Advances in Mathematics and Statistical Sciences. On Positive Definite Solution of the Nonlinear Matrix Equation

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1 Advace i Mathematic ad Statitical Sciece O Poitive Defiite Solutio of the Noliea * Matix Equatio A A I SANA'A A. ZAREA Mathematical Sciece Depatmet Pice Nouah Bit Abdul Rahma Uiveity B.O.Box 9Riyad 6 SAUDI ARABIA azaea@pu.edu.a _zaea@yahoo.com Abtact: - I thi pape a efficiet iteative method i peeted to olve a ew oliea matix equatio * A A I with eal matice ad. Some popetie of the poitive defiite olutio fo the oliea matix equatio ae deived. Moeove eceay ad ufficiet coditio fo the exitece of the poitive defiite olutio ae deived. The eo etimatio of the iteative method i alo give. Fially two umeical example ae give to demotate the efficiecy of the algoithm. Key-Wod: - : Noliea matix equatio poitive defiite olutio exitece iteative method fixed-poit theoem. Itoductio The oliea matix equatio aie i a wide vaiety of applicatio ad eeach aea icludig automatic cotol ladde etwo dyamic pogammig tochatic filteig tatitic ad othe field of pue ad applied mathematic [6]. Fo example coide the oliea matix equatio of the fom * A A Q () whee A i a quae eal matix ad Q i a poitive defiite matix. whe i a pecial cae of algebaic dicete type Riccati equatio []. Equatio () ha bee ivetigated by may autho [-]. Some umeical method uch a fixed poit iteatio [67]. Newto method [7] SDA algoithm [] LR algoithm [] ad Buttefly SZ algoithm [] have bee popoed to compute the poitive defiite olutio. Baed o the fixed-poit theoem the ufficiet ad eceay coditio fo the exitece of a poitive defiite olutio of the matix () have bee give whe i egative itege []. Recetly fo i poitive itege equatio ( ) ha bee tudied i [7]. I thi pape we coide ew oliea matix equatio of the fom * A A I () R whee i eal umbe i uow matix I i the idetity matix ad A R i oigula matix. The matix equatio () could be viewed a a pecial cae of the ymmetic liea matix equatio [6]. Thi pape i ogaized a follow. I Sectio ome popetie of poitive defiite olutio of the matix equatio () ae give. I Sectio a iteative method i cotucted to compute the uique poitive defiite olutio. The eo etimatio of thi iteative method i alo give. Fially Sectio iclude two umeical example to illutate the theoetical eult. I thi pape we will ue mathematical iductio techique i the mot poof. The aim of thi pape i to fid the poitive defiite olutio of equatio (). Thoughout thi pape we wite A ( A ) mea that A i poitive emidefiite (poitive defiite) A * deote the tapoe of A ad I i the idetity matix. Moeove A B( A B ) i ued a a diffeet otatio fo A B A B ( ) deote the equece.... We deote by (A) A the pectal adiu of A. The om ued i thi pape i the pectal om of the matix A ule othewie oted. ISBN:

2 Advace i Mathematic ad Statitical Sciece Some Popetie of the Solutio I thi ectio we'll dicu ome popetie of poitive defiite olutio of the matix equatio (). Theoem.. If m ad M ae the mallet ad the laget eigevalue of a olutio of equatio () epectively ad i a eigevalue of A the m M. m M Poof. Let v be a eigevecto coepodig to a eigevalue of the matix A ad v. Sice the olutio of equatio () i a poitive defiite matix the m v v v v v v m m v v v v v v v v v v v v v v m m m m Theoem. If equatio () ha a poitive defiite olutio the ad.. Poof. Sice be a poitive defiite olutio of equatio () ad ad. The Iteative Method I thi ectio the iteative method fo obtaiig a poitive defiite olutio of equatio () i etablihed. A iteative method to compute the uique poitive defiite olutio i popoed the two lemma ae biefly eviewed ad the ued to etablih eceay ad ufficiet coditio fo the exitece of a poitive defiite olutio of Equatio ().. Algoithm. Tae. Fo... compute * B ( )B () whee B B * *. Lemma. [] Let f be a opeato mootoe fuctio o ( ) ad let AB be two poitive opeato that ae bouded below by a;i.e.a ai ad B ai fo the poitive umbe a. The fo evey pectal om f ( A ) f ( B ) f ( a) A B. Lemma. [](Lowe-Heiz iequality) p The fuctio t i ode-peevig fo p p p i.e. A B A B. The ext theoem give the eceay ad ufficiet coditio fo the exitece of a olutio of equatio (). Theoem. Let the equece be detemied by the Algoithm. I B B I ad if equatio () ha a poitive defiite olutio the covege to poitive defiite olutio Moeove if fo evey ad I B B I the equatio () ha a poitive defiite olutio. Poof. Let equatio () ha a poitive defiite olutio. We pove the cocluio by iductio. Fom algoithm. we have * B B by uig Lemma. (Lowe-Heiz iequality) we get B ( I )B B ( )B * *. Aume that the cocluio hold fo... that i the fo = + by uig Lowe-Heiz iequality we have B ( I )B B ( )B * *. The equece i iceaig. To pove it i bouded fom above let I i.e. * B B I I ad I. ISBN:

3 Advace i Mathematic ad Statitical Sciece To pove that I.... Let the cocluio I hold fo the fo * * I B ( I ) B B B I I thu I... i mootoic iceaig ad bouded fom above by I. Moeove it limit exit that i lim. The equece by taig the limit of () we get A A I ad the olutio atifie Equatio (). Suppoe fo evey we poved that the limit of exit. Let A A I by taig the limit of both ided a we have A A I Coequetly equatio () ha poitive defiite olutio. Theoem. Let be the iteate i Algoithm. ad I B B I. If q. The q whee i a poitive defiite olutio of equatio (). Poof. Fom Theoem. it follow that the equece () i coveget to a poitive defiite olutio of equatio (). We compute the pectal om of the matix we obtai * * B ( I ) B B ( I ) B R P * whee R B ( I ) B P B * ( I ) B. By uig lemma. with a mootoe opeato f ( x) x ; ad fo * *... the R B ( I ) B B B I povided a R ai ad by the ame mae we get P ai. Sice f ( a) a f ( ) B let q we have q afte -tep q q. Coollay.. Aume that Equatio () ha a olutio. If q the covege to with at leat the liea covegece ate. Poof. A we have q. The chooe a eal umbe that atifie q. Sice thee exit a N uch that fo ay N q. Hece. Theoem. If equatio () ha a poitive defiite olutio ad afte iteative tep of Algoithm. ad we have the i-. ad ii- whee i the iteate i Algoithm.. Poof. i- Fom algoithm. ad lemma. the tae the om of both ide * * B ( I )B B ( I )B B. I Sice ad a. Coequetly ad ISBN:

4 Advace i Mathematic ad Statitical Sciece I a the. ii- the tae the om of both ide <. Numeical Expeimet I thi ectio we give two umeical example to illutate that the matix equece geeated by iteative method (.) covege to the uique poitive defiite olutio of equatio (). The uique olutio i computed fo diffeet matice A. All pogam ae witte i MATHAMATICA. Fo the followig example the pactical toppig citeio ae ad the olutio i.. Example Example Coide the oliea matix equatio * A A I whee... A A Fo..8 Afte 6 iteatio we get epectively. See Table. Example Coide the oliea matix equatio * A A I whee A Fo Afte 999 iteatio we get epectively. See Table. ISBN:

5 Advace i Mathematic ad Statitical Sciece. Table I the followig table we deote * A A I. Table : Eo aalyi fo Example fo diffeet value of.87e-6.987e E-.68E-6.7E-6.6E-8.89E-.9E E-6.8E-8. 6E-9.9E E-.699E E E-9. 79E-.789E E-6.E E E-.8E E-8.E-8.9E-..9.7E-.96E-6.676E-8.88E E-.87E-.8E-8.87E-8.86E- Table : Eo aalyi fo Example fo diffeet value of K E-.7E-7.8E-9.696E E-.E-7.89E E E-.9888E-7.987E-9 9.E E E-9.96E E-.7E-6.6E E E-.9 E-6.8 E-6.769E-8.9 E E-.E E-.76E-.96E-6.898E-6.779E-8.777E E-.8E- Cocluio I thi pape a ew fom of oliea matix equatio () wa coideed. Some popetie of a poitive defiite olutio equatio () have bee tudied. Baed o the fixed poit theoem a iteative method i popoed to compute the uique poitive defiite olutio. Alo the eceay ad ufficiet coditio fo the covegece to poitive defiite olutio wee deduced. Fially umeical example how that the iteative method i feaible to compute the uique poitive defiite olutio. Refeece: [] W.N. Adeo J. T.D. Moley ad G.E. * Tapp Poitive Solutio to A B B Liea Algeba ad It Applicatio Vol. 99 pp. 6. [] P. Bee ad H. Fabede O the Solutio of the Ratioal Matix Equatio Q L* L EURASIP Joual o Advace i Sigal Poceig Vol.7 7 pp.. [] R. Bhatia Matix Aalyi Gaduate Text i Mathematic 69 Spige Velag 997. [] S. M. El-Sayed ad A. M. Al-Dbiba O Poitive Defiite Solutio of Noliea Matix Equatio A A I Applied Mathematic ad Computatio Vol. pp. -. [] J. Egweda O the Exitece of A Poitive Defiite Solutio of the Matix Equatio T A A I Liea Algeba ad it Applicatio Vol.999 pp [6] S. Fital ad C.H. Guo A Note o the Fixed Poit Iteatio fo the Matix Eqatio A* A I Liea Algeba ad it Applicatio. Vol.9 8 pp. 98. [7] C. H. Guo ad W. W. LiThe Matix Equatio A* A Q ad it Applicatio i Nao Reeach SIAM Joual o Scietific Computig Vol. No. pp. -8. [8] C.H. Guo ad P. Lacate Iteative Solutio of Two Matix Equatio Mathematic of Computatio Vol pp [9] V.I. Haaov ad I.G. Ivaov O Two Petubatio Etimate of the Exteme Solutio to the Equatio A* A Q Liea Algeba ad it Applicatio Vol. 6 pp [] I.G. Ivaov V.I. Haaov ad F. Ulig Impoved Method ad Statig Value to Solve the Matix Equatio A* A I Iteatively Mathematic of Computatio Vol. 7 pp ISBN:

6 Advace i Mathematic ad Statitical Sciece [] P. Lacate ad L. Rodma Algebaic Riccati Equatio Oxfod Sciece Oxfod 99. [] W.W. Li ad S.F. u Covegece Aalyi of Stuctue-Peevig Doublig Algoithm fo Riccati-Type Matix Equatio SIAM Joual o Matix Aalyi ad Applicatio Vol. 8 6 pp [] B. Meii Efficiet Computatio of the Exteme Solutio of A* A Q ad Mathematic of Computa- A* A Q tio Vol. 7 pp. 89. [] B. Meii Matix Equatio ad Stuctue: Efficiet Solutio of Special Dicete Algebaic Riccati Equatio Poceedig of the WLSS- COO Bulgaia. [] G.K. Pedee Some Opeato Mootoe Fuctio Poceedig of the Ameica Mathematical Society Vol pp.9. [6] A. RaM. C. Reuig The Symmetic Liea Matix Equatio The Electoic Joual of Liea Algeba Vol.9 pp.9-7. [7] S. A. Zaea S. M. El-Sayed O Poitive Defiite Solutio of the Noliea Matix Equatio A A I Applied Mathematical Sciece Vol. 9 No. pp. 7-. ISBN:

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