MALAYSIAN JOURNAL OF MATHEMATICAL SCIENCES. Effat Golpar-Raboky Department of Mathematics, University of Qom, Iran

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1 Malaysa Joural of Mathematcal Sceces 8(): (04) MALAYSIAN JOURNAL OF MAHEMAICAL SCIENCES Joural homepage: ABS Algorthms for Iteger WZ Factorzato Effat Golpar-Raboky Departmet of Mathematcs, Uversty of Qom, Ira E-mal: ABSRAC Classes of teger ABS algorthms have bee troduced for solvg lear Dophate equatos. he algorthms are powerful methods for developg all matrx factorzatos. Here, we provde the codtos for the exstece of the teger WZ ad ZW factorzatos of a teger matrx. he, we preset algorthms based o the teger ABS algorthms for computg the teger WZ ad ZW factorzatos of a teger matrx as well as the teger Z XZ ad W XW factorzatos of a totally umodular symmetrc postves defte matrx. Keywords: ABS algorthm, Umodular matrx, Iteger factorzato, WZ factorzato, X factorzato.. INRODUCION Implct matrx elmato schemes for the soluto of lear systems were troduced by Evas(993) ad Evas ad Hatzopoulos (979). hese schemes propose the elmato of two matrx elemets smultaeously (as opposed to a sgle elemet Gaussa Elmato) ad s emetly sutable for parallel mplemetato (Evasad Abdullah (994)). ABS class of algorthms was costructed for the soluto of lear systems Ax = b utlzg some basc deas such as projecto ad rak oe update techques (Abaffy ad Broyde (984); Abaffy ad Spedcato (989)).he ABS class later exteded to solve optmzato problems (Abaffy ad Spedcato (989)) ad systems of lear Daphate equatos (see Esmael et al. (00); Khorramzadeh ad Mahdav-Amr (009); Khorramzadeh ad Mahdav-Amr (008)). A scaled verso of the lear

2 Effat Golpar-Raboky ABS class was descrbed Abaffy ad Spedcato (989). Revews of ABS methods ca be foud Spedcato et al. (003) ad Spedcato et al. (00). A basc ABS algorthm starts wth a osgular matrx H R (Spedcato s parameter), as a bass for the ull space correspodg to the empty coeffcet matrx (o equatos). Gve the Abaffa matrx H wth rows geeratg the ull space of the frst equatos, the ABS algorthm computes H + as a ull space geerator of the frst equatos. Cosder the followg lear system, Ax = b, x R, A R, b R () where rak(a) s arbtrary. Obvously, the system () s equvalet to the followg scaled system, V Ax = V b, () where V, the scale matrx, s a arbtrary osgular matrx. Let a be the th row of A. A talored scaled ABS algorthm as appled to A ca be descrbed as follows, where the output varable r gves the rak of A. Algorthm.he scaled ABS (SABS) algorthm. Step: Let H R be arbtrary ad osgular ad v R be a arbtrary ozero vector. Set = ad r = 0. Step: Compute s = H A v. Step3: If s = 0 the set H + = H o the frst rows). ad go to Step 5 (the th row s depedet Step4:{ s 0} compute p = H f, where f R (Broyde s parameter), s a arbtrary vector satsfyg s f 0 ad update H by H A vq H H = (3) + H, q H A v 70 Malaysa Joural of Mathematcal Sceces

3 where q ABS Algorthms for Iteger WZ Factorzato R (Abaffy s parameter) s a arbtrary vector satsfyg s q 0. Let r = r +. Step5: If = the Stop (colums of H + geerates the ull space of A) else defe v + R, a arbtrary vector learly depedet of v,, v, let = + ad go to Step. he matrces H are geeralzatos of projectos matrces ad have bee amed Abaffas sce the Frst Iteratoal Coferece o ABS Methods (Luyoyag (99)). hey probably frst appeared a book by Wedderbur (934). A mportat result of the ABS algorthms s the establshmet of a mplct matrx factorzato V AP = L, (4) Where L s a lower tragular matrx (see Abaffy ad Spedcato (989)). Choces of the parameters H, v, f ad q determe partcular methods wth the class. he basc ABS class s obtaed by takg v = e (Abaffy ad Spedcato (989)), the th ut vector R. All matrx factorzatos ca be produced by usg the scaled ABS algorthm wth proper deftos of the parameters (Galata (00)). From (Abaffy ad Spedcato (989)) we recall some propertes of the Basc ABS algorthms for LU factorzato. P. he mplct LU algorthm s defed by the followg choces, whch are well defed f A s regular (all leadg prcpal submatrces are osgular) H Hae H = H = H +, p H e. e H a = (5) I, Malaysa Joural of Mathematcal Sceces 7

4 Effat Golpar-Raboky P. Let H = I, the δ =e Ha satsfes δ = a,, det( A ), δ =,, >, (6) det( A ) where A, s the th leadg prcpal submatrx of A. P3. Let the codtos of P be satsfed. he, the followg propertes hold: (a) he frst rows of H+ are detcally zero. (b) he last colum of H+ s equal to the last colum of H. he block ABS algorthm, s due to Abaffy ad Galata (986) for the scaled ABS class, ad further developed several papers by Galata(00, 003, 004), s a block form of the ABS algorthm (Abaffy ad Spedcato (989)). Let A be full rak row ad,..., s be postve teger umbers so that s =. Assume that osgular matrx V s parttoed by V = V,..., V ] wherev R. he block scale ABS algorthm s as follows. [ s () Determe F R such that F H A V s osgular ad set P = H F () Update the Abaffa matrx H by H = H H A V ( Q H A V) Q H, (7) + whereq R s a arbtrary matrx so that Q H A V s osgular. he remader of our work s orgazed as follows. I Secto, we dscuss the teger ABS class of algorthms. I Secto 3, we preset a exstece codto for the teger WZ factorzato. he, we preset a algorthm for computg the teger WZ factorzato as well as the Z XZ factorzato of a totally umodular symmetrc postve defte matrx usg the block teger ABS algorthm. I Secto 4, we compute the teger ZW factorzato by approprately settg the parameters of the block teger 7 Malaysa Joural of Mathematcal Sceces

5 ABS Algorthms for Iteger WZ Factorzato ABS algorthm. We also compute the teger W XW factorzato of a totally umodular symmetrc postve defte matrx. A exstece codto for the teger ZW factorzato based o the teger ABS algorthm s gve. Secto 5 llustrates a example for computg the ZW factorzato. Cocludg remarks are gve Secto 6.. INEGER ABS ALGORIHM he teger ABS (IABS) class algorthms for lear Dophate equatos preseted by Esmael et al.(00) to compute the geeral teger soluto of lear Dophate equatos. Codtos for the exstece of a teger soluto ad determato of all teger solutos of a lear Dophate system are gve Esmael et al. (00). Frst we recall some results from umber theory ad the preset the IABS algorthm. Defto.. A R s a umodular matrx ff det( A ) =. If A s umodular, the A s also umodular. Defto..A matrx A s called totally umodular f each square submatrx of A has determat equal to 0, +, or -. I partcular, each etry of a totally umodular matrx s 0, +, or -. heorem..(fudametal theorem of the sgle lear Dophate equato). Let a,..., a ad b be teger umbers. he Dophate lear equato ax a x = b has a teger soluto f ad oly f gcd ( a,..., a ) b (f >, the there are a fte umber of teger solutos). Proof. See Pohst (993). he teger ABS algorthm (IABS) has the followg structure, wth gcd( ) u the greatest commo dvsor of a vector u. Malaysa Joural of Mathematcal Sceces 73

6 Effat Golpar-Raboky Algorthm.he teger ABS (IABS) algorthm. Step: Let H Z be arbtrary ad umodular matrx. Set = ad r = 0. Step: Compute s = H A v. Step3: If s = 0 the set H + = H ad go to Step 5 (the th row s depedet o the frst = rows). Step4:{ s 0} compute gcd( s )= δ ad p = H f, where vector satsfyg s f = δ ad update H by f Z s a arbtrary H H A vq H = H +, q H A v where q Z s a arbtrary vector satsfyg s q = δ. Let r = r +. Step5: If = the stop (colums of H + geerates the ull space of A) else let = + ad go to Step. Let V Z be a umodular matrx. he, the scaled teger ABS algorthm s computed by applyg Algorthm o V A wth A replacg a. heorem.. If all the prcpal submatrces of A are umodular, the teger LU algorthm s well defed. Proof. See Corollary 4. Zou ad Xa (005). Corollary..If A s totally umodular of full rak. he there exsts a row permutato matrx Π so that Π A = LU, where L ad U are teger lower ad upper tragular matrx respectvely. Corollary.. Every totally uomodular symmetrc postve defte matrx has a teger LU factorzato. Furthermore, a recet work we have show that a specal verso of our approach costructs the Smth ormal form of a teger matrx, beg utlzed solvg lear Dophate systems of equatos (Golpar-Raboky ad Mahdav-Amr (0)). v 74 Malaysa Joural of Mathematcal Sceces

7 ABS Algorthms for Iteger WZ Factorzato Next, we compute the teger WZ ad the teger WZ factorzatos of a o-sgular teger matrx as well as the W XW ad the Z XZ factorzatos of a totally umodular symmetrc postve defte matrx usg the teger ABS algorthms. 3. WZ FACORIZAION USING HE BLOCK SCALED ABS ALGORIHM he well kow LU factorzato s oe of the most commoly used algorthms to solve lear systems ad WZ factorzato offers a terestg varat of the factorzato. o solve a system of lear equatos, the WZ factorzato procedure proposed Evas (993a,b) s coveet for parallel computg. he WZ factorzato offers a parallel method for solvg dese lear systems, where A s a square matrx, ad b s a -vector. Defto 3.. Let s be a real umber, ad deote by s greatest (least) teger less (greater) tha or equal to s. ( s ), the Defto 3.. We say that a matrx A s factorzed a teger WZ (IWZ) form f A = WZ, (8) where the W-matrx ad the Z-matrx are teger matrces havg followg structures: W =, Z =, (9) wth the empty bullets stadg for zero ad the other bullets stadg for possble teger ozeros. Malaysa Joural of Mathematcal Sceces 75

8 Effat Golpar-Raboky Defto 3.3. We defe a X-matrx as follows: X =. (0) he followg theorems express the codtos for the exstece of a teger WZ factorzato of a umodular matrx (see Rao (997)). Later, we gve a ew set of codtos useful for our purposes. heorem 3.. (Factorzato heorem) Let A Z be umodular. he A has a teger WZ factorzato f ad oly f for every k, k =, s, wth s =, f s eve ad s =, f s odd, the submatrx a a a a a a a a k = a a a a a a a a of A s umodular.,, k, k+, k, k, k k, k+ k, k+, k+, k k+, k+ k+,,, k, k+, k k () Proof. See heorem Rao (997). heorem 3.. If A Z s totally umodular of full rak, the the teger WZ ad ZW factorzatos ca always be obtaed by pvotg. hat s, there exsts a row permutato matrx Π ad the factors W ad Z such that Π Proof. See heorem 3 Rao (997). A= WZ. () Corollary 3.. Every totally umodular symmetrc postve defte matrx has the teger WZ ad ZW factorzatos. 76 Malaysa Joural of Mathematcal Sceces

9 ABS Algorthms for Iteger WZ Factorzato Now, we preset a ew terpretato of heorem 3. based o the block IABS algorthm wth blocksze equal to two. he, we show how to compute the teger WZ factorzato usg IABS algorthm. heorem 3.3. Let A Z be umodular. If k, k =,..., be umodular the the block scaled IABS algorthm wth parameter choces H = I, V = [ v, v ] = [ e, e + ] ad Q = [ q, q ] = [ e, e + ] s well defed ad the mplct factorzato V AP wth p = H e, p + = H e +, =,..., ad V = [ V,..., V ] leads to a teger WZ factorzato. Proof. Let H = I ad H + defed by (7). he, accordg to property P3, we have H+ = K I L (3) wth V V V,, Z. K L Let P = [ p, p ] = H [ e, e + ], P = [ P,..., P ] = [,..., ]. he, the teger block ABS algorthm produce V AP = L, where L s a block lower tragular matrx. Now, we have ad V AP = L APV = V LV AP= V LV A= WZ, (4) Where P= ( PV ) s a teger Z-matrx wth s o dagoal ad 0 s o off dagoal ad W = V LV s a teger W-matrx. We observe that the frst rows ad the last rows of H + are equal zero ad we delete the rows. I dog ths we use of the matrx E obtag from I by deletg ts the frst rows ad the last rows. Here, we preset a algorthm for computg the teger WZ factorzato. Malaysa Joural of Mathematcal Sceces 77

10 Effat Golpar-Raboky Algorthm 3.he teger WZ factorzato. Step : Let H = I ad =. Step : Let A [ a, a + ], s = H A ad = P = [ p, p ] = H [ e, e ]. + + Step 3: Let Q = [ e, e + ] ad F = Q S. Costruct E from I by deletg ts the frst rows ad the last rows. Update H by Step 4: Let = +. If H + = E( H S ( F go to Step (). ) P ). Step 5: Compute AP = W, where P = [ p,..., p ]. Stop. heorem 3.4. Let A be totally tumodular symmetrc postve defte. he, there exsts a Z XZ factorzato for A, obtaed by the ABS algorthm. Proof. Cosder the assumptos of heorem 3.3 ad let V = P, for = =,, s. he, V AP= L A= V LP = Z XZ (5) where X s a X-matrx. 4. ZW FACORIZAION USING HE BLOCK SCALED ABS ALGORIHM Now, the teger ZW factorzato s preseted as a alteratve to the teger WZ factorzato. Defto 4.. We say that a matrx A s factorzed the form teger ZW f 78 Malaysa Joural of Mathematcal Sceces

11 ABS Algorthms for Iteger WZ Factorzato A = ZW, (6) where the matrces W s a teger W-matrx ad Z s a teger Z- matrx. heorem 4.. Let A Z be umodular. he matrx A has a teger ZW factorzato f ad oly f for every k, k =,..., s, wth s=, f s eve, ad s=, f s odd, the submatrx as k+, s k+ as k+, s+ k Λ k = (7) as + k, s k+ a s+ k, s+ k of A s umodular. Proof. See heorem Rao (997) replacg by Λ. Here, we compute the teger ZW factorzato usg the block teger ABS algorthm. heorem 4.. Let A Z be umodular. If Λk, k =,..., be umodular the the block IABS algorthm wth parameter choces H I, V = [ v, v ] = [ e, e ] ad Q q q e e + + wth = + + V = [, ] = [, ] s well defed ad the mplct factorzato AP p = H e + + factorzato., p = H e, =,..., ad V = [ V,..., V ] leads to a teger ZW + + Proof. Let H = I ad H + defed by (7). he, accordg to property P3, we have I K 0 H+ = (8) 0 L I Malaysa Joural of Mathematcal Sceces 79

12 wth Effat Golpar-Raboky, K, L Z. Let P = [ p, p ] = H [ e, e ], P = [ P,..., P ] V V V + = [,, ]. he, the teger block ABS algorthm produce V AP = L, where L s a lower tragular matrx. Now, we have + ad V AP L APV = V LV AP= V LV A = = ZW (9) where, P= ( PV ) s a teger W-matrx wth s o dagoal ad 0 s o off dagoal ad Z = V LV s a teger Z-matrx. We observe that the frst ( + ) th to ( + ) th rows of H + are equal zero ad we delete the rows. I dog ths we use of the matrx E obtag from I by deletg ( + ) th utl ( + ) th rows. Here, we preset a algorthm for computg the teger ZW factorzato. Algorthm 4. he teger ZW factorzato. Step : Let H = I ad =. Step : Let A a a = [, ], + + S = H A ad P = [ p, p ] = H [ e, e ] Step 3: Let Q = [ e, e ] ad F = Q S. Costruct E from + + deletg ( + ) th utl ( + ) th rows. Update H by I by H + = E( H S ( F Step 4: Let = +. If go to Step (). ) P ). 80 Malaysa Joural of Mathematcal Sceces

13 ABS Algorthms for Iteger WZ Factorzato Step 5: Compute AP = Z, where P = [ p,..., p ]. Stop. heorem 4.3. Let A be totally umodular symmetrc postve defte. he, there exsts a W XW factorzato for A, obtaed by the ABS algorthm. Proof. Cosder the assumptos of heorem 4. ad let V = P, for =,..., s. he, V AP= L A= V LP = W XW (0) where X s a X-matrx. For computg the teger WZ (ZW) factorzato by the Algorthm 3 (4), the kth step we eed to store the ( ) ozero elemets of submatrx of P, the ( + ) ozero elemets of submatrx of S ad 4 for F, used to update H. hus the storage requred s the storage of A, for, / 4 for F plus ( ) = / for the matrx P. = We observe that o computatos are requred for evaluatg P. I the evaluato of H + o more tha ( + )( ) multplcatos are requred for computg HA, sce ut submatrx I H, multplcatos ad 4 dvsos are requred for computg F, o more tha ( ) multplcatos ad ( ) dvsos are requred for computg F P, o more tha ( + )( ) multplcatos are requred for computg the ozero elemets of SF P follows by summg all terms wth o more tha. S he the computg cost 3 + O 3 ( ). 5. A NUMERICAL ILLUSRAION Here, we preset a umercal llustrato of the Algorthm 3 for computg a teger WZ factorzato. Example: Cosderg the followg matrx: Malaysa Joural of Mathematcal Sceces 8

14 Effat Golpar-Raboky A = Upo a applcato of Algorthms 3 for computg the teger WZ factorzato we have, P = whch s a Z-matrx ad W = AP = whch s a W-matrx. herefore, 8 Malaysa Joural of Mathematcal Sceces

15 ABS Algorthms for Iteger WZ Factorzato A = CONCLUSION We provded the codtos for the exstece of the teger WZ ad the teger ZW factorzatos of a umodular teger matrx. he, we preseted effcet algorthms computato ad storage for computg the teger WZ ad ZW factorzatos of a teger matrx ad the teger Z XZ ad W XW factorzatos of a totally umodular symmetrc postves defte matrx usg the teger ABS algorthm. ACKNOWLEDGEMENS he author thaks the Research Coucl of Uversty of Qom for ts support. REFERENCES Abaffy, J. ad Broyde, C. G. (984).A class of drect methods for lear equatos.numer.math.45: Abaffy,J. ad Galata, A. (986). Cojugate drecto methods for lear ad olear systems of algebrac equatos.colloq. Math. Soc. Jaos Bolya.Numercal Methods. Mskolc.50: Abaffy, J. ad Spedcato, E. (989). ABS Projecto Algorthms, Mathematcal echques for Lear ad Nolear Equatos.Chchester:Ells Horwood. Bodo, E. (00).O the block mplct LU algorthmforlear systems of equatos.mathematcal Notes. Mskolc.(): -9. Malaysa Joural of Mathematcal Sceces 83

16 Effat Golpar-Raboky Esmael.H., Mahdav-Amr, N. ad Spedcato,E. (00). Geeratog the teger ull space ad codtos for determato of a teger bass usg the ABS algorthms.bullet of the Iraa Mathematcal Socety.7: 8. Esmael, H., Mahdav-Amr, N. ad Spedcato, E. (00). A class of ABS algorthms for Dophate lear systems.numer.math.90:0 5. Evas, D. J. ad Hatzopoulos, M. (979). A parallel lear system solver., Iteratoal Joural of Computer Mathematcs.7: Evas, D. J. (993a).Implct matrx elmato schemes. It. J. Computer Math.48:9-37. Evas, D. J. ad Abdullah,R. (994).he parallel mplct elmato (PIE) method for the soluto of lear systems.parallel Algorthms ad Applcatos.4: Evas, D. J. (993b). Parallel mplct schemes for the soluto of lear systems. : G. Wter Althaus ad E. Spedcato (eds.). Algorthms for large Scale Lear Algebrac Systems. NAO ASI Seres, Seres C. Mathematcal ad Physcal Sceces. Kluwer. 508: Evas, D. J. (999).he Cholesky QIF algorthm for solvg symmetrc lear systems. Iteratoal Joural of Computer Mathematcs.7: Galata, A. (00). Rak reducto, factorzato ad cojugato. Lear ad Multlear Algebra.49: Galata, A. (003). Projecto methods for lear ad olear equatos. Dssertato submtted to the Hugara Academy of Sceces for the degree MADoktora. Uversty of Mskolc. Galata, A. (004). Projectors ad Projecto Methods.Kluwer. Golpar-Raboky, E. ad Mahdav-Amr, N. (0).Dophate quadratc equato ad Smth ormal form usg scaled exteded teger ABS algorthms. Joural of Optmzato heory ad Applcatos. 5(): Malaysa Joural of Mathematcal Sceces

17 ABS Algorthms for Iteger WZ Factorzato Khorramzadeh, M. ad Mahdav-Amr, N. (009). Iteger exteded ABS algorthms ad possble cotrol of termedate results for lear Dophate systems.4or.7: Khorramzadeh, K. ad Mahdav-Amr, N. (008).O solvg lear Dophate systems usg geeralzed Rosser s algorthm. Bullet of the Iraa Mathematcal Socety. 34(): 5. Pohst, M. (993).Computatoal Algebrac Number heory. Bosto: BrkhauserVerlag. Rao, S.C.S. (997). Exstece ad uqueess of WZ factorzato. Parallel Comput. 3:9 39. Spedcato, E., Bodo, E., Del Popolo, A. ad Mahdav-Amr, N. (003). ABS methods ad ABSPACK for lear systems ad optmzato.a revew.4or. : Spedcato, E, Bodo, E., Zuqua, X. ad Mahdav-Amr, N. (00). ABS methods for cotous ad teger lear equatos ad optmzato. CEJOR. 8: Spedcato, E., Xa, Z. ad. Zhag, L. (997). he mplct LX method of the ABS class.optmzato Methods ad Software. 8: Wedderbur, J. H. M. (934). Lectures o Matrces. Colloquum Publcatos. New York: Amerca Mathematcal Socety. Zou, M. ad Xa, Z. (005).ABS algorthms for Dophate lear equatos ad lear LP problems. J. Appl. Math. ad Computg. 7(-): Malaysa Joural of Mathematcal Sceces 85