Scaled Toda-like Flows. Moody T. Chu 1. North Carolina State University

Transcription

1 Scaled Tda-like Flws Mdy T. Chu 1 Department f Mathematics Nrth Carlina State University Raleigh, Nrth Carlina September 21, This research was supprted in part by Natinal Science Fundatin under grants DMS and DMS

2 Abstract This paper discusses the class f isspectral ws _X =[X A X] where dentes the Hadamard prduct and [ ] is the Lie bracket. The presence f A allws arbitrary and independent scaling fr each element in the matrix X. The time-1 mapping f the scaled Tda-like w still enjys a QR-like iteratin. The scaled structure includes the classical Tda w, Brckett's duble bracket w, and ther interesting ws as special cases. Cnvergence prf is thus unied and simplied. The eect f scaling n avariety f applicatins is demnstrated by examples.

3 1. Intrductin. Fr simplicity, we will cnne ur discussin in this paper t the real case nly. It is cnvenient tintrduce tw special subsets in R nn : S(n) := fx 2 R nn jx T = Xg O(n) := fq 2 R nn jq T Q = Ig: Recent research has revealed a number f remarkable cnnectins between smth ws and discrete numerical algrithms [2, 11, 9, 10, 15, 16]. Amng these, a by nw classic result is the relatinship between the Tda lattice and the QR algrithm. That is, the time-1 mapping fx(k)g f the slutin X(t) t the initial value prblem (1) _X = [X 0 (X)] X(0) = X 0 crrespnds exactly t the sequence by applying the QR algrithm t the matrix e X(0) [11, 15, 5]. In (1), 0 (X) :=X ; ; X ;T where X ; dentes the strictly lwer triangular matrix f X. The Tda w (1) latter was generalized t the class [8] (2) _X = [X P L (X)] X(0) = X 0 where P L (X) dentes the prjectin f X nt a certain specied linear subspace L f R nn. By specifying dierent L, (2) gives rise t dierenttypes f matrix factrizatins, many f which are in abstract frms. Fr example, the fllwing therem which includes the well knwn Schur decmpsitin therem as a special case has been prved in [8] by using (2). Therem 1.1. Given a symmetric matrix A 2 R nn, there exists a real and rthgnal matrix Q such that the symmetric matrix T = Q T AQ has zer entries in any prescribed psitins f T except pssibly alng the diagnal. Anther interesting isspectral w is the s called Brckett's duble bracket w [3, 4] (3) _X = [X [X D]] X(0) = X 0 where X and D are matrices in S(n) andd is xed. The w riginally arises as a gradient w. Remarkably, it is nticed in [2] that if X is tridiagnal and (4) D =diagfn ::: 2 1g then (3) cincides precisely with (1). A gradient w hence becmes Hamiltnian [1]. Equatin (3) is a special case f a mre general prjected gradient ws [9]: (5) _X = [X [X P A (X)]] X(0) = X 0 1

4 where P A (X) dentes the prjectin f X 2S(n) nt an ane subspace A f S(n). The vectr eld in (5) represents the prjectin f the negative gradient f the bjective functin (6) F (X) := 1 2 kx ;P A(X)k 2 nt the isspectral feasible set (7) M(X 0 ):=fq T X 0 QjQ 2O(n)g: The bject is t minimize the distance between the sets M(X 0 ) anda. Taking A = fdg, wehave (3). Thus the Brckett duble bracket w gives the least squares apprximatin f D subject t the spectral cnstraint. Such a slutin, which itself has many interesting applicatins, can be characterized in terms f the spectral decmpsitin [4, 9]. Therem 1.2. Suppse bth D and X 0 are ins(n) and have distinct eigenvalues. Let the eigenvalues f D and X 0 berdered as 1 < ::: < n and 1 < ::: < n, respectively. Then the unique asympttically stable equilibrium pint f (3) is given by (8) ^X = 1 q 1 q T 1 + :::+ nq n q T n where q 1 ::: q n are the nrmalized eigenvectrs f D crrespnding respectively t 1 ::: n. In particular, if D is a diagnal matrix with distinct eigenvalues, then ^X must be a diagnal matrix whse elements are similarly rdered as thse in D. One cmmn characteristic f all the ws discussed abve is that they are always described in the s called Lax pair (9) _ X =[X k(x)] where k(x) is matrix-valued functin f X. Of particular interest is the case when X 2S(n) and k(x) is skew-symmetric. In this paper, we prpse anther Tda-like w by taking k(x) = A X where A is a cnstant matrixand represents the Hadamard prduct. In the cntext that X is being scaled cmpnentwise by A, we call the w assciated with the dierential equatin (10) _X = [X A X] X(0) = X 0 : a scaled Tda-like w. We shall shw that dierent chices f the scaling matrix A result in (1), (2) and (3) as special cases. At least in thery, the time-1 mapping f the scaled Tda-like w enjys a QR-like iteratin. Fr symmetric cases, weprvide a simple prf n the glbal cnvergence f the scaled Tda-like w. Finally the eect f scaling is discussed. 2

5 2. QR-like Iteratin. In this sectin we explain why the w X(t) f (10) evaluated at integer times still enjys a QR-like iteratin. The ntin f the Q and R matrices in the QR decmpsitin will be replaced bythel and R matrices dened in the sequel. We shall state the results withut prfs since they are very similar t thse already dne in [8], nly keeping in mind that the matrix A allws arbitrary scaling. Given any square matrix A =[a ij ], let ~ A = [~a ij ] dente the "cmplementary" matrix where ~a ij := 1 ; a ij. Assciated with (10) are the tw dierential systems (11) and (12) _L = L(A X) L(0) = I _R = ( ~ A X)R R(0) = I: Here we adpt the ntatin L and R nly as a reminder abut hw the multiplicatin is invlved in (11) and (12), respectively. Let X(t), L(t) and R(t) represent the slutin t the initial value prblems (10), (11) and (12), respectively,ver an interval [0 T]fr sme T>0. Then we have (13) Therem 2.1. X(t) =L(t) ;1 X 0 L(t) =R(t)X 0 R(t) ;1 : Therem 2.2. (14) e X 0t = L(t)R(t): Therem 2.3. (15) e X(t)t = R(t)L(t): (16) Because f (13), we may rewrite (11) as _L = L(A L ;1 X 0 L) L(0) = I: S the dierential system becmes autnmus. In a similar way we may rewrite (12). T emphasize the dependence n the initial matrix X 0,wenw dente the slutins f (10) and the assciated (11) and (12) by X(t X 0 ), L(t X 0 )andr(t X 0 ), respectively. By setting t = 1 in (14) and (15), it becmes clear that fr psitive ingeter k in the dmain f existence we have (17) (18) e X(k X 0) e X(k+1 X 0) = e X(0 X(k X 0)) = L(1 X(k X 0 ))R(1 X(k X 0 )) = e X(1 X(k X 0)) = R(1 X(k X 0 ))L(1 X(k X 0 )): That is, if LR is an abstract LR decmpsitin f e X(k),thenRL gives t e X(k+1).Such a prperty as in (17) and (18) is referred t a QR-like iteratin. 3

6 3. Cnvergence. Hencefrth we shall cnsider nly ws in S(n). In rder that X(t) =[x ij (t)] 2S(n) fr all t, the scaling matrix A in (10) is necessarily skew-symmetric. In this case, it fllws frm (11) that L(t) is rthgnal. We nw prveavery useful cnvergence prperty fr the scaled Tda-likew. Our majr result is as fllws. Therem 3.1. Suppse the strictly lwer triangular part f A =[a ij ] are nnnegative. Then (19) lim A X(t) =0: t!1 Mre precisely, whenever a ij > 0, the crrespnding entry x ij (t) (and x ji (t)) cnverges t 0 as t ges t innity. Prf. Cnsider the partial sums f 1 ::: n g f the diagnal entries f X, i.e., (20) k := kx i=1 x ii : Since L(t) is rthgnal, wehave frm (13) that jjx(t)jj F = jjx 0 jj F where jjjj F dentes the Frbenius matrix nrm and, hence, k (t) is bunded fr all t. It is nt dicult t see frm the equatin (10) that (21) and that (22) _ k =2 _ n =0 nx kx i=k+1 j=1 a ij x 2 ij fr 1 k<n. Since a ij 0 fr all i>j, (22) implies that each k (t) is a nn-decreasing functin in t. It fllws that bth lim t!1 k (t) and lim t!;1 k (t) exist. Using (22), we nd Z 1 nx kx ;1 i=k+1 j=1 a ij x 2 ij (t) dt is integrable. In particular, s lng as a ij > 0, we nd that each x ij (t) isl 2 integrable ver (;1 1). Tgether with the fact that (x 2 ij ) is unifrmly bunded, it fllws that lim t!1 x ij (t) =0(See[8]). In the next sectin we shall see hw dierent chices f A lead t a variety f interesting ws, including (1), (2) and (3). Thus we think the abve therem, unifying the prf f cnvergence, is f interest in its wn right. 4

7 4. Chices f A. We nw demnstrate hw dierent chices f A result in sme classical ws. Mre extic applicatins will be discussed in the next sectin. Example 1. Chse A =[a ij ]suchthat (23) a ij := 8 >< >: 1 if j =1andi>j ;1 if i =1andj>i 0 therwise. Let the clumns f L(t) be dented as L(t) =[l 1 (t) ::: l n (t)]. The rst clumn f L(t) is f particular interest. Frm (11), we have (24) dl 1 dt = n X i=2 x i1 l i : On the ther hand, frm (13), we have (25) Tgether, we nd that (26) nx i=1 x i1 l i = X 0 l 1 : dl 1 dt = X 0l 1 ; (l T 1 X 0l 1 )l 1 since x 11 = l T X 1 0l 1. It is easy t see that the right hand side f (26) is precisely the prjected gradient fr the prblem (27) (28) Maximize F (x) :=x T X 0 x Subject t x T x =1 and, hence, the w l 1 (t) cnverges t the eigenvectr assciated with the mst dminant eigenvalue. Indeed, the exact slutin f (26) is given by (29) l 1 (t) = ex 0t l 1 (0) jje X 0t l 1 (0)jj which is related t the Tda w [11, 13] and has been studied as the cntinuus pwer methd [7]. Frm (13) and Therem 3.1, it is bvius that x 11 (t) cnverges t the mst dminant eigenvalue f X 0. Example 2. Chse A =[a ij ]suchthat (30) a ij := 8 >< >: 1 if i>j ;1 if j>i 0 therwise. Obviusly the resulting (10) is the classical Tda lattice equatin (See (1)). The cnvergence f the classical Tda lattice and, hence, f the QR algrithm t a diagnal matrix fllws frm Therem 3.1 [11, 15] immediately. 5

8 Example 3. Let be an arbitrary subset f rdered integer pairs f(i j)j1 j< i ng. Chse A =[a ij ]such that (31) a ij := 8 >< >: 1 if (i j) 2 ;1 if (j i) 2 0 therwise. Therem 3.1 implies that x ij (t) f the crrespnding (10) cnverges t zers whenever (i j) 2 (See (2)). This re-prves Therem 1.1 In all the examples abve, the value 1 can be replaced by arbitrary numbers (except that a ij 0fri>jand A skew-symmetric) and, by Therem 3.1, we shall have similar cnvergence results. The fllwing is ne particular example. Example 4. Let D = diagfd 1 ::: d n g be an arbitrary diagnal matrix with d i d j if i j. ChseA =[a ij ]suchthat : (32) a ij := d i ; d j Then (10) becmes a Brckett duble bracket equatin (See (3) and (4)). Therem 3.1 can nw beusedtshw that the Brcket w (3) and (4) cnverges t ne f n! pssible diagnal matrices (See Therem 1.2), but it says nthing abut which ne is the stable equilbrium pint. 5. Eect f Scaling. In additin t generating dierent ws, the scaling intrduced by the matrix A has several ther interesting eects frm cmputatinal pint f view. T illustrate the idea, we shall assume that the initial value X 0 is generic, i.e., X 0 is nt an equilibrium pint f (10) which, being n algebraic curves, frms a nwhere dense set f measure zer. The mst bvius eect can be seen by cmparing the "unifrmly" scaled Tda w where a ij = c c > 1 with the classical Tda w wherea ij = 1 (See (30)). The differential system being autnmus, it is clear that X c;scaled T da (t X 0 )=X Tda (ct X 0 ). That is, the scaled Tda w is expected t reach cnvergence c times faster than the classical Tda w. A mre subtle cmparisn is t cnsider a partial rdering n skew-symmetric matrices dened by (33) (34) A 0 if a ij 0 fr all i>j A B if A ; B 0. Given the same X, it can be seen frm (22) that if A B, then the k crrespnding t A \grws" innitesimally faster then that crrespnding t B. Thus, fr example, the Brckett w X Brckett (t X 0 ) (See (32) with D dened by (4)) grws faster in the sense f majrizatin [12, Page 166] than the classical Tda w X Tda (t X 0 ) (See (30)) at least fr suciently small t>0. Figure 1 illustrate this majrizatin prperty f 6

9 10 Partial sums f diagnal entries f X(t) =Brckett flw =Tda flw t Fig. 1. Majrizatin f Brckett w versus Tda w. Brckett w versus Tda w fr the matrix X 0 = :6489 0:3286 ;0:1870 0:4121 ;1:3056 0:3286 2:0112 ;0:4956 3:4960 0:4198 ;0:1870 ;0:4956 2:1534 ;0:1380 ;1:8459 0:4121 3:4960 ;0:1380 0:5415 ;0:5440 ;1:3056 0:4198 ;1:8459 ;0:5440 ;1:3550 We nte, as is demnstrated in Figure 1, that at sme latter stage f integratin it is pssible that i T da (t) i Brckett (t) fr sme i. The chice f A in the frm (32) is f particular interest. Apparently what is imprtant in the diagnal matrix D is nt the values f d i i =1 ::: n but rather the relative spacing f these elements. Diagnal matrices with elements either fd 1 ::: d n g r fd 1 + c ::: d n + cg generate the same w. We nte then that the chice (35) D =diagf1 0 ::: 0g gives rise t the matrix (23) which leads t the cntinuus pwer methd. The chice (36) D = diagf2 1 0 ::: 0g :

10 n the ther hand, will cause the crrespnding w X(t) tcnverge, accrding t Therem 3.1, t a limit pint f the frm X(1) = ::: ::: ::: ::: where indicates sme nn-zer values. Furthermre, since X(1) is the least squares apprximatin t D, the srting prperty guarantees that x 11 (1) andx 22 (1) are the rst tw largest eigenvalues f X 0. Thisisacntinuus versin f the s called simultaneus iteratin [14, Chapter 14]. Mtivated by the abve bservatin, we nd anther applicatin f the Brckett w, which is aimed at aggregating eigenvalues int blcks. Fr example, the limit pint X(1) f the w crrespnding t the chice (37) D = diagf3 ::: 3 2 ::: 2 1 ::: 1 {z } n 1 many {z } n 2 many {z } n 3 many will be a diagnal blck matrix with three blcks. The eigenvalues f the (1 1) blck f X(1) are the rst n 1 mst dminant eigenvalues f X 0 the eigenvalues f the (3 3) blck are the last n 3 eigenvalues f X 0 and the (2 2) blck cntains the remaining eigenvalues. In this way the eigenvalues f X 0 are aggregated int three grups accrding t their rdering. In the special case when n 2 =1,weare able t single ut the (n 1 +1) th largest eigenvalue f X 0 by slving the dierential equatin. We think this feature is very interesting and useful. Of curse, the chice f values r 3 fr diagnal matrices D in (35), (36) r (37) is fr the purpse f demnstratin nly. Onemay certainly chse dierentvalues and cnsider the pssible speedup in cnvergence as we have studied earlier. Given any diagnal matrix D =diagfd 1 ::: d n g, let (D) dente the plytpe in R n whse vertices are exactly the clumns f D. Given X 0 2 S(n), let its eigenvalues be written int the diagnal matrix = diagf 1 ::: n g. Cnsider the tw plytpes (D) and (). Our thery suggests that the least squares apprximatin t D subject t the spectral cnstraint 1 ::: n is the diagnal matrix such that the plytpe () is similar t (D) asmuch as pssible in the sense f Therem 1.2. This feature shuld nd applicatins in gemetric design. Suppse we make ne r mre vertices f D mre distinguishable than ther vertices, then the crrespnding Brckett w (3) shuld shw up the crrespnding eigenvalues earlier than ther eigenvalues. This is like rattling a lng and narrw plytpe () inside a lng and narrw plytpe (D). It is expected that the lnger edges f () shuld align with thse f (D) rst while the shrter edges f () are yet t be settled. Our numerical experiment seems t cnrm this intuitin. In Figure 2 we cmpare the standard Brckett w where D is dened by (4) with the mdied Brckett w where (38) D = diagf10n n ; 1 ::: 2 1g: 8 g

11 Partial sums f diagnal entries f X(t) =Standard flw 2 =Mdified flw t Fig. 2. Majrizatin f standard Brckett w versus mdied Brckett w. Clearly, 1 M dif ied (t) cnverges signicantly faster than 1 Standard (t) while the rest are cnverging at almst the same rate. 6. Cnclusin. The structure f Tda lattice has been mdied t allw scaling in the secnd cmpnent f the Lax pair. A number f interesting facts cncerning the scaled Tdalike w (10) have been studied in this paper. With specially selected scaling matrix A, the scaled Tda-like w includes as special cases several well knwn ws that are related t imprtant numerical linear algebra algrithms. We have shwn that the time-1 mapping f the scaled Tda-like w still enjyaqr-likeiteratin except that the crrespnding QR-like decmpsitin nw becmes metaphysical. When restricted t symmetric matrices, a unied prf f glbal cnvergence is given in Therem 3.1. The eect f scaling is demnstrated thrugh numerical examples. It seems that by increasing the scaling factr ne might cut shrt the interval f integratin fr reaching the equilibrium. Of particular interest is by maneuvering the diagnal matrix D ne can lcate intermediate eigenvalues f matrix X 0 accrding t their rdering. 9

12 REFERENCES [1] A. M. Blch, R. W. Brckett and T. S Ratiu, A new frmulatin f the generalized Tda lattice equatins and their xed pint analysis via the mmentum map, Bull. Amer. Math. Sc. (N.S.) 23(1990), [2] A. M. Blch, Steepest descent, linear prgramming, and Hamiltnian ws, Cntemprary Math., 114(1990), [3] R. W. Brckett, Least squares matching prblems, Linear Alg. Appl., 122/123/124(1989), [4] R. W. Brckett, Dynamical systems that srt lists and slve linear prgramming prblems, in Prceedings f the 27th IEEE Cnference n Decisin and Cntrl, IEEE, (1988), , and Linear Alg. Appl., 146(1991), [5] M. T. Chu, The generalized Tda w, the QR algrithm and the center manifld therem, SIAM J. Alg. Disc. Meth., 5(1984), [6] M. T. Chu, On the cntinuus realizatin f iterative prcesses, SIAM Review, 30(1988), [7] M. T. Chu, Curves n S n;1 that lead t eigenvalues r their means f a matrix, SIAM J. Alg. Disc. Meth., 7(1986), [8] M. T. Chu and L. K. Nrris, Isspectral ws and abstract matrix factrizatins, SIAM J. Numer. Anal., 25(1988), [9] M. T. Chu and K. R. Driessel, The prjected gradient methd fr least squares matrix apprximatins with spectral cnstraints, SIAM J. Numer. Anal., 27(1990), [10] M. T. Chu, Matrix dierential equatins: A cntinuus realizatin prcess fr linear algebra prblems, Nnlinear Anal., TMA, t appear. [11] P. Deift, T. Nanda and C. Tmei, Dierential equatins fr the symmetric eigenvalue prblem, SIAM J. Numer. Anal., 20(1983), [12] R. A. Hrn and C. R. Jhnsn, Tpics in Matrix Analysis, Cambridge University Press, New Yrk, [13] J. Mser, Finitely many mass pints n the line under the inuence f an expnential ptential An integrable dynamical system, in Dynamical Systems Thery and Applicatin, ed. J. Mser, Springer- Verlag, New Yrk, [14] B. N. Parlett, The Symmetric Eigenvalue Prblem, Prentice-Hall, Englewd Clis, N.J., [15] W. W. Symes, The QR algrithm and scattering fr the nite nn-peridic Tda lattice, Physica, 4D(1982), [16] D. S. Watkins, Isspectral ws, SIAM Review, 26(1984),