A two-sided Iterative Method for Solving

Transcription

1 NTERNATONAL JOURNAL OF MATHEMATCS AND COMPUTERS N SMULATON Volume 9 0 A two-sided teative Method fo Solvig * A Noliea Matix Equatio X= AX A Saa'a A Zaea Abstact A efficiet ad umeical algoithm is suggested fo fidig the positive defiite solutios of the matix equatio * X = A X A whee A is a osigula eal matix if such solutios exist The suggested techique is called the "Two-sided teative Pocess" Popety of solutios is discussed theeof ecessay ad sufficiet coditios fo existece of a positive defiite solutio ae deived ad also the eo aalysis ad the covegece ate ae aalyzed Fially two umeical examples ae give to illustate the effectiveess of the algoithm Keywods covegece ate o liea matix equatio positive defiite solutio two-sided method 0BNTRODUCTON N ecet yeas studies of vaious physical stuctues of oliea equatios have attacted much attetio i coectio with the impotat poblems that aise i scietific applicatios These physical stuctues of oliea equatios have may foms such as odiay ad patial diffeetial equatios [][] matix equatios []-[9] ad oliea pogammig poblems [0][] this pape we oly focused o matix equatio The impotace of matix equatios ad its applicatios wee give i [8] ad the efeeces theei Also diffeet iteative methods fo solvig some ids of matix equatio wee give such as fixed poit iteatio [6][8][9][] [] Newto method [4] SDA algoithm[7] LR algoithm [] Buttefly SZ algoithm [] ad Two-sided teative Pocess[] [8] we studied: oliea matix equatio of the fom * X = A X A () whee A is a squae eal matix ad X is uow squae matix Two popeties of a positive defiite solutio of () wee discussed; fist oe is elated with the smallest the lagest eigevalues of a solutio X of () ad a eigevalues of A The secod oe gives the elatio betwee the tems of () A iteative method was poposed to compute the uique * positive defiite solutio whe A > fo X = A X A with eal squae matices ad The poposed method was based o the fixed poit theoem Moeove ecessay ad Saa'a A Zaea Autho is with Picess Nouah Bit Abdulahma Uivesity Riyad64 KSA (coespodig autho to povide phoe: ; ( sazaea@puedusa s_zaea@yahooocom) sufficiet coditios fo the existece of the positive defiite solutios wee deived The eo estimatio ad geeal covegece esults of the iteative method wee also povided Fo moe coveiece we will metio all esults i [8] Theoem f m ad M ae the smallest ad the lagest eigevalues of a solutio X of equatio () espectively ad λ is a eigevalues of A the m M λ m M Theoem f equatio () has a positive defiite solutio X the Α Α> ad > Α Α Algoithm Tae 0 = 0 Fo = compute * X = B ( + X )B B Α whee = B = Α Theoem 4 Let the sequece { } be detemied by the Algoithm defiite solutio the { } 0 < < B B < ad if () has a positive coveges to positive defiite solutio Moeove if > 0 fo evey ad 0 < < B B < the () has a positive defiite solutio Theoem Let be the iteates i algoithm ad 0 < < B B < f q = < the < q whee X is a positive defiite solutio of () Coollay 6 Assume that () has a solutio f q = < the { X } = 0 coveges to X with at least the liea covegece ate Theoem 7 f () has a positive defiite solutio ad afte iteative steps of Algoithm ad we have Ι < ε ε > 0 the SSN:

2 NTERNATONAL JOURNAL OF MATHEMATCS AND COMPUTERS N SMULATON Volume 9 0 i- + < ε ad ii- whee is the iteates i Algoithm * * BXB X + BB < ε 4 This pape aims to fid the positive defiite solutio of the matix equatio () this pape mathematical iductio techique will be used i the most poofs The followig otatios ae used thoughout [8] ad the est of the pape The otatio A 0 ( A > 0) meas that A is positive semidefiite (positive defiite) Fo matices A ad B we wite A B( A> B) if A B 0( A B > 0 ) ad { } ( A) deotes the sequece 0 We deote by ρ = A the spectal adius of A The om used i this pape is the spectal om of the matix A uless othewise oted PROPERTY OF THE SOLUTONS AND THE TERATVE METHOD this sectio we shall discuss the popety of positive defiite solutios of the matix equatio () A two-sided algoithm fo solvig equatio () is poposed the Lemma is biefly eviewed Aftewads it will be used to establish the coditios fo the existece of a positive defiite solutio of () whe the matix A is osigula matix ad B = Α B = Α A 4BTheoem f () has a positive defiite solutio X the ΑΑ Α Α > Poof Let X be a positive defiite solutio of equatio () fom Theoem ad Lowe-Heiz iequality [7] we get > + > + Α Α Α Α X Α ( ) Α > = + > Α Α Α + Α Hece ΑΑ Α Α > B BAlgoithm Tae X0 = Y0 = Fo = 0 compute * X B ( X )B + = + () * + = + () Y B ( Y )B whee B = Α B = Α C 6BLemma [6] Let f be a opeato mootoe fuctio o ( 0 ) ad let AB be two positive opeatos that ae bouded below by a;i e A a ad B a fo the positive umbe a The fo evey spectal om f(a) f(b) f (a) A B D 7BTheoem f thee exist umbes ad so that 0 < < ad the followig coditios ae satisfied * () i < BB< ( + ) ( + ) ( ii) q = < ( + ) The () has a positive defiite solutio X whee X is the same limit of the two sequeces { X } ad { Y } ae defied i the Algoithm Poof To pove the theoem we will show that X0 = < X < X + Y+ < Y < Y0 = = ad Y X 0 as Fom algoithm * * 0 X = B ( + X ) B = B ( + ) B * = ( + ) BB> ( + ) ( + ) = = X 0 X > X 0 Let Xi Xi > is tue whe = i thus by usig Lowe- Heiz iequality we get X ( ) i+ B Xi B B ( Xi ) B Xi = + > + = That is < = ad { X } is iceasig X X + 0 Next we will pove X+ < Y+ fo = 0 Fom algoithm Y0 = > = X0 Sice > the B ( + ) B > B ( + ) B the by usig Lowe-Heiz iequality we get Y = B ( + ) B > B ( + ) B = X whe Let Yi > Xi is tue = i thus by usig Lowe-Heiz iequality we get Y ( ) i+ B Yi B B ( Xi) B Xi+ = + > + = That is Y > X = 0 Fially we will pove that Y + < Y = 0 Fom algoithm we get * = ( + 0) = ( + ) = ( + ) Y B Y B B B B B ( + ) < = = Y0 + Let Yi Yi < is tue whe = i thus by usig Lowe- Heiz iequality we get ( ) i+ i ( i ) i Y = B + Y B < B + Y B = Y That is + < = ad { Y } is deceasig Hece we poved Y Y 0 X0 = < X < X + Y+ < Y < Y0 = = Now we shall pove that Y X 0 as fo that we have SSN:

3 NTERNATONAL JOURNAL OF MATHEMATCS AND COMPUTERS N SMULATON Volume 9 0 * * Y X = B ( + Y )B B ( + X )B = P Q * * whee P = B ( + Y )B ad Q = B ( + X )B By usig lemma with a mootoe opeato f( x) = x ad Y > X X = fo all = the 0 * ( + ) P = B ( + Y )B > B ( + )B > = = a ( + ) povided a = P > a by the same mae we pove Q > a The Y X B ( + Y )B B ( + X )B * BBY X is sice * BB< the we have + Y X < Y X + < afte -steps q Y X ( ) Y X < q Y0 X0 = q whee q = < ( + ) * lim Y = lim X = X = A X A > 0 E 8BTheoem the Y X 0 as that f > 0 ad Y > 0 fo evey 0 < < ad * < BB< ( + ) ( + ) solutio Poof Suppose 0 poved that the limits of { } 0 the () has a positive defiite X > ad Y > 0 fo = we Y ae exist Let X ad { } X + = AX A > ad Y+ = AY A > 0 by taig the limits as we have X= AX A > 0 ad Y= AY A > 0 Cosequetly equatio () has positive defiite solutio Hece the theoem is poved F 0BTheoem BLet ad Y be the iteates i algoithm * < BB< ( + ) ( + ) ad 0 < < f q = < The < ( ) ad < ( ) ( + ) Y Y q whee q is a positive defiite solutio of () Poof Fom Theoem it follows that the sequeces () ad () ae coveget to a positive defiite solutio X of () We compute the spectal om of the matix obtai X ( ) X = B + X B B ( + X) B = R P * whee R = B ( + X ) B we P = B * ( + X) B By usig lemma with a mootoe opeato f( x) = x ; > ad = fo = the X X0 ( ) ( ) R= B + X B + BB> povided a = R > a ad by the same mae we get P > a Sice f(a) = a f( ) = * X X BB X X let < X X ( + ) q = < ( + ) we have X X < q X X afte -steps < q X X 0 Fom theoem X0 = < X < X+ Y+ < Y < Y0 = = 0 ad 0 = Ι Υ 0 = Ι Cosequetly q 0 ( ) < q Similaly we ca pove q < q ( ) G BCoollay 0 Assume that () has a solutio f { X } ad { } Υ Υ q = < ( + ) the Y ae covegig to with at least the liea covegece ate Poof As we have < q = < The + ( + ) ( ) choose a eal umbe that satisfies q < θ < Sice as thee exists a N such that fo ay N q θ Hece X+ X θ X X Similaly we ca pove Y+ X θ Y X H BTheoem 4 f matix equatio () has a positive defiite solutio ad afte iteative steps of Algoithm () the iequalities Ι ε Υ Υ Ι < ε ad Y <ε < imply SSN:

4 NTERNATONAL JOURNAL OF MATHEMATCS AND COMPUTERS N SMULATON Volume 9 0 i- ii- iii- iv- v- + X ε ( ) + < + + Y+ Y < ε + ( ) + + X ε + ( ) < Β Β Β Β Β ε < Υ Β Υ Β Β Β ε Β whee { X } ad { Y } = 0 ae the iteates geeated positive defiite solutio of equatio () ad ε = mi { ε ε ε } > 0 Poof i- Fom algoithm ad lemma the tae the oms of both sides * * + X = B ( + X )B B ( + X )B X Β Β < X ( + ) Fom theoem < ad ad X X as Cosequetly 0 as the ii- Fom theoem Y + X ε ( ) + < + Y < ad Y ad Y X as By the same mae we ca pove that iii- Similaly we ca pove + Y+ Y < ε + iv- Β Β + Β Β = Β Β Β Β + = Β ( ) Β the tae the oms of both sides Β Β + Β Β = Β ( ) Β ( ) + + Y + < ε Β X < Β ε v- Similaly we ca pove Υ Β Υ Β Β Β < Β ε ( + ) NUMERCAL EXPERMENTS this sectio we give two umeical examples to illustate X ad { Y } geeated by iteative that the matix sequeces { } method ae covegig to the uique positive defiite solutio of () The uique solutio is computed fo diffeet osigula matices A All pogams ae witte i MATHAMATCA Fo the followig examples the pactical 0 stoppig citeio is max { X Y X Y } 0 ad the solutio is = 00 A 4 Examples ) Example Coside the oliea matix equatio * X = A X A whee A = Α = 47 ad Β = 0497 Fo = 04 Let = 0000 ad = 0 Afte 0 iteatios we get X = Fo = 0 Let = 08 ad = 096 Afte iteatios we get X = Fo = 74 Let = 0 ad = 096 Afte 7 iteatios we get X = See Table Fo = 00 Let = ad = 09 Afte iteatios we get X = See Table Fo = 000 Let = ad = 08 Afte 4 iteatios we get SSN:

5 NTERNATONAL JOURNAL OF MATHEMATCS AND COMPUTERS N SMULATON Volume X = See Table Fo = 0000 Let = 099 ad = Afte iteatios we get X = See Table ) Example * Coside the oliea matix equatio X = A X A whee A = 0H with size m ad 0 i< j 0 H = ( hij ) : hij = i = j j + ( i+ j) i > j - Fo m=0 Α = 4987 Β = = Let = 06 ad = 09 Afte 7 iteatios we get appoximate solutio See Table - Fo m=00 Α = 7087 Β = = 0 Let = 000 ad = 00 Afte 7 iteatios we get appoximate solutio See Table - Fo m=00 Α = 64E+6 Β = = Let = 09 ad = 09 Afte 7 iteatios we get appoximate solutio See Table B 4 Tables the followig tables we deote δ x = X X δ y = X Y ad δ xy = X Y Table : Eo aalysis fo Example fo diffeet values of K δ x 9E- 6667E- 7406E-7 64E E- 968E- 974E- 86E-6 δ y E E E-8 8E E E- 4E- δ xy E- 7444E-6 008E-8 746E-0 87E-* 7094E- 7E m E E- 469E E-7 96E E-8 89E- 774E- 9769E E E-6 664E E-9 809E-4 88E E-* 8906E- 6877E E-* E E- 687E- 9769E E- 46E-4 678E E E-4* 8469E-9 047E- 4008E- 877E E E E-* 408E- 9476E-7 064E-0 084E-4* 00E-6 9E- 9884E-6 Table : Eo aalysis fo Example fo diffeet values of K 7 7 δ x E-4 4E E E- 468E E- 6788E- 9984E-8 749E- δ y E E-7 84E-0* E- 98E-9 407E E-8 079E- δ xy E- 4409E-8 96E E E E-* E E-* V CONCLUSON this pape a two-sided iteative pocess fo the matix equatio was ivestigated The ovel idea hee is that the two sequeces wee obtaied by statig with two diffeet povided values (a) a iteval i which the solutio is located that is X < X < Y fo all ; ad (b) a bette stoppig citeio Popety of solutio was discussed as well ad sufficiet solvability coditios o a matix A wee deived Moeove geeal covegece esults fo the suggested iteatio fo equatio wee give Some umeical examples wee peseted to show the usefuless of the iteatios The two-sided iteatio method descibed above possesses some advatages We ca compute X + ad Y + i paallel [] ad if the coditios of Theoems ae satisfied we SSN:

6 NTERNATONAL JOURNAL OF MATHEMATCS AND COMPUTERS N SMULATON Volume 9 0 ca calculate the solutio X of () fo ay powe of X i () as we see i examples ad while this caot be calculated fo oe-sided iteatio method t is also easy to popose a stoppig citeia usig max Y X < Y o { } { } max Y X Y < toleace which ae ot applicable fo oe-sided iteatio methods Hee we coside the case whe A is a o sigula REFERENCES [] A Kabalaie M M Motazei M Shabai ad B Eladsso Applicatio of homo-sepaatio of vaiables method o oliea system of PDEs WSEAS Tasactios o Mathematics Vol pp [] L H You P Comios ad J J Zhag Solid Modellig with Fouth Ode Patial Diffeetial Equatio teatioal Joual of Computes Vol No4 pp [] P Bee ad H Fassbede O the solutio of the atioal matix equatio X = Q + L* X L EURASP Joual o Advaces i Sigal Pocessig Vol7 pp [4] T Cime Recet advaces i oliea optimal feedbac cotol desig Poceedigs of the 9 th WSEAS teatioal Cofeece o Applied Mathematics pp [] A Khamis ad D Naidu Noliea optimal tacig usig fiite hoizo state depede Riccati equatio (SDRE) Poceedigs of the 4 th teatioal Cofeece o Cicuits Systems cotol Sigals (WSEAS) pp [6] G vaov V Hasaov ad F Ulig mpoved methods ad statig values to solve the matix equatio X ± A* X A = iteatively Mathematics of Computatio Vol 74 pp [7] W W Li ad S F Xu Covegece aalysis of stuctuepesevig doublig algoithms fo Riccati-type matix equatio SAM Joual o Matix Aalysis ad Applicatios Vol 8 pp [8] S A Zaea O positive defiite solutios of the oliea matix equatio X= AX A Poceedigs of the ed WSEAS teatioal cofeece o Mathematical Computatioal ad Statistical Scieces pp [9] S A Zaea ad S M El-Sayed O positive defiite solutios of the oliea matix equatio X AX A= Applied Mathematical Scieces Vol 9 No pp [0] M Ettaouil M Lazaa K Elmoutaouail ad K Haddouch A ew algoithm fo optimizatio of the ohoe etwo achitectues usig the cotiuous hopfield etwos WSEAS TRANSACTONS o COMPUTERS Vol No 4 pp 6 0 [] B Qiao X Chag M Cui K Yao Hybid paticle swam algoithm fo solvig oliea costait optimizatio poblems Wseas Tasactios o Mathematics Vol No pp [] S M El-Sayed A Two-Sided iteative method fo computig positive defiite solutios of a oliea matix equatio ANZAM Joual Vol 44 pp 4 00 [] S Fital ad C H Guo A ote o the fixed poit iteatio fo the matix equatios X ± A* X A = Liea Algeba ad its Applicatios Vol 49 pp [4] C H Guo ad W W Li The matix equatio X + A* X A = Q ad its applicatio i ao eseach SAM Joual o Scietific Computig Vol No pp [] B Meii Efficiet computatio of the exteme solutios of X + A* X A = Q ad X A* X A = Q Mathematics of Computatio Vol 7 pp [6] R Bhatia Matix Aalysis Gaduate Texts i Mathematics 69 Spige Velag New Yo c New Yo 997 [7] G K Pedese Some opeato mootoe fuctios Poceedigs of the Ameica Mathematical Society Vol 6 pp SSN: