The structured distance to normality of Toeplitz matrices with application to preconditioning

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1 The structured distace to ormality of Toeplitz matrices with applicatio to precoditioig Silvia Noschese 1, ad Lothar Reichel 2 1 Dipartimeto di Matematica Guido Casteluovo, SAPIENZA Uiversità di Roma, P.le A. Moro, 2, I-185 Roma, Italy. oschese@mat.uiroma1.it. Research supported by a grat from SAPIENZA Uiversità di Roma. 2 Departmet of Mathematical Scieces, Ket State Uiversity, Ket, OH 44242, USA. reichel@math.ket.edu. SUMMARY A formula for the distace of a Toeplitz matrix to the subspace of {e iϕ }-circulat matrices is preseted, ad applicatios of {e iϕ }-circulat matrices to precoditioig of liear systems of equatios with a Toeplitz matrix are discussed. Copyright c 26 Joh Wiley & Sos, Ltd. key words: Toeplitz matrix, circulat matrix, {e iϕ }-circulat, matrix earess problem, distace to ormality, precoditioig 1. Itroductio Toeplitz matrices arise i may applicatios i sigal processig, time series aalysis, ad the umerical solutio of partial differetial equatios; see, e.g., [1, 2, 3, 4, 5, 6]. A Toeplitz matrix δ τ 1 τ 2... τ σ 1 δ τ 1 τ 2... τ 2. σ 2 σ.. 1. T = (; σ, δ, τ) = C (1) τ1 τ2 σ 2... σ 1 δ τ 1 σ... σ 2 σ 1 δ is said to be geeralized Hermitia if, for some π < ϕ π, σ l = τ l e iϕ, 1 l <. (2) Here ad below i = 1. The scalar δ C is arbitrary ad the bar i (2) deotes complex cojugatio. We deote the set of geeralized Hermitia Toeplitz matrices i C by H. A Toeplitz matrix (1) is said to be a {e iϕ }-circulat if, for some π < ϕ π, σ l = τ l e iϕ, 1 l <. (3)

2 STRUCTURED DISTANCE TO NORMALITY 1 Agai, the scalar δ C i (1) is arbitrary. We refer to the set of {e iϕ }-circulats i C with π < ϕ π arbitrary as the set of geeralized circulats ad deote it by C. Properties of geeralized circulat matrices are discussed i, e.g., [7, Sectio 3.2.1] ad [8, Sectio 3.4.2]. I particular, they ca be diagoalized efficietly with the aid of the fast Fourier trasform. Circulat matrices are obtaied whe ϕ =. We deote the set of circulat matrices i C by C. Several approaches to characterize ormal Toeplitz matrices ca be foud i the literature; see, e.g., Fareick et al. [9], Gu ad Patto [1], ad Ito [11]. The followig result by Gu ad Patto [1], which is show by usig properties of differeces of products of Toeplitz matrices, is cetral for the developmet of the preset paper. Propositio 1.1. ([1, Theorem 3.4]) Let T deote the set of ormal Toeplitz matrices i C. The T = H C. Itroduce the set of ormal matrices N C ad let F deote the Frobeius orm. The distace of matrices i C to N has bee ivestigated i, e.g., [12, 13, 14, 15, 16, 17]. We are cocered with the structured ad ustructured distaces to ormality of Toeplitz matrices (1) i the Frobeius orm. The ustructured distace is give by d F (T) = mi N N T N F. The distace of Toeplitz matrices (1) to the algebraic variety of geeralized Hermitia Toeplitz matrices is defied as H F (T) = mi T H F. H H We refer to H F (T) as the distace to geeralized symmetry. The distace of Toeplitz matrices to the algebraic variety of geeralized circulat matrices is give by C F(T) = mi C C T C F, (4) ad we refer to C F (T) as the distace to a geeralized circulat. Fially, the distace of Toeplitz matrices (1) to the algebraic variety of circulat matrices is give by C F (T) = mi T C F, (5) C C ad is referred to as the distace to a circulat. Clearly, C F (T) C F (T). The structured distace to ormality is defied as T F (T) = mi { H F (T), C F (T)}. (6) I geeral, d F (T) is strictly smaller tha T F (T). Toeplitz matrices of order two provide a exceptio. Propositio 1.2. Let T C 2 2 be a Toeplitz matrix. The T F (T) = d F(T). Proof: The closest ormal matrix to T = (2; σ, δ, τ), i the Frobeius orm, has the diagoal δi. Therefore, the closest ormal matrix is a 2 2 Toeplitz matrix. We remark that the distace to ormality of a geeral 2 2 matrix has bee ivestigated by Causey [18]; see also Higham [13] for a discussio.

3 2 S. NOSCHESE AND L. REICHEL For certai matrices, such as baded Toeplitz matrices with suitably restricted badwidth, H F (T) may severely overestimate the distace to the algebraic variety of ormal matrices, i.e., we may have d F (T) H F (T); see Sectio 4 ad [16] for illustratios. Nevertheless, we have observed that for may Toeplitz matrices the ormalized structured distace to ormality, T F (T)/ T F, is close to the ormalized distace to ormality, d F (T)/ T F. Assume for the momet that the Toeplitz matrix T C is osigular, let b C, ad cosider the solutio of the liear system of equatios Tx = b. (7) Circulat matrices, that approximate T 1 i some sese, have bee used successfully as precoditioers for iterative methods for the solutio of (7). For istace, whe T is Hermitia ad positive defiite, T. Cha [19] proposed that the iverse of the solutio C of (5) be used as a precoditioer for the cojugate gradiet method, i.e., that the cojugate gradiet method be applied to the solutio of C 1 Tx = C 1b. (8) Sice C is a circulat, so is the precoditioer C 1. A useful property of circulat precoditioers is that matrix-vector products C 1 y for ay y C ca be evaluated i oly O( log ) arithmetic floatig-poit operatios with the aid of the fast Fourier trasform; see, e.g., [7, 8, 2] for details. May approaches to determie circulat precoditioers for Hermitia Toeplitz matrices are described ad compared i [8, 2, 21]. Circulat precoditioers for geeral complex Toeplitz matrices are discussed i [8, 2, 22]. I particular, Cha ad Yeug [22] aalyze the performace of the cojugate gradiet method applied to the ormal equatios (C 1 T)H (C 1 T)x = (C 1 T)H C 1 b (9) associated with (8), where the superscript H deotes traspositio ad complex cojugatio. Let C deote the solutio of (4). We propose that C 1 be used as a precoditioer istead of C 1 whe C F (T) < C F (T). This precoditioer, so far, has ot bee ivestigated; however, we ote that Cha ad Ng [23], as well as Potts ad Steidl [24], use geeralized circulats to costruct precoditioers differet from C 1 for Hermitia Toeplitz systems. We remark that for ay y C, C 1 y ca be evaluated i oly O( log ) arithmetic floatig-poit operatios with the aid of the fast Fourier trasform; see Sectio 5 or [8] for details. This paper is orgaized as follows. Sectio 2 discusses the distace of a Toeplitz matrix to the set of geeralized circulat matrices ad Sectio 3 the distace to geeralized Hermitia matrices. A applicatio to scaled Jorda blocks is described i Sectio 4. Precoditioig is cosidered i Sectio 5, computed examples are preseted i Sectio 6, ad cocludig remarks ca be foud i Sectio Distace to a geeralized circulat matrix

4 STRUCTURED DISTANCE TO NORMALITY 3 Theorem 2.1. Let the Toeplitz matrix T = (; σ, δ, τ) satisfy h( h)σ h τ h. (1) The T = (; σ, δ, τ ), determied by ad θ = arg ( ) h( h)σ h τ h σh = ( h)σ h + hτ h e iθ, τh = hσ he iθ + ( h)τ h, h = 1 : 1, is the uique closest matrix to T i C i the Frobeius orm. Moreover, if (1) is violated ad (σ, τ) (, ), the there are ifiitely may matrices T C, which deped o a arbitrary agle θ R, at the same miimal distace from T. We have C F (T) = 1 h( h)( σ h 2 + τ h 2 ) 2 h( h)σ h τ h. (11) Proof: It suffices to determie the closest matrix T = (; σ,, τ ) C to the matrix T = (; σ,, τ). Substitute τ = z ad σ h = z h e iθ, h = 1 : 1, ito T. The the squared distace D(z, θ) = T T 2 F ca be expressed as D(z, θ) = = ( h)( z h τ h 2 + z h e iθ σ h 2 ) ( h) z h τ h 2 + h z h σ h e iθ 2. We seek to determie a vector τ ad a agle θ such that D(τ, θ ) = Sice z D(z, θ) is covex ad z D(z, θ) = for mi D(z, θ). z C k π<θ π z h = z h (θ) = ( h)τ h + hσ h e iθ, h = 1 : 1, this vector miimizes D(z, θ) for a give agle θ. The desired value of θ R is obtaied by miimizig d(θ) = D(z(θ), θ). We have d(θ) = h( h) τ h σ h e iθ 2 ( = 1 h( h)( σ h 2 + τ h 2 ) 2 Re e iθ ) h( h)σ h τ h.

5 4 S. NOSCHESE AND L. REICHEL If (1) holds, the d(θ) is miimal for θ = arg( h( h)σ h τ h ) ad achieves the value d(θ ) = 1 h( h)( σ h 2 + τ h 2 ) 2 h( h)σ h τ h. This shows (11). We tur to the situatio whe (1) is violated. The d(θ) = 1 h( h)( σ h 2 + τ h 2 ) for all values of θ R. It follows that there are ifiitely may ormal matrices T = T (θ) defied by σ h = σ h (θ) = ( h)σ h + hτ h e iθ for h = 1 : 1, at the same miimal distace from T., τh = τ h (θ) = hσ he iθ + ( h)τ h, Let C R deote the variety of real geeralized circulat matrices. Corollary 2.2. Let T = (; σ, δ, τ) R. If the sum (1) is positive, the the projectio of T oto C R is the circulat T 1 = (; σ, δ, τ ) with σ h = ( h)σ h + hτ h, τh = hσ h + ( h)τ h, h = 1 : 1. If, istead, the sum (1) is egative, the the projectio of T oto C R is the skew-circulat matrix T 2 = (; σ, δ, τ ) with σ h = ( h)σ h hτ h, τh = ( h)τ h hσ h, h = 1 : 1. Whe the sum (1) vaishes, both matrices T1 ad T2 are closest matrices to T i C i the Frobeius orm. We have C F (T) = 1 mi j( j)(σ j τ j ) 2, j( j)(σ j + τ j ) 2. j=1 Remark 2.1. Note that although the matrices T ad e iα (T βi), for ay π < α π ad β C, have the same distace to the set C, they have differet projectios oto C (at distace δ ). j=1 3. Distace to a geeralized Hermitia matrix This distace was cosidered i [16] for baded Toeplitz matrices. The followig result is a immediate geeralizatio of [16, Theorem 3.1]. The result ca be show similarly as Theorem 2.1.

6 STRUCTURED DISTANCE TO NORMALITY 5 Theorem 3.1. Let the Toeplitz matrix T = (; σ, δ, τ) satisfy ( h)σ h τ h. (12) The T = (; σ, δ, τ ), determied by σ = σ + τeiθ 2, τ = τ + σeiθ 2, θ = arg( ( h)σ h τ h ), is the uique closest matrix to T i H. Moreover, if (σ, τ) (, ) ad (12) is violated, the there are ifiitely may matrices T H, which deped o a arbitrary agle θ R, of the same miimal distace, amely We have T = (; σ + τeiθ 2, δ, τ + σeiθ ). 2 H F (T) = 1 ( h)( σ h τ h 2 ) ( h)σ h τ h. 4. A scaled Jorda block Cosider the scaled Jorda block µ µ... µ... J = R, µ. (13)... µ µ This is a upper bidiagoal Toeplitz matrix. The eigevalue is maximally defective. The circulat matrix µ µ... J1 µ... = R... µ µ µ

7 6 S. NOSCHESE AND L. REICHEL ad the skew-circulat matrix µ µ... J2 µ... = µ µ are the closest matrices to J i C with C F (J) J F = J J 1 F J F µ R = J J 2 F J F = 1. (14) Thus, the ormalized distace to the set C decreases to zero as the order of J icreases. It follows from (14) ad László [14, Theorem 2] that C F (J) = d F (J). O the other had, the symmetric matrix J (s) = µ/ µ/2 µ/2... µ/2 µ/ R µ/2 µ/2... µ/2 µ/ µ/2 ad the skew-symmetric matrix µ/ µ/2 µ/2... µ/2 µ/2... J (a) = R µ/2 µ/2... µ/2 µ/ µ/2 are the closest matrices i H to J. They are equidistat to J with H F (J) J F = J J(s) F J F = J J(a) F J F = 1 2. This distace is maximal; see [15, Theorem 3.2]. Figure 1 displays the eigevalues of J 1, J 2, J (s), ad J (a). 5. Precoditioig It is well kow that a {e iθ }-circulat matrix ca be diagoalized by the product of the Fourier matrix ad a suitable uitary diagoal matrix; see, e.g., [8]. I particular, the followig result holds.

8 STRUCTURED DISTANCE TO NORMALITY Figure 1. Let J R 3 3 be a Jorda block (13) with µ = 1. The figure displays eigevalues i the complex plae of the associated matrices J (s) (black ), J (a) (blue ), J 1 (red ), ad J 2 (gree +). The horizotal axis gives the real part of the eigevalues ad the vertical axis the imagiary part. Propositio 5.1. A {e iθ }-circulat matrix C C, with etries that satisfy σ h = τ h e iθ, 1 h <, has the spectral factorizatio where F θ has the etries ad D C is diagoal. C = F θ DF H θ, (15) [F θ ] hk = 1 e i2πh (k θ 2π ), h, k <, The spectral factorizatio (15) ca be computed with the aid of the fast Fourier trasform i O( log ) arithmetic floatig-poit operatios, ad ca be applied to evaluate C 1 y for ay y C i O( log ) arithmetic floatig-poit operatios, assumig that C 1 exists. The latter property makes it attractive to apply osigular geeralized circulats as precoditioers i the same maer as circulats are used i (9). Therefore, we propose to let C C solve the miimizatio problem (4) ad compute the solutio of the precoditioed ormal equatios (C 1 T) H (C 1 T)x = (C 1 T) H C 1 b (16) associated with the liear system of equatios (7) by the cojugate gradiet method. Some spectral properties of the miimizer C of (4) ca be established similarly as aalogous results for the circulat miimizer C of (5) show by Cha et al. [25]. Specifically, the followig results are obtaied by replacig the Fourier matrix i the proofs i [25] by the matrix F θ defied i Propositio 5.1. Theorem 5.1. The geeralized circulat C closest to a Hermitia Toeplitz matrix T is Hermitia.

9 8 S. NOSCHESE AND L. REICHEL Proof: The result ca be show by combiig Propositio 5.1 with [25, Theorem 1 (3)] ad [25, Lemma 1 (iii)]. Theorem 5.2. If T is Hermitia, the λ mi (T) λ mi (C) λ max (C) λ max (T), where λ max ( ) ad λ mi ( ) deote the largest ad the smallest eigevalues, respectively. It follows that C 2 T 2, where 2 deotes the spectral orm. Moreover, if T is Hermitia positive defiite, the so is C. I particular, i this situatio C is ot sigular ad C 1 2 T 1 2. Proof: The propositio ca be show by replacig the Fourier matrix i the proof of [25, Theorem 2] by the matrix F θ of Propositio 5.1. The bouds for the spectral orms of C ad C 1 are immediate. The above results show that if the matrix T i (7) is Hermitia ad positive defiite, the the iverse of the solutio C of (4) ca be used as a precoditioer for the cojugate gradiet iterative method. The more clustered the spectrum of C 1 T is, the faster the covergece rate will be. I order to aalyze the covergece rate of the precoditioed cojugate gradiet method, we have to geeralize a result show i [25] for the precoditioer C 1 proposed by T. Cha [19]. Theorem 5.3. Let A C be arbitrary ad let c : C C be the liear operator that maps A to the solutio of the miimizatio problem mi C A F. C C The c 2 = 1. Here c 2 deotes the operator orm of c with the domai ad rage equipped with the spectral orm. Proof: The result ca be show by replacig the Fourier matrix F i the proofs of [8, Theorems 4.8-9] by the matrix F θ of Propositio 5.1. We omit the details. We ow are i a positio to discuss the performace of geeralized circulat precoditioers for the cojugate gradiet method. I particular, we show superlier covergece. This is established by geeralizig proofs i [8, 26] of the aalogous result for circulat precoditioers. The proofs use properties of the geeratig fuctio associated with a Toeplitz matrix. Theorem 5.4. Let the fuctio f(t) = δ+ j=1 ( σj e ijt + τ j e ijt) be 2π-periodic, cotiuous, ad positive for t R, ad let the Toeplitz matrices T [f] = (; σ, δ, τ), = 1, 2, 3,..., be geerated by f. The the spectra of c(t ) 1 T are clustered aroud 1 for large. Proof: The result ca be show similarly as [8, Theorems 4.1], which establishes the aalogous result for circulat precoditioers C 1, where C solves the miimizatio problem (4). We therefore oly outlie the proof. Let ε > be arbitrary but fixed, ad let the trigoometric polyomial p M (t) = δ + M ( σj e ijt + τ j e ijt) j=1

10 STRUCTURED DISTANCE TO NORMALITY 9 satisfy sup π<t π p M (t) f(t) ε. Let c( T ), with > 2M, deotes the Toeplitz matrix geerated by p M. The structure of c( T ) may differ from that of the correspodig matrix i the proof of [8, Theorems 4.1]. We have c(t ) T = c(t ) c( T ) T + T + c( T ) T. It follows from Theorem 5.3 ad [8, Theorem 3.3] that Note that c(t ) c( T ) T + T 2 ( c 2 + 1) sup p M (t) f(t) 2ε. π<t π c( T ) T = W + U, where W ad U are Hermitia Toeplitz matrices with the first row of W equal to ad the first row of U give by [, τ 1 /, 2τ 2 /,, Mτ M /,,, ], [,,, ( M)σ M /,, ( 2)σ 2 /, ( 1)σ 1 /]e iθ with θ defied i Theorem 2.1. Hece, similarly as i the proof of [8, Theorems 4.1], we have that rak(u ) 2M. The remaider of the proof is idetical to that of [8, Theorems 4.1]. Thus, for N, where N = M(M + 1)(1 + sup π<t t f(t) /ε), we have W 2 ε. By Theorem 5.2, c(t ) 2 ad c(t ) 1 2 are bouded uiformly i. It follows that for ay ε >, there exist M ad N > such that for ay > N, c(t ) T has at most 2M eigevalues of magitude larger tha 3ε. We remark that a precoditioer C 1 ca be determied by the miimizer C of (3) also whe the Toeplitz matrix T is o-hermitia. We coclude this sectio with two special cases. Let T = (; σ, δ, τ) be a geeralized Hermitia Toeplitz matrix, i.e., its etries are give by σ h = τ h e iθ, 1 h <. The T = δi + e iθ/2 S, where S = (; σ,, σ) is Hermitia. If S is positive defiite ad δ =, the oe ca avoid to apply the cojugate gradiet method to the precoditioed ormal equatios (16) by observig that the system (7) is equivalet to the liear system of equatios Sx = e iθ/2 b with a Hermitia positive defiite Toeplitz matrix. This system ca be precoditioed. Similarly, if T = (; σ, δ, τ) is a geeralized Hermitia Toeplitz matrix with arg(δ) = θ/2, the the liear system of equatios (7) ca be expressed as a equivalet system with the Hermitia matrix δ I + S. The latter system ca be precoditioed. 6. Numerical examples The first two examples of this sectio illustrate the closeess of the spectra of the closest ormal matrix ad the matrix of least structured distace to ormality (6) to a give Toeplitz matrix commeted o i Sectio 1. The last six examples are cocered with precoditioig.

11 1 S. NOSCHESE AND L. REICHEL Figure 2. Example 1: The eigevalues of N (blue ) ad C (red ) i the complex plae. The figures display eigevalues; the horizotal axes mark the real part of the eigevalues ad the vertical axes the imagiary part. Example 1. Cosider the 7 7 petadiagoal Toeplitz matrix T geerated by the followig MATLAB code: rad( state,); delta=rad+i*rad; =7; sigma=[rad+i*rad;rad+i*rad;zeros(-3,1)]; tau=[rad+i*rad,rad+i*rad,zeros(1,-3)]; T=toeplitz([delta;sigma],[delta,tau]); We use a MATLAB implemetatio of the algorithm described i [13, 17] desiged to determie the closest ormal matrix to T. Deote the computed ormal matrix by N, which is assumed to be the closest ormal matrix. We obtai d F (T) = T N F = ad d F (T)/ T F = The distace to ormal geeralized Hermitia matrices is give by H F (T) = A ivestigatio of the latter distace for baded Toeplitz matrices T ca be foud i [16]. The distace to a geeralized circulat is C F (T) = Thus, T F (T) = C F (T). The ormalized distace to a geeralized circulat is give by C F (T)/ T F = Hece, C F (T)/ T F is quite close to d F (T)/ T F. Let C deote the closest matrix to T i C i the Frobeius orm. Figure 2 displays the spectra of N ad C. The eigevalues of these matrices are see to be close. Example 2. The followig MATLAB code determies a 7 7 septadiagoal Toeplitz matrix:

12 STRUCTURED DISTANCE TO NORMALITY 11 x x 1 6 Figure 3. Example 2: The eigevalues of N (blue ) ad C (red ) i the complex plae. The plots of the eigevalues of N ad C almost coicide. rad( state,); delta=rad+i*rad; =7; sigma=[1.e3*rad(3,1)+i*rad(3,1);zeros(-4,1)]; tau=[rad(1,3)+i*1.e6*rad(1,3),zeros(1,-4)]; T=toeplitz([delta;sigma],[delta,tau]); We proceed similarly as i Example 1 to determie the closest ormal matrix N to T, ad obtai the (ustructured) distace to ormality d F (T) = ad the distace to the set H of ormal geeralized Hermitia matrices is H F (T) = The ormalized ustructured distace to ormality is d F (T)/ T F = The distace to a geeralized circulat is give by C F (T) = Thus, T F (T) = C F (T). The correspodig ormalized distace to a geeralized circulat is C F (T)/ T F = , which shows that the ormalized distace to a geeralized circulat is about the same as the ormalized ustructured distace to ormality. Let C deote the closest matrix to T i C i the Frobeius orm. Figure 3 depicts the spectra of N ad C. The eigevalues of these matrices are distributed similarly o close curves. Example 3. Let the Toeplitz matrix T = [t j,k ] C be defied by t j,k = j k, 1 k j, 1 (1 + k j) 2 + i k j, 1 j < k, (17) for i = 1 ad a positive iteger. Table I displays the ormalized distaces to ormality, the sets C ad C, ad geeralized symmetry for several values of. The ormalized structured

13 12 S. NOSCHESE AND L. REICHEL d F (T)/ T C F (T)/ T F C F (T)/ T F H F (T)/ T F Table I. Example 3: Normalized distaces to the sets N, C, C, ad H. For all, T F (T)/ T F = H F (T)/ T F. matrix smallest eigevalue largest eigevalue 5 T H T (C 1 T) H C 1 T (C 1 C T H T (C 1 T) H C 1 T (C 1 T)H C 1 T T H T (C 1 T) H C 1 T (C 1 T)H C 1 T T H T (C 1 T) H C 1 T (C 1 T)H C 1 T T H T (C 1 T) H C 1 T (C 1 T)H C 1 T Table II. Example 3: Smallest ad largest eigevalues of matrices of uprecoditioed ad precoditioed ormal equatios. The matrices C deote the best geeralized circulats ad C the best circulat approximatios of T i the Frobeius orm. The parameter idicates the order of the matrices. distace to ormality is the miimum of the latter etries. The tabulated distaces are see to chage oly slowly with. Figure 4 displays the eigevalues of the Toeplitz matrix T C 25 25, of the closest ormal matrix N, of the closest geeralized circulat, ad of the closest geeralized Hermitia matrix. The eigevalues of the closest geeralized circulat ca be see to approximate the eigevalues of both T ad N quite well. Figure 5 illustrates the differeces i the eigevalues of the closest circulats ad the closest geeralized circulats. The eigevalues of the closest geeralized circulats are see to approximate the eigevalues of Toeplitz matrices of order 25 more accurately. This is i agreemet with the fact that C F (T) < C F (T), as show i Table I. Table II displays the extreme eigevalues of the matrices of the uprecoditioed ad precoditioed ormal equatios associated with Toeplitz matrices (17) of several sizes. The

14 STRUCTURED DISTANCE TO NORMALITY Figure 4. Example 3: The eigevalues of T C (black o), of the closest ormal matrix (blue ), of the closest geeralized circulat (red +), ad of the closest geeralized Hermitia Toeplitz matrix (gree ). The horizotal axis is the real axis ad the vertical axis is the imagiary axis of the complex plae Figure 5. Example 3: The eigevalues of T C (black o), of the closest circulat (blue ), ad of the closest geeralized circulat (red +). The horizotal axis is the real axis ad the vertical axis is the imagiary axis of the complex plae. geeralized circulat matrices are see to be better precoditioers tha the correspodig circulat matrices i the sese that the precoditioed matrices obtaied with the geeralized

15 14 S. NOSCHESE AND L. REICHEL circulats have their eigevalues i shorter itervals, i.e., are better coditioed, tha the correspodig precoditioed matrices obtaied with circulat precoditioers. d F (T)/ T C F (T)/ T F C F (T)/ T F H F A(T)/ T F Table III. Example 4: Normalized distaces to the sets N, C, C, ad H. For all, T F (T)/ T F = C F(T)/ T F. matrix smallest eigevalue largest eigevalue 5 T H T (C 1 T) H C 1 T (C 1 C T H T (C 1 T) H C 1 T (C 1 T)H C 1 T T H T (C 1 T) H C 1 T (C 1 T)H C 1 T T H T (C 1 T) H C 1 T (C 1 T)H C 1 T T H T (C 1 T) H C 1 T (C 1 T)H C 1 T Table IV. Example 4: Smallest ad largest eigevalues of matrices of uprecoditioed ad precoditioed ormal equatios. The matrices C deote the best geeralized circulats ad C the best circulat approximatios of T i the Frobeius orm. The parameter idicates the order of the matrices. Example 4. Let the Toeplitz matrices T = [t j,k ] C be give by 1 (1 + j k) 2, 1 k j, t j,k = 1 + i (1 + k j) 2, 1 j < k. The etries t j,k coverge to zero as j k icreases faster tha i Example 3. The ifiitedimesioal operator obtaied whe = is i the Wieer class. The Tables III ad IV are

16 STRUCTURED DISTANCE TO NORMALITY Figure 6. Example 4: The eigevalues of T C 5 5 (black o), of the closest ormal matrix (blue ), of the closest geeralized circulat (red +), ad of the closest geeralized Hermitia Toeplitz matrix (gree ). The horizotal axis is the real axis ad the vertical axis is the imagiary axis of the complex plae. aalogous to the Tables I ad II, respectively. The agle θ, defied i Theorem 2.1, is small for the Toeplitz matrices T of the preset example. Therefore the circulat precoditioers C of Table IV are close to the correspodig geeralized circulat precoditioers C. Table IV shows the circulat ad geeralized circulat precoditioers to perform about the same. Figure 6 displays the eigevalues of the Toeplitz matrix T C 25 25, of the closest ormal matrix N, of the closest geeralized circulat, ad of the closest geeralized Hermitia matrix. The eigevalues of the closest geeralized circulat ca be see to approximate the eigevalues of both T ad N quite well. The eigevalues of the closest circulats ad closest geeralized circulats are close; we therefore omit showig the former. Example 5. Cosider the symmetric positive defiite Toeplitz matrices T = (; σ, δ, τ) R geerated by f(t) = t 4 +1, t [ π, π]. These matrices also are cosidered i [8, Sectio 4.6]. The δ = 1 + π4 5 ; σ k = 4( 1)k k 2 (π 2 6 k 2 ), τ k = σ k, k = 1 : 1. Table V shows the umber of iteratios required for covergece of the cojugate gradiet method for the solutio of the system T x = b ad of the associated precoditioed systems with circulat precoditioer c (T ) 1 ad geeralized circulat precoditioer c(t ) 1. Here c (T ) deotes the circulat miimizer of (4) with T replaced by T. I this ad the followig examples the right-had side vector is b = [1, 1,..., 1] T R ad the iitial iterate is the zero vector. The iteratios are termiated whe the Euclidea orm of the relative residual error associated with the curret iterate is less tha Whe the dimesio is odd, the sum (1) is egative, ad c(t ) is a skew-circulat. The skew-circulat precoditioer c(t ) 1 requires oe fewer iteratio tha the circulat precoditioer c(t ) 1 to satisfy the stoppig

17 16 S. NOSCHESE AND L. REICHEL criterio. The additioal cost for computig c(t ), compared with the computatio of c (T), is just O() arithmetic floatig-poit operatios. T c(t ) 1 T c (T ) 1 T Table V. Example 5: Number of iteratios for uprecoditioed, geeralized circulat precoditioed, ad circulat precoditioed systems. The matrices c(t ) deote the best geeralized circulats ad c (T ) the best circulat approximatios of T i the Frobeius orm. The parameter idicates the order of the matrices. Example 6. Itroduce the Hermitia positive defiite Toeplitz matrices T = (; σ, δ, τ) C determied by the symbol f(t) = 2 k= si(kt) + cos(kt) (1 + k) 1.1, t [ π, π]. Precoditioers for these matrices are discussed i [2, Sectio 2.5]. We have δ = 2; σ k = 1 + i (1 + k) 1.1, τ k = σ k, k = 1 : 1. Table VI shows the umber of iteratios required for covergece by the cojugate gradiet method to solve the system T x = b, ad the precoditioed systems c(t ) 1 T x = c(t ) 1 b ad c (T ) 1 T x = c (T ) 1 b, as well as the relative residual error associated with the computed approximate solutio. The iteratios are termiated whe Euclidea orm of the relative residual vector is less tha We ote that, for all dimesios, the precoditioers c(t ) 1 are {e i π 2 }-circulats. For almost all dimesios, the geeralized circulat precoditioers c(t ) 1 yield smaller residual errors after the same umber of iteratios tha the circulat precoditioers c (T ) 1. Example 7. Cosider the symmetric positive defiite Toeplitz matrices T = (; σ, δ, τ) R give by δ = 1; σ k = cos(k) k + 1, τ k = σ k, k = 1 : 1; see the last example ivestigated i [19]. The matrices c(t ) are {e iπ }-circulats. Table VII shows the umber of iteratios required for covergece by the cojugate gradiet method to solve the uprecoditioed system T x = b ad the correspodig precoditioed systems

18 STRUCTURED DISTANCE TO NORMALITY 17 matrix iteratios residual 32 T c(t ) 1 T c (T ) 1 T T c(t ) 1 T c (T ) 1 T T c(t ) 1 T c (T ) 1 T T c(t ) 1 T c (T ) 1 T T c(t ) 1 T c (T ) 1 T T c(t ) 1 T c (T ) 1 T T c(t ) 1 T c (T ) 1 T T c(t ) 1 T c (T ) 1 T Table VI. Example 6: Number of iteratios ad spectral orm of the relative residuals for uprecoditioed, circulat precoditioed, ad geeralized circulat precoditioed systems. The matrices c(t ) deote the best geeralized circulat matrices ad c (T ) the best circulat approximatios of T i the Frobeius orm. The parameter idicates the order of the matrices. with the precoditioers c (T ) 1 ad c(t ) 1. The same stoppig criteraio as i Example 6 is used. Similarly as i the above examples, the precoditioer c(t ) 1 typically gives slightly faster covergece tha c (T ) 1. Example 8. I our last example, we cosider ill-coditioed symmetric positive defiite Toeplitz matrices T = (; σ, δ, τ) R geerated by the symbol f(t) = t 4, t [ π, π]. These matrices also are cosidered i [2, Sectio 4.4]. The δ = π4 5 ; σ k = 4( 1)k k 2 (π 2 6 k 2 ), τ k = σ k, k = 1 : 1. Table VIII shows the umber of iteratios required to solve T x = b by the uprecoditioed ad precoditioed cojugate gradiet methods. The iteratios are termiated whe the Euclidea orm of the relative residual error associated with the curret iterate is less tha We cosider the circulat precoditioer c (T ) 1 ad the geeralized circulat precoditioer c(t ) 1. For odd, c(t ) is a skew-circulat. The coditio umber of the

19 18 S. NOSCHESE AND L. REICHEL T c(t ) 1 T c (T ) 1 T Table VII. Example 7: Number of iteratios for uprecoditioed, geeralized circulat precoditioed, ad circulat precoditioed systems. matrix T, defied by κ(t ) = T 2 T 1 2, grows like O( 4 ). The coditio umbers are displayed i Table VIII. The table shows the cojugate gradiet method precoditioed with the geeralized circulat c(t ) 1 to require fewer iteratios tha the cojugate gradiet method precoditioed with the circulat c (T ) 1. T c(t ) 1 T c (T ) 1 T κ(t ) Table VIII. Example 8: Number of iteratios for uprecoditioed, geeralized circulat precoditioed, ad circulat precoditioed systems. Uprecoditioed cojugate gradiet iteratio does ot satisfy the stoppig criterio for = 123 withi 9661 iteratios. 7. Coclusio ad extesios This paper discusses the distace i the Frobeius orm of a Toeplitz matrix to the set of geeralized Hermitia Toeplitz matrices ad to the set of {e iφ }-circulats. The uderlyig miimizatio problem for {e iφ }-circulats geeralizes a miimizatio problem solved by circulat precoditioers itroduced by T. Cha. It is atural to apply {e iφ }-circulats as precoditioers. We preset some theoretical ad umerical results that shed light o the performace of these precoditioers. We are presetly ivestigatig applicatios of these precoditioers to liear systems of equatios with a matrix that does ot have Toeplitz structure. Ackowledgmet We would like to thak the referees for suggestios that improved the presetatio.

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