A Comparison of Methods for Computing the Eigenvalues and Eigenvectors of a Real Symmetric Matrix. By Paul A. White and Robert R.

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1 A Cmparisn f Methds fr Cmputing the Eigenvalues and Eigenvectrs f a Real Symmetric Matrix By Paul A. White and Rbert R. Brwn Part I. The Eigenvalues I. Purpse. T cmpare thse methds fr cmputing the eigenvalues f a real symmetric matrix fr which prgrams are readily available. The cmparisn was made n the basis f cmputing time and accuracy. II. Methds Tested. Three cdes were selected frm SHARE which represented three different methds. A. Jacbi. SHARE distributin 75 by MIT cmputing lab (FRTRAN). This is the riginal versin f the Jacbi methd [1] in which plane rtatins are used t prduce zers in all ff-diagnal psitins using the maximum ff-diagnal element as a pivt at each step. This is prbably slwer but mre accurate than the "serial" versin which pivts n the ff-diagnal elements in sequence whether r nt they are large. B. Givens. SHARE distributin 664 (AN F22) by the AEC Cmputing and Applied Mathematics Center, NYU (FRTRAN). This is the methd devised by W. Givens in 1954 at the ak Ridge Natinal Labratry [2] in which plane rtatins are used t prduce a tri-diagnal matrix with the eigenvalues as the riginal matrix. The rts f the matrix are cmputed by the use f Sturm's sequence derived frm the tri-diagnal matrix. C. Husehlder. SHARE distributin # 1385 (AN 22) by the AEC Cmputing and Applied Mathematics Center, NYU (FRTRAN). This is a variatin f the Givens methd in which the tri-diagnal matrix is prduced by an rthgnal transfrmatin that des nt depend n plane rtatins. It is described in detail in [3] by J. H. Wilkinsn. The Sturm sequences are used as befre t btain the rts frm the tri-diagnal matrix. Series verflw and/r underflw prblems ccurred in the earlier experiments because a scaling device was nt incrprated int the cdes f methds B and C. Bth methds were mdified t incrprate this feature in the decks used in the experiments described belw. Mdified listings are available frm the authrs. III. Descriptin f the Tests. Nine test matrices were used, eight 32 X 32 and ne 8X8. The eigenvalues f each were cmputed by all three methds and the accuracy and time f cmputatin were cmpared. The eight 32 X 32 matrices were cnstructed by using a tensr prduct f five 2X2 matrices. This had the advantage f being able t generate the test matrices in the machine by inserting nly five data cards. It was als pssible t calculate in a very easy way the crrect eigenvalues f these matrices. The 8X8 matrix was the well-knwn Rsser matrix m._ Received July 23, Revised Nvember 4,

2 458 PAUL A. WHITE AND RBERT R. BRWN Fr the eight 32 X 32 tensr prduct matrices, the eigenvalues were als cmputed by Jacbi prcedures using duble precisin arithmetic applied t the single precisin matrices btained frm the tensr prduct calculatins. These differed frm the exact eigenvalue by less than ne in the eighth place. This wuld indicate that n significant runding takes place in the tensr prduct r nrmalizing calculatins. IV. Summary f Results. The table belw gives the fllwing data fr each f the eight tensr prduct matrices: the Euclidean nrm, V^T/tJ; the true maximum and minimum eigenvalues; the maximum and average errr in the cmputed eigenvalues fr each methd; the cmputing time in minutes fr each methd. In the case f the Rsser matrix, all f the true eigenvalues and cmputed errrs in eigenvalues are given tgether with the cmputatin times. The data indicate that the Husehlder methd is cnsistently faster and mre accurate than the Givens methd. This is the cnclusin reached by rtega [5] based n mre extensive tests. In the Jacbi methd the times and accuracies are definitely crrelated as is t be expected, i.e., the faster the cmputatin, the mre accurate the result since the fewer number f arithmetic peratins prduces less rund ff errrs. Whenever the cmputatin time fr the Jacbi methd is less than that f the Husehlder methd the results are mre accurate. The Jacbi times and accuracies vary cnsiderably frm ne matrix t anther while the results fr the Husehlder methd remain rather unifrm. n the average the Jacbi times are a little lnger and the accuracies a little less than thse f the Husehlder methd. The results fr the Rsser matrix did nt differ appreciably frm ne methd t anther. The Husehlder methd is definitely better than the Givens and being smewhat faster and mre accurate n the average as well as being mre cnsistent than the Jacbi methd, it is t be preferred. V. Dcumentatin. The fllwing data, cdes, etc. were used in the develpment f the results in these experiments. A. The Eight Tensr Prduct Matrices. Fr each f the eight test matrices, each f the five 2X2 matrices used t frm the 32 X 32 tensr prduct is given tgether with its eigenvalues. B. The Crrect Eigenvalues f the Eight Tensr Prduct Matrices. In each f the eight cases, the crrect eigenvalues were cmputed by frming the tensr prduct f the eigenvalues f the five 2X2 matrices. A, B, C, D, and E designate the five pairs f eigenvalues fr the five 2X2 matrices in each case. C. Cmputatin f the Eigenvalues f the Eight Tensr Prduct Matrices by Givens' and Jacbi's Methds. The eigenvalues f the eight tensr prduct matrices are cmputed by Givens' and Jacbi's methds. The input cnsists f the five 2X2 matrices in each case; each 2X2 matrix is n ne card cnsisting f tw vectrs, ne fr each rw. D. Cmputatin f the Eigenvalues f the Eight Tensr Prduct Matrices by Husehlder's Methd. This is the as C, with Husehlder's methd instead f Givens' and Jacbi's.

3 THE EIGENVALUES F A REAL SYMMETRIC MATRIX 459 tu. S S ri*-!!c «3 W3 t > ^f M «5 P B H n & K 3 ^H s.a S3 TtHTtH(NCMt> ^ ^cncm^r^t^i l (M CM CM (M ccicii r)htt<!m^rtt<c(m ^immhtf í r- Tt" hnhi-(^h cc lllll+ + + r-s Kl 3 ec-*5c^ i iirthcc l L^T-H(M>(M i I ^ i I CM t^ CM 7-HCMCM^HrH s.s îijnnq 1-ICMCMl-1 7 ( 7-C 1-I 7-H 1 i (M M 3?. p ce H > t^^ctt<cmc 1CDNH1M -*T7}<Cr^T^TtlCTt< Jr^rfit^ c r- c» c i (M i l i li i I I I I++.a. g H NHHHMN cttiî^cdccc HN>NCN1 CMlCMCMCM l l M l (N tí îiii-iitt<t^a5 ) H! " th (NtH - í- < y, ci 2H CDtNTtlTH «7ÍC 1 5N NCJIÍtHiH wrhn^ln C35(M^I(M^CM ICHH1 (NHrtîNC CMCMCM"# C i i IÍ3 s -r 33 s CM CM 1-1 t CC re Q i i Ci -V cr: (M 3 S ^^iitfi-1!t)h5 5»i-II>^C» -* (M 3 N ^ th l C3ÍHH rt<^±l >.TjHi l"l t~icr--it~ c ih CM * «5 N CC CD > 3 í

4 46 PAUL A. WHITE AND RBERT R. BRWN E. Cmputatin f the Eigenvalues f the Rsser Matrix by All Three Methds. The eigenvalues f the Rsser matrix are cmputed by the three methds. The matrix as given n the input cards has eight rw vectrs. F. Prgram: 32 Eigenvalues frm Five 2X2 Matrices. This is the listing fr the prgram that prduced the data in B. G. Multimethd Eigenvalue Prgram. This is a listing fr the prgram that prduced the data in C. The five 2X2 matrices fr each case are the data. The prgram cmputes in a subrutine the 32 X 32 tensr prduct matrix which is used as data fr each f the eigenvalue cmputatins by Givens' and Jacbi's methds which are subrutines in the main prgram. H. Tensr Prduct. This rutine frms the tensr prduct f N, 2 X 2 matrices and is a subrutine in the main prgram G. I. Givens Methd. This is the Givens methd fr cmputing the eigenvalues f a real symmetric matrix. It is written as a subrutine f the main prgram G. J. Jacbi Methd. This is the Jacbi methd written as a subrutine f the main prgram G. K. Husehlder Methd. This is the Husehlder methd written as a subrutine fr the main prgram G. The listing f G as given here des nt include this methd. A revised versin was used in which it was added as "methd 3." It has als been mdified frm the SHARE versin t prevent ver- and underflw. Part II. The Eigenvectrs I. Purpse. The purpse was the as in Part I, except this time bth eigenvalues and eigenvectrs were cmputed and times and accuracies cmpared. II. Methds Tested. The three prgrams were used as befre. Each prgram prvides a methd fr the cmputatin f the eigenvectrs. The prcedure in the Jacbi prgram is the bvius ne f cmputing the prduct f the rtatins. In the prgrams that use the Givens and Husehlder prcedures t btain the tri-diagnal matrix, the methd is used. Wilkinsn's methd [6] is applied t find the eigenvectrs f the tri-diagnal matrix, and then the reflectins are applied in reverse rder t btain the riginal eigenvectrs. We shall call the Givens prcedure fllwed by the abve Methd A and Husehlder's prcedure fllwed by the abve Methd B. III. Descriptin f Tests. The eight tensr prduct matrices were used tgether with a 21 X 21 matrix which was used by Wilkinsn [6] t illustrate hw catastrphic errrs may arise in cmputing the vectrs f a tri-diagnal matrix. IV. Summary f Results. Each matrix has 32 vectrs crrespnding t the 32 eigenvalues and each vectr 32 cmpnents fr a ttal f 124 cmpnents per matrix. Hwever, the vectrs f a tensr prduct matrix are simply tensr prducts f the vectrs f its five 2X2 matrices; hence a relatively smaller number f different magnitudes is really invlved. In fact each f the 32 vectrs uses exactly the set f numbers fr its cmpnents and this set varies frm 12 t 18 different numbers. Thus there are at mst 18 different cmputed numbers invlved in the 124 cmpnents f any matrix. The next table shws the number f places f

5 THE EIGENVALUES F A REAL SYMMETRIC MATRIX 461 accuracy that are crrect in each f the nrmalized unit vectrs all f the time. Thus it is the maximum errr in a cmpnent f a vectr except that an ccasinal errr in the last accepted place is disregarded. The smaller the number f a vectr, the larger the magnitude f its eigenvalue. The ttal cmputatin time is als given fr each methd. Vectrs crrespnding t multiple eigenvalues are mitted in the table since n simple reasnable basis f cmparisn was knwn. The cmputatin time is less fr Methd B than fr Methd A, but the savings in time is in the eigenvalue cmputatin and nt in the vectr part since the tw methds agree in this part. Thus the percentage time saving is less fr the entire cmputatin than fr the eigenvalue cmputatin alne. The accuracy in the vectr cmputatin is essentially the even thugh the Methd B vectrs are based n slightly mre accurate eigenvalues. The vectrs crrespnding t the middle valued eigenvalues are mre accurate than thse at the tw ends. This is characteristic f the methd used and the distributin f the eigenvalues f the matrices used. Cmparing Methd B with the Jacbi cmputatins, we find a reversal f the results fr the eigenvalues. Althugh the times are lnger fr the Jacbi methd (.29 n the average cmpared with.22 fr Methd B), the vectrs are significantly better in almst all cases. Furthermre, the Jacbi vectrs are unifrmly gd regardless f the magnitude f the crrespnding value and nly vary frm 6 t 7 places f accuracy depending n the speed f cmputatin. These results led t the cnsideratin f ne mre matrix the ne referred t in [6]. This matrix was invented t illustrate hw very significant errrs may result frm the calculatin f the eigenvectrs in the A r B Methd if extreme care is nt taken The matrix is the tri-diagnal ne ij The eigenvalues ccur in pairs with equal magnitudes and ppsite signs tgether with. The crrespnding eigenvectrs have cmpnents with equal magnitudes. Wilkinsn gives nly the maximum eigenvalue and crrespnding vectr, the cmputed values f which are cmpared in the table belw. These data shw that the Jacbi vectrs are definitely better althugh the A and B Methd vectrs are much better than in the case f the tensr prduct matrices. Thus the catastrphic errrs that may ccur did nt, but the cnclusin seems t be that the Jacbi methd is t be preferred if the vectrs are required.

6 462 PAUL A. WHITE AND RBERT R. BRWN i CC (M HH "* --H lc (NiiN (M -* <M M "* <M (M (N (NhM (M (N iß (M (M C N c(m IM (M (MIM ** "* (MctN MN c c N (M <M M (M ^* CD"* ^* ^ M C N M cc ^ "* ißiß tm ím (M M c c-* -* ** Tf< -* ißiß (N (M NN *» C M t cc ^ > c (M (N ** c CI - CM ccn -* "* cc -*N t as lcûiq l -M li Iß iß CD iß c i c c "7NtJ -g -ÇJ ja» CD - N-TJ i iß»ß iß c K»ßciß ifjti c c c iß iß iß» Iß iß CD «=Ng g 71 3t CQ.^ cc^'s a iß P, CD X ~- ^N 4) g en tf N. >> "cs I * iß c iß»» g g iß c iß -* c-* ißiß IC» I r^ '* -* ** l T ^ N t > ** iß TTl N c c ** ccc iß CM Til N CM * N-7f l N cc '* II t^ N CM Cd * ^t* -*N-* t*n CM CM ^t< "* CMNCM * CM C N US * c-* cc * CM CIN CMNCM CM N C CMN c cc CM * rji C CM Tji CM C CM "*N CMNCM CM < «< m< &*,< «, «*! «< P3 «! pa «; «* i (M ** iß.s^-í.aí ^.scs'scs.aí'í.aí'sí b-cs-í x^--s * $ % ^ÜCJüfHÜcU^fpüCulHÜ^CúU çp',_ CD «CD'm CD «CVJ "t-. CD «CD cs^^^ á^i-5^ SÄ'-s^ 3^i-5 cs^h-ss cas^-ís cs^^ñs cö^i-ss

7 THE EIGENVALUES F A REAL SYMMETRIC MATRIX 463 Wilkinsn Matrix Vectrs Crrespnding t (±) MAX Eigenvalue Cmpnent. True Methd A & Methd B Jacbi Cmpnent True Methd A & Methd B Jacbi V. Dcumentatin. All f the prgrams are the as befre in which indicatrs are set t prduce vectrs as well as values. L. Eigenvectrs f all different 2X2 matrices invlved in any f the eight tensr prduct matrices were cmputed here. Actually they were redne by hand t btain eight crrect digits. M. The eigenvectrs f the eight tensr prduct matrices were cmputed by using H. The input cnsists f five 2X2 matrices fr each case in which each matrix cnsists f tw eigenvectrs f the crrespnding 2X2 matrix fr its rws. The rws f the tensr prduct f five such matrices are the vectrs f the crrespnding tensr prduct matrix. N. The eigenvalues and vectrs are cmputed by the three methds using C set t cmpute vectrs as well as values with the additinal subrutine D included.. The eigenvalues and vectrs f the Wilkinsn matrix given as 21 rw vectrs are cmputed using the rutine as that used in N. Aerspace Crpratin University f Suthern Califrnia Ls Angeles, Califrnia 1. C. G. J. Jacbi, "Ein leichtes Verfahren, die in der Therie der Säkularstörungen vrkmmenden Gleichungen numerisch aufzulösen," /. Reine Angew. Math., v. 3, 1846, p Wallace Givens, Numerical Cmputatin f the Characteristic Values f a Real Symmetric Matrix, ak Ridge Natinal Labratry Reprt RNL J. H. Wilkinsn, "Husehlder's methd fr the slutins f the algebraic eigen prblem," The Cmputer Jurnal, v. 3, p J. B. Rsser, C. Lanczs, M. R. Hestenes & W. Karush, "Separatin f clse eigenvalues f a real symmetric matrix," J. Res. Nat. Bur. Standards, v. 47, 1951, p James M. rtega, An Errr Analysis f Husehlder's Methd fr the Symmetric Eigenvalue Prblem, Stanfrd Univ. Appl. Math, and Statistics Lab., Tech. Reprt N J. H. Wilkinsn, "The calculatin f the eigenvectrs f cdiagnal matrices," The Cmputer Jurnal, 1958, p

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