A New Approach for Computing WZ Factorization

Transcription

1 vlble t ppl. ppl. Mth. ISSN: Vol. 7, Iue (December ), pp pplcto d ppled Mthemtc: Itertol ourl (M) New pproch for Computg WZ Fctorzto Efft Golpr-bo Deprtmet of Mthemtc he Uvert of Qom Qom, Ir g.rbo@qom.c.r eceved: prl 7, ; ccepted: November 9, btrct Ler tem re frequetl cetfc d egeerg computg. Vrou erl d prllel lgorthm hve bee troduced for ther oluto. h pper ee to compute the WZ d the ZW fctorzto of ogulr mtrx ug the rght vere of eted ubmtrce of. We troduce two ew mtrx fctorzto, the QZ d the QW fctorzto, d compute the fctorzto ug our propoed pproch. Keword: Mtrx fctorzto, Q fctorzto, LU fctorzto, WZ fctorzto, ZW fctorzto, QZ fctorzto, QW fctorzto MSC No.: D4, 5B, 53. Itroducto be m rel mtrx, d rght vere of (.e., = I). Here we preet geerl formulto for computg the mtrx fctorzto of, depedg o the choce of rght vere of ome of the ubmtrce of. Ug the geerl formulto; we preet ew method for computg the mtrx fctorzto uch WZ d ZW. We lo troduce two ew mtrx fctorzto, QZ d QW, d how how to compute the fctorzto ug our propoed pproch. 57

2 57 Efft Golpr-bo he emergece of prllel computg cued reercher to recoder m of the mot mportt d commo erl umercl lgorthm for ther uefule d vblt o prllel computer. rllel mplct elmto (IE) for the oluto of ler tem w troduced b Ev (993, 994). prllel pproch for olvg ler tem of equto provded b the WZ fctorzto. he bc de fctorzto of, clled the Qudrt Iterlocg Fctorzto (QIF) [Ev (998)]. hee QIF method eem to be potetll ttrctve ltertve to Gu elmto or Chole fctorzto for prllel computto. he propoed cheme eld the oluto of two elemet multeoul d re emetl utble for prllel mplemetto. WZ fctorzto preet effcet lgorthm for vg problem m feld [eo et l. (993), Bl (), ()]. he remder of our wor orgzed follow. I Secto, we preet geerl lgorthm for computg geerl fctorzto for mtrx ug the rght vere of the mtrx. I Secto 3, we troduce the WZ fctorzto. I Secto 4, we tud the correpodg method, relted to the WZ d ZW fctorzto d troduce two ew fctorzto the QZ d QW fctorzto. Fll, we coclude Secto 5.. he ght Ivere of Neted Submtrce Here, we preet recuro procedure for computg rght vere of mtrx. o do th, we gve expreo tht l the rght vere of mtrx to the rght vere of the ubmtrce of. Choog the ubmtrce led to the computto of vrou ew mtrx fctorzto. m (,..., m ), where be the th row of. I the equel, ule otherwe explctl tted, we ume tht m d h full r, the ug bc lgebrc m mm techque [o d Mtr (97)], we c fd mtrx uch tht vertble d o ( ). (.) umg tht B },..., m permutto of the umber,..,. m m { b,..., b deote ubmtrx of the mtrx B ( b b,..., m )..,..., } d { m Here, we preet tertve proce to etblh reltohp betwee >. Frt we recll reult from ler lgebr [Khzl ()]. d for emr.. B S. C D

3 M: Iter.., Vol. 7, Iue (December ) 573 he, N M L K S, where BN L B C D N CK D M C BD K, ) (, ) (. (.) heorem.. },..., { be uch tht the vere of the qure mtrx ext, d let be uch tht, (.3) where I. (.4) the mtrx be prttoed ), (. he,. vertble d ), ( ) (. (.5). he mtrce tf. (.6) roof: Sce K d ), (, (.7)

4 574 Efft Golpr-bo. (.8) Ug the formul (.) for the vere of mtrx, we deduce the reult gve the theorem. emr.. From the lgebrc defto of gve b I ( ). (.9) It follow tht dempotet mtrx. It oblque proecto mtrx ule,...,. I tht ce, we hve for (.) he followg properte of re el verfed: d ; for < ; d;, (.) N( ) N( )... N( ), (.) m where, N(B) deote the ull pce of B. heorem.. d t, for,..., m. he, we hve:. Sp,..., ) Sp (,..., ). (. Sp t,..., t ) Sp (,..., ). ( 3. for <. 4. t for >. 5. t,, otherwe (.3)

5 M: Iter.., Vol. 7, Iue (December ) 575 roof: : B the defto of d from (.9) we hve ( ) d. (.4) Hece, Sp(,..., ),. S be the mtrx whoe colum re,..., ; we hve S U. Coequetl Sp (,..., ) (,..., ) Sp. (.5) : he proof mlr to the precedg oe. 3, 4: rt 3 d 4 follow from (.). 5: rt 5 follow from prt 3, prt 4, d (.). Oe mportt reult of the heorem. the etblhmet of mtrx fctorzto S = F, where S (,..., m ). Now, we re red to preet lgorthm for computg the mtrx fctorzto. lgorthm : Geerl lgorthm: Mtrx Fctorzto. Iput: full row r mtrx,...,m. m, m ; d permutto,..., m of the umber () I, d =. () Chooe o tht (3) Compute, t d t. t (4). (5) = +, f m the go to (). (6) Compute the mtrx fctorzto S = F, where S,..., ). Stop. emr.3. ( m Dfferet choce of the permutto,..., m d the prmeter led to vrou mtrx fctorzto.,,,...,, d I,. he the Q fctorzto gve

6 576 Efft Golpr-bo b d the LU fctorzto gve b e, for,..., [ee Blll d Sdo (998)]. We wll how how to chooe the prmeter of the lgorthm for computg ome ew mtrx fctorzto. heorem.3. d I,. Coder permutto,..., o tht e for,..., where updte b e e (.6) he, the followg properte re true. () For, the th colum of re zero. (b) For >, the th row of re equl to the th row of. roof: () We prove, b ducto, tht for,.... For we hve,.e. the e e th colum of zero. Now, ume tht the theorem true up to <, d prove for. B the ducto hpothe, we hve e, for < d e, the e,,...,. herefore, for the th colum of re zero, provg (). (b) We hve e e (.7) Sce for, th colum of zero b propert () d b the updte formul (.7), the th row of re equl to the th row of, for >, provg ttemet (b). emr.4. o compute the vector we hve d t we do ot eed explctl. u,,...,, the

7 M: Iter.., Vol. 7, Iue (December ) 577 t u t, u t u. 3. WZ Fctorzto Implct mtrx elmto cheme for the oluto of ler tem were troduced b Ev (993) d Ev d Htzopoulo (979). hee cheme propoe the elmto of two mtrx elemet multeoul ( oppoed to gle elemet Gu Elmto) d emetl utble for prllel mplemetto [Ev d bdullh (994)]. Defto 3.. mtrx (, ) clled W-mtrx f, for ll (,) wth > d + > or wth < d +. he mtrx clled ut W-mtrx f ddto, for,..., d, for ( ) /. he trpoe of W- mtrx clled Z-mtrx. he, thee mtrce hve the followg form: W, Z (3.8) Defto 3.. We tht mtrx fctorzed the form WZ f WZ (3.9) where the mtrx W W-mtrx d Z Z-mtrx. o olve tem of ler equto, the WZ fctorzto plttg procedure propoed [Ev d Hddom (98)], coveet for prllel computg. detled l for th fctorzto gve [Ev d Hddom (98)]. he WZ fctorzto prllel method for olvg dee ler tem (.), where qure mtrx, d b - vector. he WZ fctorzto logou to the LU fctorzto d utble for prllel computer. chrcterzto for the extece of the WZ fctorzto preeted [o

8 578 Efft Golpr-bo (997)]. bcwrd error l for the WZ fctorzto gve [Shehch d Ev (98)]. pvotg trteg for modfed WZ fctorzto propoed [lmov d Ev (995)]. he mtrce W d Z hve two oppote zero qudrt. he, we refer to W d Z the terlocg qudrt fctor of. he ext theorem, gve chrcterzto for the extece of the WZ fctorzto of. heorem 3.. Fctorzto theorem be ogulr mtrx. h qudrt terlocg fctorzto QIF, =WZ f d ol f for ever,, where = f eve d = ff odd ( ( ) deote the gretet (let) teger le (bgger) th or equl to ), the ubmtrx,,,,,,,,,,,,,,,, (3.) of vertble. roof: See o (997). heorem 3.. If ogulr, the WZ fctorzto c lw be crred out wth pvotg. here ext row permutto mtrx d the fctor W d Z uch tht = WZ. roof: See o (997). heorem 3.3. he WZ fctorzto ext for the mmetrc potve defte or trctl dgoll domt mtrce. roof: See o (997).

9 M: Iter.., Vol. 7, Iue (December ) 579 Whe mmetrc potve defte mtrx, t poble to fctor the form LL for ome lower trgulr mtrx L. h ow Chole fctorzto. vrt of clcl Chole fctorzto, clled Chole QIF gve b Ev [(998)]. 4. Specl Ce of the Geerl lgorthm. It w how Blll d Sdo (998); wth prmeter choce,,..., d I,, the Q fctorzto v Grm-Schmth lgorthm of gve b d the mplct LU fctorzto of v Gu elmto techque gve e ug the geerl lgorthm. I the equel, we hll vetgte ome choce of the mtrx, d the permutto,..., for computg WZ, ZW, QZ d QW fctorzto ug the geerl lgorthm. We ume tht be eve umber. 4.. WZ Fctorzto be ogulr mtrx d the permutto,..., defed b:, f odd, f eve, (4.) f e deote the th colum of the dett mtrx, the the ecod mportt d e choce for the uxlr prmeter the geerl lgorthm the vector e. Frt, order to gurtee tht, we ume tht row wp re performed. Ideed, f ( e,..., e ) deote permutto mtrx, the uch tht (,..., ). I th ce, e, t d t e t e (4.) d the geerl lgorthm compute the fctorzto ΠS = F. Now, order to gurtee tht, we ume tht be ogulr, for,...,. heorem 4.. be eve umber, be defed b (4.) d he, there ext WZ fctorzto for, obted b the geerl lgorthm. be vertble, for =,,.

10 58 Efft Golpr-bo roof: e,,...,. he, ccordg to heorem.3, for,..., /, we hve, I (4.3), L wth, L, d, I, (4.4), L where d L. e. ccordg to heorem. we obt Z-mtrx S,..., ) wth dgol etre d W-mtrx t,..., t ) o tht ( m ( m S WZ (4.5) where Z S Z-mtrx d W = W-mtrx. 4.. ZW Fctorzto Now we compute the ZW fctorzto ug the geerl lgorthm. be ogulr mtrx, be eve umber d the permutto,..., defed b:

11 M: Iter.., Vol. 7, Iue (December ) 58, f odd. (4.6), f eve If e deote the th colum of the dett mtrx, the the ecod mportt d e choce for the uxlr prmeter the geerl lgorthm the vector e. Frt, order to gurtee tht, we ume tht be ogulr, for,...,. heorem 4.., be eve umber, defed b (4.6) d be vertble, for =,,. he, there ext ZW fctorzto for, obted b the geerl lgorthm. roof: e,,...,. he, ccordg to heorem.3, for,..., /, we hve I, I L, (4.7) wth,, L, d I, I L, (4.8) wth, d, L. e. ccordg to heorem. we obt W-mtrx S,..., ) wth dgol etre d Z-mtrx t,..., t ) o tht ( m ( m

12 58 Efft Golpr-bo S ZW, (4.9) where W S W-mtrx d Z = Z-mtrx QZ Fctorzto Defto 4... We tht fctorzed the form QZ f QZ, where the mtrx Q orthogol mtrx,.e., (983)] d Z Z-mtrx. Q Q QQ I, (4.3) [Golub d V Lo be ogulr mtrx. We how how to chooe the prmeter of the geerl lgorthm for computg the QW fctorzto for. Oe choce for the vector the geerl lgorthm. From heorem., th led to t. (4.3) Note tht, exct rthmetc. hu, the geerl lgorthm, b choog mmedtel. well defed d the ext reult obted heorem 4.3., be defed b (4.) d,,...,. he, S (,..., ) m orthogol, S W-mtrx d QZ fctorzto recogzed for. roof: B heorem., we hve for d for. (4.3) herefore, the et of vector {,..., m } tht orthogol. he mtrx (,..., m ) S uch

13 M: Iter.., Vol. 7, Iue (December ) 583 S S D dg,..., ) (4.33) ( d S W-mtrx. he, S F S F QZ QW Fctorzto Defto 4... We tht fctorzed the form QW f QW, (4.34) where the mtrx Q orthogol mtrx d W W-mtrx. heorem 4.4., defed b (4.6) d,,...,. he, S (,..., ) m orthogol, S Z-mtrx d QW fctorzto recogzed for. roof: he proof me the proof of heorem 4.3. Now, we hll how the w of mplemetg vodg the explct ue of the mtrce durg the computto. We c wrte the mtrx hu, I. (4.35). (4.36) he ew formul more tble umercll [bff d Spedcto (989)]. 5. Cocluo We computed the rght vere of mtrx ug the rght vere of ome ubmtrce of. Our cotructve pproch llow u to chooe the pecl ubmtrce d compute ome ew

14 584 Efft Golpr-bo fctorzto for. We preeted two ew fctorzto QW d QZ d how how our propoed pproch compute WZ, ZW, QW d QZ fctorzto of mtrx. he geerl lgorthm c be mplemeted for the WZ d the ZW fctorzto wth o more th 3 O( ) multplcto. he m torge for d L 3 t mot. he computtol 4 cot for computg the QZ d the QW fctorzto ug geerl lgorthm o more th 3 3 O( ) multplcto. EFEENCES bff,. d Spedcto, E. (989). BS roecto lgorthm: Mthemtcl echque for Ler d Noler Equto, Ell Horwood, Chcheter. eo,. Uldo, M. d Zpt, E.L. (993). rllel WZ fctorzto o Meh Multproceor,. Mcroproceg d Mcroprogrmmg, Vol. 38, No. 5, pp Blll, M. d Sdo, H. (998). New Iterpretto of elted Hug' Method, Ler lgebr d t pplcto, Bl, B. d Bl,. (). he complete fctorzto precodtoer ppled to the GMES(m) method for olvg Merov ch. roceedg of the Federted Coferece o Computer Scece d Iformto Stem, pp Bl, B. d Bl,. (). GU-ccelerted WZ Fctorzto wth the Ue of the CUBLS Lbrr, roceedg of the Federted Coferece o Computer Scece d Iformto Stem, pp Ev, D.. (993). Implct mtrx elmto cheme, It.. Computer Mth. Vol. 48, pp Ev, D.. d bdullh,. (994). he prllel mplct elmto (IE) method for the oluto of ler tem, rllel lgorthm d pplcto, Vol. 4, pp Ev, D.. d Htzopoulo, M. (979). prllel ler tem olver, Itertol ourl of Computer Mthemtc, Vol. 7, pp Ev, D.. (998). he Chole QIF lgorthm for olvg mmetrc ler tem, Iter.. Computer Mth. Vol. 7, pp Ev, D.. d Hddom,. (98). modfcto of the qudrt terlocg fctorzto prllel method, Iter.. Computer Mth. Vol.8, No., pp Fddeev, D. K. d Fddeev, V.N. (963). Computtol Method of Ler lgebr, Freem, S Frcco. Golub, G. d V Lo, C.F. (983). Mtrx Computto, oh Hop U.., Bltmore. o, C.. d Mtr, S. K. (97). Geerlzed Ivere of Mtrce d pplcto, Wle, New or. o, S.C. S. (997). Extece d uquee of WZ fctorzto, rllel Comput., Vol. 3, pp Shehch,. d Ev, D.. (98). Further l of the qudrt terlocg fctorzto (Q.I.F) method, Itertol ourl of Computer Mthemtc, Vol., pp lmov,. d Ev, D.. (995). he WZ fctorzto method, rllel Comput. Vol., pp. -.