QUARTERLY OF APPLIED MATHEMATICS

Transcription

1 QUARTERLY OF APPLIED MATHEMATICS Voume XLI October 983 Number 3 DIAKOPTICS OR TEARING-A MATHEMATICAL APPROACH* By P. W. AITCHISON Unversty of Mantoba Abstract. The method of dakoptcs or tearng was ntroduced by G. Kron n order to reduce computatons n the souton of certan probems arsng from arge nter-connected power dstrbuton networks. Here the method s gven a purey mathematca form whch can be used to sove arge systems of near equatons by frst sovng some smaer sub-probems and then combnng these soutons nto a compete souton. The sub-probems are formed from sets of equatons and varabes whch are strongy connected, wthn the sub-probem, but ony weaky connected to those of another sub-probem.. Introducton. The ate Gabre Kron [7, 8] ntroduced the method of dakoptcs n the 95s, and snce then t seems to have remaned cosey assocated wth eectrca power system probems (Happ [4, 5]), and other specased appcatons such as substructurng probems n cv engneerng (Przemeneck [9]). A number of authors have gven aternatve vewponts, such as Brameer [], Brann [3], Kesavan [6], A rather compcated mathematca generasaton was gven by Roth [], whe computer agorthms and mathematca anayss was gven by Steward []. On the whoe, wth the excepton of Steward's paper, dakoptcs has remaned as a compcated mxture of eectrca and mathematca concepts whch are dffcut to foow and to use for both the engneer and mathematcan. The present paper gves a purey mathematca anayss of the souton of a sparse system of a near equaton, whch paraes the dakoptcs approach of Kron. The mathematca method has the advantages of beng a numerca technque appcabe to any near system, and does not rey on a knowedge of eectrca networks. It s qute dfferent from the approach gven n the appendx to Steward's paper [], but t resuts n the same formuae as those gven by Brameer [], Secton of ths paper gves the basc agorthms and formuae, whe Sec. 3 gves a numerca ustraton and Sec. 4 contans the background theory for the method.. Mathematca formuae and agorthms. Suppose Az = c s a matrx form of a system of near equatons, where A s the order m X m coeffcent matrx, z s the order m X coumn vector of varabes, and c s the order m X coumn vector of constants. The * Receved December, Brown Unversty

2 66 P. W. AITCHISON souton procedure s gven n parts abeed (a) to (e). (a) If A s sparse, then t s possbe to rearrange the equatons and varabes so that the new coeffcent matrx, M, has most non-zero entres n square dagona bocks, as n: A = X X The agorthms for accompshng ths are not ncuded here, but are dscussed by Steward []. In the usua network-based dakoptcs approach ths dagona bock form s derved mpcty durng the anayss of the network and the resutng formuaton of the probem. From now on t w be assumed that A has the form () and that the matrx s very sparse outsde of the dagona bocks. In eectrca power dstrbuton network each dagona bock can correspond to the equatons derved from an amost sef-contaned dstrbuton network, whe the few other non-zero entres correspond to the nterconnectons between those networks. Steward [] gves a more genera verson of ths nvovng a bock-tranguar form. (b) Convert the system of equatons nto a new arger system wth coeffcent matrx, B, whch s soey of square dagona bock form. Ths can be done arbtrary, as ong as the foowng rues are observed. Each non-zero entry n A s aso an entry n B, or s a sum of entres n,. B, and a entres n B are of ths type. If two entres n A are n dfferent rows (coumns), then the correspondng entres, or parts of entres, n B must be n dfferent rows (coumns).,. [So each row (coumn) of B contans entres, or parts of entres, from just one row (coumn) of A.] The best way of satsfyng these rues s usuay to eave the orgna dagona bocks unchanged, and to move a of the other non-zero entres nto one or more new square dagona bocks (ths s ustrated n Sec. 3). Hence f there are r rows and r coumns of A whch have non-zero entres not n the dagona bocks, then the new dagona bocks must contan a tota of r rows and r coumns. If there are r rows and s coumns wth, for exampe, r > s, then n order to make the new dagona bocks square, r s of these entres can be spt as a sum of two parts, both n the same row, wth the extra parts fng the empty coumns of the bock n B. A smar stratagem can be used f r < s, and s ustrated n Sec. 3. The matrces A and B, wth orders m X m and n X n, are reated by an equaton () A = P'BQ', (4) where P' and Q' have orders m X n and n X m, and are constructed as foows. The fth

3 DIAKOPTICS ON TEARING 67 row of P' contans a n coumn j f row j n B contans an entry or parts of entres, from row n A. A other entres are zero. Notce that each coumn of P' contans a snge non-zero entry,, and so t s of fu row rank. Smary they'th coumn of Q' contans a n row f coumn n B contans entres, or parts of entres, from coumn j n A. A other entres are zero. Notce that each row of Q' contans a snge non-zero entry,, and so t s of fu coumn rank. (c) From the order m X n matrx P' construct the order n X m matrx P as the matrx of zeros except for a snge entry,, n each coumn. Specfcay, occurs n coumn j, row, where n P' the frst n row j occurs n coumn In other words P s the transpose of the matrx P' wth ony the frst non-zero entry retaned n each row of P'. Readers famar wth generazed nverse matrces (see Ben-Israe []) w recognse P' and P as generazed nverses of each other, satsfyng P'P = I. (5) Smary, defne the m X n matrx Q as the transpose of the matrx formed from Q' by retanng ony the frst non-zero entry n each coumn of Q'. Q' and Q satsfy QQ' = I. (6) (d) From P, P', Q, and Q' defne the order n X (n m) and («m) X n matrces K and L by, K I PP', L = (I Q'Q), (7) where / PP' denotes the matrx of the n m non-zero coumns n / PP', and / Q'Q\ht matrx of n m non-zero rows of I Q'Q. The matrces K and L both have rank n m. They may be constructed drecty by the foowng agorthm. Emnate the frst n each row of P', and change a of the remanng non-zero entres from to. Insert n m extra rows so that f coumn j has one of the remanng non-zero entres from P\ then row j s a new row contanng the snge non-zero entry,, n coumn j. Emnatng the m zero coumns, eaves the matrx K. Emnate the frst n each coumn of Q' and change a the remanng non-zero entres from to. Insert n m extra coumns so that f row has one of the remanng non-zero entres from Q', then coumn s a new coumn contanng the snge non-zero entry,, n row. Emnatng the non-zero rows eaves the matrx L. (e) The souton for the system Az = c can now be formed as foows (the proof that (LB~]K)~ exsts and that z s a souton s gven n Sec. 4), z = Q( ~ B-]K{LB'K)~]L)B-]Pc. (8) The advantages of ths formua over the drect souton of Az c, arse from the bock dagona structure of B. The matrces B 'A' and B '/' can be cacuated usng agorthms whch utze the ndvdua dagona bocks of B, thus reducng the compexty of the system whch must be soved. The porton (LB~xK)~xLB~xPc may be cacuated by any of the usua agorthms as the souton, y, of the matrx system (LB K)y = LB~Pc, where the coeffcent matrx, LB~K, has order (n m) X (n m). Hence the sze of ths

4 68 P. W. AITCHISON coeffcent matrx s governed by the ncrease n sze of B over A, whch n turn s governed by the number of coumns, or rows, of A, wth non-zero entres outsde of the dagona bocks (see Eq. ()). The compexty of cacuatons n sovng a sparse m varabe system wth the best drect agorthms has upper bound mt, where t s the number of non-zeros (Red []) though ths upper bound s rarey attaned n practce. Appyng ths resut to the modfed probem anaysed above gves an approxmate upper bound to the compexty of k s,t, + rt', = where k, s, and t: are, respectvey, the number of bocks n B, the sze of the / th bock and the number of non-zero eements n t, whe r and t' are the sze and number of non-zero entres n LB~ XK. If the sze of the argest bock n B s N, then ths upper bound satsfes k «JV tj + rt' = Nt + rt'. = The frst term, Nt, w usuay be much smaer than the upper bound, mt, for the orgna probem, Az = c, provded the bock sze, N, s consderaby ess than the sze, m, of the orgna matrx A. The second term, rt', depends very much on the orgna probem the number r w be sma compared wth m snce t measures the argest number of rows or coumns contanng nonzero entres not n the dagona bocks, whe t' s the number of non-zero entres n LB~XK. B~ tsef may have consderabe f-n among the dagona bocks when compared wth B, and the product LB~XK combnes ony certan rows and coumns of B~\ normay resutng n a decrease n the non-zero entres. It shoud aso be noted that the souton z, as gven n equaton (8), s cosey reated to the souton of the system, By = I K(LB~]K)LB~]Pc, whch nvoves the enarged matrx, B. In fact the soutons y and z are reated by z Qy. The souton y gves the souton of the torn system, from whch the souton z s reconsttuted. A smpe numerca exampe. Suppose the system Az c s the varabe system shown beow wth A aready n the form of Eq. (). The numerca souton s gven n a convenent fashon wthout necessary foowng the most effcent souton agorthm whch woud be used n a computer souton of a arge system. Note that a bank ndcates a zero entry. A Z Z z3 ZA Z5 z6 z7 Z8 Z9 Z Z Z

5 B s chosen, usng the rues () and (3) n part (b), as DIAKOPTICS ON TEARING 69 B = P Hence P' and Q' are deveoped, as descrbed n part (b), as p'. Q ' = Consequenty P, Q, K and L, as descrbed n parts (c) and (d), are, p = - -I

6 7 P. W. AITCHISON Q = Ony at ths pont are cacuatons carred out for the souton z as gven n equaton (8) of part (e). B" J.5 [T5 L B K = LB_K = (LB K)" = ^ Hence, Pc = 3-3 B_Pc = LB Pc (LB K)"LB~Pc= B XK(LB "He) LB = " -. 5_

7 DIAKOPTICS ON TEARING 7 Hence z = Q(B Pc-B_K(LB K)~LB~Pc) = Q - " ' -7 = L Proofs. The foowng facts need to be proved n order to justfy the agorthm gven n Sec. : (I) The nverse of LB" ]K exsts; (II) z Q(I B~xK(LB~]K)~xL)B~]Pc, as gven n Eq. (8), s a souton of Az = c. These resuts are estabshed n the foowng two emmas. Lemma. The matrx LB~ ]K s non-snguar f and ony f the matrx A s non-snguar. Proof. Frst t s shown that the coumns of Q' are a bass for the nu space of L (the set of x wth Lx ). L I Q'Q, by equaton (7), where I Q'Q, conssts of the non-zero rows of I Q'Q. Aso, (I Q'Q)Q' =, by Eq. (6), L has rank n m, and Q' has rank m, and so the coumns of Q' are a bass of the nu space of L. Smary the coumns of K span the nu space of P', and so P'K. Suppose there s a vector w wth LB~Kw, then B~xKw s n the nu space of L. If w 7^, then B~Kw =, snce K has fu coumn rank and B s non-snguar. It foows that for some order m X vector q and f w =, then q =. By equaton (4), B~*Kw = Q'q, Aq = P'BQ'q = P'BB~xKw = P'Kw =. Hence, f LB xkw for some w =t=, then Aq = for some q =, and so f A s non-snguar, then so s LB~K. Conversey, suppose q s such that Aq, then by Eq. (4), Aq = P'BQ'q =, and f q =/=, then BQ'q ^ snce Q' has fu coumn rank and B s non-snguar. Snce the nu space of P' s spanned by the coumns of K, BQ'q = Kw for some w, and f q =, then w =. Therefore LB^Kw = LB~xBQ'q = LQ'q =, snce the coumns of Q' are n the nu space of L. Hence, f A s snguar, then so s LB~K, and the resut foows.

8 7 P W AITCHISON Lemma. If A and B are non-snguar, then the matrx system, Az = c, has souton gven by z = Q( - B~]K(LB^,KyL)B Pc. Proof. Let /) = (/ B K(LB~ K)~xL)B~ xpc, for convenence. /4z = P'BQ'z, by Eq. (4) = P'BQ'QD = P'BD - P'B(I - Q'Q)D = P'BB]Pc - P'BB ]K(LB K) * LB~]Pc - /" (/ -?'?) > = c- P'B(I- Q'Q)D, snce = /, P? = I, by Eq. (5), and P'K = P'(I P'P) = P' P'PP =, by Eqs. (5) and (7) (where / P'P represents the non-zero coumns of I P'P). It remans to show that P'B(I Q'Q)D =. However, LD = LB~^Pc - (LB = LB {Pc - LB~ ]Pc =. )(LBXK )~^ LB~ xpc Snce, by defnton, I Q'Q dffers from L ony by havng some extra zero rows, t foows that f Hence, the resuts, Az = c, foows. LD =, then P'B(I - Q'Q)D =. References [] Ad Ben-Israe and T. N. E. Greve, Generazed nverses: Theory and appcatons. Wey, New York, 974, reprnted by Kreger, Huntngton, New York, 98 [] A. Brameer, M. N. John and M. R. Scott, Practca dakoptcs for eectrca networks, Chapman and Ha, London, 969 [3] F. H. Brann Jr., The reatons between Kron's method and the cassca methods of network anayss, IRE Wescon Conventon Record, 8, 3-8 (959) [4] H. H. Happ, The appcatons of dakoptcs to the soutons of power system probems. Eectrc Power Probems: The Mathematca Chaenge, SIAM, Phadepha, pp. 69-3, 98 [5] H. H. Happ, Pecewse methods and appcatons to power systems, John Wey, New York, 98 [6] H. K. Kesavan and J. Dueckman, Mut-termna representatons and dakoptcs. Unversty of Wateroo, Wateroo. Canada, 98 (a report) [7] G. Kron, Dakoptcs pecewse soutons of arge scae systems, Eect. J. (London) Vo. 58-Vo. 6 (JUne 957-Feb. 959) [8] G. Kron, Dakoptcs, Macdonad, London 963 [9] J. S. Przemeneck, Matrx structura anayss of substructures, AIAA Journa,, (963) [] J. K. Red, A survey of sparse matrx computaton, Eectrc Power Probems: The Mathematca Chaenge, SIAM, pp , 98 [] J. P. Roth, An appcaton of agebrac topoogy. Kron' s method of tearng. Quartery of App. Math., 7, -4 (959) [] D. V. Steward, Parttonng and tearng systems of equatons, SIAM J. Numer. Ana., Ser. B,, (965)