Heinrich Voss. Abstract. rational and polynomial approximations of the secular equation f() = 0, the

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1 ouds for the Miimum Eigevalue of a Symmetric Toeplitz Matrix Heirich Voss Techical Uiversity Hamburg{Harburg, Sectio of Mathematics, D{27 Hamburg, Federal Republic of Germay, voss tu-harburg.de Abstract I a recet paper Melma [2] derived upper bouds for the smallest eigevalue of a real symmetric Toeplitz matrix i terms of the smallest roots of ratioal ad polyomial approximatios of the secular equatio f() =, the best of which beig costructed by the( 2)-Pade approximatio of f. I this paper we prove that this boud is the smallest eigevalue of the projectio of the give eigevalue problem oto a Krylov spaceoft ; of dimesio 3. This iterpretatio of the boud suggests ehaced bouds of icreasig accuracy. They ca be substatially improved further by exploitig symmetry properties of the pricipal eigevector of T. Keywords. Toeplitz matrix, eigevalue problem, symmetry Itroductio The problem of dig the smallest eigevalue of a real symmetric, positive deite Toeplitz matrix (RSPDT) is of cosiderable iterest i sigal processig. Give the covariace sequece of the observed data, Pisareko [4] suggested a method which determies the siusoidal frequecies from the eigevector of the covariace matrix associated with its miimum eigevalue. The computatio of the miimum eigevalue of a RSPDT T was cosidered i, e.g. [2], [7], [8], [9], [], [], [3], [6]. ybeko ad Va Loa [2] preseted a algorithm which is a combiatio of bisectio ad Newto's method for the secular equatio. y replacig Newto's method with a root dig method based o ratioal Hermitia iterpolatio of the secular equatio, Mackes ad the preset author i [] improved this approach substatially. I[] itwas show that the algorithm from [] is equivalet to a projectio method where i every step the

2 eigevalue problem is projected oto a two-dimesioal space. This iterpretatio suggested a further ehacemet to the method of ybeko ad Va Loa. Fially, by exploitig symmetry properties of the pricipal eigevector, the methods i [] ad [] were accelerated i [6]. If the bisectio scheme i a method of the last paragraph is started with a poor upper boud for, a large umber of bisectio steps may be ecessary to get a suitable iitial value for the subsequet root dig method. Usually the domiat share of the cost occurs i the bisectio phase, ad a good upper boud for is of predomiat importace. ybeko adva Loa [2] preseted a upper boud for which ca be obtaied from the data determied i Durbi's algorithm for the Yule-Walker system. Dembo [3] derived tighter bouds by usig (liear ad quadratic) Taylor expasios of the secular equatio. I a recet paper Melma [2] improved these bouds i two ways, rst by cosiderig ratioal approximatios of the secular equatio ad, secodly, by exploitig symmetry properties of the pricipal eigevector i a similar way as i [6]. Apparetly, because of the somewhat complicated ature of their aalysis, he restricted his ivestigatios to ratioal approximatios of at most third order. I this paper we prove that Melma's bouds obtaied by rst ad third order ratioal approximatios ca be iterpreted as the smallest eigevalues of projected problems of dimesio 2 ad 3, respectively, where the matrix T is projected oto a Krylov space of T ;.Thisiterpretatio agai proves the fact that the smallest roots of the approximatig ratioal fuctios are upper bouds of the smallest eigevalue, avoidig the somewhat complicated aalysis of the ratioal fuctios. Moreover, it suggests a method to obtai improved bouds i a systematic way by icreasig the dimesio of the Krylov space. The paper is orgaized as follows. I Sectio 2 we briey sketch the approaches of Dembo ad Melma ad prove that Melma's bouds ca be obtaied from a projected eigeproblem. I Sectio 3 we cosider secular equatios characterizig the smallest odd ad eve eigevalue of T ad take advatage of symmetry properties of the pricipal eigevector to improve the eigevalue bouds. Fially, i Sectio 4 we preset umerical results. 2 Ratioal approximatio ad projectio Let T =(t ji;jj ) i j= ::: 2 IR ( ) be a real ad symmetrictoeplitz matrix. We deote by T j 2 IR (j j) its j-th pricipal submatrix, ad by t the vector t =(t ::: t ; ) T.If (j) (j) 2 ::: (j) j are the eigevalues of T j the the iterlacig property (k) j; (k;) j; (k) j,2 j k, holds. 2

3 We briey sketch the approaches of Dembo ad Melma. To this ed we additioally assume that T is positive deite. If is ot i the spectrum of T ; the block Gauss elimiatio of the variables x 2 ::: x of the system t ; t t T T ; ; I that characterizes the eigevalues of T yields x = (t ; ; t T (T ; ; I) ; t)x =: We assume that () < (;). The x 6=,ad () is the smallest positive root of the secular equatio f() :=;t + + t T (T ; ; I) ; t = () which may be rewritte i modal coordiates as f() =;t + + ; X j= (t T v j ) 2 (;) j ; where v j deotes the eigevector of T ; correspodig to (;) j From f() = ;t + t T T ; t = ;( ; ;tt T ; ) t t T ; ) t T ; =: (2) ;T ; ;t ad f (j) () > for every j 2 IN ad every 2 [ (;) ] it follows that the Taylor polyomial p j of degree j such that f (k) () = p (k) j (), k = ::: j, satises f() p j () for every < (;) ad p j () p j+ () forevery : Hece, the smallest positive root j of p j is a upper boud of () ad j+ j. For j =adj = 2 these upper bouds were preseted by Dembo [3], for j =3it is cotaied i Melma [2]. Improved bouds were obtaied by Melma [2] by approximatig the secular equatio by ratioal fuctios. The idea of a ratioal approximatio of the secular equatio is ot ew. Dogarra ad Sorese [4] used it i a parallel divide ad coquer method for symmetric eigevalue problems, while i [] it was used i a algorithm for computig the smallest eigevalue of a Toeplitz matrix. Melma cosidered ratioal approximatios of f where () := r j () =;t + + j () a b ; 2() :=a + b c ; 3() := a b ; + c d ; 3 <

4 ad the parameters a b c d are determied such that (k) j () = dk d k tt (T ; ; I) ; t = k t T T ;(k+) ; t k = ::: j: (3) = Thus, 2 ad 3, respectively, are the ( )-, ( )- ad ( 2)-Pade approximatios of () :=t T (T ; ; I) ; t (cf. raess []) For the ratioal approximatios r j it holds that (cf. Melma [2], Theorem 4.) r () r 2 () r 3 () f() for < (;) ad with the argumets from Melma oe ca ifer that for j = 2 ad j = 3 the iequality r j; () r j () eve holds for every less tha the smallest pole of r j. Hece, if j deotes the smallest positive rootofr j () = the () 3 2 : The ratioal approximatios r () adr 3 () tof() are of the form of a secular equatio of a eigevalue problem of dimesios 2 ad 3, respectively. Hece, there is some evidece that the roots of r ad r 3 are eigevalues of projected eigeproblems. I the followig we prove that this cojecture actually holds true. Notice that our approach does ot presume that the matrix T is positive deite. Lemma 2. Let T be a real symmetric Toeplitz matrix such that is ot i the spectrum of T ad T ;. Let e := ( ::: ) T 2 IR, ad deote by V` := spafe T ; e ::: T ;` e g the Krylov space of T ; correspodig to the iitial vector e. The ( ) e ::: (4) T ;t T ;t ;` is a basis of V`, ad the projected eigeproblem of T x = x oto V` ca be writte as ~y := t s T y = T y =: y s ~ (5) where ad = ::: `. ::: ` ::: 2`;. A = Proof For ` = the Lemma is trivial. Sice T ; e = v for ` = a basis of V is give i (4). 2 ::: `+. ::: `+ ::: 2`. A s=. ` A j = t T T ;j ;t: (6) () 4 ( t + t T v = t + T ; v =

5 Assume that (4) dees a basis of V` for some ` 2 IN, thet ;` e may be represeted as Hece T ;`; e = T ; T ;` e = T ;z T ; ;z z= = T ; e + T ; `; X j= j T ;j ;t: T ;z =: T ; e + w where t t T t T ; w = T ;z () ( t + t T w = t + T ; w = T ;z The secod equatio is equivalet to w = T ;2 ; ;z ; T X ; t = `; j= ad (4) dees a basis of V`+ for ` +. j T ;j;2 ; t ; T ; ; ;t 2 spaft t ::: T;`; tg ; ; Usig the basis of V` i (4) it is easily see that eq. (5) is the matrix represetatio of the projectio of the eigevalue problem T x = x oto the Krylov space V`. 2 Lemma 2.2 Let s ~ ad ~ be deed as i Lemma 2.. The the eigevalues of the projected problem ~ y = ~ y which are ot i the spectrum of the subpecil w = w are the roots of the secular equatio g`() :=;t + + s T ( ; ) ; s: (7) For F := (T ;t ::: T ;` ;t) the secular equatio ca be rewritte as g`() =;t + + t T F (F T (T ; ; I)F ) ; F T t: (8) Proof: The secular equatio i (7) is obtaied i the same way as the secular equatio f() =oft x = x at the begiig of this sectio by block Gauss elimiatio. The represetatio (8) is obtaied from = F T T ; F, = F T F ad s = F T t. 2 Lemma 2.3 Let s be deed i Lemma 2., ad let The the k-th derivative of` is give by `() =s T ( ; ) ; s: (k) ` () =k t T (F (F T (T ; ; I)F ) ; F T ) k+ t k : (9) 5

6 Proof: Let The yields G() :=(F T (T ; ; I)F ) ; : d d G() =G()F T FG() () `() = tt FG ()F T t i.e. eq. (9) for k =. = t T F (F T (T ; ; I)F ) ; F T F (F T (T ; ; I)F ) ; F T t = t T (F (F T (T ; ; I)F ) ; F T ) 2 t Assume that eq. (9) holds for some k 2 IN. The it follows from eq. () (k+) ` () =k t T d d f(f (F T (T ; ; I)F ) ; F T ) k+ gt = (k +)t T (F (F T (T ; ; I)F ) ; F T ) k d d (F (F T (T ; ; I)F ) ; F T )t = (k +)t T (F (F T (T ; ; I)F ) ; F T ) k F d d G()F T t = (k +)t T (F (F T (T ; ; I)F ) ; F T ) k FG()F T FG()F T t = (k +)t T (F (F T (T ; ; I)F ) ; F T ) k+2 t which completes the proof. 2 Lemma 2.4 Let F := (T ; ;t ::: T ;` ;t). The it holds that (F (F T T ; F ) ; F T ) k t = T ;k ;t for k = ::: ` () ad t T (F (F T T ; F ) ; F T ) k t = t T T ;k ;t for k = ::: 2`: (2) Proof For k = the statemet () is trivial. Let H := F (F T T ; F ) ; FT ; : The for every x 2 spa F, x := Fy, y 2 IR` ad T ;t 2 spa F yields i.e. eq. () for k =. Hx = F (F T T ; F ) ; F T T ; Fy = Fy = x F (F T T ; F ) ; F T t = HT ; ; t = T ; ; t 6

7 If eq. () holds for some k<`the it follows from T ;(k+) ; t 2 spaf (F (F T T ; F ) ; F T ) k+ t = (F (F T T ; F ) ; F T )(F (F T T ; F ) ; F T ) k t which proves eq. (). = (F (F T T ; F ) ; F T )T ;k ; t = (F (F T T ; F ) ; F T )T ; T ;(k+) ; t = HT ;(k+) ; t = T ;(k+) ; t Eq. (2) follows immediately from eq. () for k = ::: `.For `<k 2` it is obtaied from t T (F (F T T ; F ) ; F T ) k t = ((F (F T T ; F ) ; F T )`t) T ((F (F T T ; F ) ; F T ) k;`t) = (T ;` ; t)t (T ;(k;` ; t) = t T T ;k t: 2 We are ow ready to prove our mai result. Theorem 2.5: Let T be a real symmetric Toeplitz matrix such that T ad T ; are osigular. Let the matrices ad be deed i Lemma 2., ad let g`() =;t + + s T ( ; ) ; s =: ;t + + `() be the secular equatio of the projected eigeproblem (5) cosidered i Lemma 2.. The `() is the (` ; `)-Pade approximatio of the ratioal fuctio () =t T (T ; ; I) ; t: oversely, if`() deotes the (` ; `)-Pade approximatio of () ad (`) (`) 2 ::: are the roots of the ratioal fuctio 7 ;t + + `() ordered by magitude, the () j (`+) j (`) j (3) for every `<ad j 2f ::: `+g. Proof: Usig modal coordiates of the pecil w = w the ratioal fuctio `() may be rewritte as `X 2 j `() = j ; j= where j deotes the eigevalues of this pecil. Hece ` is a ratioal fuctio where the degree of the umerator ad deomiator is ot greater tha ` ; ad `, respectively. From Lemma 2.3 ad Lemma 2.4 it follows that (k) ` () = k t T (F (F T T ; F ) ; F T ) k+ t = k t T T ;(k+) ; t = (k) () for every k = ::: 2` ;. Hece ` is the (` ; `)-Pade approximatio of. 7

8 From the uiqueess of the Pade approximatio it follows that ` = `. Hece (`) (`) 2 ::: are the eigevalues of the projectio of problem T x = x oto V`, ad (3) follows from the miimax priciple. 2 Some remarks are i order:. The ratioal fuctios ad 3 costructed by Melma [2] coicide with ad 2, respectively. Hece, Theorem 2.5 cotais the bouds of Melma. Moreover it provides a method to compute these bouds which is much more trasparet tha the approach of Melma. 2. Obviously the cosideratios above apply to every shifted problem T ; I such that is ot i the spectra of T ad T ;. Notice that the aalysis of Melma [2] is oly valid if is a lower boud of (T ). 3. I the same waylower bouds of the maximum eigevalue of T ca be determied. These geeralize the correspodig results by Melma [2] where we do ot eed a upper boud of the largest eigevalue of T. 3 Exploitig symmetry of the pricipal eigevector If T 2 IR ( ) is a real ad symmetric Toeplitz matrix ad E deotes the - dimesioal ipmatrix with oes i its secodary diagoal ad zeros elsewhere, the E 2 = I ad T = E T E. Hece T x = x if ad oly if T (E x)=e T E 2 x = E x ad x is a eigevector of T if ad oly if E x is. If is a simple eigevalue of T the from kxk 2 = ke xk 2 we obtai x = E x or x = ;E x.wesaythata eigevector x is symmetric ad the correspodig eigevalue is eve if x = E x, ad x is called skew-symmetric ad is odd if x = ;E x. Oe disadvatage of the projectio scheme i Sectio 2 is that it does ot reect the symmetry properties of the pricipal eigevector. I this sectio wepresetavariat which takes advatage of the symmetry of the eigevector ad which essetially is of equal cost to the method cosidered i Sectio 2. To take ito accout the symmetry properties of the eigevector we elimiate the variables x 2 ::: x ; from the system where ~ t =(t ::: t ;2 ) T. t ; t ~ T t ; ~t T ;2 ; I E ;2 t ~ t ; t ~ T E ;2 t ; A x = (4) 8

9 The every eigevalue of T which is ot i the spectrum of T ;2 is a eigevalue of the two-dimesioal oliear eigevalue problem t ; ; tt ~ (T ;2 ; I) ; t ~ t; ; ~ tt (T ;2 ; I) ; E ;2 t ~ x t ; ; ~ t T E ;2 (T ;2 ; I) ; t ~ t ; ; tt ~ (T ;2 ; I) ; t ~ =: x (5) Moreover, if is a eve eigevalue of T,the( ) T is the correspodig eigevector of problem (5), ad if is a odd eigevalue of T the ( ;) T is the correspodig eigevector of system (5). Hece, if the smallest eigevalue () is eve, the it is the smallest root of the ratioal fuctio f + () :=;t ; t ; + + ~ tt (T ;2 ; I) ; (~ t + E;2 ~ t) (6) ad if () is a odd eigevalue of T the it is the smallest root of f ; () :=;t + t ; + + ~ t T (T ;2 ; I) ; (~ t ; E;2 ~ t): (7) Aalogously to the proofs give i Sectio 2, we obtai the followig results for the odd ad eve secular equatios. Theorem 3. Let T be a real symmetric Toeplitz matrix such that is ot i the spectrum of T ad of T ;2. Let t := t ~ E;2 t, ~ ad let V` := spa e T ; e ::: T ;` be the Krylov space of T ; correspodig to the iitial vector e := ( ::: ) T. The 8 > < is a basis of V`. >: e T ; ;2t A ::: T ;` ;2t The projectio of the eigeproblem T x = x oto V` ca be writte as ~ y := t t ; s T y = T y =: s ~ y (8) e o 9 >= A> where = ::: `. :::. ` ::: 2`; A = 2 ::: `+. :::. `+ ::: 2` A s =. ` A (9) ad j =:5~ t T T ;j ;2 ~ t =(~ t E;2 ~ t) T T ;j ;2 ~ t: (2) 9

10 The eigevalues of the projected problem (8) which are ot i the spectrum of the subpecil w = w are the roots of the secular equatio g () =;t t ; + + s T ( ; ) ; s =: ;t t ; + + ` () =: (2) Here, ` () is the (` ; `)-Pade approximatio of the ratioal fuctio () :=~ tt (T ;2 ; I) ; (~ t E;2 ~ t): oversely, if` () deotes the (` ; `)-Pade approximatio of () ad (`) is the smallest root of the ratioal fuctio 7 t t ; ; + ` () = the () mi( (`+) + (`+) ; ) mi( (`) + (`) ;): As i the prvious sectio, for ` =ad` = 2 Theorem 3. cotais the bouds which were already preseted by Melma [2] usig ratioal approximatios of the eve ad odd secular equatios (6) ad (7). 4 Numerical results To establish the projected eigevalue problem (7) oe has to compute expressios of the form j = t T T ;t ;j j = ::: 2`: For ` = the quatities ad 2 are obtaied from the solutio z of the Yule- Walker system T ; z = ;t which ca be solved ecietly by Durbi's algorithm (cf. [6], p. 95) requirig 2 2 ops. Oce z is kow = t T z ad 2 = kz k 2 2. To icrease the dimesio of the projected problem by oe we have to solve the liear system T ; z`+ = z` (22) ad we have to compute two scalar products 2`+ =(z`+ ) T z` ad 2`+2 = kz`+ k 2 2. System (22) ca be solved ecietly i oe of the followig two ways. Durbi's algorithm for the Yule-Walker system supplies a decompositio LT ; L T = D where L isalower triagular matrix (with oes i its diagoal) ad D is a diagoal matrix. Hece, for every ` the solutio of eq. (22) requires 2 2 ops. This method for (22) is called Leviso-Durbi algorithm. For large dimesios eq. (22) ca be solved usig the Gohberg-Semecul formula for the iverse T ; ; (cf. [5]) T ; ; = ; y T t( : ; 2) (GGT ; HH T ) (23)

11 where G := ::: y ::: y 2 y ::: y ;2 y ;3 y ;4 ::: A ad H := ::: y ;2 ::: y ;3 y ;2 ::: y y 2 y 3 ::: A are Toeplitz matrices ad y deotes the solutio of the Yule-Walker system T ;2 y = t( : ; 2). The advatages associated with eq. (23) are at had. Firstly, the represetatio of the iverse of T ; requires oly storage elemets. Secodly, the matrices G, G T, H ad H T are Toeplitz matrices, ad hece the solutio T ; z` ca be calculated i oly O( log ) ops usig fast Fourier trasform. Experimets show that whe 52 this approach is actually more eciet tha the Leviso-Durbi algorithm. I the method of Sectio 3 we alsohavetosolveayule-walker system T ;2 z = ~ t by Durbi's algorithm, ad icreasig the dimesio of the projected problem by oe we have to solve oe geeral system T ;2 z`+ = z` usig the Leviso-Durbi algorithm or the Gohberg-Semecul formula. Moreover, two vector additios z`+ E ;2 z`+ ad 4 scalar products have to be determied, ad 2 eigevalue problems of very small dimesios have to be solved. To summarize, agai O() ops are required to icrease the dimesio of the projected problem by oe. If the gap betwee the smallest eigevalue () ad the secod eigevalue () 2 is large, the sequece of vectors coverges very fast to the pricipal eigevector of z` T ad the matrix becomes early sigular. I three of 6 examples that we cosidered the matrix eve became (umerically) ideite. However, i all of these examples the relative error of the eigevalue approximatio of the previous step was already ;8. I a forthcomig paper we will discuss a stable versio of the projectio methods i Sectios 2 ad 3. Example To test the bouds we cosidered the followig class of Toeplitz matrices T = m X k= k T 2k (24) where m is chose such that the diagoal of T is ormalized to t =, T =(T ij )=(cos((i ; j))) ad k ad k are uiformly distributed radom umbers i the iterval [ ] (cf. ybeko adva Loa [2]). Table cotais the average of the relative errors of the bouds of Sectio 2 i test problems for each of the dimesios = 32, 64, 28, 256, 52 ad 24. Table 2 shows the correspodig results for the bouds of Sectio 3. I both tables

12 TALE. Average of relative errors bouds of Sectio 2 dim ` = ` =2 ` =3 ` =4 32 :5 E + 4:29 E ; 2 8:38 E ; 3 :82 E ; 3 64 :64 E + 6:4 E ; 2 :38 E ; 2 4:2 E ; :76 E + 7:6 E ; 2 :88 E ; 2 5:9 E ; :3 E + 9:25 E ; 2 :78 E ; 2 6:2 E ; :5 E + : E ; 2:47 E ; 2 6:8 E ; 3 24 :65 E + :5 E ; 2:43 E ; 2 6:6 E ; 3 TALE 2. Average of relative errors bouds of Sectio 3 Dimesio ` = ` =2 ` =3 ` =4 32 5:8 E ; 8:33 E ; 3 8:54 E ; 4 3:2 E ; :39 E ; 2:3 E ; 2 :25 E ; 3 3:65 E ; 4 28 :79 E + 2:4 E ; 2 :6 E ; 3 6:4 E ; :27 E + 4:25 E ; 2 4:58 E ; 3 7:5 E ; : E + 5:43 E ; 2 4:9 E ; 3 8:77 E ; 4 24 : E + 5:45 E ; 2 4:8 E ; 3 7:42 E ; 4 2

13 TALE 3. Average of commo logarithm of relative errors bouds of Sectio 2 dim ` = ` =2 ` =3 ` =4 32 ;: (:32) ;:93 (:98) ;3:76 (2:5) ;6:3 (3:8) 64 :4 (:26) ;:65 (:89) ;3:5 (2:9) ;5:26 (3:43) 28 :4 (:9) ;:63 (:3) ;3:4 (2:62) ;5:5 (3:65) 256 :64 (:22) ;:4 (:84) ;2:64 (:8) ;4:42 (3:9) 52 :8 (:27) ;:35 (:77) ;2:23 (:9) ;3:74 (2:28) 24 :4 (:24) ;:37 (:76) ;2:37 (:4) ;3:93 (2:7) TALE 4. Average of commo logarithm of relative errors bouds of Sectio 3 dim ` = ` =2 ` =3 ` =4 32 ;:45 (:37) ;3:38 (:76) ;6:88 (3:65) ;:28 (4:2) 64 ;:29 (:3) ;2:72 (:46) ;5:88 (2:93) ;9:27 (3:85) 28 :7 (:23) ;2:79 (:8) ;5:84 (3:37) ;9:27 (3:93) 256 :44 (:24) ;2:32 (:4) ;5:7 (2:96) ;8:42 (4:) 52 :65 (:27) ;2:5 (:35) ;4:88 (2:93) ;8: (4:26) 24 :97 (:23) ;2:5 (:4) ;5:3 (3:5) ;8:2 (4:47) the rst two colums cotai the relative errors of the bouds give by Melma. The experimets clearly show that exploitig symmetry of the pricipal eigevector leads to sigicat improvemets of the bouds. The mea values of the relative errors do ot reect the quality of the bouds. Large bouds are take ito accout with a much larger weight tha small oes. To demostrate the average umber of correct leadig digits of the bouds i Table 3 ad Table 4 we preset the mea values of the commo logarithms of the relative errors. I parethesis we added the stadard deviatios. Ackowledgemet Thaks are due to Wolfgag Mackes for stimulatig discussios. Refereces [] D. raess, Noliear Approximatio Theory. Spriger Verlag, erli 986 [2] G. ybeko ad. Va Loa, omputig the miimum eigevalue of a symmetric positive deite Toeplitz matrix. SIAM J. Sci. Stat. omput. 7 (986),

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