Circulant and skewcirculant matrices for solving Toeplitz problems

Thomas Huckle Institut f¨ur Angewandte Mathematik und Statistik Universit¨at W¨urzburg Am Hubland D-8700 W¨urzburg, Federal Republic of Germany

0. Introduction

In recent papers G. Strang, R. Chan and T. Chan [15,4,5,3] studied the use of circulant matrices C for solving systems of linear equations Tnx = b with ⎛ ⎞ t0 t1 ...... tn−1 ⎜ ⎟ ⎜ . ⎟ ⎜ t1 t0 t1 . ⎟ ⎜ . . ⎟ ⎜ ...... ⎟ Tn := T (t0,t1, ..., tn−1):=⎜ . t1 . . . ⎟ ⎜ . .. .. ⎟ ⎝ . . . t1 ⎠

tn−1 ...... t1 t0 a symmetric positive definite Toeplitz matrix; thereby a symmetric C is de- fined by C = T (c0,c1, ..., ck−1,ck,ck−1, ..., c1) for even n and C = T (c0,c1, ..., ck,ck, ..., c1) for odd n with k = n/2. Applying the preconditioned conjugate gradient method [10] for solving the system −1 Tnx = b one has to find a matrix C such that the eigenvalues of C Tn are clustered and Cy = d can be solved very fast. For circulant C the second condition is fulfilled by [7]. Thus G. Strang choose C := Cs := T (t0,t1,t2, ..., t2,t1) [15], and with R. Chan he proved in [4], that under certain assumptions Cs is again positive definite −1 and the spectrum of Cs Tn is clustered around 1. T. Chan introduced

(n − 1)t1 + tn−1 tn−1 +(n − 1)t1 C := T (t0, , ..., ). f n n

Cf minimizes C − TnF over all symmetric circulant matrices, where F denotes the Frobenius norm. The spectrum of Cf is again clustered around 1 [see 3].

In this paper we introduce new circulant/skewcirculant approximations to Tn and study their properties. Thereby a S is said to be skewcirculant,if S = T (s0,s1, ..., sk, −sk, ..., −s1)foroddn and S = T (s0,s1, ..., sk−1, 0, −sk−1, ..., −s1) for even n [7,8].

If Tn is indefinite any circulant approximation will be indefinite in general; but pre- conditioning is only possible for positive definite C. However from Cs and Cf one can

1 −1 always construct positive definite circulant matrices such that the spectrum of C Tn is approximately clustered around ±1.

Every n × n circulant matrix C has the same eigenvectors; thus approximating Tn by C can be considered as approximating the eigenvalues of Tn and ordering the eigenvectors of C such that they are near to those of Tn. Therefore these eigenvectors are efficient start vectors in the Lanczos algorithm for determining any, especially the maximal or minimal eigenvalue of Tn in O(nlog(n)) arithmetic operations. Finally in section 4 for some numerical examples we compare the clustering properties of the circulant/skewcirculant approximations and test the Lanczos method with the start vectors described above.

1. A new circulant/skewcirculant approximation to T

For symmetric Toeplitz matrices T,T1,T2 let us define a scalar product by (T1,T2)E := T T e1 T1T2e1 with e1 =(1, 0, ..., 0) , and the corresponding norm

n−1 2 2 2 T E := Te1 2 = ti . i=0

Similar to the best circulant Frobenius norm approximation to T one can search for a circulant matrix Ce with

Ce − T E =minC − T E. C circulant

The solution of this problem is

t1 + tn−1 tn−1 + t1 C := T (t0, , ..., ). e 2 2 In the same way we can define the skewcirculant matrix

t1 − tn−1 tn−1 − t1 S := T (t0, , ..., ), e 2 2 which solves min S − T E. S skewcirculant

Then for Ca := Ce +(a − t0)I and Sb := Se +(b − t0)I with a + b = t0 it holds

Ca + Sb = T. (1)

Let us define T := {T/T real symmetric n × nToeplitzmatrix},

C := {C ∈ T/C circulant}, C0 := {C ∈ C0/c0 =0},

2 S := {S ∈ T/S skewcirculant}, S0 := {S ∈ S0/s0 =0}.

Then the matrices T (0, 1, 0, ..., 0, 1), T (0, 0, 1, 0, ..., 0, 1, 0), ... , form a of C0,and T (0, 1, 0, ..., 0, −1), T (0, 0, 1, 0, ..., 0, −1, 0), ... , form a basis of S0, and it holds

C⊥S0, C0⊥S, C ⊕ S0 = T = C0 ⊕ S relative to (., .)E . The equation (1) can be used to derive estimations for the minimal eigenvalue μ1(T ). Following Kato [12,I.-(6.79) and II. Problem (6.2)] it holds

μ1(T )=μ1(Ca + Sb) ≥ μ1(Ca)+μ1(Sb)=μ1(Ce)+μ1(Se) − t0; furthermore for

T (x):=2Sb +(x − 1)(Sb − Ca)=2Ca +(x +1)(Sb − Ca)=(1+x)Sb +(1− x)Ca

the minimal eigenvalue μ1(T (x)) =: μ1(x) is a concave, piecewise holomorphic function of x,andT (−1) = 2Ca, T (0) = T and T (1) = 2Sb.DenotebyxS,resp. xC , the normed eigenvector corresponding to μ1(1), resp. μ1(−1). Then we have the series expansions

− T − μ1(x)=2μ1(Sb)+(x 1)xS (Sb Ca)xS + ... T − μ1(x)=2μ1(Ca)+(x +1)xC(Sb Ca)xC + ... . T − Thus μ1(1) is the maximum of μ1(x)iffxS (Sb Ca)xS = 0; this implies T − − b =(xS CexS + t0 μ1(Se))/2,a= t0 b. (2)

Since μ1(x) is concave this gives an upper bound for μ1(T ):

≤ T − μ1(T ) 2μ1(Sb)=μ1(Se)+xS CexS t0

with b defined by (2). − T − Similarly μ1( 1) is the maximum iff xC (Sb Ca)xC =0;thisgives ≤ T − μ1(T ) μ1(Ce)+xC SexC t0.

Thus we have proved Theorem 1. For the minimum eigenvalue of a symmetric Toeplitz matrix T with circulant and skewcirculant approximations Ce and Se it holds

− ≤ ≤ { T T }− μ1(Ce)+μ1(Se) t0 μ1(T ) min μ1(Se)+xS CexS ,μ1(Ce)+xC SexC t0. (3)

These bounds can be used as an initial intervall in the Cybenko-Van Loan algorithm for computing the minimum eigenvalue of a symmetric positive definite Toeplitz matrix [6]; they can be determined in O(nlog(n)) operations.

3 2. Circulant and skewcirculant preconditioners for Toeplitz systems

For preconditioning the system Tnx = b , also the matrix Ce can be used, but in most examples Cs and Cf seem to have better clustering properties. Another al- ternative is preconditioning with the skewcirculant approximations Ss and Sf ,where Ss := T (t0,t1,t2, ..., −t2, −t1)and

(n − 1)t1 − tn−1 tn−1 − (n − 1)t1 S := T (t0, , ..., ), f n n with Sf − TnF =minS − TnF . S skewcirculant

Now let T = T (t0,t1, ...) be a real single infinite positive definite Toeplitz matrix in ∞ ikΘ ∞ | | ∞ the Wiener class, this means 0 < k=−∞ tke and k=0 tk

(D − RJ)y = λ−(D − RcJ)y and (D + RJ)z = λ+(D + RcJ)z (5) and (D − RJ)y = λ−(D − RsJ)y and (D + RJ)z = λ+(D + RsJ)z. (6)

Defining Hc := (Rc − R)J and Hs := (Rs − R)J,weget ⎛ ⎞ ⎛ ⎞ h1 h2 · 0 g1 g2 · gm ⎜ h2 0 ⎟ ⎜ g2 0 ⎟ H = ⎝ ⎠ and H = ⎝ ⎠ c · 0 s · 0 0000 gm 00 0

4 with hj = tj − tn−j ,gj = −tj − tn−j for j ≤ m. Thus (5) and (6) are equivalent to

(1 − λ+) (1 − λ−) (D + RJ)z = Hcz, (D − RJ)y = −Hcy (7) λ+ λ− and (1 − λ+) (1 − λ−) (D + RJ)z = Hsz, (D − RJ)y = −Hsy. (8) λ+ λ−

As the order n increases, D, Hc and Hs approach singly infinite Toeplitz and Hankel matrices D˜ = T (to,t1,t2, ...), ⎛ ⎞ t1 t2 t3 · ⎜ t2 t3 · ⎟ H˜c = ⎝ ⎠ , t3 · ·

H˜s = −H˜c,andRJ converges against 0. Now in view of (7) and (8) the clustering around −1 −1 1ofCs Tn and Ss Tn depends on the clustering around 0 of the eigenvalues of the infinite Hankel-Toeplitz problems

H˜cz˜ =˜νD˜z˜ and H˜cy˜ = −μ˜D˜y˜

H˜sz˜ =˜νD˜z˜ and H˜sy˜ = −μ˜D˜y.˜

Because of H˜s = −H˜c the proof in [4] for the clustering around 0 ofν ˜ andμ ˜ shows the −1 −1 −1 clustering of λ−(Ss Tn)andλ+(Ss Tn). Thus for large n the eigenvalues of Ss Tn are clustered around 1, too.

The eigenvalues of Ss have the form

j j m−1 j m+1 j n−1 λj = t0 + t1σw + ... + tm−1(σw ) +0− tm−1(σw ) − ... − t1(σw ) =

m−1 j j −1 (2j+1)kπi/n = t0 + t1(σw +(σw ) )+... = tke k=1−m

iπ/n 2 −1 with σ = e and w = σ ;thusSs and Ss are positive definite and uniformly bounded for positive definite T (see [4, equation (3)] for the circulant case and [7]). Hence, we have proved Theorem 2. Suppose T is real and positive and in the Wiener class. Then for large −1 n the skewcirculants Ss and Ss are uniformly bounded and positive definite, and the −1 eigenvalues of Ss Tn are clustered around 1.

In[3]R.Chanproved,thatthespectralradiusof Cs − Cf tends to 0 for n →∞and −1 that for large nCf and Cf are also positive definite and uniformly bounded, and that −1 −1 the spectra of Cs Tn and Cf Tn are assymptotically equal; his proof can be transfered nearly without changes to the skewcirculant case and thus the same holds for Ss, Sf and −1 Ss .

5 In order to show, that skewcirculant matrices are efficient preconditioners, it remains to prove, that the system Sy = d can be solved very fast. The eigenvalues of an n×n circulant matrix Cn and a skewcirculant matrix Sn are given by [see 7]

2jπi/n 2jπi2/n 2jπi(n−1)/n T xj (Cn)=(1,e ,e , ..., e )

and (2j+1)πi/n (2j+1)πi(n−1)/n T xj (Sn)=(1,e , ..., e ) . Thus it holds

2(2j+1)πi/(2n) 2(2j+1)πi2/(2n) T T x2j+1(C2n)=(1,e ,e , ...) =(xj (Sn) , ∗ , ∗ ,...,∗ ) ,

T T and C2n and Sn have diagonalizations C2n = U2nDcU2n and Sn = VnDsVn with Vn a submatrix of the Fourier matrix U2n. Hence, a linear system with Sn canbesolvedby FFT of order 2n. Another way to get a solution of a linear skewcirculant system may be the use of the skewcirculant version of the FFT [9].

All in all the skewcirculant approximations to Tn have the same properties as the circulant approximations and are efficient preconditioners. It depends on the special structure of Tn, which of the two alternatives is to prefer, e.g. in some examples the skewcirculant matrix Ss is positive definite while Cs is indefinite [4, example 1] and in some other examples the −1 −1 eigenvalues of Ss Tn are better clustered than that of Cs Tn.

In [11, 7.6] Szeg¨o and Grenander have introduced another circulant matrix, that approximates a Toeplitz matrix Tn and can√ be used for preconditioning. Set U := n 2πikj/n (u1, ..., un):=(uk,j)k,j=1 with uk,j := e / n and D := diag(d1, ..., dn)with

p j 2πjk d := t0 +2 (1 − )t cos (9) k p j n j=1

H for 0 ≤ p ≤ n.ThenCg := U DU is a circulant matrix with double eigenvalues dk = dn−k, k =1, ..., n/2, and one or two simple eigenvalues dn and dn/2 (for even n). For positive definite T (9) shows, that Cg is positive definite and bounded for large p and n.The eigenspace to dk, k<n/2,isequaltospan(uk,un−k)=span(vk,vn−k)with

n T T T √2 2πkj vk := uk + un−k = cos( ) , n n j=1 n−1 T −2πik/n T 2πik/n T √2 2πkj vn−k := e uk + e un−k = cos( ) . n n j=0

Then vk ± vn−k are the reciprocal, resp. antireciprocal, eigenvectors of dk [see 2]. Thereby a vector x is said to be reciprocal (antireciprocal), if Jx = x (Jx = −x)withJ defined in (4).

6 In [8, 7.3] Szeg¨o and Grenander introduced the norm

1 n |T |2 = λ (T )2 n 2π j n j

with λj (Tn) the eigenvalues of Tn, and proved, that

∞ p 2p n 2j2(p − j) |T − C |2 ≤ t2 +2 t2 + t2. n g n j j np2 j j=1 j=p+1 j=1

 Thus for Cg to be a good approximation to Tn we have to choose p such that p n n 2  p 2 and j=p+1 tj j=1 tj . Hence, especially for Toeplitz band matrices with bandwith  p nCg is√ a good circulant approximation. In the general case, where T is in the Wiener class, p = n is a possible choice, which for n large enough guarantees, that Cg is a good approximation to Tn.

T If Tn is indefinite, any circulant approximation C = VDV with D diagonal and −1 V unitary will be indefinite, too; so the eigenvalues of C Tn are in general complex. An obvious way to generate a positive definite circulant matrix for preconditioning is the T choice of C˜ := V |D|V .IfC is a good approximation to Tn, the same will be true for the eigenvalues and eigenvectors, and we get − 1 − 1 − 1 − 1 − 1 − 1 I1 0 C˜ 2 T C˜ 2 = V |D| 2 V T W ΔW T V |D| 2 V T ≈ V |D| 2 Δ|D| 2 V T ≈ V V T 0 −I2

T −1 with Tn = W ΔW , and thus the eigenvalues of C˜ T seem to be clustered around ±1. Hence C˜ can be a good choice for preconditioning the indefinite linear system Tx = b.

3. Computing eigenvalues and eigenvectors of a symmetric Toeplitz matrix

Computing the eigenvalues of a symmetric Toeplitz matrix Tn, especially the minimum eigenvalue, is a problem of considerable interest [14,6,16]. By a suitable shift we can assume, that Tn is positive definite. For the following let C and S be any circulant approximation to Tn. All eigenvectors of Tn -andC,resp. S - can be chosen to be reciprocal or antireciprocal. Hence assuming, that the eigenvectors of C or S are good approximations to those of Tn, leads to the following algorithm for computing the minimum eigenvalue of Tn:

(i) Compute the smallest eigenvalues of C and S and the corresponding reciprocal (an- tireciprocal) eigenvectors vr(C), vr(S), va(C), va(S), and set Ur := (vr(C),vr(S)) , Ua := (va(C),va(S)); if the minimum eigenvalue of C or S is a simple eigenvalue and has no reciprocal or antireciprocal eigenvector, then one can complete Ur,resp. Ua,byan eigenvector of the next smallest eigenvalue.

7 T T (ii) Compute λr := λmin(Ur TUr)andλa := λmin(Ua TUa) with corresponding eigenvec- tors xr and xa.

(iii) Define yr := Urxr, ya := Uaxa, and apply Lanczos algorithm [13,10] to the matrix −1 A = Tn and vectors yr,resp.ya.

The algorithm generates approximations to the eigenvectors of Tn, that are either reciprocal or antireciprocal; thus, if the eigenvector to λmin(Tn) is reciprocal, then λr will tend to λmin(Tn), otherwise λa will tend to λmin(Tn).

The algorithm can be improved replacing (iii) by

(iii’) Set yr := Urxr and ya := Uaxa;letμr and μa be chosen such that 0 <μr ≤ λr and −1 −1 0 <μa ≤ λa; apply the Lanczos method to Ar := (Tn − μrI) ,resp. Aa := (Tn − μaI) . If we define μr = λr, then the matrix Ar is indefinite and solving a linear system with the matrix Tn − μrI may get unstable [see 1]; on the other side if we choose μr ≈ 0, we get no improvement over (iii).

In all numerical experiments with (iii’) λmin(Tn) is computed to sufficient accuracy by only one Lanczos step. Thereby μr can be defined by λr or in a similar way. Computing the eigenvectors and eigenvalues of C and S needs O(nlog(n)) operations. The cost of each Lanczos step depends on the method used for solving (Tn − μI)x = b,e.g. solving Yule-Walker takes in VLSI architecture O(n) operations, the new ”fast” algorithms take O(nlog(n)2) operations [1]; another method may be the preconditioned cg algorithm. In view of the low number of arithmetic operations required for our algorithm, it may be also efficent, to use the first or second computed estimate for λmin(Tn) together with the lower bound of (3) as an initial intervall for the Cybenko-Van Loan method.

Obviously the algorithm can be generalized to compute λmax(Tn)orλi(Tn), 1 ≤ i ≤ n by starting with the i-th eigenvalues and eigenvectors of C and S.

8 4. Examples

For comparing the circulant/skewcirculant approximations to Tn and for testing the Lanczos algorithm described above we consider the following examples:

2 1. ti =1/(i +1) for i =0, ..., 14 , cf. [15,5];

2. t0 =1 and ti = random(−.2,.2) for i =1, ..., 14 ;

3. ti = cos(i)/(i +1) for i =0, ..., 14 , cf. [5];

4. t0 =11.7122 and ti = random(−1, 1) for i =1, ..., 19 ;

5. t0 =6.2, ti = random(−1, 1) for i =1, ..., 10 and ti =0 for i =11, ..., 19 ; 20 × − 6. T = k=1 wkT2πΘk a20 20 matrix with (TΘ)i,j = cos(Θ(i j)) for i, j =1, ..., 20 and wk and Θk are uniformly distributed random numbers taken from [0, 1], cf. [6]; 40 × 7. T = k=1 wkT2πΘk a40 40 matrix defined in the same way like 6.

n−1 | | The first table demonstrates the lower bound of (3); thereby r := k=1 tk .

Nr. μ1(Ce)+μ1(Se) − t0 μ1(T ) μn(T ) t0 r 1 0.6463 0.6475 1.9645 1 0.58 2 0.2397 0.6822 1.6376 1 0.98 3 0.4514 0.4536 2.3584 1 1.39 4 5.3373 7.3197 17.210 11.7122 10.7 5 2.0180 2.3841 11.139 6.2 5.2 6 -2.6052 1.3627 27.017 8.67 26.1 7 -12.3068 0.0475 71.16 17.1 82.1 Table 1

Figure 1 shows the eigenvalue distribution of the various circulant/skewcirculant approxi- mations for the first example;√ thereby the y-values 1, 2, ..., 10 are related to T , Ce, Se, Cs, Ss, Cf , Sf , Cg with p = n, Cg with p = n/2, and Cg with p = n − 1.

9 Fig. 1. Eigenvalue distribution for example 1.

The following 7 figures show the spectrum of X−1T with preconditioner X;therebythe −1 −1 −1 −1 −1 −1 −1 y-values√ 1, 2, ..., 9 are related to Ce T , Se T , Cs T , Ss T , Cf T , Sf T , Cg T with −1 −1 − p = n, Cg T with p = n/2, and Cg T with p = n 1.

Fig. 2. Clustering for example 1.

Fig. 3. Clustering for example 2.

10 Fig. 4. Clustering for example 3.

Fig. 5. Clustering for example 4.

Fig. 6. Clustering for example 5.

11 Fig. 7. Clustering for example 6.

Fig. 8. Clustering for example 7.

The worse eigenvalues in Figure 7 and 8 are related to negative eigenvalues of the corre- sponding circulant/skewcirculant approximations; thus to avoid indefinite preconditioners it may be efficient, to use always C˜ asdefinedinsection2. The positive defineteness is in all examples preserved by the Frobenius norm approxima- tions Cf and Sf and by the Szeg¨o-Grenander approximations Cg.ForCg it seems, that the optimal choice of p should be n − 1.

Next let us demonstrate the clustering around ±1 in the indefinite case for the Toeplitz matrices of example 1. For that purpose we determine Cs for the indefinite Toeplitz − − − ˜−1 − matrices T I , T 0.7I and T 0.9I , and compute the spectrum of Cs T αI . Thereby in Figure 9 the y-values 1,2,3 are related to α =1, 0.7, 0.9.

12 ˜−1 − Fig. 9. Spectrum of Cs (T αI)forα =1, 0.7, 0.9.

Table 2 shows the behaviour of the Lanczos algorithm described above for computing the maximum eigenvalue μn(T )usingCf and Sf as circulant/skewcirculant approximations to T .Therebyun denotes the eigenvector of Euclidean length 1 corresponding to μn(T ), and ucn, usn, ucsn, ucsn,n−2 are the projections b of un on the subspaces spanned by the eigenvectors related to μi(Cf )andμi(Sf )fori = n, n−2withthesameJ-symmetry as un. In the μn-columns of Table 2 the Lanczos estimations for μn(T ) are listed corresponding to the Krylov spaces (b, T b, T 2b, .., T (i−1)b). Table 3 demonstrates the Lanczos algorithm with step (iii) as described in section 3.

Nr. un − ucn μn un − usn μn un − ucsn μn un − ucsn,n−2 μn μn(T ) 2 0.58 1.4834 0.63 1.3508 0.5314 1.4908 0.0586 1.4908 1.6134 1.6050 1.6072 1.6124 1.6131 3 0.5324 2.0444 0.2076 2.3096 0.0349 2.3568 0.0349 2.3568 2.3584 6 0.1534 26.753 0.0339 26.993 27.017 26.992 27.017 27.016 7 0.0495 72.052 72.160 72.160 Table 2. Lanczos algorithm for computing the maximum eigenvalue of T .

Nr. u1 − uc1 μ1 u1 − us1 μ1 u1 − ucs1 μ1 u1 − ucs1,3 μ1 μ1(T ) 4 0.69 8.2486 0.9486 9.0213 0.6737 8.2486 7.3197 5 0.3955 2.8370 0.3126 2.6472 0.1031 2.4195 2.3841 7 0.1657 0.3978 0.1442 0.3925 0.0475 Table 3. Lanczos algorithm for computing the minimum eigenvalue of T .

13 Using the Lanczos algorithm with (iii’) in all examples the eigenvalues of T are com- puted in sufficient accuracy by only one Lanczos step.

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