Avalable at http://pvamu.edu/aam Appl. Appl. Math. ISSN: 93-9466 Vol. 5, Issue (June 00) pp. 67-80 (Prevously, Vol. 5, No. ) Applcatons and Appled Mathematcs: An Internatonal Journal (AAM) Developng an Improved Shft-and-Invert Arnold Method H. Saber Najaf and M. Shams Solary Department of Mathematcs aculty of Scences & Computer Center Gulan Unversty P.O. Box 94 Rasht, Iran hnajaf@gulan.ac.r shamssolary@gmal.com Receved: May 6, 009; Accepted: Aprl, 00 Abstract An algorthm has been developed for fndng a number of egenvalues close to a gven shft and n nterval [ Lb, Ub ] of a large unsymmetrc matrx par. The algorthm s based on the shft-andnvert Arnold wth a block matrx method. The block matrx method s smple and t uses for obtanng the nverse matrx. Ths algorthm also accelerates the shft-and-nvert Arnold Algorthm by selectng a sutable shft. We call ths algorthm Block Shft-and-Invert or BSI. Numercal examples are presented and a comparson has been shown wth the results obtaned by Sptarn Algorthm n Matlab. The results show that the method works well. Keywords: genvalue, Shft-and-Invert, Arnold method, Inverse, Block matrx, LDV decomposton MSC (000) No.: 65N5, 655 66
68 Saber Najaf and Shams Solary. Introducton The egenvalue problem s one of the most mportant subjects n Appled Scences and ngneerng. So ths encouraged scentsts nto ganng new methods for ths problem. or standard problems, powerful tools are avalable such as QR, Lanczos, Arnold Algorthm and etc. see Datta (99) and Saad (994). Computng the egenvalues of the generalzed egenvalue problem ( Ax Bx ) s one of the most mportant topcs n numercal lnear algebra. The shft-and-nvert Arnold method has been popularly used for computng a number of egenvalues close to a gven shft and/or the assocated egenvectors of a large unsymmetrc matrx par ( Ax Bx ). We consder the large unsymmetrc generalzed egenproblem A B, (*) where A and B are n n large matrces. An obvous approach s to transform (*) to a standard egenproblem by nvertng ether A or B but f A or B are sngular or ll-condtoned ths manner wll not work well. We are nterested n computng some nteror egenvalues of ( A, B ) n the complex plane or some egenvalues that are stuated n the nterval [ Lb, Ub ]. We wll descrbe ths problem and show a new method for solvng ths problem. One of the most commonly used technques for ths knd of problem s the shft-and-nvert Arnold method Ja and Zhang (00), whch s a natural generalzaton of the shft-and-nvert Lanczos method for the symmetrc case rcsson and Ruhe (980). When A B s nvertble for, the egenvectors of the matrx par ( A, B ) are the same as those of the matrx( A B) B. Therefore, we can run the Arnold method on the matrx ( A B) B. If the shft s sutably selected, we set C ( A B) B so the Arnold method appled to the egenproblem of the shft- and- nvert matrxc. It may gve a much faster convergence wth egenvalues n nterval ncludng shft. Instead of a fxed or constant shft, Ruhe provded an effectve technque Ruhe (994) for selectng the shft dynamcally, also can see Saber and Shams (005). The shft-and-nvert can be used when both A and B are sngular or near sngular. Snce the shft-and-nvert Arnold method for problem (*) s mathematcally equvalent to the Arnold method for solvng the transformed egenproblem, the former has the same convergence problem as the latter; t s descrbed n Secton. Ths motvates us to derve a Block Shft-and-Invert Algorthm and to develop correspondng more effcent algorthms. In secton 3, we wll dscuss on Block Shft-and-Invert Algorthm by block matrx and LDV decomposton. Then, we try to fnd egenvalues for system A B n nterval [lb, ub] by selectng a sutable shft. Secton 4 descrbes Sptarn functon n Matlab and some propertes of t. Secton 5 reports several numercal examples and compares Block Shft-and-Invert Algorthm wth Sptarn Algorthm.
AAM: Intern. J., Vol. 5, Issue (June 00) [Prevously, Vol. 5, No. ] 69. Shft-and-Invert We start ths secton wth a defnton n generalzed egenvalue problem descrbe shft-and-nvert method. A B and then Defnton.: In the generalzed egenvalue problem ( A B ), the matrx ( A B ) s called a matrx pencl. It s convenently denoted by ( A, B ). The par ( A, B ) s called regular f det( A B) s not dentcally zero; otherwse, t s called sngular. We would lke to construct lnearly transformed pars that have the same egenvalues or egenvectors as ( A, B ) and such that one of the two matrces n the par s nonsngular. We have a theorem, see Golub and Van Loan (989), Saad (988) that shows when the par ( A, B ) s a regular par, then there are two scalars, * * such that the matrx A * * B s nonsngular. When one of the components of the par ( A, B ) s nonsngular, there are smple ways that generalzed problem transfer to a standard problem. or example A B B A or BA, or when A, B are both Hermtan and, n addton B s postve defnte, we T T have B LL (Cholesky factorzaton) L AL. None of the above transformatons can be used when both A and B are sngular. In ths partcular stuaton, a shft can help for solvng the equaton... Reducton to Standard orm We know that for any par of scalars, the par ( A B B, A ) has the same egenvectors as the orgnal par ( A, B ) n Ax Bx or Ax Bx. An egenvalue (, ) of the transformed matrx par s related to an egenvalue par (, ) of the orgnal matrx par by,. Shft-and-nvert for the generalzed problems corresponds through two matrces (A, B), typcally the frst. Thus, the shft-and-nvert par would be as follows: ( I,( A B) ( B A )). The most common choce s 0 and whch s close to an egenvalue of the orgnal matrx Saad (99).
70 Saber Najaf and Shams Solary.. The Shft- and- Invert Arnold Method If the matrx A B s nvertble for some shft, the egenproblem (*) can be transformed nto the standard egenproblem A B. () A B B B. ( A B) ( ) B ( AB) B. Hence, C, () where. It s easy to verfy that (, ) s an egenpar of problem (*) f and only f (, ) s an egenpar of the matrxc. Therefore, the shft- and- nvert Arnold method for the egenproblem () s mathematcally equvalent to the standard Arnold method for the transformed egenproblem (). It starts wth a gven unt length vector (usually chosen randomly) and bulds up an orthonormal bass V m for the krylov subspace k m ( c, v ) by means of the Gram- Schmdt orthogonalzaton process. In fnte precson, reorthogonalzaton s performed whenever same sever cancellaton occurs Ja and Zhang (00), Saad (988). Then the approxmate egenpars for the transformed egenproblem () can be extracted from k m ( c, v ). The approxmate solutons for problem () can be recovered from these approxmate egenpars. The shft- and- nvert Arnold process can be wrtten n matrx form ( A B) BV V H h v e, (3) * m m m m, m m m and A B BV V H (4) ( ) m m m, th where e m s the m coordnate vector of dmenson m, Vm ( Vm, vm ) ( v, v,, vm ) s an n ( m ) matrx whose columns form an orthonormal bass of the ( m ) dmensonal krylov subspace km (, c), and H ~ m s the ( m ) m upper Hessenberg matrx that s same as H m expect for an addtonal row n whch the only nonzero entry s h m, m n the poston ( m, m ).
AAM: Intern. J., Vol. 5, Issue (June 00) [Prevously, Vol. 5, No. ] 7 Suppose that ~ (, ~ y ),,,, m are the egenpars of the matrx H m m. Then, H y y, (5) and and V y. (6) m ~ When the shft-and-nvert Arnold method uses (, ~ ) to approxmate the egenpars (, ) of the problem (), the ~ and ~ are called the Rtz values and the Rtz vectors of A wth respect to k (, ) m c. or detals, refer to rcsson and Ruhe (980). Defnng the correspondng resdual r ( A B). (7) Then, we have the followng theorem: Theorem.: ~ The resduals r~ correspondng to the approxmate egenpars (, ~ ) by the shft-and-nvert Arnold method satsfy: r h AB e y (8) * m, m m Proof: rom relatons (3), (4) and (6), we obtan
7 Saber Najaf and Shams Solary r ( A B) ( A B) V y (( AB) ( ) B) V y m m ( A B)( I ( )( A B) B) Vmy ( A B)(( A B) B I) Vmy AB V ( H I) y m m h AB e y * m, m m. 3. A Technque for Computng genvalues ( A B ) Below, we try to show the nverse matrx by LDV decomposton and block matrx. Then by selectng a sutable shft we try to fnd egenvalues for A B n specal nterval [Lb,ub]. In the last secton, we descrbed that the shft-and-nvert Arnold method for the egenproblem A B s mathematcally equvalent to the standard Arnold method for the transformed egenproblem ( A B) ( ) B ( AB) B C, or A B ( A B) B,, where s a shft. or computng ( A B), we can use block matrx method as follows: In ths method the matrx dvded n block block matrx and by applyng LDV decomposton, Datta (994) the nverse s computed M 0 0 ( A B ) M A B I A I A B C D C A I 0 S 0 I S ( D C A B ) M A 0 I 0 A A BS C A A BS. I A B 0 I 0 S CA I S CA S We can show that f the nverse of M exsts, then the matrces A and Sare nvertble. In fact, by ths decomposton nstead of fndng M we compute the nverse of L, D, and V, whch s
AAM: Intern. J., Vol. 5, Issue (June 00) [Prevously, Vol. 5, No. ] 73 much easer and faster than fndng the nverse of M drectly. In Matlab nv functon (t 3 calculates nverse matrx) requres n operatons for a matrx wth dmenson n but we can see block nverse needs only n 3 operatons. When we can fnd M setc M B and Arnold Algorthm can be used for solvng C. Lb Ub We choose. If ( ) M A B s not nvertble we set Lb Ub Ub and. So, we bsect the nterval [Lb, ub] to fnd a sutable shft. Algorthm (Block Shft-and-Invert [BSI]): Step : Input A, B, Lb, Ub ; Step : Whle ( Lb Ub) do Lb Ub (a), M A B ; (b) If M s sngular Ub ; go to (a) ; lse go to the next step; Step 3: Use Block Inverse method for computng M, M A B I 0A 0I A B, C D C A I 0 S 0 I S ( D C A B ), M A 0 I 0 A A BS C A A BS ; I A B 0 I 0 S CA I S CA S Step 4: C M * B ; Step 5: Gan egenvectors and egenvalues ( V, lm ) of matrx C by Arnold Algorthm; Step 6: or to (rank matrx) ln( ) ; lm( )
74 Saber Najaf and Shams Solary Step 7: or = to (rank matrx) Step 8: Stop. If ( Lb ln( ) Ub ) ln( ) s the egenvalue. lse ( There are not any egenvalues for nput argument ); 4. Sptarn In ths functon the Arnold algorthm wth spectral transformaton s used. [ xv, lmb, result] sptarn( A, B, Lb, Ub). Ths command fnds egenvalues of the equaton ( A B) x 0 n the nterval [ Lb, Ub]. A, B are n n matrces, Lb and Ub are lower and upper bounds for egenvalues to be sought. A narrower nterval makes the algorthm faster. In the complex case, the real parts of lmb are compared to Lb and Ub. x are egenvectors, ordered so that norm ( A xv B xv dag(lmb) ) s small. lmb s the sorted egenvalues. If result 0 the algorthm succeeded and all egenvalues n the ntervals have been found. If result 0 the algorthm s not successful, there may be more egenvalues, try wth a smaller nterval. Normally the algorthm stops earler when enough egenvalues have converged. The shft s chosen at a random pont n the nterval [ Lb, Ub ] when both bounds are fnte. The number of steps n the Arnold run depends on how many egenvalues there are n the nterval. After a stop, the algorthm restarts to fnd more Schur vectors n orthogonal complement to all those already found. When no egenvalues are found n Lb lmb Ub, the algorthm stops. If t fals agan check whether the pencl may be sngular. 5. Numercal Tests and Comparsons Sptarn Algorthm and Block Shft-and-Invert (BSI) Algorthm are tested for varous matrces by Matlab Software. All tests are performed on a Intel(R) Celeron(R) M, CPU.46 GHZ Laptop, Matlab Verson 7.5. We save egenvalues n box [ Lb, Ub ] for dfferent Matrx wth dfferent condtons. Sptarn Algorthm and BSI Algorthm are marked wth () and (), for example r shows the value result n Sptarn functon and r shows the number of egenvalues by BSI Algorthm.
AAM: Intern. J., Vol. 5, Issue (June 00) [Prevously, Vol. 5, No. ] 75 r, r, denote the number of egenvalues n nterval [ Lb, Ub ] t,t are the CPU tmes n seconds, are the smallest egenvalue n [ Lb, Ub ], are the largest egenvalue n [ Lb, Ub ] n.c falure to compute all the desred egenvalues r, r are resdual ( norm( Ax Bx ) ) for the smallest egenvalue xample : We were nterested n fndng the egenvalues of Ax Bx. A, B are sprandom, and unsymmetrc matrces of dfferent rank. Set a regon [ Lb, Ub ] wth Lb 5, and Ub 5. Dmenson r r Table : Results of xample t () s t () s r r 0x0 9 9 0.054 0.00 8.755x0-5 4.739x0-5 -0.347+0-0.347+0 -.750700 -.75070 0x0 9 9 0.0493 0.0053 9.755x0-5 8.6045x0-5 0.06+0 0.06+0 3.4984+0 3.4984+0 50x50 50 50 0.98 0.0673 9.6086x0-4 3.6083x0-4 -0.509+0.0758-0.509+0.0758-4.608+0-4.608+0 00x00 98 98 0.550 0.470.8376x0-3.88x0-3 -0.09+0-0.09+0 4.33-4.8307 4.33-4.8307 300x300-68 90 8.8775 0.363 n.c.6x0 - -0.86+0.647 0.07+0.0435 n.c 0.7588+9.8360 500x500-8 486 404.9588 49.560 n.c 8.8866x0 - -0.4+0.3783 0.006+0 n.c -.65-7.3638 000x000-80 977.7393x0 3 395.786 n.c 9.60x0 - -0.03+0.654 0.05+0 n.c -3.3883-6.865 It s seen from Table that BSI Algorthm s much more effcent than Sptarn Algorthm n all cases. or example Sptarn Algorthm fals for some matrces such as n=300, 500, 000. xample : In ths example we assume A to be an ll condtoned matrx such as Hlbert and B s Identty matrx on regon [0, ].
76 Saber Najaf and Shams Solary Dmenson Cond A r r Table : Results of xample t () s t () s r r 0x0.065x0 3 9 9 0.5635 0.3804 3.6456x0-6 3.3x0-6.0930x0-3.095x0-3 0.349 0.349 0x0.9373x0 8 9 9 0.9 0.000.3004x0-4 5.988x0-6 3.3307x0-6 3.3307x0-6 0.4870 0.4870 50x50.6060x0 9 0 39 0.684 0.0070 7.375x0-6 3.470x0-6 5.55x0-7 5.55x0-7 0.6797 0.6797 00x00 4.76x0 9 6 74 0.485 0.0694 8.9034x0-6 3.735x0-6.70756x0-7.70756x0-7 0.84 0.84 00x00.57x0 0 0 5 0.374 0.500.4683x0-5 3.5940x0-6.934x0-4.934x0-4 0.957 0.957 500x500 5.045x0 0 86 0.6536 4.90 5.043x0-5 3.7338x0-6.5535x0-5.5535x0-5 0.4056 0.4056 000x000 9.97x0 0 5 590.9654 38.0649 5.96x0-5.504x0-5.0x0-6.0x0-6 0.495 0.495 000x000 3.786x0 6 9 7.000 365.7796 8.389x0-5 8.70x0-6 3.697x0-4 3.697x0-0.5809 0.5809 We can see that BSI Algorthm works better than Sptarn Algorthm for large and ll condtoned matrces. or example we can compare columns r and r for large matrces. Numbers of egenvalues that are ganed by BSI Algorthm are more than the egenvalues that are ganed of Sptarn Algorthm n regon [0, ], for matrx wth dmenson 000 Sptarn Algorthm fnds only 6 egenvalues n regon [0, ] but BSI Algorthm fnds 9 egenvalues n ths nterval. xample 3: Has been taken from Ba and Barret (998), Consder the constant coeffcent conventon dffuson dfferental equaton u( x, y) p u ( x, y) pu ( x, y) p3u( x, y) u( x, x y y On a square regon [ 0,] [0,]. Wth the boundary condton u ( x, y) 0, where p, p and p 3 are postve constants dscretzaton by fve pont fnte dfferences on unform n n grd ponts usng the row wse natural orderng gves a block trdagonal matrx of the form ) T ( ) I A ( ) I T ( ) I ( ) I ( ) I T wth
AAM: Intern. J., Vol. 5, Issue (June 00) [Prevously, Vol. 5, No. ] 77 4 4 T, 4 where ( ) p h, ( ) ph, p 3h and h. The order of A s N n. By ( n ) takng p, p 0 p3 and B = Identty matrx, for dfferent order on the regon [5,7] we have Table 3. Dmenson r r Table 3. Results of xample 3 t () s t () s r r 9x9 3 3 0.507 0.79 5.968x0-6 4.4409x0-6 5.44 5.44 5.44 5.44 36x36 0.069 0.0043.767x0-4.6398x0-5 5.47 5.470 5.809 50.809 00x00 30 30 0.47 0.004.88x0-5.0940x0-5 5.3097 5.3097 5.990 5.990 5x5 7 75.599 0.69 5.0565x0-5.588x0-5 5. 5. 5.966 5.966 400x400 95 39 9.6975.5786 8.505x0-5 3.9x0-5 5.466 5.0000 5.9777 5.9777 900x900 99 300 37.765 8.97 4.66x0-5 5.5997x0-5 5.645 5.0579 5.9897 5.9897 600x600 6 50 57.387 6.056.6665x0-5 9.047x0-5 5.087 5.087 5.3307 5.994 500x500 73 834 369.7465 55.750 3.948x0-5.5793x0-4 5.047 5.0000 5.053 5.996 As we can see BSI Algorthm gves more egenvalues n less tme and wth hgher accuracy than Sptarn Algorthm for dfferent matrces. xample 4: Has been taken from Ba and Barret (998) Delectrc channel wavegude problems arse n many ntegrated crcut applcatons. Dscretzaton of the governng Helmholtz equaton for the magnetc feld H H k n ( x, y) H H x x x H k n ( x, y) H H. y y y
78 Saber Najaf and Shams Solary By fnte dfference leads to an unsymmetrc matrx egenvalue problem of the form C C C C H H x y B H x, B H y where C and C are fve- or- trdagonal matrces, C and C are (tr-) dagonal matrces, B and B are nonsngular dagonal matrces. The problem has been tested n the regon [0, 0]. Dmenson r r Table 4: Results of xample 4 t () s t () s r r 0x0 0 0 0.0504 0.008 7.770x0-5 6.03x0-5.369- (0.087).3690- (0.087) 8.3760 8.3760 50x50 50 50 0.0678 0.08.60x0-4.67x0-4.045- (0.70).045- (0.70) 8.9395 8.9395 00x00 00 00 0.743 0.334 5.575x0-4.6864x0-4.0038-0.087.0038-0.087 8.9634 8.9634 300x300 n.c 300 47. 3.8444 n.c.79x0-4 n.c.0004-0.094 n.c 8.9708 500x500 n.c 500 463.583 8.5093 n.c 3.354x0-4 n.c.000-0.077 n.c 8.974 We can see Sptarn Algorthm faled to compute the desred egenvalues for some matrces. The results of ths example are plotted as gure and gure. The broken lnes are shown the results of BSI Algorthm and connected lnes are shown the results of Sptarn Algorthm.
AAM: Intern. J., Vol. 5, Issue (June 00) [Prevously, Vol. 5, No. ] 79 6. Conclusons In ths paper, we have consdered Block Shft-and-Invert (BSI) Algorthm. In ths method we compute M ( A B). Ths computaton has been done by block matrx and a sutable shft. As we have shown that f we need all egenvalues n nterval [Lb, ub] or close to a gven shft for sngular matrx, the BSI Algorthm obtans them wth very hgh speed and accuracy. All numercal examples have been compared wth correspondng results from Sptarn Algorthm n Matlab. It has been shown that BSI results were much more effcent than Sptarn results. Acknowledgments The authors would lke to thank the referees for ther comments whch mproved the presentaton of the paper. RRNCS Ba, Z., Barret, R., Day, D., Demmel, J., and Dongarra (998). Test matrx collecton for non- Hermtan genvalue problems matrx market. Datta, B. N. (994). Numercal Lnear Algebra and Applcatons, ITP, an Internatonal Thomson Company.
80 Saber Najaf and Shams Solary rcsson, T. and Ruhe, A. (980). The spectral transformaton Lanczos method for the numercal soluton of large sparse generalzed symmetrc egenvalue problems, Math. Com., Vol. 35, pp. 5-68. Golub, G. H. and Van Loan, C.. (989). Matrx Computatons, The John Hopkns Unversty press USA. Ja, Z. and Zhang, Y. (00). A Refned shft- and- nvert Arnold Algorthm for large unsymmetrc Generalzed gen problems, Computers and Mathematcs wth Applcatons, Vol. 44, pp. 7-7. Ruhe, A. (994). Ratonal Krylov algorthms for nonsymmetrc egenvalue problems. Matrx pars. Ln. Alg. and ts Appl., Vol. 97, pp. 83-95. Saber Najaf, H. and Shams Solary, M. (005). Shftng algorthms wth maple and mplct shft n the QR algorthm, Appled Mathematcs and Computaton, Vol. 6, pp. 947-96 Saad, Y. (988). Varatons on Arnold s method for computng egen elements of large unsymmetrc matrces, Lnear Algebra and ts Applcatons, Vol. 34, pp. 69-95. Saad, Y. (99). Numercal methods for large egenvalue problems, Theory and Algorthms, Wley, New York.