Symmetric centrosymmetric matrix vector multiplication

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1 Linar Algbra and its Applications 320 (2000) Symmtric cntrosymmtric matrix vctor multiplication A. Mlman 1 Dpartmnt of Mathmatics, Univrsity of San Francisco, San Francisco, CA , USA Rcivd 15 April 1999; accptd 15 July 2000 Submittd by R.E. Hartwig Abstract W prsnt a mthod for th multiplication of an arbitrary vctor by a symmtric cntrosymmtric matrix, rquiring 5 4 n2 + O(n) floating-point oprations, rathr than th 2n 2 oprations ndd in th cas of an arbitrary matrix. Combining this mthod with Trnch s algorithm for Toplitz matrix invrsion yilds a mthod for solving Toplitz systms with th sam complxity as Lvinson s algorithm Elsvir Scinc Inc. All rights rsrvd. AMS classification: 65F05 Kywords: Cntrosymmtric matrix; Matrix vctor multiplication; Toplitz systm; Trnch s algorithm; Lvinson s algorithm 1. Introduction Th multiplication of an arbitrary vctor by a matrix on a squntial machin rquirs, in gnral, 2n 2 floating-point oprations (following [5], w dfin a floating-point opration, or flop, as ithr an addition/subtraction or multiplication/division). This rmains tru vn for a symmtric matrix. Howvr, if, in addition to bing symmtric, th matrix is also prsymmtric (symmtric with rspct to th southwst northast diagonal), thn th complxity can b rducd significantly. Matrics, which ar both symmtric and prsymmtric, ar calld symmtric cntrosymmtric. In this papr, w prsnt a mthod for multiplying an arbitrary vctor addrss: mlman@uclid.math.usfca.du (A. Mlman). 1 On lav from Bn-Gurion Univrsity, Br-Shva, Isral /00/$ - s front mattr 2000 Elsvir Scinc Inc. All rights rsrvd. PII:S (00)

2 194 A. Mlman / Linar Algbra and its Applications 320 (2000) by such a matrix with 5 4 n2 + O(n) flops. Additional multiplications by th sam matrix cost only an additional n 2 + O(n) flops pr multiplication. W also brifly considr an application of this mthod by combining it with Trnch s algorithm [5,7] for Toplitz matrix invrsion to produc a mthod for solving positiv-dfinit symmtric Toplitz systms, which has th sam complxity as Lvinson s algorithm [5,6]. Th papr is organizd as follows: Sction 2 is dvotd to a fw basic dfinitions and rsults; in Sction 3, w prsnt our mthod and discuss som implications, whras in Sction 4, w brifly show an application to Toplitz systms. 2. Prliminaris AmatrixM R (n,n) is said to b prsymmtric if it is symmtric about its southwst northast diagonal. For such a matrix M, this is th sam as rquiring that JM T J = M,whrJisamatrix with ons on its southwst northast diagonal and zros vrywhr ls (th xchang matrix). It is asy to s that th invrs of a prsymmtric matrix is also prsymmtric. A matrix Q R (n,n) is said to b cntrosymmtric if it satisfis JQJ = Q. Its invrs is also cntrosymmtric. A matrix which is both symmtric and prsymmtric is thrfor calld symmtric cntrosymmtric,ordoubly symmtric. An vn vctor v is dfind as a vctor satisfying Jv = v and an odd vctor w as on that satisfis Jw = w. A symmtric matrix T R (n,n) is said to b Toplitz if its lmnts T ij satisfy T ij = ρ j i,whr{ρ j } n 1 j=0 ar th componnts of a vctor (ρ 0,r) T R n, with r = (ρ 1,...,ρ n 1 ) T R n 1,sothat ρ 0 ρ 1 ρ 2... ρ n 1 ρ 1 ρ 0 ρ 1... ρ n 2 T = ρ n 2 ρ n 3 ρ n 4... ρ 1 ρ n 1 ρ n 2 ρ n 3... ρ 0 Th matrix T is symmtric cntrosymmtric and so is its invrs. Such matrics appar in many applications and w rfr to [2] for a good ovrviw. W dnot th xchang matrix by J without spcifically indicating its dimnsion, which is assumd to b clar from th contxt. 3. Symmtric cntrosymmtric matrix vctor multiplication Considr th multiplication M n b (n),whrm n is a ral n n symmtric cntrosymmtric matrix and b (n) is a ral n-dimnsional vctor. Th matrix M n can b partitiond as follows:

3 A. Mlman / Linar Algbra and its Applications 320 (2000) α t T β M n = t M n 2 Jt, β t T J α whr M n 2 R (n 2,n 2),t R n 2 and α, β R. W not that M n 2 is also symmtric cntrosymmtric. Th mthod w ar about to construct is basd on this partition of M n and on th fact that b (n) = b (n) + b o (n),whr b (n) = 1 2 (b(n) + Jb (n) ) and b o (n) = 1 2 (b(n) Jb (n) ). Jb (n) o = b (n). W comput M n b (n) as M n b (n) Th vn part of b (n) satisfis Jb (n) = b (n) o b (n) = (σ,b,σ ) T and b o (n) = (σ o,b o, σ o ) T, rspctivly, w hav α t T β M n b (n) = t M n 2 Jt β t T J α σ b σ, whras for th odd part w hav + M n b o (n). Writing (α + β)σ + t T b = σ (t + Jt) + M n 2 b, (α + β)σ + t T b which computs th first and last componnts of M n b (n),and α t T β σ o (α β)σ o + t T b M n b o (n) = t M n 2 Jt b o o = σ o (t Jt) + M n 2 b o, β t T J α σ o (α β)σ o t T b o which computs th first and last componnts of M n b o (n). W thn rpat this procdur for M n 2 b and M n 2 b o, add th rsult to that of th prvious stp, and continu in this mannr until th whol product is computd. This lads to th algorithm, labld Algorithm 1, for computing y = M n b (n). W hav usd similar notation as in [5], with th subscripts and o rfrring to vn and odd quantitis, rspctivly. For positiv numbrs (as is th cas hr), th notation mans roundd to th narst smallr intgr. For th sak of clarity, som rdundant quantitis wr introducd in th algorithm. Ths do not affct th complxity. Algorithm 1. m = n+1 2 b (1 : m) = (1/2) (b(1 : m) + b(m : 1 : 1)) b o (1 : m) = (1/2) (b(1 : m) b(m : 1: 1)) if n is vn for j = 1 : m 1 α = q(j,j) β = q(j,n j + 1)

4 196 A. Mlman / Linar Algbra and its Applications 320 (2000) s = q(j,j + 1 : n j) s (1 : m j) = s(1 : m j)+ s(n 2j : 1: m j + 1) s o (1 : m j) = s(1 : m j) s(n 2j : 1 : m j + 1) z (j) = (α + β)b (j) + s (1 : m j) T b (j + 1 : m) z (j + 1 : m) = z (j + 1 : m) + b (j)s (1 : m j) z o (j) = (α β)b o (j) + s o (1 : m j) T b o (j + 1 : m) z o (j + 1 : m) = z o (j + 1 : m) + b o (j)s o (1 : m j) nd ls (n is odd) for j = 1 : m 1 α = q(j,j) β = q(j,n j + 1) s = q(j,j + 1 : n j) s (1 : m j) = s(1 : m j)+ s(n 2j : 1: m j) s o (1 : m j) = s(1 : m j) s(n 2j : 1 : m j) z (j) = (α + β)b (j) + s (1 : m j) T b (j + 1 : m) s (m j)b (m) z (j + 1 : m) = z (j + 1 : m) + b (j)s (1 : m j) z o (j) = (α β)b o (j) + s o (1 : m j) T b o (j + 1 : m) z o (j + 1 : m) = z o (j + 1 : m) + b o (j)s o (1 : m j) nd nd y(1 : m) = z (1 : m) + z o (1 : m) if n is vn y(m + 1 : n) = z (m : 1: 1) z o (m : 1: 1) ls (n is odd) y(m + 1 : n) = z (m 1 : 1: 1) z o (m 1 : 1: 1) nd W strss that this is a concptual algorithm only, as th final implmntation dpnds on th architctur of th machin on which it is supposd to run. Th algorithm could also b xcutd backwards, by starting with M 1 b (1) for n odd, or M 2 b (2) for n vn. Furthrmor, th computations of th vn and odd parts of y ar indpndnt of ach othr and could b procssd in paralll. At ach stp of Algorithm 1, two componnts ar computd for both b (n) and b o (n), from th outr componnts down to th middl ons as it procds from stp to stp.

5 A. Mlman / Linar Algbra and its Applications 320 (2000) Lt us now hav a look at th numbr of flops xcutd by Algorithm 1. W first considr M n b (n) whn n is vn. At th stag whr M j b (j) (j <n) is carrid out, w hav γ s T δ ρ M j b (j) = s M j 2 Js b (j 2) (γ + δ)ρ + s T b (j 2) = ρ (s + Js) + M j 2 b (j 2), δ s T J γ (γ + δ)ρ + s T b (j 2) ρ whr γ,δ R and s R j 2. Taking into account th symmtry of b (j) and (s + Js), w nd to xcut, at this stag, (j 1) additions and (j 1) multiplications. Excpt for j = n, this must thn b addd to th solution vctor found thus far, which rquirs an additional j/2 additions. Summing ovr all stps, this givs (n + (n 2) + +2) n (( n ) ( n ) ) additions and (n + (n 2) + +2) n 2 multiplications. This yilds ( 3 8 n2 1 4 n) additions and ( 1 4 n2 ) multiplications. Analogously, th sam complxity is obtaind for M n b o (n), so that if w apply th matrix to an arbitrary vctor b (n) and also tak into account th cost of computing th vn and odd parts of b (n) and of adding th vn and odd parts of th solution, w obtain for th computation of M n b (n) a complxity of ( 3 4 n2 + n) additions and ( 1 2 n2 + n) multiplications. This mans that th total complxity of th mthod is 5 4 n2 + O(n) flops. Th sam complxity is obtaind whn n is odd. If th vctors { vj + Jv j } n+1 2 j=1 and { vj Jv j } n+1 2 j=1, whr th v j s ar th columns of M n, ar stord (at a cost of roughly n 2 /4), thn ach subsqunt multiplication by an arbitrary vctor costs only n 2 + O(n) additional flops. No xtra storag is rquird if th matrix M n, which rquirs th sam amount of storag, is ovrwrittn by ths data. 4. Positiv-dfinit Toplitz systms of quations In this sction w brifly illustrat an application of symmtric cntrosymmtric matrix vctor multiplication in th contxt of Toplitz systms by showing that it lads to yt anothr O(n 2 ) algorithm for such problms. Th spcial systm of quations Tx = r, with T and r as in Sction 2 and T positiv-dfinit, is calld th Yul Walkr (YW) systm of quations. It frquntly appars in digital signal procssing problms and can b solvd by Durbin s mthod with 2n 2 + O(n) flops (s [4]). Its solution is usd in an fficint algorithm for th

6 198 A. Mlman / Linar Algbra and its Applications 320 (2000) invrsion of a positiv-dfinit Toplitz matrix, du to Trnch [7]. If Durbin s algorithm is usd for solving th YW quations, thn Trnch s algorithm rquirs a total of 13 4 n2 + O(n) flops. Howvr, in [3], an algorithm ( th split Lvinson mthod ) is drivd, rquiring only 3 2 n2 + O(n) flops to solv th YW quations. This rducs Trnch s algorithm s complxity to 11 4 n2 + O(n) flops. W rfr, onc again, to [5] for an ovrviw of algorithms and rfrncs rlatd to Toplitz matrics. Thr ar othr algorithms for ths sam problms with bttr complxity for larg matrics and w rfr to, i.., [1] and rfrncs thrin. A classical mthod for solving positiv-dfinit ral symmtric Toplitz systms of linar quations with arbitrary right-hand sid (of th form Ty = b) andwhich dos not involv matrix invrsion, is Lvinson s algorithm [5,6]. Its complxity is 4n 2 + O(n) flops. An altrnativ to Lvinson s algorithm can b obtaind by first invrting th cofficint matrix using Trnch s mthod and thn applying th rsulting matrix to th right-hand sid, i.., y = T 1 b. Howvr, this would not b advantagous in trms of numbr of oprations if rgular matrix multiplication wr usd. On th othr hand, th invrs of a symmtric Toplitz matrix is symmtric cntrosymmtric and w can thrfor, givn T 1, apply our algorithm to comput T 1 b. Lt us hav a look at th complxity of such a procdur. Trnch s algorithm with th split Lvinson algorithm rquirs 11 4 n2 + O(n) flops. Multiplying th invrs matrix and th right-hand sid using Algorithm 1 thn adds 5 4 n2 + O(n) flops, yilding a total of 4n 2 + O(n) flops, which is prcisly th sam as for Lvinson s algorithm. In fact, it may b mor dsirabl to us our mthod if th systm nds to b solvd for mor than on right-hand sid, spcially sinc ach subsqunt solv rquirs only n 2 + O(n) flops, onc th ncssary quantitis hav bn stord. Such an algorithm is, in gnral, likly to b lss accurat than Lvinson s algorithm, but itrativ rfinmnt of th solution could always b usd, if ncssary. Howvr, ths ar practical considrations (as is th siz of th matrix, which may giv an advantag to th so-calld suprfast solvrs in [1]) and th choic of algorithm will dpnd on th circumstancs. Rfrncs [1] G.S. Ammar, W.B. Gragg, Th gnralizd Schur algorithm for th suprfast solution of Toplitz systms, in: J. Gilwicz, M. Pindor, W. Simaszko (Eds.), Rational Approximations and its Applications in Mathmatics and Physics, Lctur Nots in Mathmatics, 1237, Brlin, 1987, pp [2] J.R. Bunch, Stability of mthods for solving Toplitz systms of quations, SIAM J. Sci. Stat. Comput. 6 (1985) [3] P. Dlsart, Y. Gnin, Th split Lvinson algorithm, IEEE Trans. Acoust. Spch, Signal Procssing ASSP-34 (1986) [4] J. Durbin, Th fitting of tim sris modl, Rv. Inst. Int. Stat. 28 (1960) [5] G. Golub, C. Van Loan, Matrix Computations, Johns Hopkins Univrsity Prss, Baltimor, MD, [6] N. Lvinson, Th Winr RMS (root man squar) rror critrion in filtr dsign and prdiction, J. Math. Phys. 25 (1947) [7] W.F. Trnch, An algorithm for th invrsion of finit Toplitz matrics, J. SIAM 12 (1964)