Applied Mathematical Sciences, Vol. 9, 2015, no. 3, HIKARI Ltd,

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1 Applied Mathematical Sciece Vol 9 5 o HIKARI Ltd wwwm-hiaricom O Poitive Defiite Solutio of the Noliear Matrix Equatio * A A I Saa'a A Zarea* Mathematical Sciece Departmet Faculty of Sciece Price Nourah Bit Abdul Rahma Uiverity Riyadh Saudi Arabia *Correpodig author Salah Mohammed El-Sayed Departmet of Scietific Computig Faculty of Computer ad Iformatic Beha Uiverity Beha Egypt Copyright 4 Saa'a A Zarea ad Salah Mohammed El-Sayed Thi i a ope acce article ditributed uder the Creative Commo Attributio Licee which permit uretricted ue ditributio ad reproductio i ay medium provided the origial wor i properly cited Abtract I thi paper we have dicued ome propertie of a poitive defiite olutio of the o liear matrix equatio A * A I where i a poitive iteger Two effective iterative method for computig a poitive defiite olutio of thi equatio are propoed The eceary ad ufficiet coditio for exitece of a poitive defiite olutio are derived Some umerical example will be preeted for illutrative purpoe Mathematic Subject Claificatio: 5A4; 65F35 Keyword: Noliear matrix equatio Propertie of olutio Iterative algorithm

2 8 Saa'a A Zarea ad Salah Mohammed El-Sayed Itroductio We coider the matrix equatio: A * AI () where i a poitive iteger The oliear matrix equatio () arie i a wide variety of applicatio ad reearch area icludig automatic cotrol ladder etwor dyamic programmig tochatic filterig ad tatitic [4 7 9] If i a egative real umber the poitive defiite olutio of () have bee tudied i ome pecial cae [-6 8-5] I thi paper we'll maily dicu the baic propertie ad ome aalyi of the poitive defiite olutio of () The ret of the paper i orgaized a follow: I Sectio we how that () alway ha a poitive defiite olutio ad we give the eceary ad ufficiet coditio for the equatio to have olutio I Sectio 3 ad 4 we'll dicu two iteratio method for olvig () Fially we'll give three umerical example i Sectio 5 to illutrate our theoretical reult The aim of thi paper i to fid the poitive defiite olutio of () I thi paper we will ue mathematical iductio techique i the mot proof The followig otatio are ued throughout the ret of the paper The otatio A (A ) mea that A i poitive emidefiite (poitive defiite) A * deote the complex cojugate trapoe of A ad I i the idetity matrix Moreover A B(A B ) i ued a a differet otatio for A B ( A B ) We deote by (A) A the pectral radiu of A The orm ued i thi paper i the pectral orm of the matrix A ule otherwie oted Some propertie of the olutio I thi ectio we'll dicu ome propertie of poitive defiite olutio of the matrix () Theorem If m ad M are the mallet ad the larget eigevalue of a olutio of () repectively ad i a eigevalue of A the m M m M Proof Let v be a eigevector correpodig to a eigevalue of the matrix A ad v Sice the olutio of () i a poitive defiite matrix the m v v v v v v

3 O poitive defiite olutio of the oliear matrix equatio 9 v v v v v v v v v v v v v v m m M M Coequetly m M m M Theorem If () ha a poitive defiite olutio the Proof Sice be a poitive defiite olutio of () I A A A A 3 The firt iteratio method (Fixed poit iteratio method): I thi ectio we'll etablih the firt iterative method for obtaiig a poitive defiite olutio of () Algorithm 3 Tae where For compute * () We'll prove that the equece () coverge to a poitive defiite olutio will atifie () Lemma 3 [3] Let ad Z be poitive defiite matrice for which ad hold The Now we'll give ome propertie of the term of the equece: Propoitio 3 If the matrix A i ormal a ( ) the where i the iterate i Algorithm 3 Proof Sice the Sice we have

4 Saa'a A Zarea ad Salah Mohammed El-Sayed Aume that for each i atified We will prove for all From we compute for all Propoitio 3 If the matrix A i ormal the where i the iterate i Algorithm 3 Proof Sice ad the We compute from we obtai Hece Aume that for each (3) i atified We 'll prove for each Sice we compute (4)

5 O poitive defiite olutio of the oliear matrix equatio Accordig to Propoitio 3 ad equality (3) we have From the equality (4) we get Hece The ext theorem give the eceary ad ufficiet coditio for the exitece of a olutio of () Theorem 3 Let the equece real umber ad ad be determied by the Algorithm 3 ad two I A A I for which (5) If () ha a poitive defiite olutio the coverge to which i a poitive defiite olutio of () for all umber ad for which I A A I Moreover if for every the () ha a poitive defiite olutio Proof From algorithm 3 we have I A A I ( ( )I ) I I I I ad ie To prove uppoe that for all Accordig to Lemma 3 we have that A A A A I A A I A A The equece i decreaig To prove it i bouded let I I I A A I A A I ( ( ) I ) I ie I ad I To prove that I for all aume that I we wat to prove

6 Saa'a A Zarea ad Salah Mohammed El-Sayed I I I I A * A I A * IA I ( ( ) I ) I thu I ad I i mootoic decreaig ad bouded from The equece below by I Moreover it limit exit that i lim by taig the limit of 3 we get I A A ad the olutio atifie () Suppoe exit Let for all we proved that the limit of I A A by taig the limit of both ided we have I A A Coequetly () ha poitive defiite olutio Theorem 3 Let be the iterate i Algorithm 3 ad there exit two real umber & for which (i) I A A I (ii) q The q q olutio of () where i a poitive defiite Proof From Theorem 3 it follow that the equece () i coverget to a poitive defiite Solutio of () We compute the pectral orm of the matrix we obtai I A A I A A A A A j j A From Theorem 3 we have & q A Coequetly A q after -tep q q Corollary 3 Aume that () ha a olutio If j coverge to with at leat the liear covergece rate q let the

7 O poitive defiite olutio of the oliear matrix equatio 3 Proof A we have q The chooe a real umber that atifie q Sice there exit a N uch that for ay N q Hece Theorem 33 If () ha a poitive defiite olutio ad after iterative tep of Algorithm 3 ad we have the where i the iterate i Algorithm 3 Proof Sice the tae orm i both ide Sice ad coequetly ad a I the 4 The ecod iteratio method (Two ided iteratio method of the fixed poit iteratio method): I thi ectio we'll etablih the ecod iterative method for obtaiig a poitive defiite olutio of () Algorithm 4 Tae I ad Y I For compute where I A A ad Y I A Y A (6) & are two poitive real umber Our theorem give ufficiet coditio for the exitece of a olutio of () Theorem 4 Let the two equece Υ are determied by the Algorithm 4 ad two real umber ad for which I A A I ad ad q (7) ad Υ coverge to the If () ha a poitive defiite olutio the

8 4 Saa'a A Zarea ad Salah Mohammed El-Sayed ame limit which i a poitive defiite olutio of () for all umber ad for which I A A I Moreover if ad Y for every ad for which I A A I the () ha a poitive defiite olutio Proof From algorithm 4 we have i icreaig ForΥ ad that i Aume that i true To prove that we have I A A I A A thu The equece From equece (6) we have Y I A Y A I A A I Y ad Y I A Y A I A Y A Y that i Y Y Y Suppoe that i true To prove that we have Y I A Y A I A Y A Y the equece i decreaig To prove that Y for From algorithm 4 Y I I I Y I A Y A I A A A Y A A A that i Y for Y Suppoe that Y Y ad Y I A Y A I A A Hece We have ad Y for all To prove the two equece ad Y have the ame limit that i Y a from algorithm 4 we have j j j Y A Y A Y A Y Y j j A Y Y j A Y let q A after -tep q Y q ( )

9 O poitive defiite olutio of the oliear matrix equatio 5 From (7) q we get a We have proved that the two equece ad Y are mootoic bouded ad wet to the ame limit a ay that i lim lim Y By taig the limit of 4 we got I A A ad the olutio atifie () Moreover Suppoe ad Y for all we proved that the limit of ad Y exit Let I A A ad Y I A Y A by taig the limit a for the equece ad Y we have I A A Coequetly () ha poitive defiite matrix Theorem 4 Let the two equece ad Υ are determied by the Algorithm 4 ad two real umber ad for which I A A I ad q the ad Y Y q Y q q q where i a poitive defiite olutio of () ad are defied i (6) Proof From Theorem 4 it follow that the equece (6) i coverget to a poitive defiite olutio of () We'll compute the pectral orm of the matrix the I A A I A A A A j j A A From Theorem 4 we had & q A Coequetly A q after -tep q q Similarly we ca prove that Y q Y q let j

10 6 Saa'a A Zarea ad Salah Mohammed El-Sayed Theorem 43 If the equatio () ha a olutio ad after - iterative tep of the Algorithm 4 I ad Y Y I ad the where ad the iterate i Algorithm 4 Y are Proof Sice Y Y Y A Y Y A Y A Y A Y A = A Y Y A Tae orm i both ide Y Y Y A Y A I A Y Y A Y Y For A Y I Y Y A Y Y Y that i for every thu Y a Coequetly Y Y a Y Y for ad from theorem 4 Y Sice Y A Y A I A From Theorem 4 we had the ad 5 Numerical Experimet I thi ectio the umerical experimet are ued to diplay the flexibility of the algorithm The olutio are computed for ome differet matrice with differet ize For the followig example the practical toppig criterio i 9 ad the olutio i 5 5 Numerical experimet for Algorithm 3 I table we deote q where i the olutio which i obtaied by the iterative method 3

11 O poitive defiite olutio of the oliear matrix equatio 7 Example : Let A a ij : ee Table a ij m i i j i j 5 Numerical experimet for Algorithm 4: I table ad 3 we deote q where i the olutio which i obtaied by the iterative method 4 Example : Let 3 ad A A A i ormal ee Table 5 4 Example 3: Let 5 5 A 7375 ad 8 5 A 8 A i ot ormal ee Table Cocluio I thi paper we have tudied ome propertie of a poitive defiite olutio of the matrix equatio () The two effective iterative method were propoed for computig a poitive defiite olutio of thi equatio Some propertie of the term of the equece () were give Alo the eceary ad ufficiet coditio for the covergece to poitive defiite olutio were deduced Fially we gave three umerical example to illutrate the efficiecy of the propoed algorithm

12 8 Saa'a A Zarea ad Salah Mohammed El-Sayed N Table : Relatiohip Betwee N q K M A ad for Differet Size of A E E-6 88E E E E E E E E E E E E E E E E E- q N K 9 6 Table : Relatiohip Betwee N q E- 383E- 4384E- 9978E E- 949E- 466E E E E- 98E- 87E- 4 ad E- 69E- 3E- 8797E E- 4697E- 387E- 887E- N K Table 3: Relatiohip Betwee N q E E E- 5757E- 5848E- 464E-4 789E E E- 34E E-4 766E E- 8789E E- 4 ad E E E E- 9389E E-4 57E E- 4565E E- Acowledgemet Thi wor wa fiacially upported by Price Nourah Bit Abdulrahma Uiverity Referece [] B Zhou G-B Cai ad J Lam Poitive defiite olutio of the oliear H matrix equatio A A I Applied Mathematic ad Computatio 9 (3) [] C H Guo ad P Lacater Iterative olutio of two matrix equatio Mathematic of Computatio 68 (999) T [3] C-H Guo ad W-W Li The matrix equatio A A Q ad it applicatio i Nao reearch SIAM Joural o Scietific Computig 3(5) ()

13 O poitive defiite olutio of the oliear matrix equatio 9 [4] D Hua ad P Lacater Liear matrix equatio from a ivere problem of vibratio theory Liear Algebra ad it Applicatio 46 (996) [5] J Egwerda O the exitece of a poitive defiite olutio of the matrix T equatio A A I Liear Algebra ad it Applicatio 94 (993) [6] L Hug The explicit olutio ad olvability of the liear matrix equatio Liear Algebra ad it Applicatio 3 () [7] P Lacater ad L Rodma Algebraic Riccati Equatio Oxford Sciece Publiher Oxford 995 [8] Q Li ad P Liu Poitive Defiite Solutio of a Kid of Noliear Matrix Equatio Joural of Iformatio & Computatioal Sciece 7 () [9] R Bhatia Matrix Aalyi Graduate Text i Mathematic 69 Spriger -Verlag New Yor [] S A Zarea S M El-Sayed ad A A Al-Eida Iteratio method to olve the r equatio A A I Aiut Uiverity Joural of Mathematic & Computer Sciece 38 () (9) - 4 [] S A Zarea ad S M El-Sayed O the Matrix Equatio + A* -a A = I Iteratioal Joural of Computatioal Mathematic ad Simulatio () (8) [] S M El-Sayed Two iteratio procee for computig poitive defiite olutio of the matrix equatio A A Q Computer & Mathematic with Applicatio 4 () [3] S M El-Sayed ad A M Al-Dbiba O poitive defiite olutio of oliear matrix equatio +A * - A=I Applied Mathematic ad Computatio 5 (4) [4] Yi S Liu ad T Li O poitive defiite olutio of a matrix equatio +A * -q A=Q ( q ) The Taiwaee Joural of Mathematic 6(4) () [5] Zha Computig the extremal poitive defiite olutio of a matrix

14 Saa'a A Zarea ad Salah Mohammed El-Sayed equatio SIAM Joural o Scietific Computig 7 (996) Received: November 4; Publihed: December 4