Internatonal Mathematcal Forum, Vol 7, 2012, no 52, 2549-2554 An Applcaton of the Dulmage-Mendelsohn Decomposton to Sparse Null Space Bases of Full Row Rank Matrces Mostafa Khorramzadeh Department of Mathematcal Scences Shraz Unversty of Technology Shraz 71555-313, Iran mkhorram@sutechacr Abstract In ths paper we present an effcent algorthm for computng a sparse null space bass for a full row rank matrx We frst apply the deas of the Markowtz s pvot selecton crteron to a rank reducng algorthm to propose an effcent algorthm for computng sparse null space bases of full row rank matrces We then descrbe how we can use the Dulmage-Mendelsohn decomposton to make the resultng algorthm more effcent 1 Introducton Let A =(a 1,,a m ) T R m n, m<n The set of all x R n, satsfyng a T t x =0,1 t m, s called the null space of A If every vector n the null space of A can be wrtten as a lnear combnatons of the columns of N, then N s called a null space generator of A If the columns of N, a generator of A, are lnearly ndependent, then N s called a null space bass of A The sparse null space bass problem (SNBP) s to fnd a bass wth fewest nonzeros for the null space of a sparse matrx A SNBPs appear n varous branches of mathematcs, engneerng and computer scence, and ts effectve soluton s a key element for the success of varous algorthms such as constraned nonlnear programmng algorthms [1], some specal nteror pont algorthms for optmzaton problems [10], the dual varable method for solvng the Naver-Stokes equatons [8] and force methods for structural optmzaton [9] To explan our deas, at frst we need to brefly descrbe a rank reducng algorthm for computng a null space bass and the Markowtz s pvot selecton crteron 11 Rank reducng algorthm Assume that A =(a 1,,a m ) T R m n, m<n, has full row rank By defnton N(a T 1 )={y R n : a T 1 y =0}
2550 M Khorramzadeh and the orthogonal complement of N(a T 1 )sr(a 1)={αa 1 : α R} Therefore, f G 1 R n 1 n satsfes G 1 y =0 y R(a 1 ) for every y R n, then, G 1 s a bass for N(a T 1 ) For 1 m, let A = (a 1,,a ) and s = M a, where M T R n (n +1) s a bass for N(A T 1 ) Moreover, let G R (n ) (n +1) satsfes G y =0 y R(s ), for every y R n +1, and M +1 = G M R (n ) n In the followng we show that M+1 T s a bass for N(AT ) Snce A has full row rank, we have dm(n(a T )) = n So t s suffcent to show that the columns of M+1 T generate N(A T ) Indeed, let x N(AT ), then snce x N(AT 1 ) there exsts some z R n +1 so that x = M T z Moreover, we have 0=a T x = at M T z = st z Snce R n+ 1 = R(G T ) N(G ), we can wrte z = z R + z N, where, z R R(G T ) and z N N(G ) In the followng we wll show that z N = 0 By defnton there exsts some y R R n so that z R = G T y RIfz N 0 then snce G z N = 0, there exsts some α 0 so that z N = αs Moreover, we have s T z = αs T s +s T G T y R Snce A has full row rank we have s 0 By defnton of G we have G s =0 Therefore, s T z = α s 2 0 Ths contradcton shows that z N = 0 and hence z = z R = G T y R Ths shows that x = M T z = M T GT z R = M T +1 z R and therefore, M T +1 s a bass for N(A T ) The above consderatons suggests the followng rank reducng algorthm for computng a null space bass of A Algorthm 1 Rank reducng algorthm for computng null space bass Step 1: Let M 1 R n n be the dentty matrx n R n n Set =1 Step 2: Compute s = M a Step 3: Compute M +1 = G M, where G R (n ) (n +1) s such that we have G y =0f and only f y = αs, for some α R Step 5: If = m then stop (Mm+1 T s the null space bass for A) else let = +1 and go to Step 2 12 Markowtz s pvot selecton crteron Here, we descrbe Markowtz s pvot selecton crteron [11] Let r t and c j be the number of nonzero elements n row t and column j of the remanng (n ) (n ) matrx after an applcaton of teratons of the Gaussan elmnaton to the matrx A The Markowtz pvot selecton crteron s a local greedy strategy that selects from the remanng submatrx a nonzero element a tj that corresponds to mnmum Markowtz count, (rt 1)(c j 1) In practce the mnmum s taken over all entres satsfyng the nequalty, (1) a tj u max{ a tl, l },
or the nequalty, (2) Applcaton of the Dulmage-Mendelsohn decomposton 2551 a t tj u max{ a lj, l }, where u, 0<u 1, s a constant The element a tj s forced to satsfy (1) or (2) to nsure the numercal stablty of the algorthm as well Ths s done to mnmze the number of fll ns n the next teraton [5, 6] Here, we frst apply a smlar crteron for the selecton of the G n rank reducng algorthm to preserve sparsty Moreover, we consder the matrx à = (M a +1,,M a m ) T as the remanng matrx of the th teraton and then apply the Markowtz s pvot selecton crteron along wth nequalty (1) or (2) to à to choose G effectvely The resultng algorthm preserves sparsty and snce n every teraton we only need to compute a sparse matrx vector and a sparse matrx matrx product, the resultng algorthm generate the sparse null bass effectvely Then, we use the Delmuge Mendelsohn decomposton to make our proposed algorthm more effcent Indeed, we frst apply ths decomposton to the full row rank matrx A to obtan a sparse submatrx of A whose null space bass completely determne a sparse null space bass of A Then, we apply the sparse rank reducng algorthm to the resultng submatrx Fnally, we examne the numercal performance of our proposed algorthm and justfy ts effcency In secton 2, we descrbe the applcaton of the Markowtz pvot selecton crteron to the rank reducng algorthm to preserve sparsty In secton 3, we explan how the Dulmege Mendelsohn decomposton can be used to make our proposed algorthm more effcent In secton 4, we consder the numercal performance of our proposed algorthm and justfy ts effcency 2 Sparse rank reducng algorthm Here, we propose an effectve algorthm for the SNBP We ntend to compute a sparse null space bass for the matrx A R m n In the begnnng of the algorthm, we let M 1 be the dentty matrx n R n n Suppose that we are at the th teraton of the rank reducng algorthm and let h T k,1 k n +1, be the kth row of M, A m =(a,,a m ) T and à R (m +1) (n +1) be gven by a T M T à = A m M T = = ( ) A m h 1 A m h n +1 a T m M T Smlar to the Markowtz s pvot selecton crteron, one may thnk of the selecton of the parameters of the rank reducng algorthm, correspondng to a mnmal product ( r t 1)( c j 1), over all entres satsfyng the nequalty, (3) ã tj u max{ ã tl, l },
2552 M Khorramzadeh where u, 0<u 1, s a constant, r t and c j are the number of nonzero elements n row t and column j of Ã, respectvely, and ã tj denotes the element n the tth row and jth column of à Moreover, let ã t j be the element, correspondng to a mnmal product ( r t 1)( c j 1), over all entres satsfyng the nequalty (3), and then at the th teraton of the rank reducng algorthm, let s = M a t and set 1 s 1 /sj G = 1 s j 1 /s j s j +1 /s j 1 s n +1 /s j 1 Note that the matrx G s an (n ) (n + 1) matrx, obtaned by adjonng the vector s =( s1 /sj,, sj 1 /s j, sj +1 /s j,, sn +1 /s j (4) )T, as a new kth column of the dentty matrx n R (n ) (n ) By performng smple algebrac multplcatons, we can verfy that for every y, G y =0f and only f y = αs, for some scalar α R Snce the computaton of the matrx à n every teraton of the algorthm s costly and tme consumng, we choose t and j so that r t, c j and hence the product ( r t 1)( c j 1) are expected to be small Snce the tth row of à s a T t H T and r t s the number of nonzeros of a T t HT, we let t be the ndex of the row of A that, among all rows of A not consdered so far, has the mnmum number of nonzeros and compute s = M a t Smlarly, snce the jth column of à s A m h j, we let j be the ndex of the column of à that, among all nonzero elements of s satsfyng (3), corresponds to the row of M wth a mnmal number of nonzeros Consderng the above argument, to determne the matrx G, we frst let a t be the row of A that, among all rows of A not consdered so far, has a mnmal number of nonzeros Then, we compute the vector s = M a t and let j, among all nonzero elements of s that satsfy (3), correspond to the row of M wth a mnmal number of nonzeros Fnally, we let G be the adjoned dentty matrx n R (n ) (n ), wth s as n (4) added as a new j th column Our proposed algorthm s expected to generate a sparse null space bass The resultng sparse rank reducng algorthm follows next Algorthm 2 SRRP (Sparse Rank Reducng algorthm) Step 1: Let a T t 1,,a T t m be the rows of A n ascendng order wth respect to ther number of nonzero elements Let M 1 be the dentty matrx n R n n and u [0, 1] Set =1 Step 2: Compute s = M a t
Applcaton of the Dulmage-Mendelsohn decomposton 2553 Step 3: Let s j be a nonzero element of s, whch, among all nonzero elements of s, satsfyng s j (5) u max{ sk, 1 k n +1}, corresponds to the row of M wth a mnmal number of nonzeros (select the frst one f there are multple mnma) Step 5: Let s =(s 1,,s j 1,s j G =,sj +1 1 s 1 /sj 1 s j 1 s j +1,,s n +1 ), s j 0, s n +1 /s j /s j 1 /s j 1 and compute M +1 = G M Step 6: If = m then stop (M m+1 s a bass for the null space of A) else let = +1 and go to Step 2 3 Makng the algorthm more effcent In ths secton we explan how we can mprove the effcency of SRRP usng Delmuge Mendelsohn decomposton Suppose that we can permute the columns and rows of A, so that the resultng permuted matrx has n the followng form: ( ) B A 0r n r = C H where, B R r r, C R m r r, H R m r n r and 0 r n r s the zero matrx n R r n r It can be ( easly) verfed that rank(n(h)) = n r Moreover, 0r 1 f y N(H), then N(A y ) Therefore, we can obtan a bass for N(A )(= N(A)) by computng a bass for N(H) The above consderaton can be utlzed to mprove the effcency of SRRP Indeed, we can apply the SRRP to H nstead of A IfA has full row rank, then the Delmuge Mendelsohn decomposton [2, 3, 4] provde us wth a permutaton of the columns and rows of A so that the resultng permuted matrx has the form of A Therefore, to mprove the effcency of SRRP we frst compute the Delmuge Mendelsohn decomposton and then apply the SRRP on a submatrx of A and compute the sparse null space of A as descrbed above
2554 M Khorramzadeh 4 Concludng remarks In ths paper we descrbed an effcent algorthm for sparse null space bass problem and then explaned how we can mprove the effcency of our proposed algorthm by usng the Dulmage-Mendelsohn decomposton Acknowledgment: The author thank the Research Councl of Shraz Unversty of Technology for ts support References [1] TS Coleman and JJ More, Estmaton of sparse Jacoban matrces and graph colorng problem SIAM J Num Anal 20, 187 209 (1983) [2] A L Dulmage and N S Mendelsohn, Covernfgs of bpartte graphs Canadan Journal of Mathematcs 10, 517 534 (1958) [3] A L Dulmage and N S Mendelsohn, A structure theory of bpartte graphs of fnte exteror dmenson Transactons of the royal socety of canada secton III 53, 1 13 (1959) [4] A L Dulmage and N S Mendelsohn, Two algorthms of bpartte graphs Journal of the socety of ndustral and appled mathematcs 11, 183 194 (1963) [5] T A Davs, Algorthm 832: UMFPACK, an unsymmetrc-pattern multfrontal method ACM Trans Math Softw 30(2), 196 199 (2004) [6] TA Davs, Drect methods for sparse lnear systems SIAM, Phladelpha, (2006) [7] JR Glbert and MT Heath, Computng a sparse bass for the null space SIAM Journal on Algebrac and Dscrete Methods 8(3), 446 459 (1987) [8] C Hall, Numercal soluton of Naver-Stokes problems by the dual varable method SIAM Journal on Algebrac and Dscrete Methods 6, 220 236 (1985) [9] MT Heath, RJ Plemmons and RC Ward, Sparse orthogonal schemes for structural optmzaton usng the force method SIAM J Sc Sta Comput 5, 514 532 (1984) [10] K Km and JL Nazareth, A prmal null-space affne-scalng method ACM Transactons on Mathematcal Software 20, 373-392 (1994) [11] H M Markowtz, The elmnaton form of the nverse and ts applcaton to lnear programmng Management Sc 3, 255-269 (1957) Receved: May, 2012