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MATHEMATICS OF COMPUTATION Volume 68, Number 226, April 1999, Pages 793–803 S 0025-5718(99)01045-5

SUPERLINEAR PCG METHODS FOR SYMMETRIC TOEPLITZ SYSTEMS

STEFANO SERRA

Abstract. In this paper we deal with the solution, by means of preconditioned conjugate gradient (PCG) methods, of n × n symmetric Toeplitz systems An (f )x = b with nonnegative generating function f . Here the function f is assumed to be continuous and strictly positive, or is assumed to have isolated zeros of even order. In the ?rst case we use as preconditioner the natural and the optimal τ approximation of An (f ) proposed by Bini and Di Benedetto, and we prove that the related PCG method has a superlinear rate of convergence and a total arithmetic cost of O(n log n) ops. Under the second hypothesis we cannot guarantee that the natural τ matrix is positive de?nite, while for the optimal we show that, in the ill-conditioned case, this can be really a bad choice. Consequently, we de?ne a new τ matrix for preconditioning the given system; then, by applying the Sherman–Morrison–Woodbury inversion formula to the preconditioned system, we introduce a small, constant number of subsidiary systems which can be solved again by means of the previous PCG method. Finally, we perform some numerical experiments that show the e?ectiveness of the devised technique and the adherence with the theoretical analysis.

1. Introduction In this paper, we discuss the solution of Toeplitz systems An x = b by using the preconditioned conjugate gradient (PCG) method. Here we suppose that the Toeplitz matrix An is symmetric and is generated by a continuous 2π-periodic function f , de?ned in the fundamental interval I = [?π, π] and periodically extended to the whole real axis, in the sense that the entries along the diagonals are given by the Fourier coe?cients of the function f . Since we are interested in the symmetric case, we have to suppose that f is also an even function; more precisely we have 1 π f (x) cos((j ? k)x)dx, 0 ≤ j, k ≤ n ? 1. π 0 We point out that the generating function f is given in some applications of Toeplitz systems. Classical examples are the kernels of the Wiener–Hopf equations [21], the spectral density functions in stationary stochastic processes [25, 24], and the point– spread functions in image processing [29]. When the generating function is continuous and positive there are many types of preconditioners [10, 14, 4, 18, 19, 5] chosen in the circulant algebra [16], in the τ (1) [An ]j,k = a|j?k| =

Received by the editor February 7, 1996 and, in revised form, October 15, 1996 and July 18, 1997. 1991 Mathematics Subject Classi?cation. Primary 65F10, 65F15. Key words and phrases. Toeplitz matrix, τ algebra, conjugate gradient method.

c 1999 American Mathematical Society

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class [3], or in the Hartley one [5], where each matrix operation costs O(n log n) ops and O(log n) parallel steps [39, 6, 5]. These preconditioners lead to superlinearly convergent PCG methods [10, 14, 4, 5, 9] in the case where f is assumed to belong to the Wiener class, i.e., the previously de?ned Fourier coe?cients give rise to an absolutely summable sequence. Under more restrictive hypotheses about the regularity properties of f , it is also shown that the related PCG methods have a quadratic rate of convergence [10, 14, 4, 5]. More recently, in the circulant and the Hartley case, the proof of superlinear convergence has been extended under the weakened assumption of continuity and positivity of f (see [12, 27]). The ?rst result we present here is this extension for the optimal τ preconditioner and a similar result for the natural circulant and τ preconditioners. The main tool is the Jackson characterization [26] of the Weierstrass approximation theorem like in the Chan, Yeung paper [12]. However, by counting the number of outliers [2] of the preconditioned matrices [12, 27], we observe that the spectral clustering around 1, achieved by the τ preconditioner, is slightly better. When f has zeros, we know that the related matrices are asymptotically illconditioned [25, 32, 36]. Unfortunately, in this case, circulant and Hartley preconditioners fail; in [18], Di Benedetto has shown that the natural circulant preconditioner [10] is singular when, for instance, f vanishes in x = 0. Moreover, this author has proved that the optimal circulant preconditioner [14] and the optimal Hartley preconditioner [14] are positive de?nite, but the preconditioned matrices have a spectral cluster around 1 and also O(nβ ) eigenvalues in any open interval (0, ) with β depending on the order of the zeros of f (for a richer analysis see [38, 20]). By virtue of the powerful convergence theory for PCG methods due, mainly, to Axelsson and Lindskog [2], it follows that the PCG methods related to these Hartley and circulant preconditioners are not competitive with the superfast methods [1, 17] and with the τ based or band-Toeplitz based PCG methods (see [19, 18, 7, 11, 37, 32, 36, 33]). In this paper, under the nonnegative assumption, we propose a new τ preconditioner which joins the good computational features of the τ algebra and a strict approximation property with respect to the matrix An (f ). We obtain an algorithm which is based on the use of the Sherman–Morrison–Woodbury formula and on superlinear PCG methods, and which has an arithmetic cost of O(n log n) ops and O(log n) parallel steps. We observe that, for the case of f having zeros, the only superlinear PCG method previously devised was the one obtained by using band-Toeplitz preconditioners (see [37]). The paper is organized as follows. In Section 2 we prove that the optimal τ preconditioner, introduced and discussed in [4], assures a superlinear convergence speed in the continuous, positive case. A similar analysis is carried out for the natural circulant and τ preconditioners. In Section 3, we introduce a new symmetric positive de?nite (SPD) τ preconditioner for which we deduce good approximation properties with respect to An (f ). In Section 4, we devise an algorithm associated to the former preconditioner and we discuss the related arithmetic and parallel costs. In Section 5, we perform some numerical experiments which plainly con?rm the e?ectiveness of the proposed ideas, and we make a practical and theoretical comparison among the best strategies. Finally, in Section 6, we discuss some “peculiar” and apparently strange features of the PCG methods related to the optimal preconditioners.

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2. The τ preconditioners In this section we de?ne the τ matrix algebra, and we introduce the natural and optimal τ preconditioners [4]. For any integer n, the n × n τ algebra is generated by the symmetric Toeplitz matrix whose ?rst row is the second vector of the canonical base. Given a symmetric Toeplitz matrix An whose ?rst row is (a0 , a1 , . . . , an?1 ), the natural τ matrix τ (An ) of An can be obtained by means of the Hankel correction: (2) τ (An ) = An ? HC(An ),

where HC(An ) is the Hankel matrix whose ?rst row is (a2 , a3 , . . . , an?1 , 0, 0). On the other hand, the optimal approximation τopt of An is de?ned by minimizing the Frobenius distance from An over τ . The ?rst row of τopt is de?ned by (φ1 , φ2 , . . . , φn ) with φj = n?j+3 aj?1 ? n?j?1 aj+1 , j = 2, . . . , n ? 2 and φ1 = n+1 n+1 4 3 a0 ? n?2 a2 , φn?1 = n+1 an?2 , φn = n+1 an?1 [4, 23]. n+1 From a spectral point of view it is possible to prove the following lemma. Lemma 2.1. Let An (f ) be the symmetric Toeplitz matrix as de?ned in (1). By supposing that the function f is continuous and mf < Mf are the minimum and the maximum of f , respectively, the following relations hold: 1. The eigenvalues λi (An (f )) belong to (mf , Mf ) [25].

jπ 2. The eigenvalues of the natural τ approximation are generated by sn (f ) n+1 , where sn (f ) is the n-th degree Fourier expansion of f [3, 4, 6] and j = 1, . . . , n. 3. The eigenvalues of the optimal τ approximation are related to the Cesaro sums of f and are contained in the interval (mf , Mf ) [20].

As a simple consequence, we may obtain information about the extreme spectral properties of the matrix τ (An (f )). De?nition 2.1. The class of continuous functions f for which the modulus of continuity ω(f, δ) is o(| log δ|?1 ) is the Dini–Lipschitz class and is denoted by C? . The complementary set is called the W class. Corollary 2.1. If f ∈ C? , then for any > 0 there exists an integer value n such ? that ?n ≥ n the eigenvalues λi (τ (An (f ))) belong to (mf ? , Mf + ). The same is ? true for the optimal τ matrix under the weaker assumption that f ∈ C with = 0 and n = 1. ? Proof. It follows plainly from the ?rst part of Lemma 2.1 and from the convergence of the Fourier series of f ∈ C? to f (see [30, 40]). Therefore, when the generating function f is strictly positive, by virtue of Corollary 2.1, we deduce that the natural τ approximation of An (f ) is asymptotically positive de?nite and the spectra of An (f ) and τ (An (f )) are asymptotically the same. Under the nonnegative assumption, the situation is more complicated. The presence of points xs in which the function f vanishes implies that we cannot guarantee the positive de?niteness of τ (An (f )) since we cannot guarantee the positiveness of the related Fourier expansion. In Section 5, we will present cases where the natural τ preconditioner is positive de?nite and cases where it is inde?nite. Therefore, this observation justi?es the search for a new τ preconditioner (see Section 3) when we consider the asymptotically ill-conditioned case, namely the case where f has zeros [25, 28, 34, 35].

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Now, let us come back to the Toeplitz linear system with a positive continuous generating function. The aim of the following statements is the proof of a superlinear rate of convergence for the PCG method whose preconditioner is the natural τ correction or the optimal τ approximation of An (f ). First we give a simple result about the global localization of the preconditioned 1 opt matrices Pτ (f ) = (τ (An (f )))?1 An (f ) and Pτ (f ) = (τopt )?1 An (f ). ? Lemma 2.2. If f ∈ C? , then, for any > 0, there exists an integer value n such m Mf 1 that, ?n ≥ n, we have that all the eigenvalues of Pτ (f ) belong to Mf ? , mf + . ? f

opt The same is true for the Pτ (f ) under the weaker assumption that f ∈ C with = 0 and n = 1. ?

Proof. It is a direct consequence of the ?rst part of Lemma 2.1 and Corollary 2.1. The subsequent step is to show the clustering property around 1 (see [10, 4]) for the 1 spectrum of Pτ (f ). Let us denote the space of all n-th degree real trigonometric c polynomials by Pn , and all n-th degree cosine polynomials by Pn . We have

n

Pn =

k=?n c Pn

bk eikx : bk = ??k , ?|k| ≤ n , b

i2 = ?1,

is the real linear space described by Pn , where the coe?cients bk are assumed and to be real. Since Pn is a convex closed set of the Banach space of continuous periodic functions, it follows that the in?mum inf p∈Pn f ? p ∞ is really attained by a polynomial p? of Pn . In the proof of the main result (Theorem 2.1) we would like to associate p? with a τ matrix. Unfortunately, the τ algebra is a real symmetric class of matrices related to even trigonometric sums (see part 2 of Lemma 2.1), that is, to cosine trigonometric polynomials. Therefore, we need to prove that p? belongs c to the linear real subspace Pn and this fact, actually, holds true in the light of the following elegant characterization due to Jackson [26]. Lemma 2.3. Let f be a continuous and 2π-periodic function. Suppose that, in addition, f is also even. Then for any positive n, there exists p? belonging to π c Pn ? Pn such that f ? p? ∞ ≤ ω f ; n+1 . Theorem 2.1. Let f be a C? and 2π-periodic even function. Then, for any positive , there exist N and M > 0 such that for all n > N , the di?erence matrix An (f ) ? τ (An (f )) has, at most, M eigenvalues outside (? , ). The same is true for the optimal τ matrix under the weaker assumption that f ∈ C. Proof. For any > 0, there exists δ > 0 such that, when |x1 ? x2 | < δ, one π ?nds |f (x1 ) ? f (x2 )| < . Let K = π ; then ω f ; K+1 ≤ . Hence, by δ Lemma 2.3, there is a cosine trigonometric polynomial p? of degree K which veri?es f ? p? ∞ ≤ . Therefore, by exploiting the linearity of the Fourier operator and of the τ transformation, the di?erence matrix An (f ) ? τ (An (f )) may be viewed as (3) An (f ) ? τ (An (f )) = An (f ? p? ) ? τ (An (f ? p? )) + An (p? ) ? τ (An (p? )). Finally, by using Lemma 2.1, Corollary 2.1, and Lemma 2.3, we ?nd that there exists N such that if n > N , the matrix An (f ? p? ) ? τ (An (f ? p? )) has a Euclidean norm bounded by 3 . Since the matrix An (p? ) ? τ (An (p? )) has rank 2(K ? 1), by virtue of equation (2) and of the Courant–Fischer minimax theorem [22], the thesis

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follows. Concerning the matrix τopt the proof is identical and basically exploits the convergence of the Cesaro sum to f for any continuous function f (see also Theorem 1 of [12]). Observe that the proof of Theorem 2.1 is slightly simpler than the one given by Chan and Yeung [12] for the optimal circulant preconditioner Copt [14]. Basically, the reason is that the preconditioner τ (An (f )) has a simpler de?nition. Moreover, the optimal τ preconditioner (as the optimal circulant preconditioner) guarantees an approximation of An (f ) not only in the C? class, but in the general class of continuous functions. However, it should be stressed that this is only an academic di?erence, since, in practice, the functions belonging to W = C\C? are pathological examples to be de?ned ad hoc, like the Lebesgue one presented in [40]. By joining the previous remarks and results, we easily ?nd Corollary 2.2. Corollary 2.2. Let f1 be a C? positive and 2π-periodic even function and f2 be a continuous positive and 2π-periodic even function. Then, for any positive , there exist N and M > 0 such that for all n > N , we have that, at most, M eigenvalues 1 opt of the matrices Pτ (f1 ) ? I and Pτ (f2 ) ? I have absolute value larger than . Finally, for completeness, it is interesting to point out that the natural circulant preconditioner [10] assures clustering of the spectra of the preconditioned matrices under the same hypotheses used for the natural τ preconditioner. The proof is exactly the same as in Theorem 2.1. Now, in the light of the analysis given in [10], we deduce that the related PCG methods, when applied to the preconditioned system T ?1 An (f )x = T ?1 b, T = τ (An (f )) or T = τopt ,

converge superlinearly. In particular, the number of required iterations, to achieve a preassigned accuracy δ, is uniformly bounded by an absolute constant and, therefore, the total cost is dominated by c(δ, f )n log n with a c(δ, f ) positive constant value independent of the dimension n. This is the same result obtained by Chan and Yeung using circulant preconditioners and Jin using Hartley preconditioners. We observe that our choice is more restrictive, because we consider the symmetric case and not the Hermitian case. However, in the symmetric case considered we take advantage of using τ preconditioners, because they produce a better clustering (see also [4]) and are associated with the sine transform that has a cost practically equal to the Fourier or Hartley transforms [6]. 3. The new τ preconditioner When the generating function f is nonnegative and possesses zeros, the class of symmetric Toeplitz matrices {An (f )}n is asymptotically ill-conditioned. In this case, the classic iterative solvers, generally, fail in the sense that they do not converge or converge very slowly. More precisely, the rate of reduction of the error tends to 1 (no reduction of the error) as the dimension n tends to in?nity. For this kind of problem, a good preconditioner is not only welcome but necessary. In the light of the second part of Lemma 2.1, it is trivial to conclude that we are not guaranteed the positive de?niteness of the natural τ preconditioner. An example of this is given by An (x4 ): actually τ (An (f )) is not positive de?nite, since, for instance, its ?rst 8 minimal eigenvalues, for n = 512, have a magnitude of ?10?4 (see also Section 6).

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The alternative classical choice, i.e., the optimal τ preconditioner is guaranteed to be positive de?nite but it is not able to “match” the zeros of f and then is not suitable in order to obtain a proper cluster around 1 of the preconditioned spectrum (see Section 6). The new SPD τ preconditioner is constructed as follows. Let g be the minimal polynomial matching all the zeros of f with their order (which is supposed to be even). It is worth noting that this minimal polynomial g is di?erent from the one proposed and analysed in [7, 11, 37] owing to the requirement that g has to be an even function like f . Then we may view f as g · h where h is a strictly positive function, and we obtain the new preconditioner as τ (An (g))τ (An (h)). This matrix, at the present moment, is de?ned only theoretically while we are interested in computational algorithms. In the following scheme we propose how to calculate this preconditioner. 1. We suppose that we know the zeros of the even function f , namely x1 , . . . , xs ∈ [0, π] and the related orders 2k1 , . . . , 2ks . Observe that, if x = 0, then the polyno? x mial gx = (2 cos x ? 2 cos(?))2 is nonnegative and is clearly the even polynomial of ? minimal degree which has a zero in x of order 2. If x = 0 the minimal nonnegative ? ? even polynomial is trivially g0 = 2 ? 2 cos x. Consequently g is easily constructed as

s

g=

i=1

k gxi . ?i

By using FFTs of constant order (depending on the number and the order of the zeros), it is not expensive (O(l log l) ops with l independent of n) to calculate the Fourier coe?cients of g and, therefore, we have economically computed the matrix An (g) and its τ correction τ (An (f )). 2. Since we have to calculate the function h, in the sense of computing its Fourier coe?cients, we employ the relation f = g · h. We face the inverse problem of recovering these coe?cients of h, by using the Fourier representation of the functions f and g. This problem can be solved in O(n) ops by using a deconvolution algorithm [8], involving the vectors xg and xf , where we have stored the Fourier coe?cients of the considered functions. Notice that the matrix τ (An (h)) is easily obtained from the entries of An (h) by exploiting the simple equation (2). Finally, we point out that τ (An (g))τ (An (h)) is symmetric since it is, by construction, a τ matrix and the τ algebra is structurally symmetric. In addition, as τ (An (g)) and τ (An (h)) are positive de?nite, the resulting product has positive eigenvalues and therefore it is an SPD matrix. 4. A new algorithm in the ill-conditioned case In this section we ?rst perform an analysis of the “approximation properties” of the matrix τ (An (g))τ (An (h)) with regard to the problem matrix An (f ). Then, by exploiting some information from this analysis, we propose a new procedure to solve the given linear system. Theorem 4.1. Let τ (2) be the preconditioner introduced in Section 3 and let 2 2 Pτ (f ) = (τ (2))?1 An (f ) be the second preconditioned matrix. Then Pτ (f ) can be 1 viewed as the sum of two contributions Pτ (h) and LR. Here, as h is strictly positive 1 and continuous, Pτ (h) has a spectrum clustered around 1 (see Corollary 2.2), while LR is a matrix having a rank independent of n.

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Proof. Preliminarily, we observe that, because of the relation f = g · h where g is polynomial of ?xed degree, we ?nd An (f ) = An (g)An (h) + LR1 with the rank of LR1 depending only on the degree of g. Moreover, by (2), we have τ (An (g)) = An (g) + LR2 , rankLR2 = constant and therefore by applying the Sherman–Morrison–Woodbury (SMW) formula (see equation (5)) we deduce τ (An (g))?1 = (I + LR3 )A?1 (g), n with rankLR3 = constant. Then, by indicating with LRj “low” and constant rank matrices, the following relations hold true:

2 Pτ (f ) = = = = =

(τ (2))?1 (An (g)An (h) + LR1 ) τ (An (h))?1 (I + LR3 )A?1 (g)(An (g)An (h) + LR1 ) n τ (An (h))?1 (I + LR3 )(An (h) + LR4 ) τ (An (h))?1 (An (h) + LR5 ) τ (An (h))?1 An (h) + LR6 .

1 The theorem is proved by putting LR = LR6 and by observing that Pτ (h) = ?1 τ (An (h)) An (h).

Unfortunately, the matrix LR is not symmetric and cannot be symmetrized 1 by the same transformation that symmetrizes Pτ (h). Therefore, we cannot apply the Courant–Fischer minimax characterization [22] but we can deduce a proper clustering around 1 of the singular values and a superlinear rate of convergence for the related PCG method when applied to the normal equations. 2 An alternative possibility is given by the representation of Pτ (f ) derived in Theorem 4.1. The original system An (f )x = b is equivalent to (4) ? (An (h) + LR5 )x = b, ? b = (τ (An (g)))?1 b,

? where b is computable sequentially in O(n) ops by using a band-solver or in parallel in O(log n) steps by means of fast sine transforms. By following the relations proved in Theorem 4.1, it is also easy to verify that the matrices LRi , i = 1, . . . , 6, may be formally and numerically computed in O(n log n) ops and O(log n) parallel steps. Finally, the solution of the original system is obtained by applying the SMW inversion formula to the system (4). We recall that the SMW formula can be applied in the following way: given B = A + LR and LR = c = we obtain x = B ?1 b as (5) B ?1 b = A?1 b ? A?1 X(Ic + Y A?1 X)?1 Y A?1 b. X · Y, X, Y T ∈ Rc×n , rank(LR) = constant

The cost of the calculation x is “reduced” to the solution of some systems with coe?cient matrix A (which are assumed easy to solve) and to the inversion of the c×c matrix Ic +Y A?1 X. In our case, we observe that the subsidiary systems having as coe?cient matrix An (h), h positive even continuous function, can be solved with the superlinear PCG method devised in [4] and extended to the continuous case in Section 2. Therefore the total cost is bounded by c1 n log n ops and c2 log n parallel steps where the coe?cients cj are independent of the dimension n and depend only on the functions f, g and on the preassigned accuracy δ.

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Table 1. f (x) = H ? (x), superlinear PCG n Iter (Prec=I) Iter (Prec=τ (An (f ))) 16 32 64 128 256 512 8 15 20 31 45 74 6 7 8 9 10 9

2

Table 2. f (x) = 1 ? e?x n I Copt 128 42 10 512 143 17

min Bn 17 17

Bn,5 3 3

τ (1) 4 4

τ (2) 4 4

5. Numerical experiments and comparisons In order to show experimentally the validity of the analysis of Section 2, we need an even function f which is positive and continuous but not in the Wiener class. As pointed out in [12], a possible candidate is the Hardy–Littlewood series [40] given by ∞ eik log k ikx e?ik log k ?ikx H(x) = e + e , x ∈ [?π, π]. k k

k=1

Unfortunately, this function is not positive and is not even. Consequently, we de?ne H ? (x) as H(x)+H(?x) + 3.02. This function is continuous, even, positive and does 2 not belong to the Wiener class. For n = 512, the Euclidean condition number of An (H ? ) is equal to 2.37 · 103 and it increases as n grows. Table 1 shows the number of steps, in order to reach an accuracy of δ < 10?7 , when the identity and our τ preconditioner is applied to the original system with bi = 1 for any i. Now, we consider the ill-conditioned case by considering nonnegative generating functions having zeros. We compare the convergence rate of the minimal band Toeplitz preconditioner [7], with the near optimal band Toeplitz preconditioner [37], with the optimal circulant preconditioner [14], with the natural τ preconditioner [4], and with our τ preconditioner for two di?erent generating functions. They are 2 1 ? e?x and x4 and are associated to ill-conditioned matrices An having Euclidean condition numbers equal to O(n2 ) and O(n4 ), respectively (see for instance [34, 35]). The matrices An are formed by evaluating the Fourier coe?cients of the generating functions by using FFTs (see [11]). In the considered tests, the vector of all ones is the right-hand side vector, the zero vector is the initial guess, and the stopping criterion is rq 2 / r0 2 ≤ 10?7 , where rq is the residual vector after q iterations. All computations are done by using Matlab. In Tables 2 and 3, I denotes that no preconditioning is used, Copt is the T. Chan optimal circulant preconditioner [14], Bn,l is the near optimal band-Toeplitz premin conditioner [37] (l indicates the semibandwidth), Bn is the minimal band-Toeplitz preconditioner [7], τ (1) is the Bini–Di Benedetto preconditioner, and τ (2) is our τ preconditioner. Notice the “strange” behavior of the natural τ preconditioner τ (1). In the case of 2 f (x) = 1?e?x , it leads to a very fast PCG method, while, in the case of f (x) = x4 , the convergence is not so good and depends strongly on the size n of the problem. The reason is very simple: as proved in Corollary 2.1, the minimal eigenvalues can be less than mf , which is equal to zero for the two considered functions. Fortunately,

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Table 3. f (x) = x4 n I Copt 128 587 77 512 7457 406

min Bn 24 29

Bn,5 11 13

τ (1) 22 88

τ (2) 8 10

Table 4. Operation count for the i-th iteration P = Bn,5 Multiplying An (f ) by a vector 18n log n + qn Solution of P y = ci 2ki n Other operations 2n + k P = τ (2) 18n log n + qn 4n log n + n 2n + k

for the ?rst generating function, τ (1) is an SPD matrix, while in the second case, for n = 128, τ (1) has 3 eigenvalues in [?2.4 · 10?3 , ?1.2 · 10?3 ] and, for n = 512, has 8 eigenvalues in [?3.1 · 10?5 , ?1.0 · 10?4 ]. Evidently, the presence of a “few” negative eigenvalues causes a dramatic slowdown in the convergence rate of the considered PCG method. 5.1. Operation count. First we observe a qualitative di?erence between the behavior of the PCG methods presented. The optimal circulant preconditioner leads min to a sublinear PCG method, while Bn and Bn,5 are optimal preconditioners, and τ (2) is superoptimal. On the other hand, the behaviour of the PCG method associated with τ (1) strongly depends on the speci?c generating function. However, the operation count for each iteration is asymptotically the same (O(n log n)), but it is better for the PCG methods related to the band Toeplitz preconditioners. For instance, the precise count for τ (2) and Bn,5 are shown in Table 4, where ki = 25 for the ?rst iteration (the LU factorization), and ki = 10 otherwise [22]. Therefore, by taking into account the number of iterations, the PCG related to Bn,5 is slightly less expensive than the one related to τ (2). However, it should be stressed that the construction of τ (2) requires the Fourier coe?cients of f and g, while the construction of Bn,5 needs also a precise evaluation of f and g in the Chebyshev points. Whence, in some cases it is easy to practically de?ne τ (2) but we may have di?culties in de?ning Bn,5 . 6. Why are optimal preconditioners so bad? We did not make an explicit comparison with the optimal τ preconditioner, because, in the ill-conditioned case, even if it is always SPD, it does not have good approximation properties with respect to the extremes of the spectrum of An (f ). Let us consider the following elementary example: the ?nite di?erence discretization of the fourth derivative on the unit interval [0, 1]. The resulting matrix is the band Toeplitz operator generated by p(x) = 16 sin4 x . By a direct calculation we obtain 2 that the eigenvalues of the optimal circulant and τ approximations are λi = p(xi ) + and λi = p(zi ) + 2 12 ? 4 cos(xi ) p(xi ) + , n n 2 sin2 (zi ), n+1 xi = 2(i ? 1)π , i = 1, . . . , n n

zi =

iπ , i = 1, . . . , n, n+1

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Table 5. f (x) = (2 ? 2 cos(x))2 n I Copt 32 18 14 128 162 31 τopt 10 16 Hopt 22 55 Aopt 14 31 τ (1) = τ (2) 2 2

respectively. It follows that ?g(n) increasing function of n with limn→∞ g(n) = 0, n ?i = 1, . . . , g(n), λi (An (p)) and λi (G) (G the optimal τ or circulant or anticirculant or Hartley preconditioner) behave in a very di?erent way when n tends to in?nity. In particular, for i constant wrt n, λi (An (p)) ? 1/n4 and λi (G) ? 1/ns with s = 1 in the circulant case and s = 3 in the τ case. Then the condition number of the preconditioned matrices grows, at least, as n4?s . So while the natural τ preconditioner can be good or bad (see Tables 2 and 3), the optimal preconditioners, in the presence of zeros of high orders, are generally bad. For evidence of this, see [23] and Tables 2, 3, and 5, where Aopt and Hopt indicate the optimal anticirculant and Hartley preconditioners. By generalizing the argument used for Table 5, we can say that the “averaging schemes”, which are inherent to minimization in the Frobenius norm, destroy the information about the in?nitesimal order of the small eigenvalues and this is the main reason for the observed slowdown of the related PCG methods. However, for symmetric Toeplitz problems, the optimal τ approach is the best, among the optimals, owing also to the structural symmetry of the τ algebra. References

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