Modified SMS method for computing outer inverses of Toeplitz matrices

Applied Mathematics and Computation 218 (2011) 3131–3143

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Modified SMS method for computing outer inverses of Toeplitz matrices q Marko Miladinovic´ ⇑, Sladjana Miljkovic´, Predrag Stanimirovic´ University of Niš, Department of Mathematics, Faculty of Science, Višegradska 33, 18000 Niš, Serbia

a r t i c l e

i n f o

Keywords: Toeplitz matrix Displacement rank Successive matrix squaring Outer inverse Convergence rate

a b s t r a c t We introduce a new algorithm based on the successive matrix squaring (SMS) method. This algorithm uses the strategy of e-displacement rank in order to find various outer inverses with prescribed ranges and null spaces of a square Toeplitz matrix. Using the idea of displacement theory which decreases the memory space requirements as well as the computational cost, our method tends to be very effective for Toeplitz matrices. Ó 2011 Elsevier Inc. All rights reserved.

1. Introduction The classical Newton’s iteration

X iþ1 ¼ UðX i Þ ¼ 2X i  X i MX i

i ¼ 0; 1; 2; . . . ;

ð1:1Þ

for the inversion of a matrix M was proposed by Schultz in 1933 [29] and it is well studied in [26]. If we look at Schultz’s paper, we immediately find that he does not suggest this method for arbitrary matrices (this would be a bit too expensive). It is interesting that he considers an example of a Toeplitz matrix since its structure admits very cheap matrix operations. Thus, the Schultz iteration is advocated only in case of agreeably structured matrices. Kailath et al. [17] introduce the concept of displacement rank as well as displacement operator for close-to-Toeplitz matrices, which is systematically studied in the general case in [13]. It is shown that the displacement theory approach is very powerful in designing fast inversion algorithms for structured matrices, especially Toeplitz and Hankel matrices. A number of different displacement operators have been introduced and analyzed in the literature [2,14,17,19,20,25,32]. The initial guess X0 is chosen such that it has low displacement rank. The limit X1, which represents the inverse or the generalized inverse of M, has also low displacement rank, see [13,14]. However, the iteration matrices Xi in the Newton’s method may not have low displacement rank. In the worst case, the displacement rank of Xi increases exponentially until it reaches the n i.e., the dimension of the matrix. Parallel with the increase of the displacement rank, the computational cost of the Newton’s process grows significantly while i increments. It follows that the growth of the displacement rank of Xi must be controlled for the purpose of developing an efficient algorithm. In order to make out of the Newton’s iteration a useful tool for computing the inverse as well as generalized inverses of a Toeplitz matrix, the authors in [2] introduce the concept of the orthogonal displacement representation and e-displacement rank of a matrix. In this way they control the growth of the displacement rank of Xi. At each step, via the truncation, the iteration matrix is approximated by an appropriate matrix with a low displacement rank. This strategy belongs to the class of compression techniques, for inversion of structured matrices which are described in [7,8,25,28]. Because of the displacement structure of the iteration matrix, the matrix-vector multiplication involved in Newton’s iteration can be done efficiently.

q

This paper is supported by the research Project 174013 of the Serbian Ministry of Science. The authors are grateful for this type of support.

⇑ Corresponding author.

E-mail addresses: [email protected] (M. Miladinovic´), [email protected] (S. Miljkovic´), [email protected] (P. Stanimirovic´). 0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.08.046

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Numerical properties of the Shultz method powered by the concept of orthogonal displacement representation as well as

e-displacement rank show the effectiveness of the method applied on structured matrices. The entire computation uses much less memory and CPU time with respect to the classical method, especially for large scale matrices. Let Cmn and Crmn denote the set of all complex m  n matrices and the set of all complex m  n matrices of rank r, respectively. By A⁄, RðAÞ; rankðAÞ and N ðAÞ we denote the conjugate transpose, the range, the rank and the null space of A 2 Cmn , respectively. Recall that, for an arbitrary matrix A 2 Cmn , the set of all outer inverses (or also called {2}-inverses) is defined by the following

Af2g ¼ fX 2 Cnm jXAX ¼ Xg:

ð1:2Þ (2)

With A{2}s we denote the set of all outer inverses of rank s and the symbol A stands for an arbitrary outer inverse of A. If A 2 Cmn ; V is a subspace of Cn of dimension t 6 r and U is a subspace of Cm of dimension m  t, then A has a {2}-inverse X r ð2Þ such that RðXÞ ¼ V and N ðXÞ ¼ U if and only if AV  U ¼ Cm ; in which case X is unique and it is denoted by AV;U . The outer generalized inverse with prescribed range V and null-space U is the generalized inverse of special interest in matrix theory. The reason of the importance of this inverse is the fact that the most common generalized inverses: the Moore–Penrose inverse A , the weighted Moore–Penrose inverse AyM;N , the Drazin inverse AD, the group inverse A#, the ð1Þ ðþÞ Bott-Duffin inverse AðLÞ and the generalized Bott-Duffin inverse AðLÞ are all {2}-generalized inverses of A with prescribed range and null space. ð2Þ In order to compute the generalized inverse AV;U many authors have proposed various numerical methods (see, for example, [22,36,37]). The Newton’s iterative scheme (1.1) has been frequently modified for the purpose of computing various generalized inverses of the structured matrices. Main results are comprised in [2–4,31,32]. The authors in the previously mentioned papers use various strategies in choosing the starting approximation X0, in defining the iterations as well as in the usage of displacement operators. The paper aims at presenting a uniform algorithm applicable in computation of outer inverses with prescribed range and null space of an arbitrary Toeplitz matrix. Our algorithm is based on the successive matrix squaring (SMS) algorithm, which is an equivalent to the Shultz method. Besides the classical modifications of the Shultz method available in the literature, corresponding to the orthogonal displacement representation and the e-displacement rank, we use several adaptations of the SMS algorithm which enable the uniform approach for various generalized inverses. The algorithm is an adaptation of the SMS algorithm proposed in [30] and uses the concept of the approximate orthogonal displacement representation introduced in [2,4]. Since the SMS algorithm for computing outer inverses from [30] comprises known SMS techniques from [6,33,34], it seems reasonable to expect that the algorithm presented in the current paper generalizes known results from the papers [2–4,31,32]. Indeed, our method can be considered as a universal algorithm which computes the outer generalized inverses of a square Toeplitz matrix. In particular cases of the general algorithm, we derive iterative procedures for computing reflexive g-inverses, the Moore–Penrose, the group and the Drazin inverse, suitable for square Toeplitz matrices. Therefore, our method includes algorithms investigated in [2,4] as particular cases. Furthermore, in the case m = n we get an alternative approach in computation of the Moore–Penrose inverse, taking into account that the algorithms derived in [3,32] are based on the different concepts of displacement operator. The organization of the remainder of this paper is the following. In the next section we state some known definitions related to a displacement operator and e-displacement rank. Also, the unified algorithm is presented. Convergence properties and the computational complexity of the algorithm are investigated in the Section 3. Numerical examples and some applications are presented in the fourth section. 2. Algorithm construction The concept of displacement operator, introduced in [17], has been studied in many papers [8,12,14,18,19,27] and its ð2Þ many different forms are presented. Displacement structure of the Drazin inverse, the group inverse and the M V;U inverse is considered in [4,9,21,35]. In the present paper we keep our attention on the displacement operator of Sylvester type, given by

DA;B ðMÞ ¼ AM  MB;

ð2:1Þ

where A, B and M are matrices of appropriate sizes. The next proposition gives some arithmetics regarding the defined displacement operator DA,B. Proposition 2.1. The following holds

DA;B ðMNÞ ¼ DA;B ðMÞN þ M DA;B ðNÞ þ MðB  AÞN: Proof. Follows immediately from the definition of the displacement operator DA,B.

h

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The rank of the displacement matrix DA, B(M) is called the displacement rank of M and is denoted by drk (M). For a Toeplitz matrix M and appropriate choices of matrices A and B the following holds drk (M) 6 2 [13]. Here we recall the definition of edisplacement rank as a rank of perturbed displacement. Definition 2.1 [2]. For a given e > 0 the relative e-displacement rank of a matrix M is defined as

drke ðMÞ ¼ min rankðDA;B ðMÞ þ EÞ; kEk6ekMk

where kk denotes the Euclidian norm. The main purpose of the displacement operator is to enable representation of the original matrix through a matrix that has low rank and from which one can also easily recover the original matrix. Besides the fact that the displacement matrix DA,B(M) has low rank it is well-known that its singular value decomposition can be easily computed. Remark 2.1. The computation of the singular value decomposition of a matrix, in general, is a very expensive tool. In our case, what makes it cheaper, as it is also explained in [2], is the fact that it can be calculated using the QR factorization of appropriate matrices. Let r1 P    P rk be the nonzero singular values of DA,B(M) and let

DA;B ðMÞ ¼ U RV T ¼

k X

ri ui v Ti ;

ð2:2Þ

i¼1

be the singular value decomposition of DA,B(M). Suppose that the original matrix M can be recovered by the formula

M¼c

k X

ri f ðui Þgðv i Þ;

ð2:3Þ

i¼1

where f and g, generally speaking, represent the products of matrices which can be generated by the vectors ui and vi respectively and where c is a constant. The representation (2.3) of the matrix M is called orthogonal displacement representation (odr) (with respect to DA,B) and the corresponding 3-tuple (U, r, V) orthogonal displacement generator (odg) where r = (r1, r2, . . . , rk) [2]. For example, the odr of an arbitrary Toeplitz matrix is given in [2]. P Proposition 2.2. Suppose that M is a Toeplitz matrix. Let DA;B ðMÞ ¼ U RV T ¼ ki¼1 ri ui v Ti ðr1 P r2 P    P rk > 0Þ be the singular value decomposition of DA,B(M) and let e be such that 0 < e < r1. Then drke(M) = r if and only if rr > ekMk P rr+1. Proof. See Theorem 2.1 from [2]. h For a given e > 0, if drke(M) = r one can get an approximate Me of M

Me ¼ c

r X

ri f ðui Þgðv i Þ; r < k;

i¼1

which is called an approximate orthogonal displacement representation (aodr) of the matrix M and the associated generator b r b Þ is called an approximate orthogonal displacement generator (aodg), where ^; V ð U;

r^ ¼ ðr1 ; r2 ; . . . ; rr Þ; Ub ¼ ½u1 ; u2 ; . . . ; ur ; Vb ¼ ½v 1 ; v 2 ; . . . ; v r : We also use the definition of the operator trunce() defined on the set of matrices as follows Me = trunce(M) [2]. In the present paper we derive an algorithm for finding the outer inverse with prescribed range and null space of an arbitrary Toeplitz matrix M 2 Cnn . The entries of the matrix M, defined by the general rule mi,j = mij, are constant complex numbers along the main diagonal parallels. In the remainder of the paper the notation toeplitz[c, r] stands for the Toeplitz matrix which is determined by its first column c and its first row r. As usual, by C+ and C we denote the Toeplitz matrices defined by C þ ¼ Z þ e1 eTn ; C  ¼ Z  e1 eTn ; where Z = toeplitz[(0, 1, 0, . . . , 0), (0, . . . , 0)] and ei is the ith column of the identity matrix. Let C+(x) and C(x) be circulant and anti-circulant matrices which are defined by

C þ ðxÞ ¼ toeplitz ½ðx1 ; x2 ; . . . ; xn Þ; ðx1 ; xn ; . . . ; x2 Þ; C  ðxÞ ¼ toeplitz ½ðx1 ; x2 ; . . . ; xn Þ; ðx1 ; xn ; . . . ; x2 Þ: For an arbitrary Toeplitz matrix M we make the choice A = C and B = C+ in the definition of the displacement operator (2.1). The parameters from (2.3) become [2]:

1 c¼ ; 2

f ðui Þ ¼ C  ðui Þ;

gðv i Þ ¼ C þ ðJ v i Þ;

where J is the permutation matrix having 1 on the anti-diagonal. For the sake of simplicity, in the rest of the paper we use the notation D instead of DC  ;C þ .

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Remark 2.2 [2]. For the approximation of the matrix M 2 Cnn given by its aodg Me, the matrix-vector product can be efficiently done by Oðrn log nÞ FFT’s, where r = drke(M). The following result provides an error bound of the aodr of M. Proposition 2.3 [2]. Suppose that M is a Toeplitz matrix. Let r = drke(M) 6 drk (M) = k, Me be an aodr of M, and let r1, r2, . . . , rk, be the nonzero singular values of DA,B(M). Then it holds

kM  M e k 6

k 1 X 1 n ri 6 nðk  rÞe: 2 i¼rþ1 2

ð2:4Þ

Many authors have built a number of iterative methods for finding generalized inverses of an arbitrary Toeplitz matrix by modifying the Newton’s method [2–4,31,32]. For that purpose, they use operators of the form DA,B(M), as well as the concept of displacement representation and e-displacement rank of matrices. It is shown that these methods are very effective and that they significantly decrease computational cost. In the following part of this section we construct an algorithm for finding outer inverses with prescribed range an null space of an arbitrary square Toeplitz matrix. Let M 2 Cnn and R 2 Cnn ; 0 6 s 6 q be arbitrary matrices. We start with the following iterative scheme (see [30]) s q

X 1 ¼ Q; X kþ1 ¼ PX k þ Q ;

ð2:5Þ

k 2 N;

where P = I  bRM, Q = bR and b presents the relaxation parameter. The original idea of the iterative scheme (2.5), named the successive matrix squaring method, was previously proposed by Chen et al. [6] in the case R = M⁄. The method proposed by Chen et al. is used for computing the Moore–Penrose inverse of the matrix M. Instead of the iterative scheme (2.5) it is possible to use the 2n  2n matrix

 T¼

P

Q

0

I

 :

The kth matrix power of the matrix T is given by

2

k

6P T ¼4 0 k

kP 1

3

Pi Q 7 5; i¼0 I

k > 1;

Pk1 i where the n  n matrix block i¼0 P Q is actually the kth approximation Xk of the pseudoinverse of the matrix M from (2.5), Pk1 i whence X k ¼ i¼0 P Q : The matrix power Tk can be computed by means of the repeated squaring of the matrix T, therefore instead of using the iterative scheme (2.5) we can observe the iterative process

T 0 ¼ T; T kþ1 ¼ T 2k ;

ð2:6Þ

k 2 N0 :

P Afterwards, the block i P i Q generated after k steps of the successive squaring (2.6) is equivalent with the approximation P k produced by 2 steps of the iterative process (2.5). Namely, we consider the sequence {Yk} given by Y k ¼ X 2k ¼ i P i Q . For the sake of completeness we restate the following well-known result. Lemma 2.1 [15]. Let H 2 Cnn and e > 0 be given. There is at least one matrix norm kk such that

qðHÞ 6 kHk 6 qðHÞ þ e;

ð2:7Þ

where q(H) denotes the spectral radius of H. The authors in [30] prove that the iterative scheme given in terms of the block matrix (2.6) converges to the outer inverse of the matrix M with prescribed range and null space, determined by an appropriately chosen matrix R. Proposition 2.4 [30]. Let M 2 Cnn and R 2 Cnn ; 0 6 s 6 q be given such that s q

MRðRÞ  N ðRÞ ¼ Cn :

ð2:8Þ

The sequence of approximations

Yk ¼

k 2X 1

ðI  bRMÞi bR;

ð2:9Þ

i¼0 ð2Þ

converges, in the matrix norm kk to the outer inverse Y ¼ M RðRÞ;N ðRÞ of the matrix M if b is a fixed real number such that

max j1  bki j < 1; 16i6t

where rank(RM) = t, ki, i = 1, . . . t are eigenvalues of RM, and kk satisfies condition (2.7) for H = I  bMR.

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Since the usage of the partitioned 2n  2n matrix in the SMS acceleration scheme is not appropriate for application of a displacement operator, our intention is to avoid such an approach. For that purpose, we observe that the sequence of approximations given by (2.9) can be rewritten by the following steps

Y kþ1 ¼ ðSk þ IÞY k ;

k 2 N0 ;

Skþ1 ¼ I  Y kþ1 M;

ð2:10Þ k

where the initial choices are Y0 = Q, S0 = P. With the rules given by (2.10) one can easily verify that Y k ¼ X 2k and Sk ¼ P 2 . Because of ineffective manipulation with high displacement rank matrix sequence {Yk} whose displacement rank, in the worst case, can increases exponentially, we introduce the following modification of the method (2.10). Namely, using the concept of the displacement representation, we modify the successive matrix squaring method from [30] by approximating the matrix Yk+1 with a matrix with a low displacement rank. We define the sequence {Zk} by

Z kþ1 ¼ ððPk þ IÞZ k Þek Pkþ1 ¼ I  Z kþ1 M;

k 2 N0 ;

ð2:11Þ

with the starting values Z0 = Q, P0 = P. For notational simplicity let us denote

Z 0k ¼ ðPk þ IÞZ k ;

ð2:12Þ drk(Z0k )

Z0k

which satisfies the inequality 6 drk(Pk) + drk(Zk). After we use the aodr of we have that the displacement rank of the sequence {Zk} is low during the iterative process. The low displacement rank of the sequence of matrices Pk is also retained, since drk(Pk) 6 drk(Zk) + m, where m = drk(M). In the rest of this section we present an algorithm for computing the outer inverse with prescribed range and null space of an arbitrary Toeplitz matrix. In our modified SMS method we compute and store the SVD of D(Zk) instead of Zk and later determine Zk using (2.3). From (2.12) and Proposition 2.1 we obtain

DðZ 0k Þ ¼ DðPk ÞZ k þ Pk DðZ k Þ þ DðZ k Þ þ 2Pk e1 eTn Z k : If we denote the SVD of D(Zk) by U Z R the following matrix form

T ZVZ

and the SVD of D(Pk) by U P R

2

DðZ 0k Þ ¼ U Z0 RZ0 V TZ0 ¼ ½ U P

ð2:13Þ

ðPk þ IÞU Z

RP

0

6 Pk e1 4 0

RZ

0

0

0

32

T PVP

V TP Z k

then the extended form of

D(Z0k )

can be rewritten by

3

76 7 0 54 V TZ 5: T 2 en Z k

ð2:14Þ

On the other hand, in order to compute the SVD of D(Pk+1), which we need for the next iteration, we have the following

DðPkþ1 Þ ¼ DðIÞ  DðZ kþ1 ÞM  Z kþ1 DðMÞ  2Z kþ1 e1 eTn M: We denote the SVD of D(Zk+1) by U Z RZ V TZ and the SVD of D(M) by U M RM V TM . The extended form of the previous identity can be written by

2

DðPkþ1 Þ ¼ U P RP V TP ¼ ½ U Z

Z kþ1 U M

Z kþ1 e1

RZ

6 0 6 e1 6 4 0 0

0

RM 0 0

32 T 3 VZM 6 T 7 6 0 07 76 V M 7 7 T 7 7: 2 0 56 4 en M 5 0 2 eTn

0

0

Therefore the SVD of D(Zk+1) can be computed by Algorithm 2.1. Algorithm 2.1. The odg of Zk+1 and the odg of Pk+1 Require: The odg of Zk, the odg of Pk, the odg of M and the truncation value e. 1: Determine the matrices U Z 0 ; RZ 0 ; V TZ 0 according to (2.14). 2: Compute the QR factorizations U Z 0 ¼ Q 1 R1 and V Z 0 ¼ Q 2 R2 . 3: Compute the SVD of R1 RZ 0 RT2 ¼ U RV. b ¼ Diagðr1 ; r2 ; . . . ; rr Þ, such that r is defined in relation to the truncation value e:rr+1 6 er1 < rr. 4: Set r b RZ ¼ R b , where U b and V b present the first r columns of U, V respectively. b and V Z ¼ Q 2 V 5: Set U Z ¼ Q 1 U; kþ1

kþ1

kþ1

6: Determine the matrices U P ; RP ; V TP according to (2.15). 7: Compute the QR factorizations UP = Q3R3 and VP = Q4R4. 8: Compute the SVD of R3 RP RT4 ¼ U RV. 9: Set U Pkþ1 ¼ Q 3 U; RPkþ1 ¼ R and V Pkþ1 ¼ Q 4 V.

ð2:15Þ

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The computational cost of Algorithm 2.1 is about OðmhÞ FFT’s, where h = maxkdrk(Zk). If h is small enough, or even it is independent of n, the algorithm requires Oðn log nÞ operations per step. To complete our modified SMS method we present Algorithm 2.2 for finding the outer inverses of a Toeplitz matrix given in terms of odr for the approximate matrices with respect to the operator D. Algorithm 2.2 Modified SMS algorithm Require: A Toeplitz matrix M and its displacement operator D, a matrix R, a relaxation parameter b and a residual error bound n. 1: Set k = 0, Zk = bR and Pk = I  bRM 2: Compute the SVD of DðPk Þ ¼ U Pk RPk V TPk ; DðZ k Þ ¼ U Z k RZ k V TZ k and DðMÞ ¼ U M RM V TM . 3: Determine the truncation value ek. 4: Compute the odg ðU Z kþ1 ; RZ kþ1 ; V Z kþ1 Þ of Zk+1 and ðU Pkþ1 ; RPkþ1 ; V Pkþ1 Þ of Pk+1 according to Algorithm 2.1 with input   ðU Z k ; RZ k ; V Z k Þ; ðU Pk ; RPk ; V Pk Þ; U M RM V TM and e = ek. 5: Determine the norm kres(Zk+1)k. If kres(Zk+1)k < n goto step 7, otherwise continue 6: Set k = k + 1, goto step 3 7: Return the matrix Zk+1.

The value of kres(Zk)k in Step 5 presents the norm of the residual kres(Zk)k = kZkMZk  Zkk. Since, in general, computing kres(Zk)k is an expensive operation, a cheaper way is to compute the norm of the vector kres(Zk)e1k. This operation can be efficiently computed by using a few FFT’s. 3. Convergence properties It is well-known that matrices with an AB-displacement structure possess generalized inverses with a BA-displacement structure. Investigations in this field started in the paper [14] for the Moore–Penrose inverse. In the recent paper [21] these ð2Þ investigations are extended to the generalized inverse MV ;U . ð2Þ

Proposition 3.1 [21]. Let M 2 Cnn be an arbitrary matrix and M V ;U be its {2}-inverse. Then

   ð2Þ rank DB;A M V;U 6 rankðDA;B ðMÞÞ þ rankðDB;A ðRÞÞ:

In the case when the input matrix M has a low displacement rank with respect to DA,B and R has a low displacement rank ð2Þ with respect to DB,A, the generalized inverse M V;U also possesses a low displacement rank with respect to DB,A. As a consequence, it is evident that choosing the matrix R to be a matrix with low displacement rank (with respect to DB,A) is of crucial importance to keep the displacement rank of the outer inverse with prescribed range and null space low. Then, it is possible to use the following result from [12] in order to show the convergence of the algorithm. Theorem 3.1 [12]. Let V be a normed space and consider a function f : V ? V and M 2 V. Assume that B = f(M) can be obtained by the iteration of the form

X k ¼ UðX k1 Þ;

k ¼ 1; 2; . . . ;

ð3:1Þ

where U is a one-step operator. Further, assume that for any initial guess X0 sufficiently close to B, the process converges:

lim X k ¼ B:

x!1

Assume that there are constants cU,

U > 0 and a > 1 such that

kUðXÞ  Bk 6 cU kX  Bka for all X 2 V with kX  Bk 6 U :

ð3:2Þ

Let S  V be a set of structured elements and trunce : V ? S be a truncation operator such that

kX  X e k 6 cR kX  Bk for allX 2 V with kX  Bk 6 U :

ð3:3Þ

Then there exists d > 0 such that the truncated iterative process Yk = (U(Yk1))e converges to B so that

kY k  Bk 6 cRU kY k1  Bka

with cRU ¼ ðcR þ 1ÞcU ;

k ¼ 1; 2; . . . :

ð3:4Þ

for any starting value Y0 = (Y0)e satisfying kY0  Bk < d. For our iterative process, the condition (3.2) in Theorem 3.1 is undoubtly satisfied for the choices cU = kMk and a = 2. The choice of the truncation value ek at each step of the iterative process (2.11) is very important, since it indicates what will be the value of the displacement rank r, which is actually the balance point between the convergence of the process and the low computational requirements of the algorithm. Instead of analyzing the conditions under which (3.3) is satisfied we give

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some specifics in the following alternative proof for the convergence, which characterizes the described process. In addition, the presented result also determines the distances between the values of the real process and the truncated process at each iteration. The notion of the q-Pochhammer symbol (or q-rising factorial), defined by

ða; qÞk ¼

k1 Y ð1  aqi Þ i¼0

is used for the purpose of proving the next result. and R 2 Cnn ; 0 6 s 6 q be given such that the condition (2.8) is satisfied. The sequence of Theorem 3.2. Let M 2 Cnn q s ð2Þ approximations {Zk} given by (2.11) converges to the outer inverse Z ¼ M RðRÞ;N ðRÞ of the matrix M if the following three conditions are satisfied: 1. b is a real number such that

max j1  bki j < 1; 16i6t

where rank (RM) = t, ki, i = 1, . . . , t are eigenvalues of RM. 2. The truncation values ek ; k 2 N0 are chosen such that

2  hk ; 0 n hk  hkþ1

ek ¼ 

ð3:5Þ

where

(

  0 hk ¼ drk Z 0k ;

hk ¼ drkðZ k Þ;

k P 0;

h0 ¼ min

e;

) d2  kPk2 ; kMk

and

( hk ¼ min

ekþ1 ;

) dkþ2 ð1  dk Þ ; kMk

k P 1;

ð3:6Þ

for 0 < e  1 and D = 1  e. 3. We chose a matrix norm kk such that the condition (2.7) from Lemma 2.1 is satisfied for H = I  bMR and also that the property kPk 6 1  e is obeyed. Under these assumptions we have the quadratic convergence of the algorithm, defined by the rule

 2k kZ  Z k k 6 max j1  bki j þ OðeÞ 16i6t kZk

ð3:7Þ

and the norm estimation

kRk k 6

ð1; dÞkþ1 eð1  ek Þ ¼ OðeÞ;  1e 2ð1 þ dÞ

k P 0;

ð3:8Þ

where {Yk} is the sequence generated by the SMS algorithm and Rk = Zk  Yk. Proof. Without loss of generality one can assume that kIk = 1. Otherwise, in the case of kIk > 1, instead of the norm kk which satisfies the theorem Condition 3 the induced matrix norm N(M) = supkEk=1kMEk can be used. According to Theorem 5.6.26 from [15] the new norm satisfies N(M) 6 kMk for all M 2 Cnn and N(I) = 1. Further, since the matrix Zk+1 is the aodr of Z 0k , it is possible to use Z kþ1 ¼ Z 0k þ Ek . According to (3.5) as well as (2.4) it is evident that kEkk 6 hk, k P 0. Let us prove the auxiliary result

kPk k 6 dkþ1 ;

k P 0:

In the case k = 0 we have kP0k = kPk 6 d. The proof in the case k = 1 follows from

P1 ¼ I  Z 1 M ¼ I  ðP0 þ IÞZ 0 M  E0 M ¼ I  Z 0 M  P0 Z 0 M  E0 M ¼ P20  E0 M; which produces

kP1 k 6 kP 0 k2 þ h0 kMk 6 kPk2 þ

d2  kPk2 kMk ¼ d2 : kMk

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Assuming that kPkk 6 dk+1 and taking into account P kþ1 ¼ P2k  Ek M as well as kEkk 6 hk we verify the inductive step:

kPkþ1 k 6 kPk k2 þ hk kMk 6 d2kþ2 þ

dkþ2 ð1  dk Þ kMk ¼ dkþ2 : kMk

Now we estimate the norm kRkk as in (3.8). In the case k = 0 we have kR0k = 0. To prove the statement in the case k P 1 we again use the mathematical induction. The base of the induction (case k = 1) can be verified from

Z 1 ¼ ðZ 00 Þe0 ¼ ðY 1 Þe0 ¼ Y 1 þ E0 ; which implies

kR1 k ¼ kE0 k 6 h0 6 e ¼

ð1; dÞ2 e: 2ð1 þ dÞ

Let us suppose the approximation of the matrix norm kRkk as in (3.8). Then the inductive step follows from

Z kþ1 ¼ ððPk þ IÞZ k Þek ¼ ðPk þ IÞðY k þ Rk Þ þ Ek ¼ Y kþ1 þ ðP k þ IÞRk þ Ek ; which together with (3.6) gives

kRkþ1 k 6 ðdkþ1 þ 1ÞkRk k þ hk 6 ðdkþ1 þ 1Þ

6

k Y kþ1 X s¼1 i¼2

ðdi þ 1Þes þ

kþ1 Y

k Y k X ð1; dÞkþ1 eð1  ek Þ þ hk 6 ðdkþ1 þ 1Þ ðdi þ 1Þes þ ekþ1  1e 2ð1 þ dÞ s¼1 i¼2

ðdi þ 1Þekþ1 ¼

i¼2

kþ1 Y kþ1 X ðdi þ 1Þes : s¼1 i¼2

So continuing the previous inequalities we obtain

kRkþ1 k 6

kþ1 ð1; dÞkþ2 X ð1; dÞkþ2 eð1  ekþ1 Þ es ¼ ¼ OðeÞ:  1e 2ð1 þ dÞ s¼1 2ð1 þ dÞ

In this way, we verify the induction and prove kRk k ¼ OðeÞ: P i Denote by Y = limYk, Z = limZk. Since 1 i¼0 D < 1, it follows that (1, D)k converges to (1, D)1, and

kZ  Yk 6

ð1; dÞ1 e ¼ OðeÞ:  2ð2  eÞ 1  e

Therefore, the estimation (3.8) is verified and it is possible to use Z = Y. Now, following the assumptions 1 and 3 we have

 2k kY  Y k k þ OðeÞ; 6 max j1  bki j 16i6t kYk (see [30], Theorem 2.1). On the other hand we got the estimate for kRkk, which produces

kZ  Z k k ¼ kZ  Y k  Rk k ¼ kY  Y k  Rk k 6 kY  Y k k þ kRk k: Finally, we get

 2k ð1; dÞkþ1 eð1  ek Þ kZ  Z k k 6 max j1  bki j þ OðeÞ þ ;  16i6t kZk 1e 2ð1 þ dÞkZk which immediately confirms (3.7) and the quadratic convergence of the sequence (2.11). It is known that all norms on a finite dimensional vector space are equivalent and we found a norm for which the process converges. Therefore, the general convergence with respect to any norm follows. h However, based on some strategies, one can choose ek to be constant during the process or to be dynamic (to depend on some changeable parameters). A realistic strategy in choosing the value for e is according to some specific heuristics. Some heuristics for the choice of the parameter ek are presented in the next section. Remark 3.1. For the purpose of choosing appropriate truncation value we remark that the restrictive condition kPk 6 1  e in Theorem 3.2 is always satisfied in the case of the ordinary inverse, the Moore–Penrose inverse, the group and the Drazin inverse. In the first two cases we have R = M⁄, which yields P = P⁄ i.e. q(P) = kPk and in the case of the Drazin inverse R = Ml, l P ind (M) from which follows P = H.

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4. Application and numerical examples The {2}-inverses have application in the iterative methods for solving nonlinear equations [1,24] as well as in statistics [11,16]. In particular, outer inverses play an important role in stable approximations of ill-posed problems and in linear and nonlinear problems involving rank-deficient generalized inverses [23,38]. On the other hand, the theory of Toeplitz matrices arises in scientific computing and engineering, for example, image processing, numerical differential equations, integral equations, time series analysis and control theory (see, for example [5,19]). Toeplitz matrices are important in the study of discrete time random processes. The covariance matrices of weakly stationary processes are Toeplitz matrices. The triangular Toeplitz matrices provide a matrix representation of causal linear time invariant filters [10]. The Moore–Penrose inverse of a Toeplitz matrix is used in the image restoration process [3]. In the sequel we state some results which are relevant for computing outer inverses of an arbitrary Toeplitz matrix M 2 Cnn . In order to get an explicit representation of the outer inverse of M, which is a limit value of the sequence of approximations (2.11) for given M and R, we observe the full rank factorization of the matrix R. Therefore, we present the following result. Corollary 4.1. For given M 2 Cnn let us choose an arbitrary matrix R 2 Cnn ; 0 6 s 6 q, and its full rank factorization q s ns sn R ¼ FG; F 2 Cs ; G 2 Cs , such that GMF is invertible. Suppose that b is a selected real number satisfying max16i6sj1  bkij < 1, where ki are eigenvalues of FGM. Under the assumption 2 from Theorem 3.2 the sequence of approximations given by (2.11) converges to the outer inverse of M, ð2Þ

Z ¼ FðGMFÞ1 G ¼ MRðFÞ;N ðGÞ 2 Mf2gs : Proof. Follows immediately from Z = Y, Theorem 2.3 from [30] and Theorem 3.2. h Corollary 4.2. Under the assumptions of Theorem 3.2 the following holds: (i) (ii) (iii) (iv)

Z = MD in the case R = Ml, l P ind(M); Z = M  in the case R = M⁄; Z = M# in the case R = M; Z 2 M{1, 2} for s = q.

Proof. It suffices to use the following representations: ð2Þ

M D ¼ M RðMl Þ;N ðMl Þ ;

l P indðMÞ;

ð2Þ

M y ¼ MRðM Þ;N ðM Þ ;

ð2Þ

M # ¼ M RðMÞ;N ðMÞ ;

Mf1; 2g ¼ Mf2gq in conjunction with Corollary 4.1. h Next, we report some numerical results obtained from testing our method, named modified SMS method, for computing outer inverses of a Toeplitz matrix. We select 5 different types of Toeplitz matrices. During the execution of the algorithm we observe and later report three different indicators: the number of iterations (No of iter), maximum displacement rank (Mdrk), as well as the sum of the displacement ranks (Sdrk). The code is written in the programming package MATLAB on a Workstation Intel Celeron 2.0 GHz. The stopping condition for Algorithm 2.2 is kres(Zk) k 6 n = 106. Also, the value of the starting parameter is b = 1/kRMk, where the matrix R is chosen according to Theorem 3.2 and Corollary 4.2. In order to find the unified approach in choosing the parameter e for all test matrices we use the static choice e = 108, which is actually the square root of the double precision of the floating point arithmetic. In the case of computing the inverse as well as the group inverse of the matrix we present the strategy of choosing the e according to some specific heuristic. This heuristic gives improvements of the algorithm such as the lower Mdrk as well as the lower Sdrk which decrease the computational cost of the algorithm and the number of operations per step. Remark 4.1. Based on Theorem 3.2, similarly as in [30] we can also determine a priori the number of steps required for achieving predefined accuracy. It is obvious that the number of steps for SMS and modified SMS is identical. By M[pjq] we denote the submatrix of M obtained by extracting the rows indexed by p and the columns indexed by q. When q = {1, 2, . . . , n}, we denote M[pjq] by M[pj:], and in case of p = {1, 2, . . . , n}, we denote M[pjq] by M[:jq]. The sequence {1, . . . , s} is simply denoted by 1:s. Example 4.1. In this example we consider the Toeplitz matrices M 2 Rnn and R 2 Rnn (n is an odd number) given by q s

M ¼ toeplitz½ð1; 0; . . . ; 0; 1Þ;

R ¼ toeplitz½ð1; 0; . . . ; 0; 1; 0; . . . ; 0; 1Þ;

where s = (n  1)/2 < q = n  1. The vector which represents R has 1 in the middle.

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One of the possible full rank factorizations of the matrix R = FG is F = R[:j1:s], G = R[1:sj:]. In the case n = 7 we have

2

1 0 0 1 0 0 1

60 6 6 60 6 6 R ¼ 61 6 60 6 6 40

3

2

1 0 0

3

1 07 7 7 2 3 1 0 0 1 0 0 1 0 17 7 7 7 6 0 0 7  4 0 1 0 0 1 0 0 5: 7 0 0 1 0 0 1 0 1 07 7 7 0 15 1 0 0

6 1 0 0 1 0 07 7 60 7 6 7 0 1 0 0 1 07 6 60 7 6 0 0 1 0 0 17 ¼ 61 7 6 6 1 0 0 1 0 07 7 60 7 6 0 1 0 0 1 05 40

1 0 0 1 0 0 1

Based on Corollary 4.1, the explicit representation of the outer inverse is

3 0:2 0 0 0:2 0 0 0:2 6 0 0:5 0 0 0:5 0 0 7 7 6 7 6 6 0 0 0:5 0 0 0:5 0 7 7 6 7 6 0 0:2 0 0 0:2 7: ¼ FðGMFÞ1 G ¼ 6 0:2 0 7 6 6 0 0:5 0 0 0:5 0 0 7 7 6 7 6 0 0:5 0 0 0:5 0 5 4 0 0:2 0 0 0:2 0 0 0:2 2

ð2Þ

M RðFÞ;N ðGÞ

The sequence of approximations {Zk} given by Algorithm 2.2 converges to the outer inverse given by this explicit representation. In Table 1 we present numerical results obtained from computing the outer inverse of M by Algorithm 2.2 considering several different dimensions of M. Also, the number of iterations of the original SMS method is presented as comparison. As one can see from Table 1 the number of iterations of the original SMS method is the same as the number of iterative steps obtained by our method. Also, the Mdrk is very low for all dimensions of the matrix M, i.e. Mdrk = 3. In the remaining of the section, for each test matrix, we consider 8 different numerical experiments with the dimensions as follows: 16, 32, 64, 128, 256, 512, 1024, 2048. Example 4.2. The inverses of two matrices

M1 ¼ toeplitz½ð1; 1=2; . . . ; 1=nÞ;

M 2 ¼ toeplitz½ð4; 1; 0; . . . ; 0Þ;

chosen from [2], are computed in the case of R = M⁄. Numerical results obtained during the computation of the inverses of matrices M1 and M2 are given in Tables 2 and 3. Besides the static choice of the truncation value e = 108, we observe the additional heuristic e = max(res/kMk, 108). We see that the number of iterations is almost identical for the two different heuristics. On the other hand, it is evident that, observing all other indicators, the great improvement is achieved by putting e = max(res/kMk, 108) instead of e = 108. Obviously, these improvements are obtained for both matrices M1 and M2. Example 4.3. The test matrix

M ¼ toeplitz½ð1; 1=2; . . . ; 1=ðn  1Þ; 1Þ; ð1; 1=ðn  1Þ; . . . ; 1=2; 1Þ is taken from [4]. Tables 4 and 5 report the numerical results obtained during the computation of the group inverse (generated in the case R = M), as well as the Moore–Penrose inverse (R = M⁄) of a Toeplitz matrix M 2 Rnn . Let us mention that another dynamic heuristic is chosen for computing the group inverse with respect to the heuristic used in Example 4.2. It is obvious that the presented heuristic, which makes e at least as small as 106, provides significant improvements of the observed indicators in comparison to the static choice e = 108.

Table 1 Numerical results generated by computing the outer inverse of M. Dim

17 33 65 129 257 513 1025 2049

Modified SMS

SMS

No of iter

Mdrk

Sdrk

No of iter

5 5 5 5 5 5 5 5

3 3 3 3 3 3 3 3

15 15 15 15 15 15 15 15

5 5 5 5 5 5 5 5

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Table 2 Numerical results generated by computing the inverse of M1. Dim

Modified SMS

SMS

e = max (res/kAk, 108) 16 32 64 128 256 512 1024 2048

e = 108

No of iter

Mdrk

Sdrk

No of iter

Mdrk

Sdrk

No of iter

11 11 12 13 13 14 14 14

2 2 2 2 3 4 4 4

15 17 20 25 27 35 38 42

11 12 13 13 14 14 14 15

13 16 16 16 16 16 16 16

105 129 148 158 171 176 180 189

11 12 13 13 14 14 14 15

Table 3 Numerical result generated by computing the inverse of matrix M2. Dim

Modified SMS

SMS

e = max (res/kAk, 108) 16 32 64 128 256 512 1024 2048

e = 108

No of iter

Mdrk

Sdrk

No of iter

Mdrk

Sdrk

No of iter

7 7 7 7 7 7 7 7

2 2 3 3 3 4 4 4

11 11 16 18 18 20 21 21

7 7 7 7 7 7 7 7

13 13 13 13 13 13 13 13

65 69 69 69 69 69 69 69

7 7 7 7 7 7 7 7

Table 4 Numerical results for computing the group inverse. Dim

Modified SMS

SMS

e = min((1 + 106  kPkk2)/kAk, 106) 16 32 64 128 256 512 1024 2048

e = 108

No of iter

Mdrk

Sdrk

No of iter

Mdrk

Sdrk

No of iter

9 10 10 10 11 11 11 11

9 10 10 9 9 9 9 9

60 66 68 72 77 80 82 82

9 10 10 10 11 11 11 11

9 10 11 12 11 11 11 11

66 76 81 87 92 91 93 95

9 10 10 10 11 11 11 11

Table 5 Numerical results generated by computing the Moore–Penrose inverse. Dim

16 32 64 128 256 512 1024 2048

Modified SMS

SMS

No of iter

Mdrk

Sdrk

No of iter

9 10 10 10 11 11 11 11

13 14 15 15 15 15 15 16

84 98 107 112 119 122 124 128

9 10 10 10 11 11 11 11

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Table 6 Numerical results generated by computing the Drazin inverse of M. Dim

Modified SMS

16 32 64 128 256 512 1024 2048

SMS

No of iter

Mdrk

Sdrk

No of iter

8 8 8 8 8 8 8 8

6 6 6 6 6 6 6 6

48 48 48 48 48 48 48 48

8 8 8 8 8 8 8 8

It is interesting to point out that the results presented in the Tables 4 and 5 are obtained for the same starting matrix M. Also, the group and the Moore–Penrose inverse of the matrix M are identical. As one can see from the tables, although the number of iterations is the same, the Mdrk as well as the Sdrk obtained from the computation of the group inverse is essentially smaller than in the case of computing the MP inverse, for the same choice of the parameter e = 108. Example 4.4. Let us consider the matrix M 2 Rnn represented by

M ¼ toeplitz½ð0; . . . ; 0; 1; 1; 0; . . . ; 0Þ; ð0; 0; . . . ; 0; 1; 1Þ: Since the index of the matrix M is ind (M) = 2, the Drazin inverse of M is computed in the case R = Ml, l P 2. For the chosen matrix R we have that rank (R) = 6. According to Corollary 4.2, the iterative process of the modified SMS method converges to the Drazin inverse. After the use of the static choice e = 108 we get the following results. It is evident from Table 6 that Mdrk = 6, which is a good result having in mind that drk(Z0) = drk(bR) = 4.

5. Conclusion The Newton’s method originally proposed by Shultz in 1933 [29] is a very important tool for computing the inverse as well as the generalized inverses of an arbitrary structured matrix. There exist several different modifications of the Newton’s method dealing with a Toeplitz matrix. Each of them is intended for computing either the ordinary inverse, the group inverse or the Moore–Penrose inverse [2–4,32]. Various heuristics in the choice of the starting approximation, in defining the iterative rules as well as in the selection of displacement operators are used in these papers. Hence, the necessity of having a unified algorithm is obvious. In this paper we use the idea introduced in [30], accompanied by the concept of displacement operator and displacement rank of iteration matrices. Namely, we present the modification of the SMS algorithm for computing the outer inverse with prescribed range and null space of an arbitrary Toeplitz matrix. Our method is actually a generalization of the ideas from [2,4] and represent the universal algorithm for computing generalized inverses containing these results as partial cases. Furthermore, by using the circulant and anti-circulant matrices for obtaining the odr of the sequence of approximations Zk, we propose a different approach for computing the Moore–Penrose inverse of a square Toeplitz matrix, with respect to the method from [3,32]. Also, by giving the unified approach we resolve the previously mentioned diversities in strategies for choosing the starting approximations as well as the diversities in defining the iterative rules and in the usage of various displacement operators. The presented quadratic convergence and numerical results show that our algorithm is as good as the SMS algorithm for computing outer generalized inverses, when it comes to the number of iterations. Furthermore, the achieved low values of Mdrk and Sdrk evidently decrease the computational cost per step having in mind that the strategy of FFT’s and convolutions are used in matrix-vector multiplications. Additionally, three different heuristics are used for the choice of the truncation value ek. For given test matrices the numerical results presented in the paper favour strategy for the dynamic choice of the truncation parameter in comparison to the static choice. References [1] A. Ben-Israel, T.N.E. Greville, Generalized Inverses: Theory and Applications, second ed., Springer, 2003. [2] D. Bini, B. Meini, Approximate Displacement Rank and Applications, in: V. Olshevsky (Ed.), Structured Matrices in Mathematics, Computer Science and Engineering II, Contemporary Mathematics, 281, American Mathematical Society, Rhode Island, 2001, pp. 215–232. [3] D. Bini, G. Codevico, M. Van Barel, Solving Toeplitz least squares problems by means of Newton’s iteration, Numer. Algorithms 33 (2003) 93–103. [4] J.F. Cai, M.K. Ng, Y. Wei, Modified Newton’s algorithm for computing the group inverses of singular Toeplitz matrices, J. Comput. Math. 24 (2006) 647– 656. [5] R. Chan, M. Ng, Conjugate gradient methods for Toeplitz systems, SIAM Rev. 38 (1996) 427–482. [6] L. Chen, E.V. Krishnamurthy, I. Macleod, Generalized matrix inversion and rank computation by successive matrix powering, Parallel Comput. 20 (1994) 297–311.

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