iterative solutions of coupled matrix convolution

SOOCHOW JOURNAL OF MATHEMATICS

Volume 33, No. 1, pp. 167-180, January 2007

ITERATIVE SOLUTIONS OF COUPLED MATRIX CONVOLUTION EQUATIONS BY ADEM KILIC ¸ MAN AND ZEYAD ABDEL AZIZ AL ZHOUR

Abstract. In this paper the results are introduced in three ways. First, we introduce some results of convolution products of matrices that will be useful in our investigation of the applications to the iterative solutions of matrix convolution equations. Second, we present the iterative solution of coupled matrix convolution equations. Finally, we prove that the iterative solution consistently converges to the exact solution for any initial value. The proposed method is effective, robust and easily programmable.

1. Introduction Matrix equations play important roles in system theory, control theory, stability theory, communication systems, perturbation analysis and other fields of pure and applied mathematics (see [1], [2], [6] and [10]). one often comes across matrix functions which are in the form of the matrix convolution equations. Although exact solutions, which can be computed by using the nice connection between Kronecker convolution product and vector operator are important but the computational efforts rapidly increase with the sizes of the matrix functions to be solved. For some applications, it is often not neccessary to compute exact solutions; approximat solutions are sufficient. In the field of matrix algebra and system identification, iterative methods have received much attention. For example, [5] gave new estimates for solutions of Lyapunov equations, [9] established new iterative algorithm for solving algebraic Received November 18, 2005. AMS Subject Classification. 44A35, 15A24, 65F10, 15A69. Key words. convolution product, Kronecker convolution product, vector-operator. 167

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ADEM KILIC ¸ MAN AND ZEYAD ABDEL AZIZ AL ZHOUR

Riccati equation related to H∞ control problem of singularly perturbed systems, [3] introduced a more general alternating iterative method than the classical alternating iterative method for the solution of linear systems, [11] presented an iterative method for solutions of Sylvester equations by using the SOR technique; [7] derived an aproximate solution of the coupled Sylvester matrix equations, and [4] presented a general family of iterative methods to solve coupled Sylvester matrix equations and general coupled matrix equations. To our best knowledge, exact and numerical solutions of matrix and general matrix convolution equations have been not fully investigated, especially the iterative solutions of coupled matrix convolution equations and the convergence of these solutions involved, which are the focus of this work. In the present paper, we present approximate solutions of coupled matrix convolution equations by using iterative methods and we prove that these solutions consistently converges to the exact solutions for any initial value. The iterative methods may extend to solve the general coupled matrix convolution equations. First, we use some notations. The notations Mm,n is the set of all m × n I matrices over the field M , and the notation Mm,n is the set of m × n matrices

whose entries are integrable on some fixed finite interval 0 ≤ t < T and closed under convolution. If m = n, we write Mn instead of Mm,n and MnI instead of I . The notations A{−1} (t) and det A(t) are the inverse and determinant of Mm,n

matrix function A(t), respectively, with respect to convolution. The notations AT (t) and V ecA(t) are transpose and vector-operator of matrix function A(t), respectively. The term ”V ecA(t)” will be used to transform a matrix A(t) into a vector by stacking its column one underneath. The notations δ(t) and Qn (t) = In δ(t) are the Dirac delta function and Dirac identity matrix, respectively, where In is the identity scalar matrix of order n × n. Finally, the notations A(t) ⊛ B(t) and A(t) ∗ B(t) are the convolution and Kronecker convolution products, respectively.

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2. Results on Matrix Convolution Products In this section, we introduce the convolution products of matrices, namely the convolution product, Kronecker convolution product. Some new results of these products are established that will be useful in our investigation of the applications to the solution of matrix convolution equations. I I . The and B(t) = [gjr (t)] ∈ Mn,p Definition 1. Let A(t) = [fij (t)] ∈ Mm,n convolution and Kronecker convolution products of A(t) and B(t) are matrix functions defined for 0 ≤ t < T as follows:

(i) Convolution product ([8]) A(t) ∗ B(t) = [hij (t)] with hij (t) =

n Z X

t

fik (t − x)gkr (x)dx =

k=1 0

n X

(1)

fik (t) ∗ gkr (t)dx.

(2)

k=1

(ii) Kronecker convolution product A(t) ⊛ B(t) = [fij (t) ∗ B(t)]ij .

(3)

Here, fij (t) ∗ B(t) is the ij-th submatrix of order n × p, A(t) ⊛ B(t) is of order mn × np and A(t) ∗ B(t) is of order m × p. Definition 2. Let A(t) = [fij (t)] ∈ MnI and Qn (t) = In δ(t). The determinant and inverse of A(t) respect to the convolution are defined for 0 ≤ t < 0 as follows ([8]). (i) Determinant det A(t) =

n X

(−1)j+1 f1j (t) ∗ D1j (t).

(4)

j=1

Here, Dij (t) is the determinant of (n − 1) ×(n − 1) matrix function obtained from A(t) by deleting row i and column j of A(t). We call Dij (t) the minor A(t) of corresponding to the entry of fij (t) of A(t). (ii) Inversion A{−1} (t) = [qij (t)] with qij (t) = (det A(t)){−1} ∗ adjA(t).

(5)

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Here, det A(t) exists, (det A(t)){−1} ∗ det A(t) = δ(t) and note that A{−1} (t) ∗ A(t) = A(t) ∗ A{−1} (t) = Qn (t). Theorem 3. Let A(t), B(t) and C(t) ∈ MnI . Then for any scalars α and β (i) (αA(t) + βB(t)) ∗ C(t) = αA(t) ∗ C(t) + βB(t) ∗ C(t)

(6)

(ii) (A(t) ∗ B(t)) ∗ C(t) = A(t) ∗ (B(t) ∗ C(t))

(7)

(iii) A(t) ∗ Qn (t) = Qn (t) ∗ A(t) = A(t), Qn (t) = In δ(t)

(8)

(iv) (A(t) ∗ B(t))T = B T (t) ∗ AT (t)

(9)

(v) (A(t)∗B(t)){−1} = B {−1} (t)∗A{−1} (t), if A{−1} (t) and B {−1} (t) are exist.(10) Proof. Straightforward by the definition of the convolution product of matrices. I , B(t) ∈ M I , C(t) ∈ M I and C(t) ∈ M I . Theorem 4. Let A(t) ∈ Mm,n p,q n,r q,s Then for 0 ≤ t < T ,

(A(t) ⊛ B(t)) ∗ (C(t) ⊛ D(t)) = (A(t) ∗ C(t)) ⊛ (B(t) ∗ D(t)) .

(11)

Proof. The (i, j)-th block of (A(t) ⊛ B(t)) ∗ (C(t) ⊛ D(t)) is obtained by taking the convolution product of i-th row block of A(t) ⊛ B(t) and the j-th column block of C(t) ⊛ D(t), i.e.,   h1j (t)∗D(t) n     X ..  = [fi1 (t)∗B(t) · · · f in (t)∗B(t)]∗  f (t)∗h (t)∗B(t)∗D(t) ik . kj   k=1 hnj (t)∗D(t)

and the (i, j)-th block of the right hand side (A(t) ∗ C(t))⊛(B(t) ∗ D(t)) obtained by the definition of the Kronecker convolution product xij (t)∗(B(t) ∗ D(t)) where xij is the (i, j)-th element in A(t) ∗ C(t). But the rule of convolution product xij (t) =

n X

(fik (t) ∗ hkj (t)) .

k=1

Lemma 5. Let u(t) and v(t) be m-vectors. Then
V ec u(t) ∗ v T (t) = v(t) ⊛ u(t).

(12)

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Proof. Straightforward by the definitions of convolution and Kronecker convolution products. I , X(t) ∈ M I I Theorem 6. Let A(t) ∈ Mm,n n,p and B(t) ∈ Mp,q . Then for 0≤t 0 is the step-size or convergence factor. The following lemma is straightforward. Lemma 7. If we set G(t) = A{−1} (t) ∈ MnI in (15), then the iterative algorithm converges to x(t) xk (t) = xk−1 (t) + µA{−1} (t) ∗ {b(t) − A(t) ∗ xk−1 (t)} .

(16)

Here, 0 < µ < 2, k = 1, 2, . . .. I If A(t) ∈ Mm,n is a non-square full column-rank matrix, for all 0 ≤ t < T ,

then we have limk→∞ xk (t) = x(t) in the following: n o
{−1} xk (t) = xk−1 (t) + µ AT (t) ∗ A(t) ∗ A(t) ∗ {b(t) − A(t) ∗ xk−1 (t)} . (17) Here, 0 < µ < 2, k = 1, 2, . . ..

I If A(t) ∈ Mm,n is a non-square full row-rank matrix, for all 0 ≤ t < T, then

we have limk→∞ xk (t) = x(t) in the following:   {−1}  T ∗ {b(t) − A(t) ∗ xk−1 (t)} . (18) xk (t) = xk−1 (t) + µ A(t)∗ A(t) ∗ A (t) Here, 0 < µ < 2, k = 1, 2, . . .. It is easy to prove that when µ = 1, the iterative solutions xk (t) in (16) converges to A{−1} (t) ∗ b(t), the iterative solutions xk (t) in (17) converges to  T {−1} A (t) ∗ A(t) ∗ A(t) ∗ b(t) and the iterative solutions xk (t) in (18) converges  {−1} T to A (t) ∗ A(t) ∗ AT (t) ∗ b(t).

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The iterative algorithm in (17) and (18) are also suitable for solving nonsquare convolution systems and very useful for finding the iterative solutions of coupled matrix convolution equations to be studied later; the convergence factor µ in (16)-(18) do not rely on the matrix A(t) and it is easy to choose, although the algorithms in (16)-(18) require computing matrix inversion only with respect to convolution at the first step. 4. Coupled Matrix Convolution Equations In this section, we will extend the iterative method to solve more coupled matrix convolution equations of the form: A(t) ∗ X(t) + Y (t) ∗ B(t) = C(t), D(t) ∗ X(t) + Y (t) ∗ E(t) = F (t).

(19)

I , B(t), E(t) ∈ M I and C(t), F (t) ∈ M I Here, A(t), D(t) ∈ Mm n m,n are given

matrix functions, X(t), Y (t) ∈ Mm,n are the unknown matrix functions to be solved. Lemma 8. Eq.(19) has a unique solution if and only if the matrix " # Qn (t) ⊛ A(t) B T (t) ⊛ Qm (t) I H= ∈ M2mn Qn (t) ⊛ D(t) E T (t) ⊛ Qm (t) is non-singular; in this case the unique solution is given by " # " # V ecX(t) V ecC(t) {−1} =H (t) ∗ V ecY (t) V ecF (t)

(20)

(21)

and the corresponding homogeneous matrix convolution equations A(t) ∗ X(t) + Y (t) ∗ B(t) = 0,

D(t) ∗ X(t) + Y (t) ∗ E(t) = 0

has a unique solution: X(t) = Y (t) = 0. Proof. If we use the Vec-notation of (19) and apply (13), we have " # " # " # Qn (t) ⊛ A(t) B T (t) ⊛ Qm (t) V ecX(t) V ecC(t) ∗ = . Qn (t) ⊛ D(t) E T (t) ⊛ Qm (t) V ecY (t) V ecF (t)

(22)

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ADEM KILIC ¸ MAN AND ZEYAD ABDEL AZIZ AL ZHOUR

This system has a unique solution if and only if the matrix H defined in (20) is non-singular and the unique solution is given as in (21). In order to derive the iterative solution to (19), we need to introduce the intermediate matrices b1 (t) and b2 (t) as follows " # " # A(t) ∗ X(t) C(t) − Y (t) ∗ B(t) = (23) b1 (t) = D(t) ∗ X(t) F (t) − Y (t) ∗ E(t) h i b2 (t) = Y (t) ∗ B(t) Y (t) ∗ E(t) h i = C(t) − A(t) ∗ X(t) F (t) − D(t) ∗ X(t) . (24) Then from (19), we obtain the following two subsystems

G1 (t) ∗ X(t) = b1 (t) and Y (t) ∗ H1 (t) = b2 (t).

(25)

Here, S1 : G1 (t) =

"

A(t) D(t)

#

h i and S2 : H1 (t) = B(t) E(t) .

(26)

Let Xk (t) and Yk (t) be iterative solutions of X(t) and Y (t), respectively. Refering to Lemma 7, it is not diffecult to get the iterative solutions to S1 and S2 as follows: " #T  T {−1} A(t) Xk (t) = Xk−1 (t) + µ G1 (t) ∗ G1 (t) D(t) ( " # ) A(t) ∗ b1 (t) − ∗ Xk−1 (t) (27) D(t) n h io Yk (t) = Yk−1 (t) + µ b2 (t) − Yk−1 (t) ∗ B(t) E(t) h iT  {−1} ∗ B(t) E(t) ∗ H1 (t) ∗ H1T (t) . (28) Substituting (23) into (27) and (24) into (28) gives " #T  T {−1} A(t) Xk (t) = Xk−1 (t) + µ G1 (t) ∗ G1 (t) ∗ D(t) (" #) C(t) − Y (t) ∗ B(t) − A(t) ∗ Xk−1 (t) ∗ F (t) − Y (t) ∗ E(t) − D(t) ∗ Xk−1 (t)

(29)

ITERATIVE SOLUTIONS OF COUPLED MATRIX CONVOLUTION EQUATIONS

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nh io Yk (t) = Yk−1 (t) + µ C(t) − A(t) ∗ X(t) F (t) − D(t) ∗ X(t) h i h iT n o{−1} −Yk−1 (t) ∗ B(t) E(t) ∗ B(t) E(t) ∗ H1 (t) ∗ H T1 (t) h i = µ C(t)−A(t)∗X(t)−Y k−1 (t)∗B(t) F (t)−D(t)∗X(t)−Yk−1 (t)∗E(t) h iT n o{−1} ∗ B(t) E(t) ∗ H1 (t) ∗ H T1 (t) + Yk−1 (t). (30) Here, a difficulty arises in that the expressions on the right-hand sides of (29) and (30) contain the unknown parameter matrix X(t) and Y (t), respectively, so it is impossible to realize the algorithm in (29) and (30). The unknown variables Y (t) in (29) and X(t) in (30) are replaced by their estimates Yk−1 (t) and Xk−1 (t). Thus, we obtain the iterative solutions Y (t) and X(t) of the coupled matrix convolution equations in (19) as: " #T  T {−1} A(t) Xk (t)=Xk−1 (t)+µ G1 (t) ∗ G1 (t) ∗ D(t) " # C(t) − A(t) ∗ Xk−1 (t) − Yk−1 (t) ∗ B(t) ∗ F (t) − D(t) ∗ Xk−1 (t) − Yk−1 (t) ∗ E(t)

(31)

Yk (t)=µ [C(t)−A(t)∗X k−1 (t)−Y k−1(t)∗B(t) F (t)−D(t)∗X k−1(t)−Y k−1(t)∗E(t)] o{−1} h iT n + Yk−1 (t). (32) ∗ B(t) E(t) ∗ H1 (t) ∗ H T1 (t) Here, µ =

1 m+n .

The iterative algorithm in (31) and (32) requires computing the matrix con {−1}  {−1} volution inversions GT1 (t) ∗ G1 (t) and H1 (t) ∗ H1T (t) only once at the first step. To initialize the algorithm, we take X(0) = Y (0) = 0 or some small real matrix. Theorem 9. If the coupled matrix convolution equation defined in (19) has a unique solution X(t), Y (t), then the iterative solution Xk (t) and Yk (t) given by the algorithm in (31) and (32) converges to X(t), Y (t) for any finite initial values X(0) and Y (0), i.e., lim Xk (t) = X(t)

k→∞

and

lim Yk (t) = Y (t).

k→∞

(33)

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ADEM KILIC ¸ MAN AND ZEYAD ABDEL AZIZ AL ZHOUR

Proof. Define two error matrices ∼

∼

Xk (t) = Xk (t) − X(t) and Yk (t) = Yk (t) − Y (t).

(34)

By using (19), (31) and (32), it is not difficult to get ∼ Xk (t)

∼

" #T  T {−1} A(t) = X k−1 (t) + µ G1 (t) ∗ G1 (t) ∗ D(t)   ∼ ∼ −A(t) ∗ X k−1 (t) − Y k−1 (t) ∗ B(t)  ∗ ∼ ∼ −D(t) ∗ X k−1 (t) − Y k−1 (t) ∗ E(t) ∼

(35)

∼

Yk (t) = Y k−1 (t) + h i ∼ ∼ ∼ ∼ µ −A(t)∗X k−1 (t)−Y k−1 (t)∗B(t) −D(t) ∗ X k−1 (t)−Y k−1 (t) ∗ E(t) h iT n o{−1} ∗ B(t) E(t) ∗ H1 (t) ∗ H T1 (t) . (36) Define the 2-convolution norm for all 0 ≤ t < T as


kX(t)k22 = tr X T (t) ∗ X(t) .

(37)

We need to compute:

  2  {−1}

∗ Y (t)

G1 (t) ∗ X(t) + GT1 (t) ∗ G1 (t) 2 n T   T {−1} = tr X(t)+ G1 (t) ∗ G1 (t) ∗Y (t) ∗ GT1 (t) ∗ G1 (t)  o  {−1} ∗ X(t) + GT1 (t) ∗ G1 (t) ∗Y (t) n  = tr X T (t)∗ GT1 (t) ∗ G1 (t) ∗X(t) + 2X T (t) ∗ Y (t) o  {−1} +Y T (t)∗ GT1 (t) ∗ G1 (t) ∗Y (t) n o = tr (G1 (t) ∗ X(t))T ∗ (G1 (t) ∗ X(t)) nn oT T n oT o  {−1} {−1} +2tr X T (t) ∗ Y (t) +tr G1 (t) ∗Y (t) ∗ G1 (t) ∗ Y (t)

2

 T n {−1} oT 2

(38) = kG1 (t) ∗ X(t)k2 + 2tr X (t) ∗ Y (t) + G1 (t) ∗ Y (t)

. 2

ITERATIVE SOLUTIONS OF COUPLED MATRIX CONVOLUTION EQUATIONS

177

Now taking the 2-convolution norm in (33) and using the formula in (36) give  ∼ 

2 ∼ ∼

T T

G1 (t) ∗ X k (t) = tr Xk (t) ∗ G1 (t) ∗ G1 (t) ∗ X k (t) 2

2

∼

= G1 (t) ∗ X k−1 (t) 2   " #T  ∼ ∼  ∼ A(t) −A(t)∗X k−1 (t)−Y k−1 (t) ∗ B(t)  T  +2µtr Xk−1 (t)∗ ∗ ∼ ∼  D(t) −D(t)∗X k−1 (t)−Y k−1 (t) ∗ E(t) 

  2 ∼ ∼

n

oT

−A(t)∗X k−1 (t)−Y k−1 (t) ∗ B(t)  {−1} 2 T

 +µ G1 (t) ∗G1 (t)∗ ∼ ∼

−D(t)∗X k−1 (t)−Y k−1 (t) ∗ E(t) 2

2 ∼

≤ G1 (t)∗X k−1 (t) 2 n T  o ∼ ∼ ∼ −2µtr A(t)∗X k−1 (t) ∗ A(t)∗X k−1 (t)+Y k−1 (t) ∗ B(t) n T  o ∼ ∼ ∼ −2µtr D(t)∗X k−1 (t) ∗ D(t)∗X k−1 (t)+Y k−1 (t) ∗ E(t)

2 n ∼ ∼

+µ2 m A(t)∗X k−1 (t) + Y k−1 (t) ∗ B(t) 2

2 o ∼ ∼

+ D(t)∗X k−1 (t)+Y k−1 (t) ∗ E(t) , 2

(39)

similarly, we have

2

∼ n∼ ∼ o

Y k (t) ∗ H1 (t) = tr Yk (t) ∗ H1 (t) ∗ H1T (t) ∗ Y T k (t) 2

∼

2

= Y k−1 (t) ∗ H1 (t) 2 nh i ∼ ∼ ∼ ∼ +2µtr −A(t)∗X k−1 (t)−Y k−1 (t) ∗ B(t) −D(t)∗X k−1 (t)−Y k−1 (t) ∗ E(t) h iT ∼ o ∗ B(t) E(t) ∗ Y T k (t)

h i ∼ ∼ ∼ ∼

+µ2 −A(t)∗X k−1 (t)−Y k−1 (t) ∗ B(t) −D(t)∗X k−1 (t)−Y k−1 (t) ∗ E(t) n oT 2

{−1} ∗H1T (t)∗ H1 (t) 2

∼

2 n ∼ T

≤ Y k−1 (t) ∗ H 1 (t) − 2µtr Y k−1 (t) ∗ B(t) 2

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ADEM KILIC ¸ MAN AND ZEYAD ABDEL AZIZ AL ZHOUR

 o ∼ ∼ ∗ A(t)∗X k−1 (t)+Y k−1 (t) ∗ B(t) n ∼ T  o ∼ ∼ −2µtr Y k−1 (t) ∗ E(t) ∗ D(t)∗X k−1 (t)+Y k−1 (t) ∗ E(t)

2 n ∼ ∼

+µ2 n A(t)∗X k−1 (t) + Y k−1 (t) ∗ B(t) 2

2 o ∼ ∼

+ D(t)∗X k−1 (t) + Y k−1 (t) ∗ E(t) .

(40)

2

Defining a non-negative definite function:

2 ∼

2 ∼



Wk (t) = G1 (t) ∗ X k (t) + Y k (t) ∗ H1 (t) 2

(41)

2

and using (39) and (40), we have

2 n ∼ ∼

Wk (t) ≤ Wk−1 (t)−2µ A(t)∗X k−1 (t)+Y k−1 (t) ∗ B(t) 2

2 o ∼ ∼

+ D(t)∗X k−1 (t)+Y k−1 (t) ∗ E(t) 2

2 n ∼ ∼

2 +µ (m + n) A(t)∗X k−1 (t)+Y k−1 (t) ∗ B(t) 2

2 o ∼ ∼

+ D(t)∗X k−1 (t)+Y k−1 (t) ∗ E(t) 2 n ∼

≤ Wk−1 (t)−µ {2 − µ (m + n)} A(t)∗X k−1 (t)

2

2 o ∼ ∼ ∼

+Y k−1 (t) ∗ B(t) + D(t)∗X k−1 (t)+Y k−1 (t) ∗ E(t) 2

2

k−1 n

2 o nX ∼ ∼

≤ W0 (t)−µ {2 − µ (m + n)}

A(t)∗X i (t)+Y i (t) ∗ B(t) 2

i=1

2 oo ∼ ∼

+ D(t)∗X i (t)+Y i (t) ∗ E(t) . 2

If the convergence factor is chosen to satisfy: 0 < µ