Eigenvalues of tridiagonal matrix using Strum Sequence and Gerschgorin theorem

Transcription

1 Eigenvalues o tridiagonal matrix using Strum Sequene and Gershgorin theorem T.D.Roopamala Department o Computer Siene and Engg., Sri Jayahamarajendra College o Engineering Mysore INDIA roopa_td@yahoo.o.in S.K.Katti Researh Supervisor S..J.C.E. Mysore University, Mysore City, Karnataka INDIA Mysore University, Mysore City, Karnataka INDIA skkatti@indiatimes.om Astrat: In this paper, omputational eiient tehnique is proposed to alulate the eigenvalues o a tridiagonal system matrix using Strum sequene and Gershgorin theorem. The proposed tehnique is appliale in various ontrol system and omputer engineering appliations. Keywords- Eigenvalues, tridiagonal matrix, Strum sequene and Gershgorin theorem. I.INTRODUCTION Solving tridiagonal linear systems is one o the most important prolems in sientii omputing. It is involved in the solution o dierential equations and in various areas o siene and engineering appliations suh ontrol system [] and omputer siene [, ]. It is also our in a wide variety o appliations, suh as the onstrution o ertain splines and the solution o oundary value prolems. There are various numerial tehniques availale in the literature, whih are useul or determining eigenvalues o real symmetri matries [4]. In most o these methods, given system matrix is onverted into tridiagonal orm. There are various methods also given in the literature or determining eigenvalues o a tridiagonal matrix [4]. In this method, strum sequene and isetion method is used to determine the eigenvalues o a given real symmetri triangular matrix. It is oserved that it require large numer o iterations to ompute the eigenvalues. It is oserved that these iterations an e redued y using well known Gershgorin theorem [4]. In [5], tehnique is presented to identiy the eigenvalues on the right hal the s-plane using Gershgorin theorem [4]. The extension to this approah is given in [6]. One o the leading methods or omputing the eigenvalues o a real symmetri matrix is Given s method [4]. In that method, ater transorming the matrix into diagonal orm say, ' S ', the leading prinipal minors o S λi orm a strum sequene. Then, using isetion approah, hange o sign in various strum sequene is oserved. Further, ased on this, eigenvalue an e determined y repeatedly using isetion method. In [7], various appliations are presented ased on Gershgorin theorem. Here, in this paper, similar to these existing appliations, we have used Gershgorin theorem in Strum sequene to determine eigenvalues in omputationally eiient manner. In order to show the omparative result, we have onsidered the example whih was earlier onsidered in [4]. II. Givens Method or Symmetri Matries [4]: Let A e a real, symmetri matrix. The Givens method uses the ollowing steps: (a) Redue A to a tridiagonal orm using plane rotations () Form a strum sequene, study the hanges in sign in the sequenes and ind the eigenvalues. The redution to a tridiagonal orm is ahieved y using the orthogonal transormation. Suppose the orthogonal matrix is given as ISSN : Vol. No. Deemer 0 7

2 [ B ] = () n n n n n The numer o plane rotations required to ring a matrix o order n to its tridiagonal orm is ( )( ) n n. We know that A and B have the same eigenvalues. I i 0, i =,..., n then, the eigenvalues are distint. Now, we deine = λi B = n λ λ n λ n n n λ n () Expanding y minors, the sequene { n } satisies =, = λ () 0 and = ( ) ; r n (4) r λ r r r r ISSN : Vol. No. Deemer 0 7

3 I none o the,,..., n vanish, then { } is a Strum sequene. That is, i V( x) denotes the numer o n hanges in sign in the sequene or a given numer x, then the numer o zeros o n in [ a, ] is V( a) V( ), provided a or is not a zero o n. In this way, one an approximately ompute the eigenvalues and y repeated isetions, one an improve these estimates. This method is explained in many text ooks on numerial analysis. III. Proposed approah In this strum sequene approah, seletion o λ is very muh important. But in exiting approah, seletion o λ is random approah, hene existing approah needs more omputations to ompute the eigenvalues. In suh ases, Gershgorin theorem [4] will e useul. It stated as ollow: th P e the sum o the moduli o the elements along the a. Then Let k k row exluding the diagonal elements every eigenvalue o A lies inside or on the oundary o at least one o the irles λ a = P, k =,..., n. (5) kk k Using this theorem, we otain ounds on the real axis. The eauty o these ounds is suh that all the eigenvalue lie etween these ounds. So, we onsider Gershgorin ound λ as initial approximation in this strum sequene method. Suppose, these ounds are denoted y E and D. Bound E always remains to the right side o D on the real axis in the s-plane whatever may e the value o E and D [6]. Thus, equation () and (4) modiied as =, = D (6) 0 and kk = ( D ) ; r n (7) r r r r r IV. Numerial Example Now we alulate the eigenvalues o a given tridiagonal matrix as given elow [4]. - 0 [ A ] = - - (8) 0 - In [4], the eigenvalues are alulated using onventional method. It as ollows: 4. Conventional method: Step : Assume λ =, then rom eq. (6) and (7), the strum sequene eomes =, = λ ; = ( λ ) = ( λ ) ( λ ) 0 = ( λ ) 0 = ( λ ) (9) Step : Using strum sequene approah, we ormulate the ollowing array to get the eigenvalues. ISSN : Vol. No. Deemer 0 74

4 Tale : Existing approah o alulating eigenvalues using strum sequene From aove tale, there exist an eigenvalue, at λ = etween in (0, ) and also eigenvalue etween (, 4). We alulate atual eigenvalue y repeated isetion method. The exat eigenvalue is alulated as Similarly, y inreasing the value o λ, we an estimate other eigenvalues o the matrix. 4. Proposed approah Step : By applying Gershgorin theorem to aove matrix, we alulate ound as D=0 and E=4. It is shown in Fig.. Fig. Gershgorin irles and ounds o a system matrix Step : Using strum sequene approah, using eq. (6) and eq. (7), we ormulate the ollowing array to get the eigenvalues. ISSN : Vol. No. Deemer 0 75

5 Tale : Proposed approah o alulating eigenvalue using using strum sequene in the interval (0, ) From aove tale, there exist an eigenvalue, etween in (0, ). We alulate atual eigenvalue y repeated seant method. Similar to aove approah, the exat eigenvalue is alulated as Tale : Proposed approah o alulating eigenvalue using using strum sequene in the interval (, ) From the aove tale there exist an eigenvalue, is at λ = Tale 4: Proposed approah o alulating eigenvalue using using strum sequene in the interval (, 4) From aove tale, there exist an eigenvalue, etween in (, 4). We alulate atual eigenvalue y repeated seant method. Similar to aove approah, the exat eigenvalue is alulated as From aove tale, and 4 it is onluded that proposed approah is omputationally eiient in omparison to existing approah. Total Computation: Existing method: Bisetion method: 855 Seant method: 5 V.CONCLUSION In this paper, a omputationally eiient approah is proposed or determination o eigenvalues o a given real symmetri triangular matrix using Gershgorin theorem. In the existing method as we oserve rom the tale the initial values are randomly hosen and the strum sequene method is applied to ind the existene eigenvalues. Whereas in the proposed approah the strum sequene methods are applied at the point o intersetion o the irles to deide the existene o eigenvalues in the Gershgorin ound and then repeated seant method has een applied in these intervals to ompute the exat eigenvalues. The seant method omputes the eigenvalues exatly and takes less numer o omputations ompared with the Bisetion method whih is used in the ISSN : Vol. No. Deemer 0 76

6 existing method. The symmetri matrix is appliale to various Computer Engineering appliations like Image proessing, Pattern Reognition, Finger print Reognition and so on. Reerenes: [] L. Wang et. al, Gloal staility or monotone tridiagonal systems with negative eedak, IEEE Conerene on deision and ontrol, pp. 008,pp [] X. H. Sun et al., Eiient tridiagonal solvers on multiomputer, IEEE Transations on Computers, Vol. 4, no., 99, pp [] Y. Juang and W. F. MColl, A two-way BSP algorithm or tridiagonal systems, Future Generation Computer Systems, Vol., Issues 4-5, 998, pp [4] M. K. Jain, S. R. K. Iyengar and R. K. Jain, Numerial Methods or Sientii and Engineering Computation, Wiley Eastern Limited, 99. [5] V. S. Pusadkar and S. K. Katti, A New Computational Tehnique to identiy real eigenvalues o the system matrix A, via Gershgorin theorem, Journal o Institution o Engineers (India), Vol. 78, De. 997, pp. -. [6] Y.V. Hote, New Approah o Kharitonov and Gershgorin theorem in Control Systems, Ph. D thesis, Delhi University, 009. [7] Y. V. Hote D. Roy Choudhury, and J. R. P. Gupta, Gershgorin theorem and its appliations in ontrol system prolems, IEEE International Conerene on Industrial Tehnology, 006, pp ISSN : Vol. No. Deemer 0 77