AN ITERATIVE ALGORITHM FOR THE BEST APPROXIMATE (P,Q)-ORTHOGONAL SYMMETRIC AND SKEW-SYMMETRIC SOLUTION PAIR OF COUPLED MATRIX EQUATIONS

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1 AN ITERATIVE ALGORITHM OR THE BEST APPROXIMATE P,Q-ORTHOGONAL SYMMETRIC AND SKEW-SYMMETRIC SOLUTION PAIR O COUPLED MATRIX EQUATIONS atemeh Panjeh Ali Beik, and Davod Khojasteh Salkuyeh Department of Mathematics, Vali-e-Asr University of Rafsanjan, P O Box 518, Rafsanjan, Iran fbeik@vruacir, beikfatemeh@gmailcom aculty of Mathematical Sciences, University of Guilan, Rasht, Iran salkuyeh@gmailcom Abstract This paper deals with developing a robust iterative algorithm to find the least-squares P, Q-orthogonal symmetric and skew-symmetric solution sets of the generalized coupled matrix equations To this end, first, some properties of these type of matrices are established urthermore, an approach is offered to determine the optimal approximate P, Q-orthogonal skew-symmetric solution pair corresponding to a given arbitrary matrix pair Some numerical experiments are reported to confirm the validity of the theoretical results and to illustrate the effectiveness of the proposed algorithm Keywords: Matrix equation; Least-squares problem; P, Q-orthogonal symmetric matrix; P, Q-orthogonal skew-symmetric matrix; Iterative algorithm 1 AMS Subject Classification: 651, 15A4 1 Introduction Let us first introduce some symbols exploited throughout the paper The notation R m n stands for the set of all m n real matrices The trace and the transpose of a given matrix A are respectively denoted by tra and A T The symbol SOR n n refers to the set of all symmetric orthogonal matrices in R n n, ie, SOR n n = {P R n n P T = P, P = I} or two given matrices P SOR n n and Q SOR n n, the matrix X R n n is called P,Q-orthogonal symmetric if PXQ T = PXQ and is said to be P,Q- orthogonal skew-symmetric if PXQ T = PXQ The set of all P,Q-orthogonal symmetric and P, Q-orthogonal skew-symmetric matrices of order n are respectively represented by SR PQ n n and SSRPQ n n A special kind of permutation matrix with ones along the secondary diagonal and zeros elsewhere is called the exchange matrix Definition 11 Golub and Loan, 1996 Let A = a ij be an n n matrix The matrix A is said to be a persymmetric matrix if Corresponding author a ij = a n j+1,n i+1 1

2 P A Beik and D K Salkuyeh for all 1 i,j n Or equivalently, A is called persymmetric if AJ = JA T where J is the exchange matrix Assume that A R n n and J is the exchange matrix of order n It can be verified that JA T,JAJ and A T J are versions of the matrix A that have been rotated anticlockwise by 9,18 and 7 degrees Moreover, JA, JA T J, AJ and A T are versions of the matrix A that have been reflected in lines at, 45, 9 and 135 degrees to the horizontal measured anti-clockwise Remark 1 Assume that A R n n and J R n n is the exchange matrix Note that A is persymmetric if AJ T = AJ In view of the fact that J SOR n n, it can be concluded that every persymmetric matrix is a I, J-orthogonal symmetric matrix where I stands for the identity matrix of order n Definition 13 A n n matrix H is said to be skew-hamiltonian if ĴH is skew- symmetric where n = k and Ik 11 Ĵ = I k Hamiltonian and skew-hamiltonian matrices have a key role in engineering, such as in linear quadratic optimal control, H optimization and solving algebraic Riccati equations; see Laub, 1991 and Zhou et al, 1995 Eigenvalues and eigenvectors of a Hamiltonian matrix are important for describing a physical system, in fact the real eigenvalues correspond to the possible energy levels of the system and the related eigenvectors express the corresponding state; eg, see Reichl, 13 In view of the importance of constructing skew-hamiltonian matrices specially in inverse problems, solving matrix equations and the structured inverse eigenvalue problem over Hamiltonian matrices have been investigated by some researchers; see Qian and Tan, 13, Zhang et al, 5 and the references therein Remark 14 Note that ĴT = Ĵ 1 = Ĵ and ĴH is skew-symmetric if and only if HĴ is skew-symmetric Without loss of generality, we say H is a skew-hamiltonian matrix if HĴT = HĴ HĴT = HĴ Remark 15 We would like to comment here that for a given matrix H, it is not difficult to see that H is a skew-hamiltonian matrix if HJ is I, J-orthogonal skew-symmetric matrix where Ik J = I k and J = Ik I k This follows from the fact that JĴ = J Notice that J, J SOR n n n = k Linear matrix equations play a cardinal role in various fields, such as control and system theory, stability theory and some other fields of applied and pure mathematics; see Ding and Chen 6, Ding et al 8, Hajarian and Dehghan 11, Wu et al 13 and references therein In recent years, several research works have discussed the way of obtaining explicit forms of the solution for different kinds of matrix equations or example, Wu et al 11b have presented an approach to resolve a general class of Sylvester-polynomial-conjugate matrix equations which include the Yakubovich-conjugate matrix equation as a special case In

3 Wu et al, 1, several explicit parametric solutions have been derived for the generalized Sylvester-conjugate matrix equation In Zhao et al, 1, an analytical expression is derived for the optimal approximate solution in the least-squares P,Q-orthogonal symmetric solution set of the matrix equation A T XB = Z with respect to a given matrix In addition, the authors have obtained an explicit expression of the minimum-norm least-square P, Q-orthogonal symmetric solution for the matrix equation A T XB = Z In Song et al, 14 by using of the coefficients of the characteristic polynomial of matrix ormula and the Leverrier algorithm, the explicit solutions to the Sylvester-conjugate matrix equation AX XB = C have been constructed Recently, Li et al 14 have derived the general expression of the least-squares solution of the matrix equation AXB+CYD = E with the least-norm over symmetric arrowhead matrices Nevertheless, in practice, applying the iterative algorithms is superior to computing explicit forms due to the required high computation costs of the explicit forms As a matter of fact, for obtaining solutions by means of the derived explicit forms, Kronecker products of matrices are required to be explicitly formed In addition, we need to compute the Moore-Penrose pseudoinverse which would be expensive for large-scale least-squares problems In the literature, the applications of iterative algorithms have been successfully examined for solving several types of matrix equations so far; for further details see Beik and Salkuyeh 11, Beik et al 14, Ding and Chen 6, Ding et al 8, Hajarian 14a, Salkuyeh and Totounian 6, Salkuyeh and Beik 14, Wu et al 11a and references therein The extension of the LSQR method for solving constrained matrix equations have been widely studied so far or instance, Li and Huang 1 have presented matrix form of the LSQR method to solve the following coupled matrix equations Recently, Hajarian 14b have proposed matrix form of the LSQR algorithm for determining least-squares solutions of linear operator equation AX = B More recently, Karimi and Dehghan 15 have developed the LSQR method to solve general constrained coupled matrix equations containing the transpose of the unknowns The proposed strategy for constructing the algorithm inkarimi and Dehghan, 15 incorporates those given inli and Huang, 1 andhajarian, 14b In addition, the performance of the proposed algorithm have been numerically compared with some of previously examined iterative schemes There is also a growing interest for using the idea of conjugate gradient method to construct iterative algorithms for determining the approximate least-squares solutions of various kinds of matrix equations; see for instance Beik and Salkuyeh 13, Cai and Chen 9, Cai et al 1, Hajarian 14, Li et al 14, Peng et al 7, Peng and Liu 9, Peng 1, Peng 13, Peng and Xin 13, Peng 15, Sheng and Chen7, Sheng and Chen1, Su and Chen1 Lately, Peng 15 has developed the conjugate gradient least-squares CGLS method to determine the R, S-symmetric solutions of the following least-squares problem, 3 p i=1 q j=1 A ijx j B ij C i = min

4 4 P A Beik and D K Salkuyeh More recently, Li et al 15 have employed the CGLS method to construct a hybrid algorithm for solving the following minimization problem 1 AX B = min, subject to the matrix inequality constraint CXD E over R, S-symmetric solutions Before stating the main goal of the current work, we recollect some definitions and properties which are required throughout this paper The inner product of X,Y R m n is defined by X,Y = try T X The induced norm is the wellknown robenius norm Suppose that A = [a ij ] m s and B = [b ij ] n q are two real matrices, the Kronecker product of the matrices A and B is defined as the mn sq matrix A B = [a ij B] The vec operator transmutes a matrix A of size m s to a vector a = veca of size ms 1 by stacking its columns In Bernstein, 9, it is demonstrated that the following relation holds 1 vecabc = C T AvecB, for arbitrary given matrices A, B and C with suitable dimensions 11 Motivation and main contribution More recently, Sarduvan et al 14 have focused on the solution set of the following least-squares problem 13 AXB C = min, over P, Q-orthogonal skew-symmetric matrices and studied the matrix nearness problem associated with 13 More precisely, the explicit form for ˆX S EX is determined such that 14 ˆX X = min X S EX X X, where X R n n is given and S EX stands for the solution set of 13 By invoking 1 and establishing some theoretical results, Algorithm 1 is presented to obtain the explicit form of ˆX As seen the computation costs of Algorithm 1 is high due to using the Kronecker product and vec operator As a matter of fact, the dimension of the matrix A defined in Step 3 of Algorithm 1 can become large even in the case that the size of coefficient matrices A and B are moderate Consequently, Steps 4, 5 and 6 consume high CPU time sec to be performed In addition, the extension of the presented approach for general types of matrix equations would be onerous These drawbacks inspirit us to examine an iterative algorithm for solving coupled matrix equations and their corresponding least-squares problems over P, Q-orthogonal symmetric and P, Q-orthogonal skew-symmetric matrices To this end, we first investigate some properties of the P, Q-orthogonal symmetric and P, Q-orthogonal skewsymmetric matrices which are utilized for constructing and analyzing the proposed iterative scheme In order to elaborate our results for more general cases, we focus on the following least-squares problem 15 M A1 XB 1 +A X T B +C 1 YD 1 +C Y T D N E 1 X 1 +E X T +G 1 YH 1 +G Y T = min H

5 5 Algorithm 1: Proposed scheme to find the best approximation P, Q- orthogonal skew-symmetric solutions of AXB = C Sarduvan et al, 14 Step 1 Input the matrices A R m n, B R n r C R m r, P,Q SOR n n and X R n n Set = + = for finding the P,Q-orthogonal skew-symmetric solution Step Compute the vectors 1 c 1 = vecc, c = vecc T, y = vec PX Q PX Q T Step 3 Compute QB A = T AP AP QB T vecc, g = vec C T Step 4 orm the spectral decomposition A 1 as follows: D A 1 = V A1 V A T 1,, A 1 = A T A, g 1 = A T g where D is a diagonal matrix with diagonal entries consisting of positive eigenvalues of A 1 and V A1 is an n n orthogonal matrix formed by the corresponding linear independent eigenvectors of A 1 Step 5 Compute s = ranka T A, k = dima T A and Step 6 Compute M = zeross zeross, k s zerosk s,s eyek s ŷ = A g +V A T AM V A T AM y A g, where the symbol refers to the well-known Moore-Penrose generalized inverse Step 7 Compute Ŷ such that ŷ = vecŷ Step 8 Compute ˆX = PŶQ Here we point out that matrix equations which include both unknown matrix and its transpose, for instance, arise in solving time-invariant Hamiltonian systems; for further information see Lancaster and Rozsa, 1983 They also appear in reducing a block anti-triangular matrix to a block anti-diagonal one, a reduction which is exploited in palindromic generalized eigenvalue problems Kressner et al, 9 The generalized inverse eigenvalue problem for persymmetric matrices consists of determining nontrivial persymmetric matrices A and B such that 16 AX = BXΛ, where X and the diagonal matrix Λ are known such that the columns of X are given eigenvectors and the diagonal entries of Λ are the given associated eigenvalues As expressed in Remark 1, a persymmetric matrix belongs to the set of P, Q-orthogonal symmetric matrices Note that the least-squares problem 15 incorporates the least-squares problem associated with the matrix equation 16 Therefore, our proposed algorithm can be applied to solve generalized persymmetric inverse eigenvalue problems GPIEPs Recently, Julio and Soto 15 derived sufficient conditions for persymmetric nonnegative inverse eigenvalue problempniep

6 6 P A Beik and D K Salkuyeh to have a solution or a comprehensive survey on the applications of structured inverse eigenvalue problems in control theory over different kinds of matrices including persymmetric matrices, one may refer to Chu and Golub, and the references therein A matrix is said to be a generalized skew-hamiltonian matrix if HJ 1 is a skew- symmetric matrices where J 1 R n n is an arbitrary orthogonal skew-symmetric matrix; ie, J1 1 = J1 T = J 1 Here it should be noticed that, without any changing, our algorithm can compute the generalized skew-hamiltonian solutions of the mentioned matrix equations so that P 1 and P are identity matrices and Q 1 and Q are given real orthogonal skewsymmetric matrices This follows from the fact that for an arbitrary given matrix X, we may write X = H 1 +S 1 such that H 1,S 1 = where H 1 = 1 X +J1 X T J 1 and S 1 = 1 X J1 X T J 1 It is not difficult to see that H 1 and S 1 are generalized Hamiltonian and generalized skew-hamiltonian matrices; respectively The convergence properties of the algorithm for determining least-squares generalized skwe-hamiltonian solutions can be established with similar strategies used throughout this work, we do not discuss the details and leave them to reader The ensuing subsection is pertained to stating the main problems which the current paper is concerned with 1 Problem reformulation or simplicity, throughout the paper, we exploit the linear operators K : R p p R r r R m n, L : R p p R r r R ν n, K : R m n R ν n R p p and L : R m n R ν n R r r defined as follows: KX,Y = A 1 XB 1 +A X T B +C 1 YD 1 +C Y T D, LX,Y = E 1 X 1 +E X T +G 1 YH 1 +G Y T H, KZ,W = A T 1 ZBT 1 +ET 1 WT 1 +B Z T A + W T E, LZ,W = C T 1 ZDT 1 +GT 1 WHT 1 +D Z T C +H W T G In this paper, our goal is to propose an iterative algorithm to solve the following problems Problem 1 Let the linear operators K and L be defined as before Assume that P 1 SOR p p, Q 1 SOR p p, P SOR r r and Q SOR r r are given ind X SR P 1Q 1 p p and Ỹ SRP Q r r such that M N K X, Ỹ L X,Ỹ = min {X,Y X,Y SR P 1 Q 1 p p SRP Q r r } M N KX,Y LX,Y Problem Let S s be the solution set of Problem 1 or two given matrices X R p p and Y R r r, find X,Y S s such that { } X X + Y Y = min X X + Y Y X,Y S s Problem 3 Let the linear operators K and L be defined as before Assume that P 1 SOR p p, Q 1 SOR p p, P SOR r r and Q SOR r r are given ind

7 X SSR P 1Q 1 p p and Ỹ SSRP Q r r such that M K X, Ỹ N L X,Ỹ = min {X,Y X,Y SSR P 1 Q 1 p p SSRP Q r r } M N 7 KX,Y LX,Y Problem 4 Let S ss be the solution set of Problem 3 or two given matrices X R p p and Y R r r, find X,Y S ss such that { } X X + Y Y = min X X + Y Y X,Y S ss The remainder of this paper is organized as follows In Section, we prove some properties of the P, Q-orthogonal symmetric and P, Q-orthogonal skewsymmetric matrices Using the established characteristics, Section 3 is devoted to constructing and analyzing an iterative algorithm to solve Problems 1,, 3 and 4 Numerical results are reported in Section 4 which reveal the effectiveness of the algorithm to solve our mentioned problems inally the paper is ended with a brief conclusion in Section 5 Useful theoretical results This section is concerned with establishing some properties and theorems which provide fundamental tools for presenting our algorithm to solve Problems 1-4 and analyzing its convergence properties rom the following theorem, it turns out that R m m can be written as a direct sum of the subspaces SR PQ m m and SSRPQ m m where P SORm m and Q SOR m m Theorem 1 Suppose that the matrices P SOR m m and Q SOR m m are given and the subspaces SR PQ m m and SSRPQ m m are signified as before Then R m m = SR PQ m m SSRPQ m m, where stands for the orthogonal direct sum with respect to the robenius inner product, Proof Before proving the validity of the assertion, we point out that straightforward computations reveals that X SR PQ m m if and only if X = PQXT PQ X SSR PQ m m if and only if X = PQX T PQ Here it should be noted that PQ is in general a nonsymmetric matrix satisfies PQ T PQ = PQPQ T = I m or simplicity, the proof is separated into the following two steps: Step 1 It should be noticed that for an arbitrary given X R m m, we may write X = W 1 +W where W 1 = 1 X +PQX T PQ and W = 1 X PQX T PQ, and it is not difficult to see that W 1 SR PQ m m and W SSR PQ m m Therefore, an arbitrary m m matrix X can be written as the summation of two matrices W 1 and W which are respectively P,Q-orthogonal symmetric and P,Q-orthogonal skew-symmetric matrices Step In this step, it is shown that for two arbitrary given matrices Z 1 SR PQ m m and Z SSR PQ m m, we have Z 1,Z = Note that PZ 1 Q and PZ Q are respectively symmetric and skew-symmetric matrices It is well-known that symmetric and

8 8 P A Beik and D K Salkuyeh skew-symmetric matrices are orthogonal to each other, therefore the result follows from the following equality immediately, trz T 1 Z = trq T Z T 1 P T PZ Q Now the result follows immediately from the given explanations in Steps 1 and It is not difficult to establish the following proposition by using the well-known properties of the trace of matrices Proposition Assume that the linear operators K, L, K and L are defined as before, then 1 KX,Y,Z + LX,Y,W = X, KZ,W + Y, LZ,W, for all X R p p, Y R r r, Z R m n and W R ν n Proof It is well-known that KX,Y,Z + LX,Y,W = tr Z T A 1 XB 1 +A X T B +C 1 YD 1 +C Y T D Using the commutative property, we have +tr W T E 1 X 1 +E X T +G 1 YH 1 +G Y T H KX,Y,Z + LX,Y,W = tr B 1 Z T A 1 X +tr X T B Z T A +tr D 1 Z T C 1 Y +tr Y T D Z T C +tr 1 W T E 1 X +tr X T W T E +tr H 1 W T G 1 Y +tr Y T H W T G Invoking the fact that the trace of a matrix is equal to the trace of its transpose, it can be derived that tr B 1 Z T A 1 X = tr X T A T 1 ZBT 1, tr D1 Z T C 1 Y = tr Y T C1 T ZDT 1, tr 1 W T E 1 X = tr X T E1 T WT 1 and tr H 1 W T G 1 Y = tr Y T G T 1 WHT 1 The results follows immediately by substituting the above four equalities into In view of Theorem 1, the following two propositions can be concluded from Proposition immediately We only present the proof of the first proposition, the second one can be established in a similar manner Proposition 3 Given the matrices P 1 SOR p p, Q 1 SOR p p, P SOR r r and Q SOR r r, assume that the linear operators K, L, K and L are defined as before, then KX,Y,Z + LX,Y,W = X, KZ,W+P1 Q 1 KZ,W T P 1 Q 1 for all X SR P 1Q 1 p p, Y SRP Q r r, Z R m n and W R ν n Y, LZ,W+P Q LZ,W T P Q,

9 9 Proof It is obvious that KZ,W = 1 KZ,W+P1 Q 1 KZ,W T P 1 Q KZ,W P1 Q 1 KZ,W T P 1 Q 1 and LZ,W = 1 LZ,W+P Q LZ,W T P Q + 1 LZ,W P Q LZ,W T P Q Substitute two preceding equations in 1 By the assumption, X SR P 1Q 1 p p and Y SR P Q r r Consequently, the assertion holds from the second step in the proof of Theorem 1 Proposition 4 Given the matrices P 1 SOR p p, Q 1 SOR p p, P SOR r r and Q SOR r r, assume that the linear operators K, L, K and L are defined as before, then KX,Y,Z + LX,Y,W = 1 X, KZ,W P1 Q 1 KZ,W T P 1 Q Y, LZ,W P Q LZ,W T P Q, for all X SSR P 1Q 1 p p, Y SSRP Q r r, Z R m n and W R ν n The following lemma is called Projection Theorem which is utilized to present some of our succeeding results, we refer the reader to Wang, 3 for its proof Lemma 5 Let W be a finite dimensional space, U be a subspace of W and U be the orthogonal complement subspace of U or a given w W, always, there exists u U such that w u w u for any u U More precisely, u U is the unique minimization vector in U if and only if w u U Here is the norm corresponding to the inner product defined on W The following two useful propositions can be established by Lemma 5 Proposition 6 Given the matrices P 1,Q 1 SOR p p and P,Q SOR r r, let X R p p and Y R r r be arbitrary P 1,Q 1 -orthogonal symmetric and P,Q - orthogonal symmetric matrices, respectively Then the matrix pair X,Y SR P 1Q 1 p p SR P Q r r is the solution of Problem 1 if and only if for all X,Y SR P 1Q 1 p p SRP Q X, KR 1,R + P 1 Q 1 KR 1,R T P 1 Q 1 + Y, LR 1,R +P Q LR 1,R T P Q =, r r, in which R 1 = M KX,Y and R = N LX,Y where the linear operators K, L, K and L are defined as before Proof Presume that W = { V1 V Consider the following subspaces of W, V 1 R m n and V R ν n }

10 1 P A Beik and D K Salkuyeh U = By Lemma 5, for we have M N if and only if { KX,Y LX,Y KX,Y LX,Y KX,Y LX,Y } X SR P 1Q 1 p p and Y SR P Q r r M N W, = min L M N M KX,,Y N LX,Y for X,Y L := SR P 1Q 1 p p SRP Q r r Or equivalently, KX,Y LX,Y =, KX,Y,R 1 + LX,Y,R =, X,Y SR P 1Q 1 p p SRP Q r r, which completes the proof in view of Proposition 3 In a similar manner used in the proof of Proposition 6, the following proposition can be established Proposition 7 Given the matrices P 1,Q 1 SOR p p and P,Q SOR r r, let X R p p and Y R r r be arbitrary P 1,Q 1 -orthogonal skew-symmetric and P,Q -orthogonal skew-symmetric matrices, respectively Then the matrix pair X,Y SSR P 1Q 1 p p SSRP Q r r is the solution of Problem 3 if and only if for all X,Y SSR P 1Q 1 p p SSRP Q r r, X, KR 1,R P 1Q 1 KR 1,R T P 1 Q 1 + Y, LR 1,R P Q LR 1,R T P Q =, in which R 1 = M KX,Y and R = N LX,Y where the linear operators K, L, K and L are defined as before 3 Proposed algorithm and its convergence analysis It is known that the conjugate gradient least-squares CGLS method is an effective algorithm to solve the following least-squares problem 31 min x R n b Ax, where is the Euclidean vector norm, A is a given large and sparse m n matrix and b R m ; for further details one may refer to Björck 1996 and Saad 1995 In this section, the main goal is to examine the idea of the CGLS method to construct an iterative algorithm for solving Problems 1-4 by exploiting the elaborated results in the previous section More precisely the current section is divided into three subsections In the first part we propose an iterative scheme to solve Problems 1 and 3 The second subsection is devoted to proving the fact that the approximate solution pairs computed by the handled algorithm satisfy an optimality property at each iterate In the last subsection it is momentarily described that how the presented algorithm can be employed to solve Problems and 4,

11 31 Solving Problems 1 and 3 Utilizing the established facts in Section, we may present an extended form of the CGLS method for solving Problems 1 and 3 in Algorithm In the sequel, the convergence properties of the algorithm are discussed Algorithm : The CGLS method for solving Problems 1 and 3 1 Data: Input A i,b i,c i,d i,e i, i,g i,h i,m,n and the symmetric orthogonal matrices P i and Q i for i = 1, Set η = η = 1 for solving Problem 1 Problem 3 Initialization: or solving Problem 1 Problem 3, choose the initial guess X SR P 1Q 1 p p and Y SR P Q r r X SSR P 1Q 1 p p and Y SSR P Q r r 3 begin 4 Compute 5 R 1 = M KX,Y ; 6 R = N LX,Y; 7 P x = 1 KR1,R + 1 η P 1 Q 1 KR 1,R T P 1 Q 1 ; 8 P y = 1 LR1,R + 1 η P Q LR 1,R T P Q ; 9 Set 1 Q x = P x ; 11 Q y = P y ; 1 for k =,1,, until the convergence do 13 K = KQ x k,q y k; 14 L = LQ x k,q y k; 15 Xk +1 = Xk+ Pxk + Pyk K + L 16 Yk+1 = Yk+ Pxk + Pyk K + L 17 R 1 k +1 = R 1 k Pxk + Pyk K + L 18 R k +1 = R k Pxk + Pyk K + L 19 P x k +1 = 1 P y k +1 = 1 Q x k; Q y k; K; L; KR1 k+1,r k η P 1 Q 1 KR 1 k +1,R k +1 T P1 Q 1 ; LR1 k +1,R k η P Q LR1 k+1,r k +1 T P Q ; 1 Q x k +1 = P x k +1+ Pxk+1 + Pyk+1 P xk + Pyk Q y k +1 = P y k +1+ Pxk+1 + Pyk+1 P xk + Pyk 3 end 4 end Q x k; Q y k; 11

12 1 P A Beik and D K Salkuyeh The following theorem presents some properties of the sequences produced by Algorithm These properties have a basic role for studying the convergence behavior of the algorithm to resolve Problems 1-4 Theorem 31 Suppose that k steps of Algorithm have been performed The sequences P x l, P y l, Q x l and Q y l l =,1,,k produced by Algorithm satisfy for i,j =,1,,,k i j P x i,p x j + P y i,p y j = KQ x i,q y i,kq x j,q y j + LQ x i,q y i,lq x j,q y j =, Q x i,p x j + Q y i,p y j =, Note that the previous theorem can be established by mathematical induction and its proof is given in Appendix Remark 3 Without loss of generality, assume that we apply the proposed algorithm to solve Problem 1 Suppose that k k > 1 steps of Algorithm have been performed To continue the algorithm at k+1-th step, we must have P x k or P y k in Lines 1 and Note that if P x k = and P y k = then Proposition 6 implies that Xk,Yk is a solution pair of Problem 1 Considering Lines 15-18, it is required to presume that KQ x k,q y k or LQ x k,q y k Otherwise, using Proposition 3 and Eq 34, we have = KQ x k,q y k,r 1 k + LQ x k,q y k,r k = Q x k,p x k + Q y k,p y k = P x k,p x k + P y k,p y k, which indicates that P x k = and P y k =, as discussed earlier, this ensures that Xk,Yk is a solution pair of Problem 1 Remark 33 In view of the orthogonality relation 3 presented in Theorem 31, it reveals that Algorithm can solve Problems 1 and 3 within finite number of steps Theorem 34 Assume that P 1 SOR p p, Q 1 SOR p p, P SOR r r and Q SOR r r are given Suppose that X,Ỹ is a solution pair of Problem 1 Problem 3 If ˆX,Ŷ is an alternative solution pair of Problem 1 Problem 3, then there exist X 1 SR P 1Q 1 p p and Y 1 SR P Q r r X 1 SSR P 1Q 1 p p and Y 1 SSR P Q r r such that X = ˆX +X 1 and Ỹ = Ŷ +Y 1, and 35 KX 1,Y 1 = and LX 1,Y 1 = Proof We prove the assertion only for the solutions of Problem 1, the same strategy can be handled to show that the conclusion is also valid for the solutions of Problem 3 Presume that X 1 = X ˆX and Y 1 = Ỹ ˆX, it is not difficult to see that X 1 SR P 1Q 1 p p and Y 1 SR P Q r r By Propositions 3 and 6 and using some straightforward computations, it turns out that 36 M K ˆX,Ŷ,KX 1,Y 1 + N L ˆX,Ŷ,LX 1,Y 1 =

13 By the assumption, X,Ỹ and ˆX,Ŷ are solutions of Problem 1 This hypophysis guarantees that M K X, Ỹ N L X,Ỹ = M K ˆX, Ŷ N L ˆX,Ŷ, which is equivalent to say that M K X,Ỹ N + L X, Ỹ M = K ˆX, Ŷ N + L ˆX, Ŷ Considering 36, we derive the following relation M KX 1,Y 1 + LX 1,Y 1 = K X, Ỹ N + L X, Ỹ M K ˆX, Ŷ N + L ˆX, Ŷ, which completes the proof By Theorem 34, in the rest of this subsection, it is demonstrated that Algorithm converges to the least-norm solution pair X,Y of Problem 1 Problem 3 or two arbitrary given matrices Z R m n and W R ν n, we set X = 1 KZ,W+ 1 η P 1 Q 1 KZ,W T P 1 Q 1, Y = 1 LZ,W+ 1 η P Q LZ,W T 38 P Q, where the value of η is respectively taken to be zero and one for solving Problem 1 and Problem 3 By applying Algorithm with X and Y as shown by 37 and 38, Remark 33 implies that there exist Z R m n and W R ν n such that the solution pair X,Y solves Problem 1 for η = and Problem 3 for η = 1 where X = 1 KZ,W + 1 η P 1 Q 1 KZ,W T 39 P 1 Q 1, 31 Y = 1 LZ,W + 1 η P Q LZ,W T P Q Now if X,Ỹ isasolution pairofproblem1problem3thentheorem34 indicates the existence of X 1,Y 1 SR P 1Q 1 p p SRP Q r r X 1,Y 1 SSR P 1Q 1 p p SSRP Q r r where X = X +X 1 and Ỹ = Y +Y 1, and the relations given in 35 are satisfied Evidently, we have X +X 1,X +X 1 + Y +Y 1,Y +Y 1 = X,X + Y,Y + X,X 1 + Y,Y 1 + X 1,X 1 + Y 1,Y 1 In view of 35, 39 and 31, Proposition 3 Proposition 4 implies that X,X 1 + Y,Y 1 = Consequently, it can be deduced that X, X Ỹ, + Ỹ = X +X 1,X +X 1 + Y +Y 1,Y +Y 1 = X,X + Y,Y + X 1,X 1 + Y 1,Y 1

14 14 P A Beik and D K Salkuyeh Therefore, we obtain X + Y X + Ỹ It should be noticed here that the inequality in the above relation holds strictly as long as X 1 or Y 1 As a result X,Y is the unique least-norm solution pair of Problem 1 Problem 3 which can be reached via Algorithm by choosing X and Y as shown by of 37 and 38 with η = η = 1 3 Optimality property of the proposed algorithm The minimization property of Algorithm can be deduced from the following proposition Proposition 35 Suppose that k steps of Algorithm have been performed Then the sequence {Rl} k l=1 satisfies 311 R 1 k,kq x i,q y i + R k,lq x i,q y i =, for i = 1,,,k 1, where R 1 k = M KXk,Yk, R k = N LXk,Yk and Xk, Yk is the k-th approximate solution pair computed by Algorithm Proof Assume that η = η = 1 In view of Proposition 3 Proposition 4, it is deduced that R 1 k,kq x i,q y i + R k,lq x i,q y i = P x k,q x i + P y k,q y i Now the result follows immediately by the property 34 given in Theorem 31 Considering that m-th step of Algorithm has been performed, we elucidate the subspaces K m and U m as follows: { } Qx Qx m 1 31 K m = span,,, Q y Q y m { KQx,Q U m = span y LQ x,q y Evidently we can see that Xm Ym KQx m 1,Q,, y m 1 LQ x m 1,Q y m 1 X Y +K m By Proposition 35, it indicates that M KXm,Ym U N LXm,Ym m The following theorem reveals that the approximate solutions produced by Algorithm satisfy an optimality property which demonstrates the minimization property of the algorithm The result is a direct consequence of Lemma 5, hence we omit its proof Theorem 36 Suppose that m steps of Algorithm have been performed Presume that the subspaces K m and U m are defined by 31 and 313, respectively Then M KXm,Y m = min M KX,Y N LXm,Ym L, N LX,Y }

15 where L := { X Y X Y X Y +K m } 33 On the solution of Problems and 4 In the current subsection we discuss an approach to solve Problems and 4 Assume that P 1 SOR p p, Q 1 SOR p p, P SOR r r andq SOR r r aregiven LetusdenotethesolutionsetofProblems and4bys s ands ss, respectively Evidentlythesets S s ands ss arebothnonempty or two given X R p p and Y R r r, Theorem 1 ensures the existence of Z i and W i i = 1, where X = Z 1 +Z and Y = W 1 +W such that Z 1 SR P 1Q 1 p p, Z SSR P 1Q 1 p p, W 1 SR P Q r r and W SSR P Q r r Remark 37 It is not difficult to see that the following statements hold If X,Y S s then X X,X X + Y Y,Y Y = Z 1 X,Z 1 X + W 1 Y,W 1 Y Z,Z + W,W If X,Y S ss then X X,X X + Y Y,Y Y = Z X,Z X + W Y,W Y Z 1,Z 1 + W 1,W 1 Consequently, from 314, it can be concluded that 316 { } { } min X X + Y Y = min Z 1 X + W 1 Y X,Y S s X,Y S s In addition, 315 implies that 317 { } min X X + Y Y X,Y S ss { } = min Z X + W Y X,Y S ss rom Remark 37, we infer that for solving Problem and Problem 4, it is sufficient to evaluate the least-norm solutions of the following new problems, respectively Problem 5 Let the linear operators K and L be defined as before Assume that P 1 SOR p p, Q 1 SOR p p, P SOR r r and Q SOR r r, X R p p and Y R r r are given ind X SR P 1Q 1 p p and Ỹ SRP Q r r such that M Ñ K X, Ỹ L X,Ỹ = min {X,Y X,Y SR P 1 Q 1 p p SRP Q r r } M Ñ 15 KX,Y LX,Y in which M = M KZ 1,W 1 and Ñ = N LZ 1,W 1 where Z 1 = 1 X +P 1 Q 1 X T P 1 Q 1 and W 1 = 1 Y +P Q Y T P Q Problem 6 Let the linear operators K and L be defined as before Assume that P 1 SOR p p, Q 1 SOR p p, P SOR r r and Q SOR r r, X R p p and Y R r r are given ind X SSR P 1Q 1 r r such that M Ñ K X, Ỹ L X,Ỹ p p and Ỹ SSRP Q = min {X,Y X,Y SSR P 1 Q 1 p p SSRP Q r r } M Ñ KX,Y LX,Y,,

16 16 P A Beik and D K Salkuyeh in which M = M KZ,W and Ñ = N LZ,W where Z = 1 X P 1 Q 1 X T P 1Q 1 and W = 1 Y P Q Y T P Q To determine the solution of Problem Problem 4, we first compute X,Ỹ as the least-norm solution pair of Problem 5 Problem 6 using the approach described at the end of Subsection 31 Then the solution pair X,Y of Problem Problem 4 is obtained by X = X+Z 1 and Y = Ỹ +W 1 X = X+Z and Y = Ỹ +W 4 Numerical experiments In this section, some numerical instances are examined to demonstrate the validity of the elaborated theoretical results and to illustrate the practicability of the proposed algorithm for solving Problems 1-4 More precisely, we consider matrix equations in the cases that they are consistent or inconsistent over P 1,Q 1 -orthogonal symmetric and skew-symmetric and Hamiltonian matrices All of reported experiments were performed on a 64-bit 45 GHz core i7 processor and 8GB RAM using some MATLAB codes on MATLAB version 8353 Example 41 Consider the following least-squares problem 41 M K X = min M KX, X SR P 1 Q where KX = A 1 XB 1 in which A 1 = , B 1 = M = , and the matrices P 1,Q 1 SOR 5 5 are given by 1 1 P 1 = 1 1, Q 1 = In order to solve 41 by Algorithm, we exploit a zero matrix as an initial guess and the following stopping criterion, = Xk Xk 1 < 1,

17 The algorithm converges in iterations and the approximate solution is obtained by 6 1 X = 1 6, 1 6 with the residual norm η = 86e 11, where η k = Rk = M A 1 XkB 1 It is noted that the matrix X = , 1 6 is a solution of KX = M which is P 1,Q 1 -orthogonal symmetric This reveals that the algorithm has provided an approximate solution to KX = M which is also an approximate solution to 41 Convergence history of the method is plotted in igure 1 for more clarification We now utilize Algorithm to compute matrix X such that 4 X X = min X S s X X, where S s is the solution set of 41 and X = According to Problem 5, we set and Z 1 = 1 X +P 1 Q 1 X T P 1Q 1 = 1 M = M KZ 1 = Applying Algorithm to solve the ensuing least-squares problem, 44 M K X = min M KX, X SR P 1 Q 1 5 5, 17

18 18 P A Beik and D K Salkuyeh we derive X3 = as an approximation of X Hence, the approximate solution of 4 is determined by X 6 1 Z 1 +X3 = or more details, log 1 has been displayed in igure 1 Todemonstrate theeffectiveness of theproposedalgorithm tocomputeap 1,Q 1 - orthogonal skew-symmetric solution of the investigated problem, let us consider the following problem 45 M K X = min M KX, X SSR P 1 Q with and M = X X = min X S ss X X, wheres ss is the set of solutions of the problem 45 With asimilar manner used for finding the solution in the orthogonal symmetric case, Algorithm is implemented to compute approximate solutions to the problems 45 and 46 All the other assumptions are as before or both of the problems, the algorithm calculates an approximate solution of the following form X13 =, for X and X in 13 iterations Here, we have η 13 = 711e 1 for Problem 3 It is mentioned that X = , 3,,

19 is P 1,Q 1 -orthogonal skew-symmetric matrix and X X13 Convergence behavior of the method to solve problems 45 and 46 is depicted in igure 1 Subsequently, we consider Algorithm to solve 46 where X is given by 43 Having in mind Problem 6, we set and Z = 1 X P 1 Q 1 X T P 1Q 1 = 1 M = M KZ = If we apply Algorithm to solve the least-squares problem, 47 M K X = min M KX, X SRR P 1 Q we get X13 =, as an approximation of X Hence, the approximate solution of 46 is determined by 3 X 3 3 Z +X13 = or more details, log 1 has been displayed in igure 1 We now set M = I, where I is the identity matrix of order n In this case it not difficult to see that the equation A 1 XB 1 = M is inconsistent over SR P 1Q 1 m n and SSR P 1Q 1 m n We employ Algorithm to solve Problems 1,, 3 and 4 and the results are given in Table 1 In this table, Iters stands for the number of the required iterations for the convergence Convergence history of the methods are presented in igure Apparently, the numerical results show that the proposed algorithm is effective, 19

20 P A Beik and D K Salkuyeh Table 1 Numerical results for Example 41 in the inconsistent case Problem 1 Problem Problem 3 Problem 4 817e e-11 95e e-1 Iters η k log 1 log Number of iterations k Number of iterations k log 1 log Number of iterations k Number of iterations k igure 1 Convergence history for Example 41 consistent case Problem 1: top-left; Problem : top-right; Problem 3: down-left; Problem 4: down-right Example 4 In this example, we consider Problem 1 with KX,Y = A 1 XB 1 +X T +C 1 YD 1 +Y T, LX,Y = X +E X T +Y +G Y T H, where A 1 = pentadiag n,,,1,1, B 1 = pentadiag n 1,,,1,1, C 1 = tridiag n 1,,7, D 1 = tridiag n, 1,4, E = tridiag n 1,3, 1, = tridiag n 1,6,3, G = pentadiag n, 1, 3,1,3, H = pentadiag n,,,3,

21 1 log 1 log Number of iterations k Number of iterations k log 1 log Number of iterations k Number of iterations k igure Convergence history for Example 41 inconsistent case Problem 1: top-left; Problem : top-right; Problem 3: down-left; Problem 4: down-right Presume that p 1 = 1,1,,1 T, p = n,n 1,,1 T, q 1 = 1,,,n T, q = 1,,, T, the matrices P 1, P, Q 1 and Q are specified as follows: P i = I p ip T i p T i p, Q i = I q iqi T i qi Tq, i = 1,, i where I signifies the identity matrix of order n It is not difficult to verify that P 1,P,Q 1 and Q are n n symmetric orthogonal matrices Suppose that X = P 1 W x + W T x Q 1 and Ỹ = P W y + W T y Q where W x = tridiag n 1,,1 and W y = tridiag n 1,1, Evidently, we have X SR P 1Q 1 n n and Ỹ SRP Q n n The matrices M and N are selected so that M = K X,Ỹ and N = L X,Ỹ Note that in this case the system { KX,Y = M, 48 LX,Y = N, is consistent over SR P 1Q 1 m n SRP Q m n Now, let us focus on the solution of Problem 1 with the above data and n = 5 In order to handle Algorithm for resolving Problem 1, we utilize X, Y =, as an initial guess and the following stopping criterion = max{ Xk Xk 1, Yk Yk 1 } < 1

22 P A Beik and D K Salkuyeh Table Numerical results for Example 4 inconsistent case Problem 1 Problem Problem 3 Problem 4 989e e e-11 87e-11 Iters η k In this case the algorithm converges in 85 iterations and we have η85 = 7e 9, where ηk = max{ M KXk,Yk, N LXk,Yk } The graph of log 1 is illustrated in igure 3 To investigate the applicability of the algorithm to solve Problem, we choose X = Y = I, where I is the identity matrix of order n All of the other assumptions are the same as before In this case, Algorithm converges in 85 iterations The convergence performance of the method is illustrated in igure 3 To demonstrate the effectiveness of the algorithm for finding the solution pairs of Problems 3 and 4, we set X = P 1 W x Wx T Q 1 and Ỹ = P W y Wy T Q where W x = tridiag n 1,,5 and W y = tridiag n 1,3, Straightforward computations reveal that X SSR P 1Q 1 n n and Ỹ SSRP Q n n The matrices M and N are chosen such that M = K X,Ỹ and N = L X,Ỹ Therefore, the system 48 is consistent over SSR P 1Q 1 m n SSRP Q m n All of the other hypophyses are assumed to be those of the orthogonal symmetric case In this case, Algorithm for both of Problem 3 and Problem 4 converges in 8 iterations or Problem 3, we have η8 = 43e 9 In igure 3, the convergence history of the method for the problems are displayed for further information inally, we choose M = tridiag n 1,1,1 and N = pentadiag n 1,1,,1,1 In this case, it can be verified that the system 48 is inconsistent over SR P 1Q 1 m n SRP Q m n and SSR P 1Q 1 m n SSRP Q m n Algorithm was handled to solve our mentioned main problems The obtained results are disclosed in Table and convergence actions are delineated in igure 4 Example 43 In this example, we work on the inverse eigenvalue problem HX = XΛ over Hamiltonian matrices, ie, the unknown matrix H is determined so the H is a Hamiltonian matrix and minimizes HX XΛ over Hamiltonian matrices This example has been previously examined in the literature, for instance see Qian and Tan 13, Table 1 or m = l, we choose l eigenpairs λ 1,y 1,,λ l,y l of the n n Hamiltonian matrix H of Example 16 from the benchmark examples for continuous algebraic Riccati equations CARE in Benner et al 1995 We set X = [Y 1, ĴY 1 ] R n m and Λ = diagλ 1, Λ 1 R m m where Y 1 = [y 1,y,,y l ], Λ 1 = diagλ 1,λ,,λ l and Ĵ is defined by 11 Here our aim in Problem 1 is to determine least-squares Hamiltonian solution H of HX = XΛ Note that Problem is equivalent to Problem 1 in Qian and Tan 13 To solve Problem, we choose A = randn,n and our goal is to find the least-squares Hamiltonian solution H of HX = XΛ so that A H is minimized

23 3 log 1 log Number of iterations k Number of iterations k log 1 log Number of iterations k Number of iterations k igure 3 Convergence history for Example 4 consistent case Problem 1: top-left; Problem : top-right; Problem 3: down-left; Problem 4: down-right Now, we use Algorithm to compute an approximate solutions of Problems 1 and It is noted that in the algorithm, the matrices P 1 and Q 1 should be respectively replaced by I n and Ĵ Numerical results for different values of n and m are presented in Tables 3 and 4 In these tables number of iterations Iters and CPU time CPU are presented and Hk is the computed solution at iteration k rom the reported numerical results in Tables 3 and 4, we can observe that Algorithm computes good approximate solutions of mentioned problems in short times and small number of iterations 5 Conclusion It has been established that the set of all real square matrices can be written as a direct sum of the set of all P, Q-orthogonal symmetric and P, Q-orthogonal skew-symmetric matrices Invoking the earlier pointed fact, we have scrutinized the extension of the well-known conjugate gradient least-squares CGLS method for finding the least-squares P, Q-orthogonal skew-symmetric solution pairs of general coupled matrix equations Some numerical examples have been examined to illustrate the good performance of the proposed algorithm to solve the considered problems

24 4 P A Beik and D K Salkuyeh log 1 log Number of iterations k Number of iterations k log 1 log Number of iterations k Number of iterations k igure 4 Convergence history for Example 4 inconsistent case Problem 1: top-left; Problem : top-right; Problem 3: down-left; Problem 4: down-right Table 3 Numerical results of Example 43 for solving Problem 1 n,m CPU Iters HkX XΛ H Hk 5, e , e , e , e , e , e , e , e , e , e , e Acknowledgments The authors would like to express their heartfelt thanks to three anonymous referees for their valuable suggestions and constructive comments which have improved the quality of the paper

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