denote the t t atrx whose dagonal eleents are the t postve egenvalues µ of (9). hen (9) and (0) can be cobned to produce the sngle atrx equaton: 0 0 X X Z X X Z 0 0 Y Y Z = Y Y Z Whch, when ultpled through on the rght by W H we get Equaton : H 0 XY = H YX 0 () 6 And we see Equaton 3: 0 Y Y 0 F F H X X 0 FF 0 = = H UΣΣ U 0 = 0 VΣ ΣV Accordngly: { } ( ) ( ( ) ) { } µ = σ = σ = σ FF = S. =,,...,q heore (3) Let KSOR and be gven, respectvely, by () and (3). hen the egenvalues µ σ( ) and β σ( KSOR ) are lnked by the functonal relaton Equaton : ( β ) + β = µβ () Moreover, the egenvalues and -nor of atrces KSOR and ( ) are related as follows Equaton 5-9: ( KSOR ) ( ) σ = σ (5) ( KSOR ) ( ) ax ( ) ρ = ρ = ρ (6) q = ( ) = ax = ( ) (7) k k k KSOR q Where: = dag( ( ), L, I ) (8) + q p q s + + ( ) = =,.q s s ( + ) + ( + ) (9)

5 Youssef, I.K. and A.I. Alzak /ournal of Matheatcs and Statstcs 8 (): 6-70, 0 where, s are the sngular values of F. Proof By usng the sngular value decoposton of the atrx F we have F = U V where U and V orthogonal atrces, then the atrx KSOR has the for Equaton 30: I P UΣV + + = ( + ) + ( + ) KSOR VΣ U I q VΣ ΣV Let the orthogonal atrces U and V be factored out then KSOR has the for Equaton 3: KSOR U 0 = 0 V I P Σ U V I Σ q Σ Σ ( + ) + ( + ) (30) (3) s + + ( ) = =, L,q (3) s s ( + ) + ( + ) where, s are the sngular values of F. We have seen that each eber of the faly of KSOR teraton atrx KSOR s untarly equvalent to a atrx ( ) havng only or atrces on the an dagonal. hat s, fro (3) and (33) Equaton 35: Q P PQ for untary QP KSOR = (35) Untary equvalent (35) ples that both the egenvalues and the -nors agree for both ( - fales of) atrces KSOR and ( ) then we have (5), (6) and (7). Fro (5) we have Equaton 36: ( KSOR ) ( ) det β I = 0 ff det β I = 0 (36) Fro the rght-hand deternant above, we see, fro (33), (3), that all β are constraned by Equaton 37: Note that (3) reveals the untarly equavalent atrx Γ wth four block subatrces, each of whch s a dagonal sub-atrx where Equaton 3: ( ) β =,ordet β I = 0; =,,,q + hen: (37) I P Σ + + Γ = ( + ) + ( + ) ) Σ I q Σ Σ U 0 U 0, Q Q = = 0 V 0 V and (3) hs ean that there s a perutaton atrx P whch pulls the two corner dagonal atrces to the an dagonal,.e., PΓ P has only or atrces along ts an dagonal. When Γ of (3) s peruted nto the block dagonal for, we obtan Equaton 33: = PΓ P = dag ( ), ( ), I + q pq where each atrx ( ) s gven by Equaton 3: (33) β s + + det( βi ( )) = = 0 s β + s ( + ) + ( + ) hus we fnd: ( I ) det β = β + + S β S + 3 = 0 ( + ) + ( + ) β + s β = 0 + ( + ) β =,or ( β + β ) + s β = 0 =,, q + Accordngly, wth the help of (5) we can wrte Equaton 38: β =,or ( β + β ) = µ β =,, q + (38) Scence Publcatons 65

6 Youssef, I.K. and A.I. Alzak /ournal of Matheatcs and Statstcs 8 (): 6-70, 0 Now the left-hand equaton, β = appears n + (33), (3) once for each occurrence of a zero egenvalue for, but β = s a specal case of the rght-hand + sde of (38), naely, when µ s set to zero. herefore, (38) s descrbed by the sngle relaton (). Fro the prevous theore we see the l -nor of the KSOR teraton atrx s equvalent to the l -nor of the ( ) then, equvalent to the square root of the spectral radus of ( ( )) ( ). hen, the proble of nzng the l -nor of the KSOR teraton atrx s equvalent to the proble of nzng the square root of the spectral radus of ( ( )) ( ). heore 3 Under the assuptons of the theore, for K = the nu of the of l -nor of the KSOR s equvalent to Equaton 39-: δ = : n δ = n R [,0] R [,0] ax [(t) + [ (t) C] ], Where: t ( + ) (,t) : = + ( + t) ( + ) ( + ) (39) (0) s + ( + ) ( ) ( ) = s s + + ( + ) s + + s s ( + ) + ( + ) ( ) ( ) = s + ( + ) ( + ) s s + 3 ( + ) ( + ) s s 3 + ( + ) ( + ) s s ( + ) + ( + ) It s easy to see Equaton 5: det( ( ) ( ) β I ) = β S + + ( + ) + ( + S ) β + Set s = t and defne Equaton 6 and 7: = ( + ) c : (3) () (5) (6) = ( + ) C : () t,t : t ( + ) ( + ) ( ) = + ( + ) (7) Wth t s the square of the spectral radus of the acob teraton atrx. Proof Fro the theore we have Equaton : = ( ) = ax ( ) KSOR q q = ax{ ρ ( ( ) ( ))} Now we go to calculate Equaton 3 and : Scence Publcatons () 66 herefore Equaton 8: det( ( ) ( )) β I ) = β (t ) β+ C (8) Solvng ths quadratc equaton, we fnd that Equaton 9: β = {(t ) ± [ (t ) C] } (9) Note that, for any t 0, R/[-,0], we have Equaton 50:

7 Youssef, I.K. and A.I. Alzak /ournal of Matheatcs and Statstcs 8 (): 6-70, 0 (t) 0 (t) C 0 (50) Note that: he egenvalues of the atrx ( ) ( ) are nonnegatve nubers and for the roots of the char-acterstc Equaton 5 and 5: β t β + C = 0 =,,...q (5) β = 0 ( + ) he largest of the two roots of (5) s gven by: L : = L ( ) {(, t ) + [ (, t) C( )] } =, q (5) Proof Fro (0) we see Equaton 56 and 57: t (,t) : = + ( + ) ( + ) l t 0 for any + R / [,0] d,t 3 t : = + l+ t 3 5 d ( + ) ( + ) hen we fnd Equaton 58: ( ) d,t d 0 for any (56) (57) (58) he axu value of each L s obtaned for the axu value of the correspondng (, t ). Note that: d(t) = ( t + ) 0,for any t 0 dt ( + ) Now for any t 0, (t) s a strctly ncreasng functon of t. Lkewse, L s strctly ncreasng functon of t, set Equaton 53 and 5: he functon (, t) ncreases strctly n the nterval (-, -). Dfferentatng L(, t) defned n (5) wth respect to, and usng (0) and () we fnd Equaton 59: d 8 5 dl d d + ( + ) = + d d ( + ) (59) L : = L = ax L =,, L,q (53) It s clear that Equaton 60: hen: dl 0 for any (, ) d (60) ( ) L : = L =,t +,t C( ) Scence Publcatons (5) Wth t = ρ ( ). he spectral radus of the atrx ( ) ( ) for any gven, s the quantty L, then fro (53) the spectral radus of the atrx ( ) ( ) s L. heore he value of, whch has nu n (39), s the unque real postve root n (0, ) of the Equaton 55: f t t 0 = + + = (55) 67 So that, the functon L(, t) ncreases strctly n the dl nterval (-,-). We wll take lt of the functon d as, we obtan Equaton 6: d 8 5 dl d + d ( + ) L = L + L 0 d 0 d 0 ( + ) hen when, dl we have 0 d lt as and obtan Equaton 6: + (6), now we take

8 Youssef, I.K. and A.I. Alzak /ournal of Matheatcs and Statstcs 8 (): 6-70, 0 d dl d d ( + ) L = L L + 0 d 0 d 0 ( + ) Set Equaton 63 and 6: (6) hat s: d d d ( ) d + d d ( ) 5 ( ) = d 8 W = + d ( + ) 5 (63) Elnatng ( d/d ) and dvdng through -, we obtan: V = ( + ) Now splfy (63) and (6) as: t + t + ( t + t) ( w + ) = V ( t t ) hen we have Equaton 65: dl L = 0 0 d (6) 0 as 0 (65) herefore, fro (6) and (65) L(, t) has a odd nuber of local nu ponts n (0, ). For any fxed t (0, ), the global nu pont of L( dl, t) s a pont n (0, ) at whch d vanshes. dl Settng = 0 then: d d d 8 = d ( + ) d ( + ) 5 ( + ) d d = d d ( + ) ( + ) It now follows that: d d = d d + ( + ) Substtutng (56) for and (57) for d/d, we obtan Equaton 66: f t t = + (66) hen, we have Equaton 67 and 68: ( + ) + + t + t r ( ) = = t t t ( + ) + t + t r ( ) = = 0 t t t + (67) (68) herefore, f ( ) has a unque zero r ( ) n that nterval. So the r ( ) s a unque real postve root n (0, ) of the equaton (66), fro that and (6), we notes that Equaton 69: ( ) L r,t L L, t (69) ± hen: d d 8 = 5 d ( + ) d ( + ) So that r ( ) s a unque real postve root n (0, ) of the equaton whch has the nu of L( ). Exaple Consder a syste wth: Scence Publcatons 68

9 Youssef, I.K. and A.I. Alzak /ournal of Matheatcs and Statstcs 8 (): 6-70, 0 Fg.. he behvour of the spectral radus of the SOR as a functon of Fg.. he behavour of the l -nor of SOR as a functon of A =, b = For splcty we adapted the rght hand sde b as t was n Young (003) and Youssef (0) so that the exact soluton s x =, x =, x 3 =, x =. It s well known that, for ths syste we have. he egenvalues of the acob teraton atrx are the roots of the equaton: he roots are: µ + 0.5µ = 0 Fg. 3. he behvour of the spectral radus of the KSOR as a functon of µ = µ = I, µ 3 = µ = I For the atrx, F = the Sngular values are s = s = It s clear that the acob teraton atrx s a skew syetrc, accordngly ther egenvalues are pure agnary coplex nubers, and satsfes µ = s. Scence Publcatons 3. RESULS We used the SVD n provng the egenvalue functonal relaton for the KSOR operator he nzaton of the l -nor s used as a good estaton for deternng the optu value of the relaxaton paraeter n the KSOR ethod as well as n the SOR ethod 69 Fg.. he behavour of the l -nor of SOR as a functon of Fro Fg. and we see that the calculated results agree wth the theoretcal results of Mlleo et al. (006)

10 Youssef, I.K. and A.I. Alzak /ournal of Matheatcs and Statstcs 8 (): 6-70, 0 Fro Fg. 3 and we see that the calculated results agree wth our theoretcal results Nuercal exaple llustratng and confrng the theoretcal relatons s consdered Scence Publcatons. DISCUSION Young (003), consdered the proble why convergence of the SOR ethod wth the optu opt n the sense of nzng the spectral radus of the teraton atrx s soe what slower than what ght expected, the spectral radus s only an asyptotc easure of the rate of convergence of a lnear teratve ethod. In hs treatent Young (003), establshed a relaton between the egenvalues of certan atrces related to A (the SOR teraton atrx, SOR ) and those of certan block atrces. Golub and Plls (990) rased the queston of deternng, for each k, a relaxaton paraeter (0, ) whch nzes the Eucldean nor of the k th power of the SOR teraton atrx, assocated wth a real syetrc postve defnte atrx wth property A. Hadjdos and Neuann (998), used the reducton of the SOR operator ntroduced by Golub and Plls (990), wth the help of the SVD of the assocated block acob teraton atrx to obtan the nzng relaxaton paraeter for the case k =. Yn and Yuan (00), used the SVD to re-derve the egenvalue functonal relatons for block skew syetrc atrces for the AOR ethod. Mlleo et al. (006), consdered systes wth block skew syetrc acob teraton atrx and used the SVD n studyng the behavor of the SOR operator fro the l -nor pont of vew they deterned theoretcally the nzng relaxaton paraeter of the l -nor. Youssef (0), defned the KSOR operator, we used the SVD n re-prove the functonal egenvalue relaton for the KSOR operator and the correspondng untary block atrx ( ). we eployed the sae arguent as n Yn and Yuan (00), also n Mlleo et al. (006) for systes wth block skew syetrc acob teraton atrx and used the SVD n studyng the behavor of the l -nor of the KSOR operator. We deterned theoretcally the nzng relaxaton paraeter of the l -nor for the KSOR operator. We confred our theoretcal results by a nuercal exaple. We wll contnue ths study n a subsequent work n whch we wll consder a generalzatons of the KSOR operator CONCLUSION We used the sae arguent defned by Golub and Plls (990), used by Yn and Yuan (00) also by Mlleo et al. (006), we proved that the KSOR teraton atrx KSOR s untary equvalent to a atrx ( ) havng only or atrces on the dagonal. We nze the l -nor of the KSOR operator for atrces whose acob teraton atrx s skew syetrc. By our results, the optal value of the relaxaton paraeter s: =. opt (p( )) 6. REFERENCES Buckley, A., 977. On the soluton of certan skew syetrc lnear systes. SIAM. Nuer. Anal., : DOI: 0.37/07035 Ehrlch, L.W., 97. Coupled haronc equatons, SOR and Chebyshev acceleraton. Math. Coput., 6: Golub, G.H. and. Plls, 990. owards an Effectve wo-paraeter SOR Method. In: Iteratve Methods for Large Lnear Systes, Kncad, D.R. (Eds.), Acadec Press, New York, ISBN-0: , pp: Hadjdos, A. and M. Neuann, 998. Eucldean nor nzaton of the SOR operators. SIAM. Matrx Anal. Appl., 9: 9-0. DOI: 0.37/S Maleev, A.A., 006. On the SOR ethod wth overlappng subsystes. Coput. Math. Math. Phys., 6: DOI: 0.3/S Mlleo, I.,.H. Yn and.y. Yuan, 006. Mnzaton of l -nors of SOR and MSOR operators. Coput. Appled Math., 9: 3-. DOI: 0.06/j.ca Woznck, I. Z., 00. On perforance of SOR ethod for solvng nonsyetrc lnear systes. Coput. Appled Math., 37: DOI: 0.06/S (00) Yn,.H. and.y. Yuan, 00. Note on statonary teratve ethods by SVD. Appled Math. Coput., 7: DOI: 0.06/S (0) Young, D.M., 003. Iteratve Soluton of Large Lnear Systes. st Edn., Dover Publcatons, Mneola, New York, ISBN-0: , pp: 570. Youssef, I.K., 0. On the successve over relaxaton ethod.. Math. Stat., 8: DOI: 0.38/jssp