Fatemeh Panjeh Ali Beik, Davod Khojasteh Salkuyeh, and Mahmoud Mohseni Moghadam

Transcription

1 GRADIENT BASED ITERATIVE ALGORITHM FOR SOLVING THE GENERALIZED COUPLED SYLVESTER-TRANSPOSE AND CONJUGATE MATRIX EQUATIONS OVER REFLEXIVE (ANTI-REFLEXIVE) MATRICES Fatemeh Panjeh Al Bek, Davod Khojasteh Salkuyeh, and Mahmoud Mohsen Moghadam Department of Mathematcs, Val-e-Asr Unversty of Rafsanjan, Rafsanjan, Iran e-mal: Faculty of Mathematcal Scences, Unversty of Gulan, PO Box 1914, Rasht, Iran e-mal: Department of Mathematcs, Islamc Azad Unversty of Kerman, Kerman, Iran e-mal: Abstract Lnear matrx equatons play an mportant role n many areas, such as control theory, system theory, stablty theory and some other felds of pure and appled mathematcs In the present paper, we consder the generalzed coupled Sylvestertranspose and conjugate matrx equatons T ν(x) = F ν, ν = 1,,, N, where X = (X 1, X,, X p) s a group of unknown matrces and for ν = 1,,, N, s p 1 s s 3 s 4 T ν(x) = A νµx B νµ + C νµx T D νµ + M νµx N νµ + =1 H νµx H G νµ, n whch A νµ, B νµ, C νµ, D νµ, M νµ, N νµ, H νµ, G νµ and F ν are gven matrces wth sutable dmensons defned over complex number feld By usng the herarchcal dentfcaton prncple, an teratve algorthms s proposed for solvng the above coupled lnear matrx equatons over the group of reflexve (ant-reflexve) matrces Meanwhle, suffcent condtons are establshed whch guarantee the convergence of the presented algorthm Fnally, some numercal examples are gven to demonstrate the valdty of our theoretcal results and the effcency of the algorthm for solvng the mentoned coupled lnear matrx equatons Keywords: Generalzed Sylvester-transpose and conjugate matrx equaton, Iteratve algorthm, Reflexve (ant-reflexve) matrx AMS Subject Classfcaton: 15A4, 65F10 1 Introducton In ths paper, the followng notatons are utlzed We use tr(a), A T, A H, A to denote the trace, the transpose, the conjugate transposed, the conjugate of the matrx A, respectvely Moreover, C m n represents the set of all m n complex matrces The set of all symmetrc Correspondng author 1

2 F P A Bek, D K Salkuyeh and M M Moghadam orthogonal matrces, also known as reflecton matrces, n C n n s represented by SOC n n, e, P SOC n n f and only f P = P H = P 1 Defnton 11 [6] Consder two arbtrary gven matrces P, Q SOC n n The matrx A C n n s called a reflexve matrx, wth respect to P and Q, f A = P AQ The set of all n n reflexve, (P, Q)-reflexve, matrces s denoted by C n n r (P, Q) Defnton 1 [6] Consder two arbtrary gven matrces P, Q SOC n n The matrx A C n n s called an ant-reflexve matrx, wth respect to P and Q, f A = P AQ The set of all n n ant-reflexve, (P, Q)-ant-reflexve, matrces s denoted by C n n a (P, Q) In the lterature, the problem of fndng solutons of several lnear matrx equatons has been nvestgated wdely, for more detals see [1, 3, 4, 1, 13, 15, 16, 17, 1,, 4, 6, 8] and the references theren Before statng the man problems of ths paper, we brefly revew some of the works whch have been recently presented n the feld of lnear matrx equatons Recently, the dea of conjugate gradent (CG) method has been developed for constructng teratve algorthms to compute the solutons of dfferent knds of lnear matrx equatons over reflexve and ant-reflexve, generalzed bsymmetrc, generalzed centrosymmetrc, mrror-symmetrc, skew-symmetrc and (P, Q)-reflexve matrces, for more detals see [7, 8, 9, 14, 15, 17, 18, 19, 3, 5] Gradent based teratve algorthm s a dfferent common approach for solvng lnear matrx equatons For nstance, Dng et al [13] have consdered the soluton of AXB = F and AXB + CXD = F Dehghan and Hajaran [10] proposed two algorthms for solvng the followng the matrx equaton p q (11) A XB + C j Y D j = F, =1 j=1 over reflexve and ant-reflexve matrces In [11], n fact, the authors have employed the dea of the Gradent based teratve method to construct two teratve algorthms for computng the generalzed bsymmetrc and skew-symmetrc solutons of the lnear matrx equaton l (1) A Y B = C =1 Moreover, Song et al [] have consdered the followng coupled Sylvester-transpose matrx equatons p ( (13) Aη X η B η + C η Xη T ) D η = F, = 1,,, N, η=1 where A η R m l η, B η R nη p, C η R m n η, D η R lη p, F R m p, = 1,, N, η = 1,, p, are gven matrces and X η R lη nη, η I[1, p] are the matrces to be determned For smplcty, we use the followng lnear operator such that T ν (X) = p =1 s 1 A νµ X B νµ + T ν : C p 1 p 1 C pp pp C λν γν s C νµ X T D νµ + s 3 M νµ X N νµ + s 4 H νµ X H G νµ

3 On the generalzed Sylvester-transpose and conjugate matrx equatons 3 where X = (X 1, X,, X p ) and ν = 1,,, N In the present work, we wll construct a gradent based algorthm for solvng the followng coupled lnear matrx equatons (14) T ν (X) = F ν, ν I[1, N], over the group of reflexve (ant-reflexve) matrces Evdently, the coupled lnear matrx equatons (14) are extremely general and nclude many nvestgated lnear matrx equatons such as the generalzed (coupled) Sylvester and Lyapunov matrx equatons, Eqs (11), (1) and (13) 11 Problem reformulaton We wll focus on the followng problems and construct an teratve algorthm for obtanng the solutons of these problems Problem 1 Assume that the matrces A νµ C λν p, B νµ C p γ ν, µ I[1, s 1 ], C νµ C λν p, D νµ C p γ ν, µ I[1, s ], M νµ C λν p, N νµ C p γ ν, µ I[1, s 3 ], H νµ C λν p, G νµ C p γ ν, µ I[1, s 4 ], F ν C λν γν and R, Q SOC p p are gven Fnd the reflexve matrx group (X 1, X,, X p ) such that X C p p r (R, Q ) and satsfy (14) where I[1, p], ν I[1, N] Problem Assume that the matrces A νµ C λν p, B νµ C p γ ν, µ I[1, s 1 ], C νµ C λν p, D νµ C p γ ν, µ I[1, s ], M νµ C λν p, N νµ C p γ ν, µ I[1, s 3 ], H νµ C λν p, G νµ C p γ ν, µ I[1, s 4 ], F ν C λν γν and R, Q SOC p p are gven Fnd the ant-reflexve matrx group (X 1, X,, X p ) such that X C p p a (R, Q ) and satsfy (14) where I[1, p], ν I[1, N] The remnder of ths paper s organzed as follows In Secton, we recall some defntons and theorems whch are used for presentng our man theoretcal results In Secton 3, frst, we nvestgate the solvablty of Problems 1 and Then, an algorthm s proposed for solvng these problems Convergence analyss of the algorthm s also dscussed In order to llustrate the valdty and applcablty of our presented results, we gve some numercal examples n Secton 4 Fnally, the paper s ended wth a bref concluson n Secton 5 Prelmnares In ths secton, we revew some necessary prncples and defntons whch are utlzed throughout ths work Defnton 1 Suppose that Y = (Y 1, Y,, Y k ) and Z = (Z 1, Z,, Z k ) where Y, Z C r s for = 1,,, k We defne the nner product <, > as follows: (1) < Y, Z >= Re[ k =1 tr(y H j Z j )] Remark For Y = (Y 1, Y,, Y k ), Y C r s for I[1, k], the norm of Y s defned by Y = Re[ k tr(y H Y )] =1

4 4 F P A Bek, D K Salkuyeh and M M Moghadam Assume that A = [a j ] m s and B = [b j ] n q defned over complex (real) number feld, the Kronecker product of the matrces A and B s defned as the mn sq matrx A B = [a j B] The vec operator transforms a matrx A of sze m s to a vector a = vec(a) of sze ms 1 by stackng the columns of A In ths paper, the followng relaton s utlzed (See []) vec(axb) = (B T A)vec(X) Lemma 3 Let X R m n be an arbtrary matrx Then vec(x T ) = P (m, n)vec(x), where P (m, n) s unquely determned by the ntegers m and n Proof See [6] Remark 4 Let X C m n be an arbtrary matrx Then and vec(x T ) = P (m, n)vec(x), vec(x H ) = vec(x T ) = P (m, n)vec(x), where P (m, n) s unquely determned by the ntegers m and n Some propertes of the matrx P (m, n) are gven as follows ([6, 17, 7]): 1 For two arbtrary ntegers m and n, P (m, n) has the followng explct form E11 T E1 T E1n T E1 T E T E T n P (m, n) = Em1 T Em T Emn T mn mn where each E j for I[1, m] and j I[1, n], s an m n matrx wth the element at poston (, j) beng one and the others beng zero For two arbtrary ntegers m and n, P (m, n) s the untary matrx, e, P (m, n)p T (m, n) = P T (m, n)p (m, n) = I mn 3 For two arbtrary ntegers m and n, P (m, n) = P T (n, m) 3 Man results Ths secton conssts of two parts In the frst part, the solvablty of Problems 1 and s dscussed In the second part, we gve an teratve algorthm for solvng the coupled lnear matrx equatons (14) over reflexve (ant-reflexve) matrces In addton, suffcent condtons are establshed whch guarantee the convergence of the proposed algorthm 31 Solvablty of the Problems 1 and By some straghtforward computatons, we can prove the followng lemmas Lemma 31 Problem 1 s solvable f and only f the system of matrx equatons (31) (T ν (X), T ν (Z)) = (F ν, F ν ), ν I[1, N], are consstent, where Z = (Z 1, Z,, Z p ) s defned such that Z = R X Q n whch the matrces R, Q SOC p p are gven for I[1, p]

5 On the generalzed Sylvester-transpose and conjugate matrx equatons 5 Proof Wthout loss of generalty, we may assume that s 1 = s = s 3 = s 4 = s Suppose that the matrx group X = (X 1, X,, X p ) satsfes n Eq (31) Therefore, p =1 p =1 s A νµ X B νµ + C νµ X T D νµ + M νµ X N νµ + H νµ X HG νµ = F ν s A νµ R X Q B νµ + C νµ Q T XT RT D νµ + M νµ R X Q N νµ + H νµ Q X HR G νµ = F ν Now, we defne the matrx group X = ( X 1, X,, X p ) such that X = X + R X Q, = 1,,, p Evdently, X s a group of reflexve matrces, e, X C p p r (R, Q ), for = 1,,, p It s easy to verfy that for ν I[1, N] p =1 s A νµ X B νµ + C νµ XT D νµ + M νµ X N νµ + H νµ XH G νµ = F ν, whch shows that the matrx groups X = ( X 1, X,, X p ) s the soluton of Problem 1 Conversely, let the matrx group X = (X 1, X,, X p ) be the soluton of Problem 1 That s X C p p r (R, Q ), = 1,,, p, and X = (X 1, X,, X p ) satsfy n Eq (14) Hence, the proof can be concluded mmedately from the fact that X = R X Q for = 1,,, p Analogous to the strategy employed n the proof of Lemma 31, we may establsh the followng lemma Lemma 3 Problem s solvable f and only f the system of matrx equatons (3) (T ν (X), T ν (Z)) = (F ν, F ν ), ν I[1, N] are consstent, where Z = (Z 1, Z,, Z p ) s defned such that Z = R X Q n whch the matrces R, Q SOC p p are gven for I[1, p] Assume that the matrces ψ 1, ψ, ψ 3 and ψ 4 are defned as follows: ψ 1 = s 1 s 1 s 1 s 1 B T 11µ A 11µ B T N1µ A N1µ B T 11µ QT 1 A 11µR 1 B T N1µ QT 1 A N1µR 1 s 1 s 1 s 1 s 1 B T 1pµ A 1pµ B T Npµ A Npµ B T 1pµ QT p A 1pµ R p B T Npµ QT p A Npµ R p,

6 6 F P A Bek, D K Salkuyeh and M M Moghadam ψ = and ψ 4 = s s s s (D T 11µ C 11µ)P (p 1, p 1 ) (D T N1µ C N1µ)P (p 1, p 1 ) (D T 11µ R 1 C 11µ Q T 1 )P (p 1, p 1 ) (D T N1µ R 1 C N1µ Q T 1 )P (p 1, p 1 ) ψ 3 = s 4 s 4 s 4 s 4 s 3 s 3 s 3 s 3 N T 11µ M 11µ N T N1µ M N1µ N T 11µ Q 1 M 11µ R 1 N T N1µ Q 1 M N1µ R 1 (G T 11µ H 11µ)P (p 1, p 1 ) (G T N1µ H N1µ)P (p 1, p 1 ) (G T 11µ RT 1 H 11µQ 1 )P (p 1, p 1 ) (G T N1µ RT 1 H N1µQ 1 )P (p 1, p 1 ) Let Π = (Ψ 1 + Ψ, Ψ 3 + Ψ 4 ), vec(x 1 ) υ = vec(x p ) vec(x 1 ), Ω r = vec(x p ) vec(f 1 ) vec(f N ) vec(f 1 ) vec(f N ) s s s s (D T 1pµ C 1pµ)P (p p, p p ) (D T Npµ C Npµ)P (p p, p p ) (D T 1pµ R p C 1pµ Q T p )P (p p, p p ) (D T Npµ R p C Npµ Q T p )P (p p, p p ) s 3 s 3 s 3 s 3 N T 1pµ M 1pµ N T Npµ M Npµ N T 1pµ Q p M 1pµ R p N T Npµ Q p M Npµ R p s 4 s 4 s 4 s 4, (G T 1pµ H 1pµ)P (p p, p p ) (G T Npµ H Npµ)P (p p, p p ) (G T 1pµ RT p H 1pµ Q p )P (p p, p p ) (G T Npµ RT p H Npµ Q p )P (p p, p p ), and Ω a = vec(f 1 ) vec(f N ) vec(f 1 ) vec(f N ) It s not dffcult to see that Eqs (31) and (3) are equvalent to the followng to the followng lnear systems, respectvely, Πυ = Ω r,,

7 On the generalzed Sylvester-transpose and conjugate matrx equatons 7 and Πυ = Ω a Now, we can conclude the followng theorems from Lemmas 31 and 3 Theorem 33 Problem 1 has a unque soluton f and only f Rank(Π, Ω r ) = Rank(Π) and Π has full column rank The unque reflexve soluton group of Problem 1 can be calculated as follows: X = X + R X Q, = 1,,, p, where X = ( X 1,, X p, X 1,, X p ) s derved such that vec( X) = (Π H Π) 1 Π H Ω r Theorem 34 Problem has a unque soluton f and only f Rank(Π, Ω a ) = Rank(Π) and Π has full column rank The unque ant-reflexve soluton group of Problem can be calculated as follows: X = X R X Q, = 1,,, p, where X = ( X 1,, X p, X 1,, X p ) s derved such that vec( X) = (Π H Π) 1 Π H Ω a 3 Proposed algorthms In ths secton, by developng the dea of gradent based teratve method, we propose an teratve algorthm to compute the reflexve ( ant-reflexve) soluton group of Eq (14) Before presentng the algorthm, for j = 1,, 3, 4, we defne the lnear operators S j (V ) as follows: S j : C λ 1 γ 1 C λ N γ N C p 1 p 1 C pp pp V S j (V ), such that V = (V 1, V,, V N ), S j (V ) = (S 1j (V ), S j (V ),, S pj (V )) and for I[1, p] S 1 (V ) = S 3 (V ) = N s 1 N s 3 A H νµv ν B H νµ, S (V ) = M νµ H V ν N H νµ, S 4 (V ) = N s N s 4 Algorthm 1 (Proposed algorthm for Problems 1 and ) D νµ V T ν C νµ, G νµ V H ν H νµ Step 1: Input the matrces A νµ C λν p, B νµ C p γ ν, µ I[1, s 1 ], C νµ C λν p, D νµ C p γ ν, µ I[1, s ], M νµ C λν p, N νµ C p γ ν, µ I[1, s 3 ], H νµ C λν p, G νµ C p γ ν, µ I[1, s 4 ], F ν C λν γν, R, Q SOC p p, for I[1, p], and ν I[1, N] Step : Set κ = 0 for computng the soluton of Problem 1, and κ = 1 for Problem Choose a tolerance ε Step 3: If κ = 0, select the ntal reflexve group of matrces X (1) = (X (1) 1, X(1),, X(1) p ) Otherwse select the ntal ant-reflexve group of matrces X (1) = (X (1) 1, X(1),, X(1) p )

8 8 F P A Bek, D K Salkuyeh and M M Moghadam Step 4: Set t = 1 For ν = 1,, 3,, N, compute R ν (1) = F ν T ν (X (1) ) Step 5: Choose the nteger values α j, for I[1, p] and j = 1,, 3, 4, such that f S j (V ) 0 for all V = (V 1, V,, V N ) where V ν C λν γν for ν I[1, N]; Set α j = 0; Otherwse set α j = 1 Step 6: For = 1,,, p, Do: If α 1 = 1, set U (t+1) Otherwse U (t+1) = 0 If α = 1, set Y (t+1) Otherwse Y (t+1) = 0 If α 3 = 1, set Z (t+1) Otherwse Z (t+1) = 0 If α 4 = 1, set W (t+1) = X (t) + ω Ns 1 (S 1 (R (t) ) + ( 1) κ R S 1 (R (t) )Q ); = X (t) + ω Ns (S (R (t) ) + ( 1) κ R S (R (t) )Q ); = X (t) + ω Ns 3 (S 3 (R (t) ) + ( 1) κ R S 3 (R (t) )Q ); = X (t) + ω Ns 4 (S 4 (R (t) ) + ( 1) κ R S 4 (R (t) )Q ); Otherwse W (t+1) = 0 Step 7: For = 1,,, p, calculate β := α 1 + α + α 3 + α 4 ; Set X (t+1) = 1 β (U (t+1) + Y (t+1) + Z (t+1) N + W (t+1) ) Step 8: Compute R (t+1) = (R (t+1) 1, R (t+1),, R (t+1) ) where R (t+1) ν = F ν T ν (X (t+1) ), ν = 1,,, N Step 9: If R (t+1) < ε then Stop; else, set t = t + 1 and go to Step 6 Theorem 35 Suppose that the lnear coupled matrx equatons (14) has unque reflexve soluton group X = (X 1, X,, X p) If the parameter ω satsfes the nequalty (33) 0 < ω < Θ, such that Θ = N p =1 Θ ν where, Θ 1ν = s 1 A νµ B νµ, Θ ν = s C νµ D νµ Θ 3ν = s 3 M νµ N νµ, Θ 4ν = s 4 H νµ G νµ Then the teratve soluton groups X (t) = (X (t) 1, X(t),, X(t) p ), t = 1,, 3,, computed by Algorthm 1 (wth κ = 0), converge for any ntal reflexve matrx group X (1) = (X (1) 1, X(1),, X(1) p )

9 On the generalzed Sylvester-transpose and conjugate matrx equatons 9 Proof Wthout loss of generalty, we may assume that α j = 1 for I[1, p] and j = 1,, 3, 4 For = 1,,, p, we set Ũ (t) = U (t) X β, Ỹ (t) = Y (t) X, β Z (t) = Z(t) X β (t), W = W (t) X β For the matrx groups U = (U 1, U,, U p ), Y = (Y 1, Y,, Y p ), Z = (Z 1, Z,, Z p ) and W = (W 1, W,, W p ), we defne T ν (U, Y, Z, W ) = T ν (L), ν = 1,,, N, where the matrx groups L = (L 1, L,, L p ) s defned such that L = (U +Y +Z +W ) for I[1, p] It s obvous that L (t) = X X (t) Therefore, T ν (Ũ (t), Ỹ (t), Z (t), W (t) ) = T ν ( L (t) ) = T ν (X ) T ν (X (t) ) = R (t) ν Suppose that the matrces R, Q SOC p p for I[1, p] are gven Assume that t steps of Algorthm 1, wth κ = 0, has been performed Evdently, the t-th approxmate soluton group X (t) = (X (t) 1, X(t),, X(t) p ) s a reflexve matrx group, e X (t) C p p (R, Q ) for I[1, p] Hence, L(t) = ( L (t) (t) (t) 1, L,, L p ) s a group of reflexve matrces, e, L (t) C p p r (R, Q ) for I[1, p] In addton, for two gven arbtrary matrces A and B, t s not dffcult to see that Re[tr(AB)] = Re[tr(BA)] = Re[tr(A H B H )] Therefore, we get Ũ (t) = Re[tr[ p =1 ω 4Ns 1 { p [ ( =1 Re tr (t) (Ũ ) H Ũ (t) ]] = 1 16s 1 ( L (t 1) L (t 1) + R A H νµ T ν(ũ (t 1), Ỹ (t 1), Z (t 1), W )]} (t 1) )Bνµ H Q ) { p + ω 16N s =1 1 N ) H ( N s1 AH νµ T ν(ũ (t 1), Ỹ (t 1), Z (t 1), W (t 1) )Bνµ H s1 AH νµ T ν(ũ (t 1), Ỹ (t 1), Z (t 1), W (t 1) )B H νµ +R A H νµ T ν(ũ (t 1), Ỹ (t 1), Z (t 1), W (t 1) )Bνµ H Q } 1 16s 1 { [ Re + Re L (t 1) ( tr ( p [ tr =1 A νµ ω N s1 4Ns 1 ( (t 1) L B νµ ) ( ( p =1 A νµ R (t 1) L Q B νµ ) { p + ω N s1 N s =1 1 T ν (Ũ (t 1), Ỹ (t 1), Z (t 1), W ) )] H (t 1) ) ( T ν (Ũ (t 1), Ỹ (t 1), Z (t 1), W ) )] } H (t 1) ) A H νµ T ν(ũ (t 1), Ỹ (t 1), Z (t 1), W (t 1) )Bνµ H

10 10 F P A Bek, D K Salkuyeh and M M Moghadam +R A H νµ T ν(ũ (t 1), Ỹ (t 1), Z (t 1), W (t 1) )Bνµ H Q } 1 { ( N s1 [tr Re ( p =1 A (t 1) νµ L B νµ ) ω Ns 1 + ω 4N s 1 { p N s1 =1 16s 1 L (t 1) ( T ν (Ũ (t 1), Ỹ (t 1), Z (t 1), W (t 1) ) A νµ Bνµ Tν (Ũ (t 1), Ỹ (t 1), Z (t 1), W (t 1) ) } ) H )] } Smlarly, we can see that Ỹ (t) 1 { 16s ω N s Ns + ω 4N s L (t 1) { p N s =1 ( [tr Re ( ( p =1 C (t 1) νµ L D νµ ) T ν (Ũ (t 1), Ỹ (t 1), Z (t 1), W ) )] } H (t 1) ) C νµ Dνµ Tν (Ũ (t 1), Ỹ (t 1), Z (t 1), W (t 1) ) }, Z (t) 1 { 16s 3 ω N s3 Ns 3 + ω 4N s 3 L (t 1) { p N s3 =1 ( [tr Re ( ( p =1 M (t 1) νµ L N νµ ) T ν (Ũ (t 1), Ỹ (t 1), Z (t 1), W ) )] } H (t 1) ) M νµ Nνµ Tν (Ũ (t 1), Ỹ (t 1), Z (t 1), W (t 1) ) }, and, W (t) 1 16s L (t 1) 4 { ( ω N s4 [tr Re ( ( p =1 H (t 1) νµ L G νµ ) T ν (Ũ (t 1), Ỹ (t 1), Z (t 1), W ) )] } H (t 1) ) Ns 4 + ω 4N s 4 { p N s4 =1 H νµ Gνµ Tν (Ũ (t 1), Ỹ (t 1), Z (t 1), W (t 1) ) } Assume that ( Λ (t) = 4N s 1 Ũ (t) + s Ỹ (t) + s Z(t) 3 + s ) 4 W (t)

11 On the generalzed Sylvester-transpose and conjugate matrx equatons 11 By some straghtforward computatons, we get 0 Λ (t) Λ (t 1) ω { N s1 Re [tr ( ( ( p =1 A (t 1) νµ L B νµ ) T ν (Ũ (t 1), Ỹ (t 1), Z (t 1), W ) )] H (t 1) ) + ( s [tr Re ( ( p =1 C (t 1) νµ L D νµ ) T ν (Ũ (t 1), Ỹ (t 1), Z (t 1), W ) )] H (t 1) ) + ( s 3 [tr Re ( ( p =1 M (t 1) νµ L N νµ ) T ν (Ũ (t 1), Ỹ (t 1), Z (t 1), W ) )] H (t 1) ) + ( s 4 [tr Re ( ( p =1 H (t 1) νµ L G νµ ) T ν (Ũ (t 1), Ỹ (t 1), Z (t 1), W ) )] } H (t 1) ) +ω { p =1 N s1 + p N s =1 + p N s3 =1 A νµ Bνµ Tν (Ũ (t 1), Ỹ (t 1), Z (t 1), W (t 1) ) C νµ Dνµ Tν (Ũ (t 1), Ỹ (t 1), Z (t 1), W (t 1) ) + p N s4 =1 Λ (t 1) ω N +ω { p =1 N s1 + p N s3 =1 N M νµ Nνµ Tν (Ũ (t 1), Ỹ (t 1), Z (t 1), W (t 1) ) H νµ Gνµ Tν (Ũ (t 1), Ỹ (t 1), Z (t 1), W (t 1) ) } T ν (Ũ (t 1), Ỹ (t 1), Z (t 1), W (t 1) ) A νµ Bνµ + q N =1 M νµ Nνµ + p N =1 T ν (Ũ (t 1), Ỹ (t 1), Z (t 1), W (t 1) ) s4 s C νµ Dνµ } H νµ Gνµ As ω satsfes n (34), by some straghtforward computatons, we have: ( { p Λ (t) Λ (t 1) ω ω N s1 =1 A νµ Bνµ + s C νµ Dνµ + s 3 M νµ Nνµ + s4 }) H νµ Gνµ N T ν (Ũ (t 1), Ỹ (t 1), Z (t 1), W (t 1) ) = Λ (t 1) ω ( ω Θ) N T ν (Ũ (t 1), Ỹ (t 1), Z (t 1), W (t 1) ) Λ (0) ω ( ω Θ) t 1 T ν (Ũ (ρ 1), Ỹ (ρ 1), Z (ρ 1), W (ρ 1) ) Therefore, 0 ω ( ω Θ) t 1 ρ=0 N ρ=0 N T ν (Ũ (ρ 1), Ỹ (ρ 1), Z (ρ 1), W (ρ 1) ) Λ (0),

12 1 F P A Bek, D K Salkuyeh and M M Moghadam whch shows that 0 ω ( ω Θ) ρ=0 N N lm t From the convergence theorem of seres, we conclude that T ν (Ũ (t), Ỹ (t), Z (t), W (t) ) T ν (Ũ (ρ 1), Ỹ (ρ 1), Z (ρ 1), W (ρ 1) ) Λ (0) < Hence, lm T ν(ũ (t), Ỹ (t), Z (t), W (t) ) = 0 t Or equvalency, lm t R(t) ν = 0, ν = 1,,, N From Theorem 33, we can conclude the result mmedately = 0 Wth a same strategy employed n the proof of Theorem 35, we may establsh the followng theorem Theorem 36 Suppose that the lnear coupled matrx equatons (14) has unque antreflexve soluton group X = (X 1, X,, X p) If the parameter ω satsfes the nequalty (34) 0 < ω < Θ, such that Θ = N p =1 Θ ν, where, Θ 1ν = s 1 A νµ B νµ, Θ ν = s C νµ D νµ Θ 3ν = s 3 M νµ N νµ, Θ 4ν = s 4 H νµ G νµ Then the teratve soluton groups X (t) = (X (t) 1, X(t),, X(t) p ), t = 1,, 3,, computed by Algorthm 1 (wth κ = 1), converge for any ntal ant-reflexve matrx group X (1) = (X (1) 1, X(1),, X(1) p ) 4 Numercal experments In ths secton, we present two numercal examples to show the effectveness of the proposed algorthm All the numercal experments presented n ths secton were computed n double precson wth some Matlab codes on a Pentum 4 PC, wth a 306 GHz CPU and 100GB of RAM Example 41 We consder the followng matrx equaton (41) AXB + CX T D + M XN + HX H G = F, where A = D = H = B = M = G = C = N =

13 On the generalzed Sylvester-transpose and conjugate matrx equatons ω=00337 ω=0050 ω=0000 log 10 δ k Number of teratons (k) Fgure 1 log 10 δ k versus k for the (R, Q)-reflexve soluton of (41) and F = The exact soluton of (41) s X = whch s (R, Q)-reflexve, where (4) R = Q = Here, we menton that R, Q SOC 3 3 We apply Algorthm 1 wth κ = 0 to solve Problem 1 correspondng to system (41) The ntal guess was taken to be the zero matrx X (0) 1 = 0 and the stoppng crteron δ k = R k F R 0 F < 10 5, was used where R k s the resdual of (41) at kth teraton For ω = 00337, 050, 0000 the method converges n 53, 73 and 9 teratons, respectvely For ω = 00337, the computed soluton s X (53) = whch s n good agreement wth the exact soluton In Fgure 1 the convergence hstory of the method s dsplayed

14 14 F P A Bek, D K Salkuyeh and M M Moghadam 0 1 ω=00 ω=00180 ω=0010 log 10 δ k Number of teratons (k) Fgure log 10 δ k versus k for the (R, Q)-ant-reflexve soluton of (41) We now consder (41) wth the rght-hand sde F = The exact soluton of the system s X = whch s n C 3 3 a (R, Q) where R and Q are defned by (4) All of the other assumptons are as before Performng Algorthm 1 to the matrx equaton wth κ = 1, the method converges n 8, 101 and 154 teratons for ω = 00, 00180, 0010, respectvely For ω = 00 the computed soluton s X (8) = We observe that the method has provded a good approxmate soluton for the system The convergence hstory s dsplayed n Fgure Example 4 In ths example, we consder the system of matrx equatons (43) { X1 + A 1 X 1 B 1 + M 1 X1 N 1 + C X T D + H X H G = F 1, X 1 + Ã1X 1 B1 + M 1 X1 Ñ 1 + C X T D + H X H G = F,

15 On the generalzed Sylvester-transpose and conjugate matrx equatons 15 where and A 1 = N 1 = H = B 1 = C = G = B 1 = C = G = M1 = D = F 1 = F = M 1 = D = Ã 1 = Ñ 1 = H = The exact soluton of Eq (43) s the matrx group (X1, X ) where X1 = 1 1 X = Here, X1 C3 3 r (R 1, Q 1 ) where R 1 = R and Q 1 = Q are as n prevous example and X C3 3 r (R, Q ), where R = Q = We apply Algorthm 1 wth κ = 0 for solvng system (43) The ntal guess was taken to be the zero matrx group (X (0) ) = (0, 0) and the stoppng crteron 1, X(0) δ k = max{ R(k) F R (0) F : = 1, } < 10 5,

16 16 F P A Bek, D K Salkuyeh and M M Moghadam 0 1 ω=0036 ω=006 ω=0016 log 10 δ k Number of teratons (k) Fgure 3 log 10 δ k versus k for the (R, Q )-reflexve of (43) was used, where R () k s the resdual of the th equaton of (43) at teraton k For ω = 0036, w = 006, 0016 the method, respectvely, converges n 79, 110 and 181 teratons For ω = 0036 the computed soluton s (X (79) 1, X (79) ), where X (79) 1 = X (79) = As seen the approxmate soluton group (X (79) 1, X (79) ) s n good agreement wth the exact soluton group The correspondng results are also depcted n Fgure 3 We now change the rght-hand sde of (43) to F 1 = 1 10 F = In ths case, the exact soluton of the system s (X1, X ), where X1 = X = We have X C3 3 a (R, Q ), for = 1, All of the assumptons are as before Algorthm 1 wth k = 1 converges n 180, 3 and 336 teraton, respectvely, for ω = 00, 00180,

17 On the generalzed Sylvester-transpose and conjugate matrx equatons ω=00 ω=00180 ω=0010 log 10 δ k Number of teratons (k) Fgure 4 log 10 δ k versus k for the (R, Q )-ant-reflexve of (43) 0010 For w = 00 the computed soluton by Algorthm 1 s (X (180) 1, X (180) ) where X (180) 1 = X (180) = The convergence hstory of the method s shown n Fgure 4 5 Concluson Ths paper has been devoted for fndng the reflexve (ant-reflexve) soluton group of a general class of complex coupled lnear matrx equatons To ths end, usng herarchcal dentfcaton prncple, an teratve algorthm has been constructed We have establshed that the proposed algorthm converges to the reflexve (ant-reflexve) soluton group of the mentoned coupled lnear matrx equatons for any ntal reflexve (ant-reflexve) soluton group In order to llustrate the feasbly and effectvely of the presented algorthm, some numercal experments have been gven References [1] F P A Bek and D K Salkuyeh, On the global Krylov subspace methods for solvng general coupled matrx equaton, Comput Math Appl, 6 (011) [] D S Bernsten, Matrx Mathematcs: theory, facts, and formulas, Second edton, Prnceton Unversty Press, 009 [3] A Bouhamd and K Jblou, A note on the numercal approxmate solutons for generalzed Sylvester matrx equatons wth applcatons, Appl Math Comput, 06 (008) [4] XW Chang and JSWang, The symmetrc soluton of the matrx equatons AX + Y A = C, AXA T + BY B T = C and (A T XA, B T XB) = (C, D), Lnear Algebra Appl, 179 (1993) [5] HC Chen, Generalzed reflexve matrces: specal propertes and applcatons, SIAM Journal of Matrx Analyss and Applcatons, 19 (1998) [6] J L Chen and X H Chen, Specal matrces, Tsnghua Unversty Press, 00 (In Chnese) [7] M Dehghan and M Hajaran, The general coupled matrx equatons over generalzed bsymmetrc matrces, Lnear Algebra Appl, 43 (010)

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