438 IIUM Engineering Journal, Vol. 14, No. 2, 2013 Rakhimov 183 ON CALCULATION OF MULTIPLE EIGENVALUES OF THE LINEAR OPERATOR-FUNCTION BY REDUCTION PSEUDO-PERTURBATION METHODS D. G. RAKHIMOV National University of Uzbekistan, Vuzgorodok, Tashkent, 700174, Uzbekistan. davranaka@yandex.com ABSTRACT: This article examines the theory of bifurcations, which applies to the problem of retaining of the approximately given multiple eigenvalues and their generalized eigenvectors. This approach allows the reduction of algebraic multiplicity of eigenvalue for one and transfers the problem to the similar one but with simple eigenvalue. The method of the false perturbation is used to construct iterative processes. ABSTRAK:Dalam artikel ini, kaedah teori bifurkasi diaplikasikan terhadap masalah untuk mengekalkan penghampiran pelbagai nilai eigen dan vektor eigen yang teritlak. Kaedah ini membenarkan pengurangan kegandaan aljabar nilai eigen kepada satu dan memindahkan permasalahan kepada yang hampir serupa tetapi dengan nilai eigen yang lebih mudah. Pengkaedahan usikan palsu digunakan untuk proses pelelaran. KEYWORDS: bifurcation theory methods; multiple eigenvalues; root number; reduction pseudo-perturbation method 1. INTRODUCTION Many of engineering tasks connected with free fluctuations are reduced to eigenvalue problems so that many of them aren't solved analytically but resort to the approached methods. The problem of retaining of the approximately given eigenvalue and its generalized eigenvectors has been considered in the articles [1-3]. The authors used method of the false perturbations which was introduced by Gavurin [4]. According to this method the operator of the false perturbation builds in a way that the known approximations of the eigenvalue and generalized eigenvectors become the exact ones but for the perturbed operator-function. Using the method of the theory of bifurcation [5], the iteration processes are then built in order to find the exact values of the eigenvalues and generalized eigenvectors of initial operator-function. The most general operator of the false perturbations was built in [3]. In its formula the generalized eigenvectors of the direct operator-function and their adjoints were used in a symmetrical way. This operator was applied to several problems of mathematical physics [6, 7]. The method of the finite-dimensional regularization [8] allows the reduction of geometric multiple of the eigenvalue to a unit (in this case operator-function has many linearly independent eigenvectors without generalized Jordan chains' elements). Here, using the results and methods developed in [8], it will be shown that algebraic multiplicity of the eigenvalue also could be reduced to a unit. The terminology and notations of [5] are used. IIUM Engineering Journal, Vol. 14, No. 2, 2013 Rakhimov 184 2. PROBLEM SETTING Let 1E and 2E - be Banach spaces, ,)(: 2010 EADEA →⊃ 2111 )(: EADEA →⊃ - are the densely defined closed linear operators, where )()( 10 ADAD ⊂ and 1A subordinate to 0 A (i.e. 122 01 EEE xxAxA +≤ on )( 0 AD ) or )()( 01 ADAD ⊂ and 0 A subordinate to 1 A (i.e. 122 10 EEE xxAxA +≤ on )( 1AD ) . We consider eigenvalue problem ( ) 0 10 =− xtAA . (2.1) Let unknown eigenvalue λ be the Fredholm point of the linear operator-function 10 tAA − with ( ) },,,{ 110 nspanAAN ϕϕλ Κ=− ( ) },,{ 1*1*0 nspanAAN ψψλ Κ=− . Then corresponding −1A and − * 1 A Jordan chains [5] are defined by formulas: ( ) ,)1( 1 )( 10 − =− s i s i AtAA ϕϕ ( ) ,)1(* 1 )(* 1 * 0 − =− s i s i AtAA ψψ i ps ,2= , ni ,1= , (2.2) 0,det )1()( 1 ≠= j p i iAK ψϕ , 0det ≠= ijLL , )()1( 1 , l k jp iij iAL ψϕ −+= , nki ,1)( = , )(,2)( ki pplj = . According to [9] elements )()( , l k j i ψϕ , )(,2)( ki pplj = , nki ,1)( = of the −1A and −* 1 A Jordan chains, could be chosen in such a way that they satisfy following biorthogonality conditions: jlik l k j i δδγϕ =)()( , , jlik l k j i z δδψ =)()( , , )(,2)( ki pplj = (2.3) )1(* 1 )( lp k l k kA −+ = ψγ , )1( 1 )( jp i j i iAz −+ = ϕ , nki ,1)( = . We assume that ,Λ , )( 0 s i ϕ )( 0 s i ψ are sufficiently good approximations to an unknowns eigenvalue λ and elements of Jordan chains: εϕϕ ≤− )( 0 )( s i s i , εψψ ≤− )( 0 )( s i s i , εϕϕ ≤− )( 0 )( s i s i . We suppose that numbers 0K and 0L (which are approximations of K and L from the formula (2.2)) are close to 1. Next lemma was proved in [3]. Lemma. Going over linear combinations, can be defined as the systems { } ,, 1, )( kpn lk l k = γ )1(* 1 )( lp k l k kA −+ = ψγ , { } ,, 1, )( kpn lk l k z = )1( 1 )( lp k l k kAz −+ = ϕ (2.4) that satisfies the following biorthogonality conditions: jlik l k j i δδγϕ =)( 0 )( 0 , , jlik l k j i z δδψ =)( 0 )( 0 , , )(,2)( ki pplj = . (2.5) Now we want to apply methods developed in [8] to the problem of retaining of the approximately given multiple eigenvalues and their generalized eigenvectors. IIUM Engineering Journal, Vol. 14, No. 2, 2013 Rakhimov 185 3. REDUCTION PSEUDOPERTURBATION METHODS Let us introduce an operator-function ( ) ( ) ∑∑∑ = = −+ = −+ ⋅+⋅+−≡ n i p k kp i k i p k kpk i izztAAtA 2 1 )1( 0 )( 0 2 )1( 10 )( 1010 ,, 1 1 γγ (3.1) which we call regularisation of the operator - function 10 tAA − . Theorem 3.1 Unknown eigenvalue λ is the simple eigenvalue of the operator-function (3.1). Moreover, the eigenvector and the defect functional of ( )tA are defined by formulas ∑∑∑∑ − == == +++= 1 1 )( 11 2 2 )( 1 1 )( 1 1 1~ p s s s n i p s s iis n i ii p ccc i ϕϕϕϕϕ , ∑∑∑ == = ++= n i ii n i p s s iis dd i 2 1 1 2 )( 1 ~ ψψψψ (3.2) Proof: If the formulas (3.2) hold then ( ) ( ) ( ) ( ) +++== ∑∑∑ − == = 1 1 )( 11 2 2 )()( 1 1 1~0 p s s s n i p s s iis p AcAcAA i ϕλϕλϕλϕλ ++++ ∑∑∑∑∑ − = = −+ = = −+ = −+ 1 2 2 )1( 10 )( 10 )( 11 1 2 )1( 10 )( 101 2 )1( 10 )( 10 )( 1 1 1 1 1 1 1 11 ,,, p s p k kpks s n i p s sps ii p s spsp zczcz γϕγϕγϕ ++++ ∑∑∑∑∑∑∑∑ = = = −+ = = −+ = = = −+ n i n s p k kp s k s s ii n i p k kp i k i p n i p s p k kpks iis s s i i i zczzc 1 2 1 )1( 0 )( 0 )( 1 2 1 )1( 0 )( 0 )( 1 2 2 2 )1( 10 )( 10 )( ,,, 1 1 1 γϕγϕγϕ ,,, 2 2 2 2 )1( 0 )( 0 )( 2 1 2 2 )1( 0 )( 0 )( 11 1 ∑∑∑∑∑∑∑ = = = = −+ = − = = −+ ++ n l n i p s p k kp l k l s iis n i p s p k kp i k i s s i l l i i zczc γϕγϕ ( ) +++== ∑∑∑ ∑∑ = = −+ = = −+ = − n i p s ssp i n i p s ssp p s s iis zdzAdA i 2 2 )( 101 )1( 101 1 2 )( 101 )1( 10 2 )1(* 1 1 1 1 1 ,, ~ *0 γψγψψψλ ∑∑∑∑∑ = = −+ = = = −+ +++ n i p k k i kp i n i p s p k sk i sp is i i i zzd 2 1 )( 01 )1( 0 2 2 2 )( 10 )()1( 10 ,, 1 1 γψγψ .,, 2 2 1 )( 0 )1( 101 2 1 1 2 )( 0 )()1( 0 1 1∑∑∑∑∑∑∑ = = = −+ = = = = −+ ++ n i p j p k k ii kp j n i p k n j p s k i s j kp ijs ii j i zdzd γψγψ Applying the functional )( 0 µ νψ to the first equality and elements )( 0 µ νϕ to the second one, we obtain the system of the linear algebraic equations to find the unknown coefficients ;,, 11 nnp cc Κ :,, 12 nnp dd Κ , 1)( 1 i Ak j jij exa =∑ − = 1)(,2 −= Aki ; (3.3) ,' 1)( 1 i Ak j jij eyb =∑ − = 1)(,2 −= Aki , (3.4) IIUM Engineering Journal, Vol. 14, No. 2, 2013 Rakhimov 186 where n ppAk ++= Λ 1 )( - is a root number of the operator-function ( )tA , jj cx 1 = , 1,1 1 −= pj , 1,21 ++ = jjp cx , 1,0 2 −= pj , 1,11 ++++ = − jkjpp cx kΛ , 1,0 −= k pj , nk ,2= , )( 1011 , l i a γϕ= , )( 10 )( 1 )1( 10 )2( 1 ,, 11 lklpkp ik za γϕψ += −+−+ , 1,2 1 −= pk , )( 10, , 11 l ippl i a γϕ= −++Λ , )( 10 )()1( 10 )1( 1, ,, 11 11 lk i lpkp kppl za i γϕψ += −+−++++ −Λ , 1,2 1 −= pk , 1,1 1 −= pl , ni ,2= , =+++ − 1,11 lpp sa Λ )1( 01 , + = l s γϕ , =+++ − klpp sa ,11 Λ )( 0 )( 1 )1( 0 )2( 1 ,,1 l s klp s kp sz γϕψ +−+−+ , 1,2 1 −= pk , = −− +++++ 1111 , is pplpp a ΛΛ )( 0 , l si γϕ= , =++++++ −− kpplpp isa 1111 , ΛΛ )( 0 )1()1( 0 )1( ,, l s k i lp s kp i siz γϕψ +−+−+ + , 1,1 −= i pk , ni ,2= , 1,1 1 −= +spl , ns ,2= ; 1,1 += jj dy , 1,1 1 −= pj , 1,11 ++++ =− jkjpp dy kΛ , 1,1 −= k pj , nk ,2= , )( 1 )1( 01, 1 11 , plp slpp s s b γϕ −++++ =−Λ , =+++ − klpp sb ,11 Λ )( 1 )()1( 1 )1( 0 ,, 1 kl s kplp s zs ψγϕ +−+−+ , 1,1 −= s pk , i l spplpp zb is ψ,)( 0, 1111 = −− +++++ ΛΛ , =++++++ −− kpplpp isb 1111 , ΛΛ )1()( 0 )1()1( 0 ,, +−+−+ + k i l s kp i lp s zis ψγϕ , 1,1 −= i pk , ni ,2= , 1,1 1 −= +spl , ns ,2= ; [ ],,, )(10)(1)1(10)2(1 11 ipipi ze γϕψ +−= −+ 1,1 1 −= pi , [ ],,, )( 20 )( 1 )1( 20 )2( 1 12 1 ipip iP ze γϕψ +−= −++ , 1,1 2 −= pi , [ ],,, )( 0 )( 1 )1( 20 )2( 1 12 11 i s pip ipP ze s γϕψ +−= −++++ −Λ , 1,1 −= spi , ns ,2= ; 1 )( 10 ,' ψi i ze = , 1,1 1 −= pi , 1 )( 0 ,' 11 ψi sipp ze s −=+++ −Λ , 1,0 −= spi , ns ,2= . It is not difficult to show that ( )εOaii += 1 , ( )εOa ii +=+ 11, , ( )εOaij = , 1, +≠ iij , 1)(,1, −= Akji , ( )εOb ii += 1 , ( )εOb ii +=+ 1,1 , 2,1 1 −= pi , ( )εOb pp += 111 , ( )εOb ii +=+ 11, , ( )εOb ii = for 1, +≠ iij , and for all other cases. Consequently, ( )εOais += 1det , ( )εOb is += 1det , i.e. systems (3.3) and (3.4) have unique solutions. Because of )(1,, )2( 10 )2( 1 )1( 10 )( 11 11 1 εψγϕ Oze pp p +=+= − − , and )(εOei = , for all 1 1 −≠ pi , we obtain )(εOc i = , 11 −≠ pi and )(111 εOc p +−=− . Therefore, )1( 10 )( 100 11~ −−= pp ϕϕϕ and 100 ~ ψψ = should be taken as initial approximations for ϕ and ψ . The solution of the equation ( ) 0~,~ 00 =ψϕtA , is taken as the initial approximation for eigenvalue λ , i.e. 001 000 0 ~,~ 1 ~ , ~ ψϕ ψϕ λ A A + = . Choosing 10 * 1 0 0 1~ ψγ A k = and 01 0 0 ~1~ ϕA k z = , we find that 1~,~ 00 =γϕ , 1 ~,~ 00 =ψz , because 0 ~ , ~ , ~ , ~ , ~ 0 )( 1010 )1( 1010 )( 1010010 111 ≠=−== − ψϕψϕψϕψϕ ppp AAAAk . We define pseudoperturbed operator using formulas ( ) ( ) 0000000 ~~*,~~, zAxAxxD ψλϕλγ += , ( ) ( ) 00000 * 0 ~*,~~,~ ψλγϕλ AyzyAxD += . IIUM Engineering Journal, Vol. 14, No. 2, 2013 Rakhimov 187 Then, ( ) 0000 ~~ ϕλϕ AD = , ( ) 000 * 0 ~ * ~ ψλψ AD = , i.e. ( )( ) { }000 ~ϕλ =− DAN , ( )( ) { } 0 * 00 ~ * ψλ =− DAN . Using the Schmidt's regularization [5] the equation ( ) 0=xtA could be reduced to the system: ( ) ( )( )[ ]     = −+Γ+= − .~, , ~ 0 0 1 000 γξ ϕλξ x AtADIx (3.5) where, ( )[ ] 1 00000 ~~., − +−=Γ zDA γλ . If we substitute x in the second equation (3.5) with the value found from the first, we construct the equation: ( ) ( ) ( )( )[ ] ,0~,~1 00 1 000 =−+Γ+−≡ − γϕλAtADItF (3.6) which is called "bifurcation equation". The exact eigenvalue λ is a simple root of bifurcation equation. Let ( )ρλ ; 0 S - be the ball of radius ρ centred at 0 λ . Theorem 3.2 If initial approximations is the sufficiently good, then there is the ball ( )rS ;0λ , where , 2 411 ρη ≤ −− = h h r ( ) ( ) 4 1 ' 0 1 0 ≤= − λλ FLFh , ( ) ( )0 1 0 ' λλη FF − = , in which equation (3.6) has unique solution λ=t . The iterations defined by the Newton's modified method, given below, ( )[ ] ( ) mmm FF λλλλ 1 01 ' − + −= , Κ,2,1,0=m , (3.7) will converge toward this solution. Proof: Firstly we verify the conditions of the Theorem 3.2 [10, p.446]: 1) ( ) [ ] ( );~,~~,~1~,~1 00000000 1 000 DODDIF =−+−=Γ+−= − Λψϕγϕγϕλ ( ) ( ) 010 DCF ≤λ ; 2) ( ) [ ] [ ] −=Γ+ΓΓ+= −− 00100 1 0010 1 000 ~,~~,~' ψϕγϕλ ADIADIF ΛΛ +Γ−=+Γ− 00000000000 ~,~2~,~2 ψϕψϕ DDkDD , ( ) ( ) 0200 ' DCkF −≥λ , ( ) ( )[ ] 1 020 1 00 ' −− −≤= DCkFM λ ; 3) ( ) ( ) ( )( )( ) 2112121121 "'' ttLtttttFtFtF −⋅≤−−+=− θ , where ( )tFL Gt "sup ∈ = . IIUM Engineering Journal, Vol. 14, No. 2, 2013 Rakhimov 188 If initial approximations are sufficiently good, we can choose ε such that 4 1 10 ≤LCM . Then, according to Theorem 3.2 [10, p.446]-where, there is unique solution of the equation (3.6) in the ball ( )rS ;0λ , where ρη ≤ −− = h h r 2 411 . The iterations calculated by formula (3.7) converge toward this solution. It should be noted that on every step of iterative process it is necessary to solve only one equation: ( )[ ] 000 ~~~, zxzA m =⋅+ γλ . Theorem 3.3 The elements of GJS { } ipn ji j i , 1,1, )( = ϕ , { } ipn ji j i , 1,1, )( = ψ are the solutions of the following recurrent systems: 0 1 0010 , i n s ss zxzAA =      ⋅+− ∑ = γλ , 0 1 00 * 1 * 0 , i n s ss yzAA γγλ =      ⋅+− ∑ = , 0,11, 1 0010 , iijij n s ss zxAxzAA +=      ⋅+− − = ∑ γλ , ,1 iix ϕ= )( j iji x ϕ= , 0,1 * 1 1 00 * 1 * 0 , iijji n s ss yAyzAA γγλ +=      ⋅+− − = ∑ , ,1 iiy ψ= )( j iji y ψ= , i pj ,2= , ni ,1= . Proof. We want to apply method of false perturbations to initial equations (2.1) and (2.2). In order to do this we are using pseudo perturbed operator 0D suggested by Loginov [8]: ∑∑∑∑ ∑∑ = == = = = +      −= n i p j j i j i n i p j n k p s s k s k j i j i j i ii k zxzxxD 1 1 )( 0 )( 0 1 1 1 1 )( 0 )( 0 )( 0 )( 0 )( 00 ,,, τψσσγ , ∑∑∑∑ ∑∑ = == = = = +−= n i p j j i j i n i p j j i n k p s s k s k j i j i ii k yzyzxD 1 1 )( 0 )( 0 1 1 )( 0 1 1 )( 0 )( 0 )( 0 )( 0 * 0 ,,, τγψσσ , which, according to theorem 1 [3], has the following properties: )( 0 )( 00 j i j i D σϕ = , )( 0 )( 0 * 0 j i j i D τψ = . Then ( ) { }n ii DAAN 10010 = =−− ϕλ , ( ) { }n ii DAAN 10 * 0 * 0 * 0 = =−− ψλ . The equation ( ) 0 10 =− xAA λ is reduced to the system [ ]      == Γ+= ∑ = − ,,1,, , 0 1 0 1 00 nsx DIx ss n s ss γξ ϕξ Where 1 1 000100 , − =       ⋅+−−=Γ ∑ n s ss zDAA γλ -is the bounded linear operator which exists according to the E. Schmidt's lemma [5]. Then IIUM Engineering Journal, Vol. 14, No. 2, 2013 Rakhimov 189 [ ] ,, 1 00 1 00∑ = − Γ+= n s kssk DI γϕξξ nk ,,2,1 Κ= . Because the initial equation has exactly n linearly independent solutions, the last system also has n linearly independent solutions. It means that the rank of the matrix of the coefficients of this system is equal to zero and therefore it is possible to assume [ ] 0 1 00 ii DI ϕϕ − Γ+= . Similarly (2.2) could be reduced to the system [ ] [ ]      == ΓΓ++Γ+= − − = − ∑ .,1,, , 0 110 1 00 1 0 1 00 nkx xADIDIx kssk s n k ksks γξ ϕξ Substituting s x in the second part of the system with the value found from the first one we can simplify the system: [ ] [ ] 0110 1 00 1 00 1 00 ,, ks n l klslsk xADIDI γγϕξξ − − = − ΓΓ++Γ+= ∑ . Because it was mentioned earlier that the rank of the matrix of this system equals 0 for each s , we can assume that [ ] [ ] ksksk xADIDIx ,110 1 000 1 00 − −− ΓΓ++Γ+= ϕ , nk ,1= , 1>s . Similarly, applying the same arguments to equations ( ) 0* 1 * 0 =− yAA λ and ( ) 1 * 1 * 1 * 0 −=− ss yAyAA λ we come to recurrent equations: [ ] 0 1* 0 * 0 kk DI ψψ − Γ+= , [ ] [ ] ksksk yADIDIy ,1 * 1 * 0 1* 0 * 00 1* 0 * 0 − −− ΓΓ++Γ+= ψ , nk ,1= , 1>s . Example: In order to illustrate these results, we consider the eigenvalue problem 0" =+ uu λ , ( ) 00 =u , ( ) ( )1'' 0 uxu = in the space ( ) ( )[ ] [ ]1;01,,0 1 00 2 CxxC ∩∪ . Detailed study of this problem could be found in [11]. If 122 122 0 ++ +− = ms ms x and 2 1 0 +<< sm , the exact value of eigenvalue is 0 µµµ == sm ( here 0 m 1 2 x m + = π µ , ( ) 0 s 1 12 x s + + = π µ ). This is the eigenvalue of multiplicity 2 and the corresponding eigenvector and Jordan chain elements are x 0 )1( sin µϕ = , xx 0 0 )2( cos 2 1 µ µ ϕ −= , IIUM Engineering Journal, Vol. 14, No. 2, 2013 Rakhimov 190       ≤≤ − ≤≤ = ,1,cos 1 8 ,0,0 002 0 0 0 )1( xxx x xx µ µψ ( )        ≤≤ − − − ≤≤ + − = .1,sin 1 14 ,0,sin 1 4 002 0 00 0)2( xxx x x xxx x µ µ ψ As initial approximations of functions )1(ϕ , )1(~ψ and elements of Jordan's )2(ϕ , )2(~ψ , have been taken the initial segments of their Taylor series with 25 terms. All calculation experiments were carried out using program Maple 11. Under 1== sm , 5 1 0 =x , 6850275,61 4 25 2 == π λ : 8321723,61 0 =λ ; 6728554,611 =λ ; 6850467,612 =λ ; 6850275,613 =λ ; 6850275,61 4 =λ . REFERENCES [1] B.V. Loginov, D.G. Rakhimov, “On pseudoperturbation method in the presence of generalized Jordan chains.” Differential equations and it is applications, ed. M.S. Salakhitdinov, Fan, Tashken, (1979): 113-25. [2] B.V. Loginov, D.G. Rakhimov, N.A. Sidorov,”Development of M. K. Gavurin's Pseudoperturbation Method.” Fields Institute Communications, 25 (2000):367-81., [3] B.V. Loginov, O.V.Makeeva, A.V. Tsiganov, “Retaining approximately givens Jordan chains of the linear operator-functions of the spectral parameter on the base of the perturbation theory.” Ulyanovsk State Pedag.University, Interunivesity collection of the science works "Functional analysis", 38(2003):52-62. [4] M.K. Gavurin , “On pseudoperturbation method for eigenvalues determination.” Journal .Computer. Mathematics. Physics., Russian, 1.5(1961):751-70. [5] M.M.Vainberg, V.A.Trenogin, “Branching Theory of Nonlinear Equations.”, Nauka, Moskow, (1969)., , 1974. [6] B.V. Loginov, O.V.Makeeva, “Pseudoperturbed iteration methods generalized eigenvalue problems.” ROMAI Journal, 1(2008):149-68. [7] B.V. Loginov, O.V.Makeeva, “The pseudoperturbation method in generalized eigenvalue problems.” Doklady Mathematics,. 77.2 (2009): 194-97. [8] D.G. Rakhimov,”On calculation of multiple Fredholm's points of discrete spectra of the linear operator-functions by pseudoperturbation methods.” Trudy Srednevolzskogo Matematicheskogo Obshchestva, 12.3(2010):106-12. [9] B.V.Loginov, Yu.B.Rusak, “Generalized Jordan structure in the branching theory.” In Right and InverseProblems for Partial Differential Equations, ed. M.S.Salakhitdinov. Fan, Tashkent,( 1978)” 133-148 [10] L.A.Lyusternik, V.I. Sobolev, “Elements of functional analysis.” Nauka, (1964). [11] O.V. Makeeva, “Pseudoperturbation method in generalized eigenvalue problems.”] Kandidat dissertation, Ulyanovsk, 2007. Russian