Symmetric Schemes for Computing the Minimum Eigenvalue of a Symmetric Toeplitz Matrix

Symmetric Schemes for Computing the Minimum Eigenvalue of a Symmetric Toeplitz Matrix

Heinrich Voss

Technical University Hamburg{Harburg, Section of Mathematics, D{21071 Ham- burg, Federal Republic of Germany, e-mail: voss @tu-harburg.de

Dedicated to Ludwig Elsner on the occasion of his 60th birthday

Abstract

In 8] and 9] W. Mackens and the present author presented two generaliza- tions of a method of Cybenko and Van Loan 4] for computing the smallest eigenvalue of a symmetric, positive de nite Toeplitz matrix. Taking advan- tage of the symmetry or skew symmetry of the corresponding eigenvector both methods are improved considerably.

Keywords. Toeplitz matrix, eigenvalue problem, projection method, symmetry

1 Introduction

Several approaches have been reported in the literature for computing the small- est eigenvalue of a real symmetric, positive denite Toeplitz matrix (RSPDT). This problem is of considerable interest in signal processing. Given the covariance se- quence of the observed data, Pisarenko 10] suggested a method which determines the sinusoidal frequencies from the eigenvector of the covariance matrix associated with its minimum eigenvalue.

Cybenko and Van Loan 4] presented an algorithm which is a combination of bisec- tion and Newton's method for the secular equation. Replacing Newton's method by a root nding method based on rational Hermitian interpolation of the secular equa- tion Mackens and the present author in 8] improved this approach substantially. In 9] it was shown that the algorithm from 8] is equivalent to a projection method where in every step the eigenvalue problem is projected to a two dimensional space.

This interpretation suggested a further enhancement of Cybenko and Van Loan's method.

1

IfTⁿ²IR⁽ⁿⁿ⁾ is a RSPDT matrix and Eⁿ denotes the (nn) ipmatrix with ones in its secondary diagonal and zeros elsewhere, then Eⁿ² =I and Tⁿ =EⁿTⁿEⁿ. Hence Tⁿx = x if and only if

Tⁿ(Eⁿx) = EⁿTⁿEⁿ²x = Eⁿx

and x is an eigenvector of Tⁿ if and only if Eⁿx is. If is a simple eigenvalue of Tⁿ then from ^kx^k2 = ^kEⁿx^k2 we obtain x = Eⁿx or x = ^;Eⁿx. We say that an eigenvector x is symmetric and the corresponding eigenvalue is even if x = Eⁿx, and x is called skew-symmetric and is odd if x =^;Eⁿx.

One disadvantage of the approximation schemes in 8] and 9] is that they do not reect the symmetry properties of the eigenvector corresponding to the minimum eigenvalue. In this paper we present variants which take advantage of the symmetry of the eigenvector and which essentially are of equal cost as the methods considered in 8] and 9].

The symmetry class of the principal eigenvector is known in advance only for a small class of Toeplitz matrices. The following result was given by Trench 11]:

Theorem 1

Tⁿ = (t^ji;jj)^ij=1:::n t^j := 1

Z

0

F()cos(j)d j = 012:::n^;1

where F : (0) ^!IR is nonincreasing and F(0+) =: M > m := F(^;):Then for every n the matrix Tⁿ has n distinct eigenvalues in (mM), its even and odd spectra are interlaced, and its largest eigenvalue is even.

IfTⁿsatises the conditions of Theorem 1 then for evenn the principal eigenvector is odd and vice versa. For general Toeplitz matricesTⁿ the symmetry class is detected by the algorithm at negligible cost.

The paper is organized as follows. In Section 2 we briey sketch the algorithms from 8] and 9]. Sections 3 and 4 describe their generalizations if the symmetry class of the principal eigenvector is taken into account. Finally, some concluding remarks are made in Section 5.

2 Nonsymmetric methods

In this section we briey review the approach to the computation of the smallest eigenvalue of a RSPDT matrix which was presented in 8] and 9].

Let Tⁿ = (t^ji;jj)^{ij=1:::n 2}IR⁽ⁿⁿ⁾

be a RSPDT matrix. We denote by T^{j 2}IR^(jj) itsj-th principal submatrix, and we assume that its diagonal is normalized by t⁰ = 1. If ^{(j)1 (j)2} :::^(j)j are the

2

eigenvalues of T^j then the interlacing property ^{(k )j;1 (k ;1)j;1 (k )j} , 2j k n, holds.

Eliminating the variables x²:::xⁿ from the system

1^; t^T

t T^n;1;I

!

x = 0 that characterizes the eigenvalue of Tⁿ one obtains

(1^;;t^T(T^n;1;I)^;1t)x¹ = 0:

We assume that ⁽ⁿ⁾¹ < ^(n;1)1 . Then x^{1 6}= 0, and ⁽ⁿ⁾¹ is the smallest root of the secular equation

f() :=^;1 + + t^T(T^n;1;I)^;1t = 0: (1) f is strictly monotonely increasing and strictly convex in the interval (0^(n;1)1 ).

Therefore for every initial guess ^{0 2} (^{(n)1 (n;1)1} ) Newton's method converges monotonely decreasing and quadratically to ⁽ⁿ⁾¹ . Since

f⁰() = 1 +^k(T^n;1;I)^;1t^k22 a Newton step can be performed in the following way:

Solve (T^n;1;kI)y =^;t for y, and set ^{k +1} =^{k; ;}1 +^{k ;}y^Tt 1 +^ky^k22

where the Yule { Walker system (T^{n;1 ;}I)y = ^;t can be solved by Durbin's algorithm (cf. 6], p. 195) requiring 2n² ops.

An initial guess ⁰ for Newton's method can be obtained by a bisection process. If is not in the spectrum of any of the submatrices T^j;I then Durbin's algorithm for (T ^;I)=(1^;) determines a lower triangular matrix L such that

1^;1L(T ^;I)L^T = diag^f1¹:::^n;1g: Hence, from Sylvester's law of inertia we obtain that

(i) < ⁽ⁿ⁾¹ if ^j > 0 for j = 1:::n^;1,

(ii) ²(^{(n)1 (n;1)1} ) if ^j > 0 for j = 1:::n^;2 and ^n;1< 0 (iii) > ^(n;1)1 if ^j < 0 for some j ^2f1:::n^;2^g.

Cybenko and Van Loan combined a bisection method with Newton's method for computing the minimum eigenvalue of Tⁿ.

Since the smallest root ⁽ⁿ⁾¹ and the smallest pole ^(n;1)1 of the rational function f can be very close to each other usually a large number of bisection steps is needed

3

to get a suitable initial approximation of Newton's method. Moreover, the global convergence behaviour of Newton's method can be quite unsatisfactory. In 8] the approach of Cybenko and Van Loan was improved substantially using a root nding method which is based on a rational model

g() := f(0) + f⁰(0) + ² b c^;

where is the current approximation of ⁽ⁿ⁾¹ , andb and c are determined such that g() = f() g⁰() = f⁰():

It is shown that for ^{k 2} (^{(n)1 (n;1)1} ) the function g(^k) has exactly one zero ^{k +1 2}(0^k) and that

⁽ⁿ⁾¹ < ^{k +1} < ^{k ;}f(^k)=f⁰(^k):

Hence, the sequence^fkgconverges monotonely decreasing to⁽ⁿ⁾¹ , the convergence is quadratic and faster than the convergence of Newton's method. The essential cost of one step are the same as for one Newton step.

In 9] it was shown that the smallest root of g() is the smallest eigenvalue of the projected eigenvalue problem

Q^TTⁿQ = Q^TQ (2)

where

Q = (q(0)q())²IR⁽ⁿ²⁾

and q() := (T^n;I)^;1e¹, e¹ = (10:::0)^T. This interpretation suggests general- izations of the method where the problem is projected to subspaces

span^fq(¹):::q(^k)^g

of the same type of increasing order k where the parameters ^j are constructed in the course of the algorithm. The resulting method was shown to be at least cubically convergent.

The representation in (2) clearly demonstrates a weakness of the approaches in 8]

and 9]: Although the eigenvector corresponding to ⁽ⁿ⁾¹ is known to be symmetric or skew-symmetric the trial vectors in the projection method have neither of these properties.

4

3 Exploiting symmetry in rational interpolation

In this section we discuss a variant of the approximation schemefrom 8] that exploits the symmetry and skew-symmetry of the corresponding eigenvector, respectively.

To take into account the symmetry properties of the eigenvector we eliminate the variables x²:::x^n;1 from the system

0

B

@

1^; ~t^T t^n;1

~t T^n;2;I E^n;2~t t^n;1 ~t^TE^n;2 1^;

1

C

Ax = 0 (3)

where ~t = (t¹:::t^n;2)^T.

Then every eigenvalue of Tⁿ which is not in the spectrum ofT^n;2 is an eigenvalue of the two dimensional nonlinear eigenvalue problem

1^;;~t^T(T^n;2;I)^;1~t t^n;1;~t^T(T^n;2;I)^;1E^n;2~t t^n;1;~t^TE^n;2(T^n;2;I)^;1~t 1^;;~t^T(T^n;2;I)^;1~t

!

x¹ xⁿ

!

= 0:

Moreover, if such a is an even eigenvalue of Tⁿ, then (11)^T is the corresponding(4) eigenvector of problem (4), and if is an odd eigenvalue of Tⁿ then (1^;1)^T is the corresponding eigenvector of system (4).

Hence, if the smallest eigenvalue ⁽ⁿ⁾¹ is even, then it is the smallest root of the rational function

g⁺() :=^;1^;t^n;1+ + ~t^T(T^n;2;I)^;1(~t + E^n;2~t) (5) and if ⁽ⁿ⁾¹ is an odd eigenvalue of Tⁿ then it is the smallest root of

g^;() :=^;1 +t^n;1+ + ~t^T(T^n;2;I)^;1(~t^;E^n;2~t): (6) If the symmetryclass of the principal eigenvector is known in advance then a straight forward generalization of the scheme in 8] can be based on (5) or (6), respectively.

In the general case it is the minimum of the smallest roots of g⁺ and g^;, and the symmetry class must be detected by the method itself.

The elimination of x²:::x^n;1 is nothing else but exact condensation of the eigen- value problem Tx = x where x¹ and xⁿ are chosen to be masters and x²:::x^n;1 are the slaves. If¹:::^n;2 denotes an orthonormal set of eigenvectors of the slave problem

T^n;2j =^{(n;2)j j} j = 1:::n^;2 then the functions g⁺ and g^; can be written as (cf. 7])

g() = g(0) +g⁰(0) + ^2n;2X

j=1

^2j

^{(n;2)j ;} (7)

5

0 0.05 0.1 0.15 0.2

−5

−4

−3

−2

−1 0 1 2 3 4 5

f f

f f g+

g+

g+ g−

g−

Fig 1: Graphs of f, g⁺ and g^; where

g(0) =^;1t^n;1+ ~t^TT^n;2;1 (~tE^n;2~t)

g⁰ (0) = 1 + ~t^TT^n;2;2 (~tE^n;2~t) = 1 + 0:5^kT^n;2;1 (~tE^n;2~t)^k22 and ^j = 1^(n;1)j (^j)^T(~tE^n;2~t):

Hence, the zeros ofg⁺ andg^; are the even and odd eigenvalues ofTⁿ, and the poles of g⁺ andg^; are the even and odd eigenvalues of T^n;2, respectively. Figure 1 shows the graphs of the functions f, g⁺ and g^; for a Toeplitz matrix of dimension 32.

If we are given an approximation of ⁽ⁿ⁾¹ then equation (7) suggests the following rational Hermitian approximation of g():

h() = g(0) +g⁰ (0) + ² b

c^; (8)

where the parameters b and c are determined from the Hermitian interpolation conditions

h() = g() h⁰() = g⁰ (): (9) Following the lines of the proof of Theorem 1 in 8] one gets the basic properties of h.

Theorem 2

Let ! be the smallest pole of g, let ² 0!), and let h be dened by equations (8) and (9). Then it holds that

6

(i) b> 0 and c> ,

whence h is strictly monotonely increasing and strictly convex in 0c). (ii) h(⁽ⁿ⁾¹ )< 0.

From Theorem 2 we deduce the following method for computing the smallest eigen- value of a RSPDT matrixTⁿ. Set = 0 as a lower bound of the smallest eigenvalue and let the variable monitor whether ⁽ⁿ⁾¹ is even or odd or the type of⁽ⁿ⁾¹ is not yet known:

=

8

>

>

<

>

>

:

;1 if ⁽ⁿ⁾¹ is odd

0 if the type of ⁽ⁿ⁾¹ is unknown

1 if ⁽ⁿ⁾¹ is even (10)

To obtain an upper bound of ⁽ⁿ⁾¹ solve the Yule { Walker system T^n;2z =^;~t, and let z⁺ :=z + E^n;2z and z^; :=z^;E^n;2z. Then

g(0) =^;1t^n;1;~t^Tz g⁰ (0) = 1 + 0:5^kz^k22

and from the monotonicity and convexity of g⁺ and g^; in 0⁽ⁿ⁾¹ ] it follows that := min^f;g⁺(0)=g⁺⁰ (0)^;g^;(0)=g^0;(0)^g

is an upper bound of ⁽ⁿ⁾¹ .

Choose ^{0 2} (0], set k := 0, and do the following steps until convergence of the sequence ^fkg:

1. Solve (T^n;2;kI)y = ^;~t using Durbin's algorithm and determine whether ^{k (n;2)1} or not.

2. If ^{k (n;2)1} then do a bisection step:

:= ^{k k +1} := 0:5( + ) otherwise obtain new bounds of ⁽ⁿ⁾¹ in the following way:

{ if >^;1 then determine g⁺(^k). If = 1 and g⁺(^k) < 0 then := ^k is an improved lower bound

{ if < 1 then determine g^;(^k). If =^;1 and g^;(^k) < 0 then := ^k is an improved lower bound

{ If = 0 and g⁺(^k) < 0 and g^;(^k) < 0 then := ^k is an improved lower bound of ⁽ⁿ⁾¹

{ if = 0 and g^;(^k)< 0 < g⁺(^k) then ⁽ⁿ⁾¹ < ^k is the smallest root of g⁺. Set := 1

{ if = 0 and g⁺(^k)< 0 < g^;(^k) then ⁽ⁿ⁾¹ < ^k is the smallest root of g^;. Set :=^;1

7

{ if > ^;1 computeg⁰⁺(^k) and determinethe smallest root⁺ofg⁺(^k) else set ⁺= 1.

{ if < 1 compute g^0;(^k) and determine the smallest root^; ofg^;(^k) else set ^;= 1.

{

To check the convergence we use the following lower bound of ⁽ⁿ⁾¹ of 8].

Lemma 3

Let 0 < ⁽ⁿ⁾¹ < < ^(n;1)1 , and let ⁽ⁿ⁾¹ be the smallest positive root of g,

2f+^;g. Letp be the quadratic polynomial satisfying the interpolation conditions p() = g() p⁰() = g⁰() p() = g():

Then p has a unique root ²() ^{and (n)1 .}

The convergence behaviour is the same as for the nonsymmetric method: ^{k0 2} (^{(n)1 (n;2)1} ) for somek⁰. Fork k⁰ the sequence^fkgconverges quadratically and monotonely decreasing to⁽ⁿ⁾¹ , and it converges faster than Newton's method forg, where ^{2 f}+^;g such that g(⁽ⁿ⁾¹ ) = 0. Notice that ^{(n;1)1 (n;2)1} . Hence, the symmetric method usually will need a smaller number of bisection steps to reach its monotonely decreasing phase than its nonsymmetric counterpart.

To test the improvement upon the nonsymmetric method we considered Toeplitz matrices

T = m^Xn

k =1

^kT^{2 k} (11)

where m is chosen such that the diagonal of T is normalized to 1, T = (t^ij) = (cos((i^;j)))

and ^k and ^k are uniformly distributed random numbers taken from 01] (cf. Cy- benko, Van Loan 4]).

Table 1 contains the average number of ops and the average number of Durbin steps needed to determine the smallest eigenvalue in 100 test problems with each of the dimensionsn = 32, 64, 128, 256, 512 and n = 1024 for the methods based on rational Hermitian interpolation. The iteration was terminated if Lemma 3 guaranteed the relative error to be less than 10^;6.

8

dimension non-symmetric method from 8] symmetric method

ops steps ops steps

32 1:071 E04 4:55 9:087 E03 (84:9%) 3:75 64 4:545 E04 5:19 3:653 E04 (80:4%) 4:12 128 1:695 E05 5:01 1:407 E05 (83:0%) 4:14 256 7:310 E05 5:50 6:046 E05 (82:7%) 4:55 512 3:297 E06 6:25 2:597 E06 (78:8%) 4:92 1024 1:352 E07 644 1:065 E07 (78:8%) 5:08

Tab. 1.

4 A symmetric projection method

The root nding method of the last section can be interpreted as a projection method where in each step the eigenvalue problem is projected to a 2 dimensional space.

Similarly as in 9] this follows easily from

Theorem 4

Let e¹ and eⁿ be the unit vector containing a 1 in its rst and last component, respectively, and for not in the spectrum of T^{n and} T^{n;2 let}

p() :=^;g()(T^n;I)^;1(e¹eⁿ):

Then

p() =

0

B

@

z1()

1

1

C

A ^where z() :=^;(T^n;2;I)^;1(~tE^n;2~t) (12) and it holds that

p()^TTp() = 2

8

>

>

<

>

>

:

;g() + g⁰ () for =

;g() + g()^;g()

^{; for 6}=

and

p()^Tp() = 2

8

>

>

<

>

>

:

g⁰ () for = g()^;g()

^{; for 6}=:

Proof

0

B

@

1^; ~t^T t^n;1

~t T^n;2;I E^n;2~t t^n;1 ~t^TE^n;2 1^;

1

C

A 0

B

@

z1()

1

1

C

A

=

0

B

@

1^; + ~t^Tz()t^n;1

~t+ (T^n;2;I)z()E^n;2~t t^n;1+ ~t^TE^n;2z()1

1

C

A=^;g()

0

B

@

10

1

1

C

A: 9

If is not in the spectrum of Tⁿ then

p()^TTⁿp() = ^;g()p()^T(T^n;I + I)(T^n;I)^;1(e¹ eⁿ)

= ^;g()p()^T(e¹eⁿ) +p()^Tp()

= ^;2g() + p()^Tp()

and for not in the spectrum of Tⁿ the symmetry of Tⁿ yields

p()^TTⁿp() =^;2g() + p()^Tp(): (13) Hence for ⁶=

p()^Tp() = 2g()^;g()

^;

and from eqn. (13) we get

p()^TTⁿp() =^;2g() + 2g()^;g()

^; :

Finally, for = one obtains from eqns. (12), (5) and (6)

kp()^k22 = 2 +^kz()^k22 = 2g⁰ () and from eqn. (13)

p()^TTⁿp() =^;2g() + 2g⁰(): ²

Theorem 4 suggests the following type of projection method for computing the smallest eigenvalue of a RSPDT matrixTⁿ:

(i) Choose parameters¹:::^k (not in the spectrum of Tⁿ) and solve the linear systems

(T^n;kI)p(^k) =^;g(^k)(e¹eⁿ)

(ii) Determine the smallest eigenvalues of the projected problems

(Q^k)^TTⁿQ^ky = (Q^k)^TQ^ky (14) where

Q^k := (p(¹):::p(^k))²IR^{(nk )}: (iii) = min^f+;g

By Theorem 4 the entries of the projected matrices A^k := (Q^k)^TTⁿQ^k and B^k:=

(Q^k)^TQ^k are given by (we divided all entries by 2) a^ij =

8

>

>

<

>

>

:

;g(ⁱ) +ⁱg⁰(ⁱ) if i = j

;g(ⁱ) + g(ⁱ)^;g(^j)

^{i;j i} if i ⁶=j (15) 10

and

b^ij =

8

>

>

<

>

>

:

g⁰ (ⁱ) if i = j g(ⁱ)^;g(^j)

^i;j if i⁶=j: (16)

In the algorithm to follow we will construct the parameters ^j in the course of the method. Increasing the dimension of the projected problem by one (adding one parameter) essentially requires the solution of one Yule { Walker system and a small number of level one operations to compute g⁺(^k) and g^;(^k). Then the matrices A^k and B^k can be updated easily from the matrices of the previous step.

Symmetric projection method

Let = 0, ¹ = 0, and dene as in eqn. (10). Solve the linear system T^n;2z =^;~t, and set z :=zE^n;2z. Compute

g(0) =^;1t^n;1;~t^Tz g⁰ (0) = 1 + 0:5^kz^k22 set A¹ := (^;g(0))²IR⁽¹¹⁾ and B¹ := (g⁰ (0))²IR⁽¹¹⁾ and := min^f;g⁺(0)=g⁺⁰ (0)^;g^;(0)=g^;0 (0)^g:

Choose any ^{2 2}(0] and set k := 2.

Repeat the following steps until convergence of the sequence^fkg: (i) Solve the system

(T^n;2;kI)z =^;~t

by Durbin's algorithm and determine whether ^k < ^(n;2)1 or ^{k (n;2)1} . (ii) If ^{k (n;2)1} then set

:= min^fkg and ^k := 0:5( + ) else

{ if >^;1 then determine g⁺(^k). If = 1 and g⁺(^k) < 0 then := ^k is an improved lower bound

{ if < 1 then determine g^;(^k). If =^;1 and g^;(^k) < 0 then := ^k is an improved lower bound

{ If = 0 and g⁺(^k) < 0 and g^;(^k) < 0 then := ^k is an improved lower bound of ⁽ⁿ⁾¹

{ if = 0 and g^;(^k)< 0 < g⁺(^k) then ⁽ⁿ⁾¹ < ^k is the smallest root of g⁺. Set := 1

{ if = 0 and g⁺(^k)< 0 < g^;(^k) then ⁽ⁿ⁾¹ < ^k is the smallest root of g^;. Set :=^;1

11

{ if >^;1 computeg⁺⁰(^k), update the matricesA^+k andB^k+and determine the smallest eigenvalue ⁺ of the k dimensional projected problem else set⁺ = 1.

{ if < 1 compute g^0;(^k), update the matricesA^;k andB^k;and determine the smallest eigenvalue ^; of the k dimensional projected problem else set^; = 1.

{

{

{

The convergence properties are obtained in the same way as in 9]: Since for ² (0^(n;2)1 ) the smallest positive root of g() is the smallest eigenvalue of the projected problem (14) (for k = 2, ¹ = 0 and ² = ) the symmetric projection method converges eventually monotonely decreasing and faster than the symmetric method from Section 3. Comparing it to the Rayleigh quotient iteration it can even be shown to be cubically convergent (cf. 9], Theorem 5).

We tested the symmetric projection method using the RSPDT matrices from (11).

In the algorithm above we took into account only vectors p(^j) if ^j < ^(n;2)1 . In this case Durbin's algorithm is known to be stable (cf. 3]). Additionally we considered a projection method (complete projection) where p(^j) was included into the projection scheme even if a bisection step was performed since^j > ^(n;2)1 . Although in the latter case Durbin's algorithm is not guaranteed to be stable we did not observe unstable behaviour. We compared the methods to the nonsymmetric counterpart of the method from Section 3 based on rational Hermitian interpolation.

dimension stable projection complete projection

ops steps ops steps

32 1:124 E04 (105:0%) 3:69 1:117 E04 (104:4%) 3:60 64 3:776 E04 (83:1%) 3:97 3:574 E04 (78:6%) 3:72 128 1:399 E05 (82:5%) 4:04 1:330 E05 (78:5%) 3:81 256 5:863 E05 (80:2%) 4:39 5:425 E05 (74:2%) 4:03 512 2:410 E06 (73:1%) 4:56 2:202 E06 (66:8%) 4:15 1024 9:982 E06 (73:8%) 4:76 8:879 E06 (65:7%) 4:21

Tab. 2.

5 Concluding remarks

We have presented symmetric versions of the methods introduced in 8] and 9] for computing the smallest eigenvalue of a real symmetric and positive denite Toeplitz matrix which improve their nonsymmetric counterparts considerably. In our numeri- cal tests we used Durbin's algorithm to solve Yule { Walker systems and to determine

12

the location of parameters in the spectrum of T^n;2. This information can be gained from superfast Toeplitz solvers (cf. 1], 2], 5]) as well. Hence the computational complexity can be reduced to O(nlog²n) operations.

References

1] G.S. Ammarand W.B. Gragg, The generalizedSchur algorithm for the superfast solution of Toeplitz systems. in J. Gilewicz, M. Pindor, W. Siemaszko, eds., Lecture Notes in Mathematics 1237, Berlin, 1987, pp. 315 | 330,

2] G.S. Ammar and W.B. Gragg, Numerical experience with a superfast real Toeplitz solver. Lin. Alg. Appl., 121 (1989), pp. 185 | 206

3] G. Cybenko, The Numerical Stability of the Levinson-Durbin Algorithm for Toeplitz Systems of Equations. SIAM J. Sci. Stat. Comput. 1 (1980), pp. 303

| 309

4] G. Cybenko and C. Van Loan, Computing the minimum eigenvalue of a sym- metric positive denite Toeplitz matrix. SIAM J. Sci. Stat. Comput., 7 (1986), pp. 123 | 131

5] F. de Hoog, A new algorithm for solving Toeplitz systems of equations.

Lin. Alg. Appl., 88/89 (1987), pp. 123 | 138

6] G.H. Golub and C.F. Van Loan, Matrix Computations. 3rd edition The John Hopkins University Press, Baltimore and London, 1996.

7] T. Hitziger, W. Mackens, and H. Voss, A condensation-projection method for generalized eigenvalue problems. in H. Power and C.A. Brebbia, eds., High Per- formance Computing in Engineering 1, Computational Mechanics Publications, Southampton, 1995, pp. 239 | 282.

8] W. Mackens and H. Voss, The minimum eigenvalue of a symmetric pos- itive denite Toeplitz matrix and rational Hermitian interpolation. SIAM J. Matr. Anal. Appl. 18 (1997), pp. 521 | 534

9] W. Mackens and H. Voss, A projection method for computing the minimum eigenvalue of a symmetric positive denite Toeplitz matrix. To appear in Lin. Alg. Appl.

10] V.F. Pisarenko, The retrieval of harmonics from a covariance function. Geo- phys. J. R. astr. Soc., 33 (1973), pp. 347 | 366

11] W.F. Trench, Interlacement of the even and odd spectra of real symmetric Toeplitz matrices. Lin. Alg. Appl. 195 (1993), pp. 59 | 68

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