A Complex Neural Network Algorithm for Computing the Largest Real Part Eigenvalue and the corresponding Eigenvector of a Real Matrix

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1 4h Ieraioal Coferece o Sesors, Mecharoics ad Auomaio (ICSMA 06) A Complex Neural Newor Algorihm for Compuig he Larges eal Par Eigevalue ad he correspodig Eigevecor of a eal Marix HANG AN, a, XUESONG LIANG, b ad LIPING WAN,c College of Physics ad Egieerig, Chegdu Normal Uiversiy, Chegdu 630, Chia a hagacdu@63.com, bliag_xuesog@sohu.com, clpwa03@6.com Keywords: Complex eural ewor; real marix; larges real par; eigevalue; eigevecor Absrac. I his sudy, we propose a ovel complex eural ewor algorihm, which exeds he eural ewor based approaches ha ca asympoically compue he larges or smalles eigevalues ad he correspodig eigevecors of real symmeric marices, o he case of direcly calculaig he larges real par eigevalue ad he correspodig eigevecor of a real marix. he proposed eural ewor algorihm is described by a group of complex differeial equaios, which is deduced from he classical eural ewor model. he proposed algorihm is a class of coiuous ime recurre eural ewor (NN), i has parallel processig abiliy i a asychroous maer ad could achieve high compuig capabiliy. his paper provides a rigorous mahemaical proof for is covergece i he case of real marices for a more clear udersadig of ewor dyamic behaviors relaig o he compuaio of eigevecor ad eigevalue. he proposed approach has obvious virues such as fas covergece speed ad o-sesiiviy o iiial value. Numerical examples showed ha he proposed algorihm has good performace. Iroducio Usig he eural ewor echology o exrac he eigevecor correspodig o he modulus maximum eigevalue of a real symmeric marix was firs preseed abou 30 years ago [], he moivaed broad ieress from egieerig ad heoreical researches [-]. I 995, Luo e al. [5, 6] proposed a very classical eural ewor algorihm for exracig o oly modulus maximum eigevalue bu also modulus miimum eigevalue ad heir correspodig eigevecors of real symmeric marices, bu he covergece proof of his algorihm o be preseed uil 004 by Zhag e al. [7]. I he recely years, some adapive geeralized eige-pairs exracio algorihms have bee preseed by some auhors [, ]. A umber of years ago, Liu e al. [3] proposed a simpler eural ewor algorihm for exracig he eigevecor correspodig o he modulus maximum eigevalue of a real symmeric marix, bu i will also diverge as he same as i he publicaio [] whe i is used o exrac he eigevecor correspodig o he modulus miimum eigevalue of a real symmeric marix. he Liu e al. [4] preseed a suble mehod o exrac he imagiary par of he eigevalue from he maximum imagiary par ad he real par of he eigevalue from he maximum real par of a geeral real marix. Uforuaely, hey could obai he correspodig eigevecor, which is someimes vial for some pracical egieerig problems. I he prese paper, we ry o o oly exrac he larges real par eigevalue bu also exrac he correspodig eigevecor of ay real marix i he complex domai. he res of his paper is orgaized as follows: I secio, we proposed he ovel complex domai eural ewor model of his paper. Some umerical examples give i secio 3 ad we summarized his paper i he las secio. he Proposed Complex Neural Newor Model he classical eural ewor algorihm for compuig he eigevecor correspodig o he modulus maximum eigevalue or modulus miimum eigevalue ca be illusraed as follows [5, 6] Copyrigh 06, he Auhors. Published by Alais Press. his is a ope access aricle uder he CC BY-NC licese (hp://creaivecommos.org/liceses/by-c/4.0/). 577

2 dv( ) v( ) v( ) Av( ) v( ) Av( )v( ), () where v( ) are he dimesio real colum vecors ha deoe he saes of euros. However, he above classical eural ewor algorihm ca oly be used o solve he eige-pair problems of real symmeric marices. We could o direcly adop i o compue he eige-pair problems of geeral real marices. Now we subsiue marix A i Eq. by he followig A : A 0 A, 0 A where A is a geeral real marix, v( ), le x( ) v( ), y ( ) oe ca ge v( ) [ x( ) y( ) ], so Eq. ca be covered io he followig forms dx( ) [ x y ] Ax( ) [ x( ) Ax( ) y ( ) Ay ( )]x( ), () dy ( ) [ x y ] Ay ( ) [ x( ) Ax( ) y ( ) Ax( )] y ( ) Add he firs equaio wih he secod equaio imes i of Eq., where i is a imagiary ui. Assume ha z ( ) x( ) y( )i, oe ges dz ( ) z ( ) zaz ( ) [ x( ) Ax( ) y( ) Ay( )]z ( ). Cosider ha x( ) Ax( ) y ( ) Ay( ) (3) z ( ) Az ( ) z ( ) Az ( ), Eq. 3 becomes dz ( ) z ( ) z ( )Az ( ) z ( ) Az ( ) z ( ) Az ( ) z ( ), (4) which is he complex eural ewor model i our sudy, where z ( ) C. he algorihm ca be used o exrac he eigevecor ad he correspodig eigevalue wih larges real par of a geeral real marix. If A is a real marix, ad le are eigevalues of A, he he correspodig ormal basic complex eigevecors are S S S. Le z ( ) x( ) iy ( ), oe ges z ( ) z ( ) S [ x ( ) iy ( )]S, (5) where z ( ) x ( ) iy ( ) deoe he proecio of z ( ) i he direcio alog wih S. heorem. Le z ( ) z ( ) z ( ) z ( ) z ( ), ad for ay o-zero iiial value z (0) C, he soluio of Eq. 4 saisfyig z ( ) z (0). Proof: Due o d z d (z z) z z z z, ad 578

3 z Az z Az z Az z Az z} z {z zaz z} z Az z Az z Az z Az {z z zaz z z z z } {z z zaz z z z z} Oe ca ge d ( z( ) ) 0, i.e. z ( ) is a cosa, i oher words, so oe ca z z z z z {z zaz obai z ( ) z ( ) z ( ) z (0) z (0) z (0), hus complee proof. heorem. If he complex vecor z ( ) is he soluio of Eq. 4 ad z ( ) x ( ) iy ( ) deoes he proecio of z ( ) alog wih he direcio S, he z ( ) ca be read as z ( ) exp[ z (0) z (0) ] z (0) z (0) exp[ z (0) z (0) ]d (6) 0 Proof: We firs eed o prove ha he proecio of z ( ) oo S ca be represeed as: z ( ) exp[ z (0) z (0) ] z (0) z (0) exp[ z (0) z (0) ]d. (7) 0 Assume ha all eigevalues of A ca be deoed as i I, where ad I are he real par ad he imagiary par of eigevalue, respecively. Le z ( ) x( ) iy ( ), ad assume ha he soluio z ( ) of Eq. 4 ca be represe as follows: z ( ) [ x ( ) iy ( )]S (8) Subsiuig Eq. 8 io Eq. 4, oe ges he followig form whe : dx ( ) dy ( ) ]S z ( x iy ) S { [ ] z } ( x iy ) S i Alog wih S, he followig equaio ca be obaied: [ dx ( ) dy ( ) i z ( x iy ) { [ ] z }( x iy ). (9) (0) Le i I, i I, i I, iser hem io Eq. 0. Afer separaig he real par ad he imagiary par, we have: dx ( ) z ( x I y ) z x, dy ( ) z ( y I x ) z y As d z d ( z z ) x dx y dy (), based o Eq., we ca obai he followig equaio: d z ( ) z ( ) z ( ) z ( ) z ( ) z ( ) z ( ) Based o he heorem, Eq. ca be covered io he followig form: () 579

4 d z ( ) z ( ) z ( ) z ( ) z ( ) z (0) z (0) z ( ) z ( ) z ( ) (3) If z ( ) 0, zr ( ) 0, he d d z ( ) zr ( ) z (0) z (0)( r ) z ( ) zr ( ) herefore (4) z ( ) d {l } z (0) z (0)( r ) zr i.e. : z z (0) exp[ z (0) z (0)( r ) ] zr zr (0) (5) If z (0) 0, herefore z ( ) 0, he Eq. 3 ca be direcly wrie as follows: z d z z (0) z (0) 4 z z z i.e. (6) z ( ) d z (0) z (0) z ( ) z ( ) z ( ) Subsiuig Eq.5 io Eq. 7, oe ges (7) z (0) d exp[ z (0) z (0) ] exp[ z (0) z (0) ] z ( ) z (0) he iegral o boh sides of Eq. 8 from 0 o reads z (0) exp[ z (0) z (0) ] exp[ z (0) z (0) ]d 0 z ( ) z (0) z (0) Direc calculaio reads z ( ) exp[ z (0) z (0) ] z (0) z (0) exp[ z (0) z (0) ]d (8) (9) (0) 0 herefore z ( ) exp[ z (0) z (0) ] z (0) z (0) 0 exp[ z (0) z (0) ]d. heorem 3: Solve Eq. 4, afer covergece, ad le v lim z ( ), he v will coverge o a eigevecor of he real symmeric marix A, which correspods o he maximum real par eigevalue ha ca be deoed as z ( ) Az ( ) z ( ) Az ( ) z ( ) z ( ). 580

5 Proof: Accordig o he equilibrium poi of he Eq. 4 oe ges Az ( ) z ( ) Az ( ) z ( ) Az ( ) z ( ) z ( ) z ( ). If we cosider z ( ) as a eigevecor of a real marix A, he he correspodig eigevalue ca be read as z ( ) Az ( ) z ( ) Az ( ) z ( ) z ( ). Now we eed o prove ha z ( ) will coverge o he eigevecor correspodig o he maximum real par eigevalue of A. Based o he heorem, we should prove ha z ( ) is ideed he liear combiaio of hose eigevecors correspodig o all eigevalues ha are ideical o. As he Eq. 7, we deoe zi ( ) wih aoher form as follows: () where i deoes he iersecio agle bewee zi ( ) ad ui vecor Si. herefore oe ge () As he aalysis of he real par ad imagiary par of z ( ) are similar, we firs cosider he aalysis of real par. Firs, simplify he exp[ z (0) z (0) i ( )] zi (0) real par i, he he real par becomes: z (0) exp[ z (0) z (0) ] cos( i ) Si (3) Le II z (0) z (0), ad assume ha z (0) is a o-zeros iiial value complex vecor, so z (0) i zi (0)Si. Defiiio l mi{l l zl (0) 0}, he here does exis a r { m} subec ha r l r. herefore he real par of z ( ) ca be read as: exp[ II i ] zi (0) real par lim i z (0) exp[ II ] r cos( i ) Si i l zi (0) r l z (0) cos( i ) Si (4) Usig he similar mehod, we have: r imagiary par i i l zi (0) r l z (0) si( i ) Si herefore r lim z ( ) [ i l zi (0) r l z (0) r cos( i ) i i l zi (0) r l z (0) si( i )]Si V r. (5) herefore, z ( ) belogs o he eige-subspace V r, which should be spaed by all eigevecors correspodig o he eigevalues r. Furhermore, if l, he zl (0) 0, i.e., he proecio of he iiial value z (0) oo he ui eigevecor S are o-zero. I his case, oly r ca saisfy he codiio ha r r. herefore he seady-sae soluio z ( ) V as before deoes he modulus maximum eigevalue. I pracice, as he dimesio of V is less ha C, so he sochasic 58

6 o-zeros iiial value is always orhogoal o he eige-subspace V. herefore we ca always obai he eigevecor correspodig o he modulus maximum eigevalue. Simulaio esuls Example. Cosider he followig 6 6 real symmeric marix A : A We use he followig sochasic iiial value z (0) for ruig he complex eural ewor algorihm Eq. 4 o obai he larges real par eigepair of marix A : i i i z (0) (6) i i i Direcly solvig he eige-pair problems of A, we ca obai he eigevalues of A use malab fucio, oe ges ~ 6 ={-.6394, -.59, , 0.453,.84,.9446}, ad he larges real par eigevecor is: v , 0.748, , 0.453, , (7) By ruig he proposed model Eq. 4 wih he iiial value z (0) as give by Eq. 6, we ca obai he eigevecor correspodig o he maximum real par eigevalues, as follows: i i i z ( ) (8) ( i)v i i i From above, we ca see ha he eigevecors ha we go from he complex eural ewor algorihm are cosa muliple of he eigevecor obaied from he direc calculaio. Oe ca ge he eigevalue by zz(())azz (()) i, which is us he larges real par eigevalue of A. Fig. illusraes he dyamic behavior of he larges real par of he symmery marix A, ad Fig. illusraes he dyamic behavior of six compoes modulus of he correspodig eigevecor. From hese figures, we oe ha he proposed algorihm also has he fas covergece propery, which is us oe virue of parallel compuig. I addiio, he proposed algorihm is o sesiive o iiial value. his virue maes acual operaio a lo more coveie. 58

7 Fig.. Dyamic behavior of maximum real par eigevalue of marix A. I should coverge o Fig.. Dyamic behavior of he modulus of he six compoes of eigevecor correspodig o he maximum real par eigevalue. I should coverge o he six compoes modulus of eigevecor correspodig o eigevalue Example. Now we cosider a geeral real marix A. he followig is a arificial real marix: A (9) By direcly solvig he eigepairs problems of A, we could obai he eigevalues of A as follows: = , = i, 3 = i, 4 =.769, 5 =.53, ad 6 = he correspodig eigevecor of he larges real par eigevalue is: v4c , 0.376, , , 0.570, (30) We ru he proposed algorihm Eq. 4 wih he iiial value z (0) also give by Eq. 6. We ca 583

8 obai he larges real par eigevalue is z ( ) Az ( ) z ( ) z ( ).769, which ideed he direcly calculaio value, ad he correspodig eigevecor as follows: i i i v4 (3) ( i)v 4 c, i i i From which, we ca see ha he eigevecor obaied from he complex eural ewor algorihm are also he cosa muliples of he eigevecor obaied from he direc calculaio. I he followig, Fig. 3 illusraes he dyamic behavior of he larges real par of real marix A, ad Fig. 4 illusraes he dyamic behavior of six compoes modulus of he correspodig eigevecor. Fig. 3. Dyamic behavior of maximum real par eigevalue of he real marix A. I should coverge o 4. Fig. 4. Dyamic behavior of he modulus of he six compoes of eigevecor correspodig o he maximum real par eigevalue. I should coverge o he six compoes modulus of eigevecor correspodig o eigevalue

9 Coclusio Based o he classical real domai eural ewor algorihm, his paper proposed a ovel complex eural ewor o direcly compue he larges real par eigevalue ad he correspodig eigevecor of real marices. wo simulaio experimes idicaed ha he proposed algorihm is effecive. Acowledgemes his wor was parially suppored by he Geeral proec of Sichua Provicial Deparme of Educaio (4ZB033), ad he Chegdu Normal Uiversiy iroduces he aleed perso scieific research sar fuds subsidizaio proec (YJC04-5). efereces [] Oa E. A: Joural of Mahemaical Biology Vol. 5 (98), p [] Che iapig: Chiese Sci Bull Vol. 4 (995), p [3] Y. Liu, Z.S. You, L.P. Cao: heoreical Compuer Sciece Vol. 367 (006), p [4] Yiguag Liu, Zhisheg You, Lipig Cao: Compuers ad Mahemaics wih Applicaios Vol. 53 (007), p [5] F.L. Luo, Y.D. Li: Neurocompuig Vol. 7 (995), p [6] F.L. Luo,. Ubehaue, Y.D. Li: Neurocompuig Vol. 8 (995), p. 3-. [7] Y. Zhag, F. Ya, ad H.J. ag: Compu. Mah. Appl. Vol. 47(004), p [8] Y. Liu, Z.S. You, ad L.P. Cao: Neurocompuig Vol. 67 (005), p [9] Y. Liu, Z.S. You, ad L.P. Cao: heoreical Compuer Sciece Vol. 367 (006), p [0] Yig ag, ad J.P. Li: Compuers ad Mahemaics wih Applicaios Vol. 60 (00), p []. D. Nguye, ad I. Yamada: IEEE ras. Sigal Process Vol. 6 (6) (03), p [] Xiaowei Feg, Xiagyu Kog, Zhasheg Dua, ad Hogguag Ma: IEEE ANSACIONS ON SIGNAL POCESSING Vol. 64 () ( 06), p