THE PUBLISHING HOUSE PROCEEDINGS OF THE ROMANIAN ACADEMY, Series A, OF THE ROMANIAN ACADEMY Volume 9, Number 1/2008, pp

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1 THE PUBLISHING HOUSE PROCEEDINGS OF THE ROMNIN CDEMY, Seres, OF THE ROMNIN CDEMY Volue 9, Nuber /008, pp ON CIMMINO'S REFLECTION LGORITHM Consann POP Ovdus Unversy of Consana, Roana, E-al: Ganfranco Cno presened for he frs e hs faous reflecon algorh n 938. He proved ha, f he ran of he proble arx s greaer han one hen, for any nal approxaon he sequence generaed by hs algorh converges o a soluon of he noral equaon assocaed o a perurbed (dagonally scaled) leas-squares proble. Sared fro hs resul we consruc a frs exenson of Cno's ehod whch generaes sequences of approxaons of leas-squares soluons for general nconssen probles. Usng soe resuls of H. Keller (965) and D. Young (97), we show ha he se of l pons of hs exenson copleely characerzes he se of leassquares soluons. The second exenson of Cno's algorh s obaned by sarng fro a prevous resul of he auhor, concernng Jacob's sulaneous proecons algorh. In hs sense, we prove ha a parcular case of Cno's ehod can be consdered as a parcular case of Jacob's ehod and ha s l pons also copleely characerze he leas-squares soluons of he nal proble. Key words: Cno s reflecon algorh, Jacob s proecons algorh, nconssen leas-squares probles. THE ORIGINL CIMMINO S LGORITHM One year afer S. Kaczarz presened for he frs e hs faous successve proecons algorh n [6], G. Cno proposed n [3] hs sulaneous reflecons ehod. Ths uses nsead of he orhogonal proecons P on he hyperplanes generaed by he syse equaons, orhogonal reflecons S, gven by x, a b P = x a, () a x, a b S ( x) = x a, () a (we denoed by a, b he -h row of he n syse arx and he -h coponen of he rgh hand sde b R and by,,, he Eucldean scalar produc and he assocaed nor on soe space R q ). Cno's reflecons S are sulaneously appled o an approxaon x R and, a convex cobnaon of he defnes he nex one x by x = ωs ( x ) = x + ω ( S( x ) x ) = ω ω (3) x, a b x ω a, ω a where

2 Consann POP ω > 0, ω = ω. (4) In [3] Cno proved he followng an convergence resul concernng he algorh (3)-(4). Theore. Le and b be such ha and he syse a 0, =,,, ran( ), (5) x = b. (6) s conssen. Then, f he weghs ω are as n (4), for any x 0 R n he sequence (x ) 0 generaed by (3) converges o a soluon of (6). Rear. The fac ha he l pons of Cno's sequence (x) 0 fro (4) (wh respec o he nal approxaon x0) "cover" he se S(; b) of all soluons of (6) wll be proved n he nex secon. The "unpleasan" aspec of he above Cno's algorh s ha he sequence (x) 0 approxaes a soluon x only n he conssen case for (6). Bu unforunaely, "real world probles" usually gve rse o nconssen syses of he for (6), (see e.g. [5], [8], [9]), whch us be reforulaed as "lnear leassquares probles": fnd x Rn such ha n x b = n{ z b, z R } (7) In hs general case, Cno's algorh or soe of s exensons sll converge, bu o soluons of "weghed" forulaons of (7) (see e.g. [], []). Ths s why we decded o analyse n hs paper soe possbles o "exend" or "adap" Cno's orgnal ehod (3)-(4) o he ore general proble (7), such ha he sequence of approxaons generaed n hs way, sll converges o one of s soluons (slar wh he resuls obaned by one of he auhors n [0], bu wh respec o Kaczarz-le proecons algorhs). Moreover, we were also neresed n he possbly of characerzng wh hese l pons, he se of all soluons of (7), denoed by LSS(;b) (see also [] and Rear before). These versons of Cno's algorh wll be descrbed n he nex wo secons of he paper.. THE FIRST EXTENSION For he consrucon of our frs verson of he algorh (3), we sared fro a rear of G. Cno, ade n he orgnal paper [3]. Ths resul (based on soe seps fro he proof of Theore ) can be brefly descrbed as follows. Corollary. If he syse (6) s no conssen and verfes (5), for any x 0 R n, he sequence (x ) 0 generaed by (3) converges o a soluon x of he noral equaon where ( ) = ( ) x b, (8) (,,, ) = ω ω ω R, (9) = D and D s he dagonal arx gven by ; b = D b (0) ω ω D dag a a = (,, ) ()

3 3 On Cno s reflecon algorh Rear. If he syse (6) s conssen, he above Corollary does no conradc he resul n Theore. Indeed, n hs case and because D s nverble (ω > 0), he (conssen) syse (6) s equvalen wh he (conssen) syse D x= D b, () whch s equvalen wh he noral equaon (8). Sarng fro he above Corollary, we oban he followng verson of Cno's algorh (3) convergen o soluons of he general leas-squares forulaon (7). Proposon. Le be as n (5), x 0 R n an arbrary nal approxaon and he weghs ω fro (4) gven by a, ω =, =,. (3) Then, he sequence (x ) 0 generaed by (3)-(4) converges o an eleen fro LSS(;b). Proof For ω as n (3) he arx D fro () s he deny, hus he noral equaon (8) s dencal wh x = b, (4).e. he noral equaon assocaed o he proble (7). Then Corollary apples and he proof s coplee. In he res of hs secon we shall prove ha he se of all l pons of he algorh (3)-(4) wh he weghs choce (3) (whch we shall denoe by LPC(;b)) concdes wh LSS(;b). For hs we shall brefly replay soe consrucons and resuls for papers [7] and []. Le B and N be n n real arces (wh N nverble) and d R n such ha he syse Bx = d (5) s conssen. For approxang s soluons, we consder he erave process: x 0 R n, wh x Tx N d = +, 0, (6) T I N B =. (7) The followng resul s proved n [7]. Theore. Le us suppose ha he arx B s syerc and nonnegave defne and consder he followng splng of B = D+ M (8) wh D syerc and nverble. Le E be anoher nverble n n arx and P E defned by Le also N be gven by P E D E D ( ) ( ) E = + B. (9) N NE E D = =. (0) Then, for any x 0 R n he ehod (6)-(7) s convergen o an eleen fro S(B;d) f and only f he arx P E s posve defne. Le now L(N;d) be he se of all l pons of he eraon (6)-(7) (wh respec o x 0 R n ). In he paper [], he followng resul s proved. Proposon. If he ehod (6)-(7) s convergen, hen he followng equaly holds LNd ( ; ) = SBd ( ; ). () We are now able o prove he prevously announced resul. Proposon 3. In he hypohess of Proposon he followng equaly holds

4 Consann POP 4 LPC( ; b) = LSS( ; b). () Proof. Le he n n arx B and d R n be defned by We hen have (see e.g. []) B =, d= b. (3) LSS( ; b) = S( B; d) (4) Moreover, Cno's algorh (3)-(4), wh ω gven by (3) becoes an algorh fro he class (6)-(7) f we defne he arces D and E, n he above Theore by wh ω gven by Then (see (0)) D = I, E = I, (5) ω ω= a (6) N and he above Proposon, (4) and (7) ell us ha = ω I (7) LPC( ; b) = L( N; d) = LSS( ; b) (8) f and only f he syerc arx P=P E, assocaed o he above Cno's algorh and gven by s posve defne or equvalenly Bu, because B s syerc and nonnegave defne and for ω fro (6) we always have P = ωi B (9) E ρ( B) ω>. (30) race( B) = a, (3) ρ( B) ω. (3) If we would have equaly n (3) hen, fro he properes of he arx B and (3) would resul ha B would have only one nonzero egenvalue, wh (algebrac) ulplcy equal o. Then, ran()= whch would conradc (5). Then (30) holds and he proof s coplee. 3. THE SECOND EXTENSION We shall sar he presenaon of hs secon by observng ha he classcal Cno s algorh (3) concdes, n he parcular case wh Jacob's sulaneous proecons ehod (see [4], [0]) ω=γ> 0, =,, (3)

5 5 On Cno s reflecon algorh Indeed, fro (3)-(4) and (7) resuls ha whch concdes wh (8) for x, a b =. (33) x x a a x, a b =, (34) x x a a The convergence condon for (8) (see agan [4], [0]) s wh E = 0 =. (35) 0 <<, (36) ( ) = aa. (37) a Proposon 4. If he assupons (5) hold for, hen he nuber 0 fro (35) sasfes (36). Proof. Because of he syery of E, we successvely ge = E aa = a a = a a, whch ogeher wh (35) gves us 0. (38) If he equaly would hold n (38), we would have (see (35)) =. (39) Bu, because he arx E fro (37) s nonnegave defne and syerc, we now ha σ(e) [ 0, ), whch eans ρ(e) σ(e). Le x R n \{0} be a correspondng egenvecor. We hen successvely oban Fro (40) and he Cauchy-Schwarz nequaly xa, x a =, = = a a. (40) x Exx x x, a x a,,,, (4) we oban he equales n (4), whch eans ha x s collnear wh a,,,. Bu, hs would ean ha ran( ), (4) whch would conradc (5). Thus, src nequaly holds n (38) and he proof s coplee. Le now ϕ, Φ(α; ) be defned by (see [0])

6 Consann POP 6 y, α b ϕ ( y) = α,,, = n, (43) α n Φα ( ; y) = y α ϕ( y), y R, (44) = where α 0 s he -h colun of and α R, α 0. Le D be he n n he arx defned by D = α α We hen consder he followng Cno Exended (CE, for shor) algorh. n. (45) = α LGORITHM CE. Le y 0 = b, x 0 R n and x an already copued approxaon. The nex one, x s gven by y + =Φ( α ; y ), (46) b = b y, (47) x, a b =, (48) x x a a where by b we denoed he -h coponen of he vecor b fro (47). Then, he followng resul holds (as n Theore 6.7 fro [0]). Theore 3. For any arx sasfyng (5) and α 0, =,, n, any vecor b R, any nal approxaon x 0 R n and any α such ha α 0, ρ( D) he sequence (x ) 0 generaed wh he algorh (46)-(48) converges o an eleen x LSS(;b). Conversely, any eleen x LSS(;b) can be obaned as he l pon of such a sequence, for an approprae choce of x 0. Moreover, for x 0 n he range of, he sequence (x ) 0 converges o he nal nor soluon of he proble (7). REFERENCES. BJÖRK,., Nuercal ehods for leas squares probles, SIM Phladelpha, CENSOR, Y., ELFVING, T., Bloc-erave algorhs wh dagonally scaled oblque proecons for he lnear feasbly proble, SIM Marx nal. and ppl., 4(00), CIMMINO, G., Calcolo approssoao per le soluzon de sse d equazon lnear, Rc. Sc. progr. ecn. econo. naz.,, pp , ELFVING, T., Bloc-Ierave Mehods for Conssen and Inconssen Lnear Equaons, Nuer. Mah., 35, pp. -, GROETSCH, C.W., Inverse probles n he aheacal scences, Veweg, Wesbaden, Gerany, KCZMRZ, S., ngenahere uflosung von Syseen lneare Glechungen, Bull. cad. Polonase Sc. e Leres.,, pp , KELLER, H., On he soluon of sngular and sedefne lnear syses by eraon, SIM J. Nuer. nal., Ser. B, (), pp. 8-90, MOHR, M., POP, C., RUEDE, U., n experenal analyss of a dfferenal nverse proble, Techncal Repor 99-, IMMD0, FU Erlangen-Nurnberg, Gerany, MOHR, M., Coparson of solvers for a boelecrc feld proble, Techncal Repor 00-, IMMD0, FU Erlangen-Nurnberg, Gerany, 00.

7 7 On Cno s reflecon algorh 0. POP, C., Exensons of bloc-proecons ehods wh relaxaon paraeers o nconssen and ran-defcen leas-squares probles, BIT, 38(), pp. 5-76, POP, C., Characerzaon of he soluons se of nconssen leas-squares probles by an exended Kaczarz algorh, Korean Journal on Cop. and ppl. Mah., 6(), pp. 5-64, YOUNG, D.M., On he conssency of lnear saonary erave ehods, SIM J. Nuer. nal., 9(), pp , 97. Receved Deceber 8, 007