On the Positive Definite Solutions of the Matrix Equation X S + A * X S A = Q

Transcription

1 The Ope Applied Mathematic Joural Ope Acce O the Poitive Defiite Solutio of the Matrix Equatio X S + A * X S A = Q Maria Adam * Departmet of Computer Sciece ad Biomedical Iformatic Uiverity of Cetral Greece Lamia Greece Abtract: I thi paper eceary ad ufficiet coditio for the exitece of the poitive defiite olutio of the oliear matrix equatio X + A * X! A = Q are preeted whe A i oigular ad a iteger a well a ome ew propertie ad boud for the eigevalue of matrice related to AQ are dicued A algebraic method for the computatio of the olutio i propoed baed o the algebraic olutio of the correpodig dicrete time Riccati equatio The exact umber of the poitive defiite olutio i alo computed Mathematic Subject Claificatio: 15A60 15A18 15A 93B0 Keyword: Numerical radiu Eigevalue Riccati equatio 1 INTRODUCTION The cetral iue of thi paper i the ivetigatio ad the computatio of the poitive defiite olutio of the matrix equatio X + A * X! A = Q (1) where the oigular matrix A!M M deote the et of all! matrice with complex or real etrie A * tad for the cojugate trapoe of A Q!M i a poitive defiite matrix ad i a iteger The matrix equatio of the form (1) arie i may applicatio i a wide variety of reearch area icludig cotrol theory ladder etwork dyamic programmig tochatic filterig ad tatitic ee [1-3] ad the referece give therei I the cae that A i oigular ad = 1 eceary ad ufficiet coditio for the exitece of a poitive defiite olutio of the matrix equatio (1) have bee ivetigated by may author [1-6] ad formula for the computatio of Hermitia ad o-hermitia olutio ca be foud i [] The more geeral equatio X + A * X!t A = Q with A oigular ha bee coidered i [7-9] (ee alo referece therei) whe t are poitive iteger ad i [10] whe! 1 0 < t 1 ad 0 <! 1t 1 ; there ome exitece coditio ad propertie of it poitive defiite olutio are obtaied Alo i ome paper umerical method for computig the extreme olutio ad everal effective iterative algorithm have bee propoed [9-1] ad a perturbatio aalyi of the poitive defiite olutio ha bee preeted To decribe our reult we itroduce ome otatio ad defiitio For A B!M the otatio A > B(A! B) *Addre correpodece to thi author at the Departmet of Computer Sciece ad Biomedical Iformatic Uiverity of Cetral Greece Lamia Greece; Tel: ; Fax: ; madam@ucggr mea that A ad B are Hermitia matrice ad A! B i a poitive defiite (emidefiite) matrix which i deoted by A! B > 0(A! B 0) For A!M (A) deote a eigevalue of A!(A) the pectrum of A!(A) = { i (A) : i (A) (A)}! A!= { (A * A): (A * A) (A * A)} deote the pectral radiu ad pectral orm of A repectively ad r(a) = { x * Ax : for each vector x!! with x * x = 1} () the umerical radiu of A Let A!M be a Hermitia matrix (A) (A) are idexed i icreaig order! mi (A) =! 1 (A)! (A)!! (A) = (A) It i alo kow [1 3-6] that whe the matrix equatio (1) for = 1 ha a poitive defiite olutio X the there exit miimal ad imal olutio X mi ad X repectively uch that 0 < X mi! X! X for ay poitive defiite olutio X The miimal ad imal olutio are referred a the extreme olutio Moreover the exitece of a poitive defiite olutio deped o the umerical radiu of the matrix Q / AQ / [5 Theorem 5] Note that ettig i (1) Y = X (3) we obtai Y + A * Y A = Q () Furthermore without lo of geerality i (1) may be aumed to be a poitive iteger Ideed i the cae that! 1 ettig Z = X the equatio (1) take the form Z! + A * Z A = Q Z 1 + A * Z! 1 A = Q for / Betham Ope

2 0 The Ope Applied Mathematic Joural 011 Volume 5 Maria Adam I thi paper we focu o the computatio of the poitive defiite olutio of the matrix equatio (1) with A!M a oigular matrix Q > 0 ad a iteger with! 1 I Sectio eceary ad ufficiet coditio for the exitece of poitive defiite olutio of (1) are obtaied a ew iequality for the pectral radiu of AQ i preeted ad i Theorem 7 a formula for computig all the poitive defiite olutio of (1) i give; whe the matrice AQ have pecial propertie more pecific eceary ad ufficiet coditio for the exitece of olutio of (1) are dicued related to ome ew iequalitie for their eigevalue I Sectio 3 boud for the eigevalue of the miimal olutio of (1) are give; a algebraic method for computig the poitive defiite olutio i propoed baed o the algebraic olutio of the correpodig dicrete time Riccati equatio ad the exact umber of poitive defiite olutio of (1) i computed whe thee exit I Sectio a umerical reult i give to illutrate the efficiecy of the propoed method ad fially cocludig remark are give i Sectio 5 EXISTENCE AND FORMULA OF THE POSITIVE DEFINITE SOLUTIONS OF X + A * X! A = Q I the firt part of thi ectio we utilize the cocept of umerical radiu i order to derive eceary ad ufficiet coditio for the exitece of poitive defiite olutio of (1) We eed the followig lemma Lemma 1 [13 Theorem 76] Let A!M be a poitive emidefiite matrix ad let! 1 be a give iteger The there exit a uique poitive emidefiite matrix B uch that B = A Theorem Let A!M be a oigular matrix ad Q!M with Q > 0 The equatio X + A * X! A = Q ha a poitive defiite olutio X!M if ad oly if for the umerical radiu of Q / AQ / hold r(q / AQ / ) 1 (5) Proof Sice the olvig of the matrix equatio (1) i equivalet to () by (3) accordig to Lemma 1 aumig that Y!M i a poitive defiite olutio of () the there exit a uique th root of Y!M which i X > 0 ad olve the equatio (1) Hece it i eeded the proof of the equivalece betwee the umerical radiu i (5) ad the olvig of () which follow eetially the ame lie a the proof of Theorem 5 of Egwerda et al [5] ad i omitted Remark 1 Coider that for the matrice AQ!M of the oliear matrix equatio (1) hold r(q / AQ / ) 1 Accordig to Theorem ad [8 Theorem 33] it i eay to be proved that for the eigevalue of A * AQ!M the followig iequality hold:! mi (A * A) 1 ( ) The followig reult preet the relatio of pectral radii of AQ i (1); the cocluio i the ame a i [7 Theorem 31] without the aumptio that the matrice atify AQ 1/ = Q 1/ A Theorem 3 Let A!M be a oigular matrix Q!M with Q > 0 If the matrix equatio X + A * X! A = Q ha a poitive defiite olutio the!(a) 1! (6) Proof Let!(A) (A) ad x it correpodig eigevector Multiplyig the equatio (1) o the left ad o the right by x * ad x repectively due to Ax =!(A)x we take : x * X x + x * A * X! Ax = x * Qx x * X x+!(a) x * X x = x * Qx (7) Sice X > 0 there exit a uitary matrix V!M ad a poitive real diagoal matrix D = diag{! 1 (X)!! (X)} (X) (X) (1 i ) uch that X = VDV * Uig the above otatio the equatio (7) ca be rewritte a x * VD V * x+!(a) x * VD V * x = x * Qx Settig the ozero vector y = V * x!! y * D y+!(a) y * D y = y * V * QVy which implie!(a) = y* V * QVy y * D y y * D y y i (! i (X)) y i! i (X) = we have y * y y * D y y * D y Coider the fuctio f (t) = t ( t ) with t![0 +) It i mootoically icreaig o (! 0 )* 1/ 1/ (8) + ad mootoically decreaig o - [ +() ad f = f ( ) =! [8 Lemma 31] Thu for every t![0 +) hold f (t)! t! t Applicatio of thi iequality for t = (X) i = 1 ad ubtitutio i (8) yield: 1/

3 O the Poitive Defiite Solutio of the Matrix Equatio X S + A * X S A = Q The Ope Applied Mathematic Joural 011 Volume 5 1 y i (! i (X))!(A) 1 y i! i (X) y i! i (X) y i! i (X) =! Sice Q > 0 clearly deote = thu the above iequality yield (6) I the followig we dicu eceary ad ufficiet coditio for the exitece of olutio of the oliear matrix equatio (1) utilizig the property of umerical radiu whe the matrice AQ have pecial propertie Propoitio Let A!M be a ormal matrix The equatio X + A * X! A = ai (a > 0) ha a poitive defiite olutio X!M if ad oly if A atifie oe from the followig iequalitie: (i)! A!! a (ii)!(a) a Proof It i well kow [1 p 5] that for a ormal matrix A hold: r(a) =!(A) =! A! (9) By Theorem the exitece of a poitive defiite olutio X!M of the matrix equatio X + A * X! A = ai i equivalet to r((ai) / A(aI) / ) = r(a A) = 1 a r(a) = 1 a r(a) 1 r(a) a (10) Hece combiig (10) ad (9) arie either (i) or (ii) iequality Remark The above Propoitio ca be viewed a a geeralizatio of [8 Corollary 1] ad [ Theorem 1113] which are pecial cae of it We eed the followig lemma i order to give a ew etimate for the eigevalue of a ormal matrix A ad Q > 0 i (1) Lemma 5 [8 Lemma ] Let A!M be a ormal ad oigular matrix If Q!M i a Hermitia matrix with AQ = QA the there exit the diagoal matrice D A D Q ad a uitary matrix V uch that V * AV = D A ad V * QV = D Q that i AQ are imultaeouly diagoalizable from a uitary matrix V Propoitio 6 Let A!M be a ormal matrix Q!M with Q > 0 ad AQ = QA If the equatio (1) ha a poitive defiite olutio the { i (A) 1!i! i i (A) (A) i }! 1 (11) Proof Suppoe that there exit a uitary matrix V!M which imultaeouly diagoalize the matrice AQ a i Lemma 5 let A = VD A V * with D A = diag{! 1 (A)! (A)} (A) (A) ad Q = VD Q V * where D Q = diag{! 1! } > 0 due to are poitive real umber The diagoal form ad the uitary imilarity ivariace property of umerical radiu of the matrice AQ [1] that i r(a) = r(vd A V * ) ad r = r(vd Q V * ) allow u to write ( ) r(q / AQ / ) = r VD Q / V * (VD A V * )VD Q / V * = r(vd A D Q V * ) = r(d A D Q ) (1) with D A D Q = diag{ (A) 1 1 (A) } The validity of (11) ow follow from (1) the iequality (5) of Theorem ad the defiitio of umerical radiu of D A D Q by () Remark 3 1 Aume that the uitary matrix V which imultaeouly diagoalize the matrice AQ i proof of Propoitio 6 give the diagoal matrice D A D Q with (A) (A) uch that the correpodig eigevalue are idexed i icreaig order 0 <! mi (A) =! 1 (A)! (A)!! (A) = (A) = (A) ad 0 <! mi =! 1!!! = = repectively Sice A ormal hold (9) thu (11) lead to iequality!(a)! 1 ie (6) ca be writte equivaletly! A!! 1 (13) Hece if the matrix equatio (1) ha a poitive defiite olutio with the matrice AQ!M atifyig the aumptio of Propoitio 6 ad the uitary matrix V i a above the for the larget eigevalue of A * AQ the iequality (13) allow u to write

4 The Ope Applied Mathematic Joural 011 Volume 5 Maria Adam (A * A) 1 ( ) (1) Sice A * A > 0 it follow 0! mi (A * A)!! (A * A) (A * A) coequetly (1) give a upper boud for!(a * A) The uitary matrix V which imultaeouly diagoalize AQ doe ot i geeral fulfil the aumptio i the precedig remark a preeted i [7 Corollary 3] ad [8 Theorem 1] 3 Note that i the pecial cae that Q = ai a > 0 the pectral radiu i! = a therefore (13) i exactly (i) or (ii) iequality of Propoitio The (1) i rewritte (A * A) a! that the eigevalue of A * A lie i the iterval 0 a I the followig theorem a formula for the poitive defiite olutio of (1) i give whe the above exit Theorem 7 Let A!M be a oigular matrix Q > 0 with r(q / AQ / ) 1 ad Y!M be a poitive defiite olutio of the equatio () The for each Y there exit a uitary matrix U!M which diagoalize Y ad a uique poitive defiite olutio X!M of (1) uch that X = U diag{! 1 1/ (Y )! 1/ (Y )! 1/ (Y )}U * (15) where (Y ) (Y ) i = 1 Proof Sice r(q / AQ / ) 1 accordig to Theorem the equatio (1) ha at leat oe poitive defiite olutio; alo the ame coditio guaratee the exitece at leat oe poitive defiite olutio of () from [5 Theorem 5] which i deoted by Y!M Accordig to the pectral theorem for Y there exit a uitary matrix U!M uch that Y = UDU * (16) where D = diag{(! 1 (Y )! (Y )} ad (Y ) (Y ) are poitive real umber Combiig the equatio i (3) with the above diagoal form of Y i (16) we defie the matrix X!M a X = UD 1/ U * = Udiag{! 1 1/ (Y )! 1/ (Y )}U * where the uique poitive th root i well defied i each cae due to (Y ) > 0 Clearly uig the diagoal form of XY from (15) (16) we derive : X + A * X! A = Udiag{ 1 1/ (Y ) 1/ (Y )} U * + A * U * ( ) diag{ 1 1/ (Y ) 1/ (Y )}! U A = UDU * + A * ( U * ) D U A = Y + A * Y A = Q Hece X coit a olutio of (1) which i a poitive defiite due to (Y ) > 0 ad uique by Lemma 1 3 RICCATI EQUATION SOLUTION METHOD FOR THE COMPUTATION OF POSITIVE DEFINITE SOLUTIONS OF X S + A * X S A = Q Let r(q / AQ / ) 1 ad Y be a poitive defiite olutio of () Workig a i [1] ad [] we are able to derive a Riccati equatio which i equivalet to the matrix equatio () I particular the matrix equatio () ca be writte a Y = Q! A * Y A whereby the followig equivalet equatio arie : Y = Q! A * Q! A * Y A A = Q! A * A A * ( ) QA! Y = Q + A * A Y + (! A * ( ) QA ) Subtitutig i the above equatio ( A * ) A ( A * A ) * F = A * A ad G =! ( A * ) QA (17) the related dicrete time Riccati equatio i derived : Y = Q + F ( Y + G) F * (18) Therefore () i equivalet to the related dicrete time Riccati equatio (18) for the olutio of which the algebraic Riccati Equatio Solutio Method ca be ued [15] More pecifically from the Riccati equatio parameter i (17) the followig matrix i formed A 1 A * A 1 QA 1! = QA 1 A * A * A 1 QA 1 QA 1 ( ( (19) O!I which atifie! * J! = J where J = ie I O a ymplectic matrix All the eigevalue of! are o-zero ( 0!() ) ad it may be diagoalized i the form! = WLW 1 where L i a! diagoal matrix with diagoal etrie the eigevalue of!! L 1 O L = O L (0)

5 O the Poitive Defiite Solutio of the Matrix Equatio X S + A * X S A = Q The Ope Applied Mathematic Joural 011 Volume 5 3 ad W cotai the correpodig eigevector! W 11 W 1 W = W 1 W (1) I the cae that () i olvable the all it olutio Y j j = 1 are give by the formula: Y j = W 1 W 11 ad Y j = W W 1 () where the block matrice W 11 W 1 W 1 W are defied by the partitio of W i (1) followig every differet permutatio of it colum [ Propoitio 1] Amog Y j there exit poitive defiite olutio for ome j = 1 therefore there exit uitary matrice U j ad D = diag{! 1 )! )} M with ) ) uch that Y j = U j diag{! 1 )! )}U j * Hece due to the fact that the aumptio of Theorem 7 hold combiig the lat relatio with (15) each poitive defiite olutio of (1) i give by X = U j diag{! 1 1/ )! 1/ )}U j * We remid that a ymplectic matrix ad it eigevalue occur i reciprocal pair Therefore we may arrage them i the diagoal matrix L i (0) o that L 1 to cotai all the eigevalue of! which lie outide the uit circle ad L = L 1 The above proce defie a correpodig (pecial) arragemet of eigevector of! i W which we deote by ˆ W =! Uig the matrix W ˆ the uique poitive defiite olutio of the dicrete time Riccati equatio i (18) coicide with the extreme olutio of () ad thee are formed : Y = W ˆ ˆ 1 W 11 ad Y mi = ˆ ˆ W W 1 Furthermore it i well kow [1 6] that the miimal olutio Y mi of () ad the imal olutio Z of the followig equatio Z + AZ A * = Q (3) atify the relatio: Y mi = Q! Z () Thu from () it become obviou that the miimal olutio of () ca be derived via the imal olutio of the equatio (3) which i of type () Note that if A i Hermitia the the matrix equatio () ad (3) coicide coequetly from () the miimal olutio i give Y mi = Q! Y (5) I the followig theorem boud for the eigevalue of the miimal olutio of (1) are derived which are related to the eigevalue of the imal olutio Theorem 8 Let A!M be a Hermitia ad oigular matrix ad Q!M with Q > 0 If the equatio (1) ha a poitive defiite olutio ad )! mi the ) (X mi )! mi ) (6)! mi )! mi (X mi )! mi! mi ) (7) where X X mi are the extreme olutio of (1) Proof Sice A i Hermitia ad (1) ha a poitive defiite olutio it i clear that there exit at leat oe poitive defiite olutio of () a well a the extreme olutio Y ad Y mi [5 6] Accordig to Weyl Theorem [13 Theorem 31] for the Hermitia matrice QY ad the formula (5) ad (3) we write :! k (Y )! k (Q Y )! k! mi (Y )! k! k )! k (X mi )! k! mi ) )! k (X mi )! k! mi ) (8) Settig i (8) k = due to X! X! Q 1/ [1 Theorem 1] it i obviou that! k ) 0 thu (6) arie immediately The validity of (7) i baed o hypothei ad (8) for k = mi Moreover it i well kow [1 ] that whe A!M i oigular the exitece of a fiite umber of Hermitia olutio of the matrix equatio () deped o the eigevalue of the matrix! I the followig V( ()) deote the eigepace correpodig to the eigevalue () ad dim(v( ())) deote it dimeio Theorem 9 Let A!M be a oigular matrix Q!M with Q > 0 ad! be the matrix i (19) ad aume that the matrix equatio X + A * X! A = Q ha at leat oe poitive defiite olutio If dim(v( ())) = 1 for every eigevalue () i = 1 with () 1 the there exit a fiite umber of poitive defiite olutio (hpd) of (1) Their umber i equal to m hpd =!( j + 1); (9) j=1 if AQ are real matrice the amog the hpd there exit real ymmetric olutio (rpd) of (1) with

6 The Ope Applied Mathematic Joural 011 Volume 5 Maria Adam p+q rpd =!( k + 1) (30) k=1 where m i the umber of the ditict eigevalue of! that lie outide the uit circle with algebraic multiplicity j j = 1 m p i the umber of real ditict eigevalue of! lyig outide the uit circle q i the umber of complex cojugate pair of eigevalue lyig outide the uit circle with algebraic multiplicity k k = 1 p + q Proof Aume that X!M i a poitive defiite olutio of (1); accordig to Theorem 7 every poitive defiite olutio X of (1) i related to a uique poitive defiite olutio Y!M of () by (15) Hece the uique ymplectic matrix (19) correpod to the two matrix equatio () ad (1) ad coequetly the total umber of Hermitia (real ymmetric) poitive olutio of (1) i the ame a i [ Theorem 9] ad give by (9)-(30) NUMERICAL RESULT I thi ectio we implemet the propoed method to olve the matrix equatio (1) ad illutrate the mai cocluio of the preet paper The propoed method compute the poitive defiite olutio uig Matlab 65 A = Example 1 Let the matrice 05 0i!03 0 Q = 1!03!03 1 i the equatio X + A * X! A = Q! 1 Obviouly A i a oigular matrix with!(a) = { 1 (A) = i (A) = i} ad! 1 (A) = 0315! (A) = 069 while Q i a poitive defiite matrix with! = { } Sice r(q / AQ / ) = 0371 Theorem guaratee the exitece of poitive defiite olutio of (1) hece the iequality (6) of Theorem 3 i verified The pectrum of matrix (19) i! () = { 1 () = i with () = i 3 () = i () = i}! 1 () = 131! () = 3095! 3 () = 0331! () = 0069 thu! ha m = eigevalue outide the uit circle the algebraic multiplicity of which i 1 = = 1 Accordig to Theorem 9 the Riccati Equatio Solutio Method ca be applied becaue it aumptio are verified; the umber of olutio i computed by (9) m hpd =!( j + 1) = j=1 The correpodig eigevector of! are the colum of the matrix W =!00030! 098i ! 01393i 0397! 00398i!011! 0110i 009! 0869i 07793!01105! 03081i i i 0067! 00108i 08051!0118! 0138i!013! 00851i 0093! 00037i The poitive defiite olutio of () are computed by (); accordig to Theorem 7 ad (15) the correpodig poitive defiite olutio of (1) are derived : 03108! i Y 1 = Y mi =!0155! 0017i ! 0108i U 1 =!053! 0108i D 1 = diag{ }! 018 X 1 = X mi = U 1/ / (miimal olutio) Y = U * ! i!0556! 0131i ! 0105i U =!05836! 0105i D = diag{ }! 1335 X = U 1/ / Y 3 = U * 0715!033! 05i! i !015! 099i U 3 = 015! 099i 0916 D 3 = diag{ }! 0905 X 3 = U 1/ / Y = Y = U 3 * 09099!063! 010i! i !066! 01038i U = 066! 01038i 0968 D = diag{ }! X = X = U 1/ / (imal olutio) U *

7 O the Poitive Defiite Solutio of the Matrix Equatio X S + A * X S A = Q The Ope Applied Mathematic Joural 011 Volume 5 5 It i alo verified that: 0 < X mi < X 5 CONCLUSIONS I thi paper we ivetigate the poitive defiite olutio of the oliear matrix equatio X + A * X! A = Q We derive eceary ad ufficiet coditio for the exitece of the poitive defiite olutio which are related to the theory ad propertie of the umerical radiu of Q / AQ / Some propertie of the olutio are dicued a well a eceary ad ufficiet coditio for the exitece of the poitive defiite olutio for pecial matrice AQ are derived A algebraic method for computig all the poitive defiite olutio i propoed which i baed o the algebraic olutio of the correpodig dicrete time Riccati equatio ad the exact umber of the poitive defiite olutio i foud The method provide imple formula for computig the olutio a verified through a umerical reult REFERENCES [1] Adam M Aimaki N Saida F Algebraic olutio of the matrix equatio X + A T X A = Q ad X! A T X A = Q It J Algebra Comput 008; (11): [] Egwerda JC O the exitece of a poitive defiite olutio of the matrix equatio X + A T X A = I Liear Algebra Appl 1993; 19: [3] Zha X Computig the extremal poitive defiite olutio of a matrix equatio SIAM J Sci Comput 1996; 17(5): [] Adam M Aimaki N Tzialla G Saida F Riccati equatio olutio method for the computatio of the olutio of X + A T X A = Q ad X! A T X A = Q Ope Appl Iform J 009; 3: -33 [5] Egwerda JC Ra ACM Rijkeboer AL Neceary ad ufficiet coditio for the exitece of a poitive defiite olutio of the matrix equatio X + A * X A = Q Liear Algebra Appl 1993; 186: [6] Meii B Efficiet computatio of the extreme olutio of X + A * X A = Q ad X! A * X A = Q Math Comput 001; 71(39): [7] Cai J Che G Some ivetigatio o Hermitia poitive defiite olutio of the matrix equatio X + A * X!t A = Q Liear Algebra Appl 009; 30: 8-56 [8] Dua X Liao A O the exitece of Hermitia poitive defiite olutio of the matrix equatio X + A * X!t A = Q Liear Algebra Appl 008; 9: [9] Liu XG Gao H O the poitive defiite olutio of the matrix equatio X ± A T X!t A = I Liear Algebra Appl 003; 368: [10] Cai J Che G O the Hermitia poitive defiite olutio of oliear matrix equatio X + A * X!t A = Q Appl Math Comput 010; 17: [11] Yi X Liu S Faga L Solutio ad perturbatio etimate for the matrix equatio X + A * X!t A = Q Liear Algebra Appl 009; 31(9): [1] Yuetig Y The iterative method for olvig oliear matrix equatio X + A * X!t A = Q Appl Math Comput 007; 188: 6-53 [13] Hor RA Joho CR Matrix aalyi 1 t ed Cambridge: Cambridge Uiverity Pre 1985 [1] Hor RA Joho CR Topic i matrix aalyi 1 t ed Cambridge: Cambridge Uiverity Pre 1991; p 607 [15] Vaugha DR A orecurive algebraic olutio for the dicrete riccati equatio IEEE Tra Automat Cotr 1970; Received: July Revied: Augut Accepted: Augut Maria Adam; Liceee Betham Ope Thi i a ope acce article liceed uder the term of the Creative Commo Attributio No-Commercial Licee (http: //creativecommoorg/licee/byc/30/) which permit uretricted o-commercial ue ditributio ad reproductio i ay medium provided the work i properly cited