LIBRARY OF THE 
 
 UNIVERSITY OF ILLINOIS 
 
 AT URBANA-CHAMPAIGN 
 
 510.84- 
 
 Vfir 
 no. 403-4-08 
 cop. 2 
 
 
The person charging this material is re- 
 sponsible for its return to the library from 
 which it was withdrawn on or before the 
 Latest Date stamped below. 
 
 Theft, mutilation, and underlining of books are reasons 
 for disciplinary action and may result in dismissal from 
 the University. 
 To renew call Telephone Center, 333-8400 
 
 UNIVERSITY OF ILLINOIS LIBRARY AT URBANA-CHAMPAIGN 
 
 Ann 7 KC\ 
 
 DEC 3 RED 
 
 L161— O-1096 
 
Digitized by the Internet Archive 
 in 2013 
 
 http://archive.org/details/testmatricesi403fosd 
 
■1/ 
 
 Report No. k03 
 
 000-1^69-0166 
 
 TEST MATRICES I 
 
 by 
 
 L. D. Fosdick 
 Y. J. Kim 
 
 June 1970 
 
 JHE LIBRARY OF THI 
 
 JUL 3 1 ' 
 
 UNIVERSITY OF ILLINOIS 
 

 
 Report No. 1+03 
 
 TEST MATRICES I* 
 
 by 
 
 L. D. Fosdick 
 Y. J. Kim 
 
 June 1970 
 
 Department of Computer Science 
 University of Illinois 
 Urbana, Illinois 6l801 
 
 Supported in part by the Atomic Energy Commission under grant 
 U. S. AEC AT(ll-l)lU69. 
 

 The following material represents the results of our work 
 this year in collecting information on test matrices. It is self- 
 explanatory except for the definition of condition numbers. The 
 notation used here is defined as follows: 
 
 CCjJ-llAMj llA- 1 !^ 
 
 where 
 
 | I A I L = max { Z la..} 
 
 | Ja[ L = {maximum eigenvalue of A*A} 
 
 ||a|L = il t |a. .| 2 } 1/£ . . 
 
 "e . a xji 
 
 ?or singular matrices we have used 
 
 C(R) = | Xmax| /| Xmin| (|Xmin| ^ 0) 
 NOTE: |Xmin| is the modulus of the smallest nonzero eigenvalue 
 
 The identification number code is to be interpreted 
 as follows : 
 
 x l x 2 x 3 x k x 5 x 6 
 
 1/2 
 
 fO real matrix 
 x 1 = { 1 complex matrix 
 K.2 either 
 
 fO Hermetian 
 
 Cp = i 1 non-aermetian 
 
 U either • 
 
 rO singular 
 x = ^ 1 ncn-singular 
 V.2 either 
 
 X' x X/- = sequence number 
 
 c -j_ x 2 x 3 = 999 designates the definition of a wide class 
 of test matrices, or statements of useful 
 properties for generating test matrices. 
 
IDENTIFICATION: OOlOOl 
 
 MATRIX : A ( I , J )= 2 
 A( I t J )=-l 
 A( I ,J )= 
 FX AMPLE (N=5) 
 
 IF I=J, 
 
 IF ABS( I -J ) = 1 
 
 OTHERWISE. 
 
 A = 
 
 2 
 -1 
 
 
 
 
 -1 
 
 2 
 
 -1 
 
 
 
 
 
 
 -1 
 
 2 
 -1 
 
 
 
 
 
 
 -1 
 
 2 
 -1 
 
 CHARACTERISTICS: REAL* SYMMETRIC NON-SINGULAR, 
 POSITIVE DEFINITE , TRIDI AGONAL. 
 
 INVERSE: AINV(I,J)= I* ( N+l-J ) / ( N+l ) IF I.LE.J, 
 
 A I N V d , J ) = A I N V ( J d ) . 
 EXAMPLE (M=5) 
 
 5 4 3 2 1 
 
 4 8 6 A 2 
 
 AINV=d/6)* 3 6 9 6 3 
 
 2 4 6 8 4 
 12 3 4 5 
 
 EIGENVALUES: EIVAL(I>= 2* ( 1-COS ( I *P I / ( N + 1 ) ) ) , 
 
 FOR 1=1,2 
 
 . »N 
 
 EXAMPLE (M=5) 
 
 
 
 
 EIVAL ( 1 )= 0.26794 
 
 91924 
 
 31122 
 
 70650 
 
 EIVAL(2)= 1.00000 
 
 00000 
 
 00000 
 
 00000 
 
 EIVAL(3)= 2.00000 
 
 00000 
 
 00000 
 
 00000 
 
 EIVAL(4)= 3.00000 
 
 00000 
 
 00000 
 
 00000 
 
 EIVAL(5)= 3.73205 08075 68877 29353 
 
 EIGENVECTORS: EIVEC(I,J)= 
 
 SIM I*J*PI/(N+1) ) 
 
 FOR I , J=l ,2, . .. »M. 
 
 FXAMPLF ( N : 
 
 = 5 ) 
 
 
 
 
 
 
 ElVECd. 
 
 d) = 
 
 0. 
 
 .50000 
 
 00000 
 
 00000 
 
 00000 
 
 EIVEC(2 
 
 ,1 ) = 
 
 0, 
 
 .86602 
 
 54037 
 
 84438 
 
 64675 
 
 EIVFC(3. 
 
 rl) = 
 
 1, 
 
 ,00000 
 
 00000 
 
 00000 
 
 00000 
 
 EIVEC (4 
 
 rl) = 
 
 0, 
 
 ►86602 
 
 54037 
 
 84438 
 
 64675 
 
 EIVEC(5. 
 
 d ) = 
 
 0, 
 
 ,50000 
 
 00000 
 
 00000 
 
 00000 
 
 EIVEC ( 1 
 
 ,2) = 
 
 0, 
 
 ,86602 
 
 54037 
 
 84438 
 
 64675 
 
 EIVEC(2. 
 
 >2) = 
 
 0, 
 
 ,86602 
 
 54037 
 
 84438 
 
 64675 
 
 EIVEC (3 
 
 ,2 ) = 
 
 0, 
 
 ►00000 
 
 00000 
 
 00000 
 
 00000 
 
 EIVEC (4. 
 
 >2)=- 
 
 -0, 
 
 ,86602 
 
 54037 
 
 84438 
 
 64675 
 
 EIVEC(5. 
 
 ,2 )=■ 
 
 -0, 
 
 ,86602 
 
 54037 
 
 84438 
 
 64675 
 
 EIVEC(1. 
 
 >3) = 
 
 1, 
 
 ,00000 
 
 00000 
 
 00000 
 
 00000 
 
 EIVEC (2 
 
 »3) = 
 
 0. 
 
 ,00000 
 
 00000 
 
 00000 
 
 00000 
 
 EIVEC (3. 
 
 r3)=- 
 
 -1, 
 
 ,00000 
 
 00000 
 
 00000 
 
 00000 
 
 EIVEC (4 
 
 r3) = 
 
 0, 
 
 ►00000 
 
 00000 
 
 00000 
 
 00000 
 
 EIVEC (5. 
 
 ,3) = 
 
 1. 
 
 ,00000 
 
 00000 
 
 00000 
 
 00000 
 
 EIVECQ 
 
 .4) = 
 
 0, 
 
 ,86602 
 
 54037 
 
 84438 
 
 64675 
 
 EIVEC (2. 
 
 >4)=- 
 
 -0, 
 
 ,86602 
 
 54037 
 
 84438 
 
 64675 
 
 EIVEC (3 
 
 .4) = 
 
 0, 
 
 ►00000 
 
 00000 
 
 00000 
 
 00000 
 
 E IV EC (4. 
 
 r4) = 
 
 0, 
 
 ,86602 
 
 54037 
 
 84438 
 
 64675 
 
 EIVEC ( 5 
 
 r4)=- 
 
 -0, 
 
 .86602 
 
 54037 
 
 84438 
 
 64675 
 
 EIVEC (1. 
 
 ,5) = 
 
 0. 
 
 ,50000 
 
 00000 
 
 00000 
 
 00000 
 
FI VFC ( 2 ,5 ) =-0.86602 54037 84438 64675 
 
 EIVFC(3,5)= 1.00000 00000 00000 00000 
 
 FIVFG (4,5 )=-0. 86602 54037 R4438 64675 
 
 FI\/EC(5,5) = 0.50000 00000 00000 00000 
 
 D E T F R M I M A M T : f ) F T ( A ) 
 
 M+l 
 
 CONDI T 
 C ( 1 ) 
 
 C ( 2 ) 
 C (F ) 
 
 EXAM 
 
 M 
 
 5 
 
 20 
 
 40 
 
 60 
 
 HO 
 
 100 
 
 ion 
 
 = s 
 
 + 
 P L F 
 
 Ml J MB 
 ( N + 1 ) 
 (M + l ) 
 
 i-cn.s 
 
 ORT( ( 
 5 3*N* 
 S 
 
 C( 1 
 
 1 .8 
 
 2 .2 
 R.4 
 1 .8 
 3.2 
 5.1 
 
 FRS: 
 
 **2-l )/2 FOR M FVEM, 
 
 **2 )/2 FOR M ODD, 
 
 ( N*PI/(N+1 ) ) ) /( 1-C0S( PI / (M+l ) ) ) , 
 
 6 * N- 2 ) * ( 2 * M * * 6+ 1 2 * M * -5 + 35 * M * * 4+60 * M * * 3 
 
 *2 + 18*IM) /ISO) / (M+l ) . 
 
 ) 
 0(1 ) 
 
 0(2 ) 
 0(2 ) 
 6(3) 
 8(3) 
 
 0(3) 
 
 C( 2 ) 
 
 39 ( 1 ) 
 78(2 ) 
 8 1(2) 
 51(3) 
 
 6 6(3) 
 13(3) 
 
 C( F ) 
 2.07( 1 ) 
 5.06( 2 ) 
 2.74(3) 
 7 ,42( 3) 
 1.51(4) 
 2.63(4) 
 
 REMARKS 
 01 SCR 
 
 THIS 
 
 'ATR I 
 WHEN 
 RFPRF 
 WHER F 
 ( ; - X C L 
 THF I 
 
 all n 
 
 (0, 4 
 
 APPRf) 
 4 . 
 
 THIS 
 F F . 
 
 I I 
 SJ ^'CF 
 THI 
 
 EOijAT 
 SIZF 
 ITI 
 1 
 PFRFO 
 
 (1,1) 
 
 • 1 ) 
 
 : MATRICES OF THIS TYPE ARISE IN THF 
 FTIZA7I0N OF THE HFAT EQUATION (REF. 1). 
 MATRIX IS A SPECIAL CASE OF A WIDE CLASS OF 
 CFS DESCRIBED BY GFAR (REF. 4). 
 
 M+l IS A POWER nF 2 THE INVERSF CAM BE FXACTLY 
 SENTFD IN A BINARY MACHINE PRPVIDFD N.LT.2**L 
 
 L IS THF NUMBER DF BITS IN THF MANTISSA 
 UDING SIGN). NOTICE THAT THF MAXIMUM ELEMENT OF 
 MVFRSF IS LFSS THAN M. 
 
 F THE EIGENVALUES ARF CONTAINED IN THE INTERVAL 
 ). AS M INCRFASFS THF SMALLEST FIGENVALUF 
 ACHES AND THE LARGEST EIGENVALUE APPROACHES 
 
 X HAS BEEN USED FREQUENTLY IM TESTING 
 
 MATRI 
 2,3) 
 
 L U OF 
 ■■ I i 
 THF 
 
 W I I. L 
 1.1] OF 
 
 1,1). 
 
 COMPOSITION OR GAUSS FL I MI NATION IS 
 
 DM THIS MATRIX MO ROW EXCHANGFS ARE MAOE, 
 
 COMPUTED PIVOTS ARF ALWAYS MAXIMUM. 
 
 I IS F OF A AS A I'EST MATRIX IN A LIMFAR 
 OUTIMF WHICH USES PARTIAL PIVOTING FOR 
 NOT I'EST ThF ROW EXCHANGE PORTION OF THF 
 
 POSITION OR GAUSS FLIMIMATIOM IS 
 
 II rHE MATRIX AIMV IT IS ALSO TRUE THAT MO 
 
 GFS ARF MADE. M0T1CF THAT THF MINOR OF THF 
 
 i MP THE MAT < IX OBTAINED BY PFR FORM IMG 
 
 P HI- GAUSS FLIMIMATIOM ON THF MATRIX 
 
 I) IS (M+l )AIMV( ORDER N- 1 ) AND THAT 
 il .A [NV( I , 1 ) FOR AMY I AND M. 
 
 . : 
 ]. P.O. i HUGH 00 1 SHAW, FOS., HANDBOOK 
 
 1, P123, MCGRAW-HILL, YR1967. 
 , R.W. SI IUGH I DN AND MARJOR I F P. 
 LIFTZKF, '. iF SFVP RAI ME THODS FOR 
 
 MF 
 
3. 
 
 4. 
 
 IDENTIFICATION: 00100 2 
 
 MATRIX: A(I,J)= 1 IF A6S(I-J)=1, 
 A(N,N )=-l 
 
 A ( I t J )= OTHERWISF. 
 EXAMPLF (M=5) 
 
 10 
 10 10 
 A= 1 1 
 10 1 
 1-1 
 
 CHARACTERISTICS: REAL, SYMMETRIC, NON-SINGULAR, 
 TRIDIAGONAL. 
 
 INVERSE: N = EVEN; AI NV( I , J ) = ( -1 ) ** ( I NT ( I /2 ) 
 
 +int( ( j-i )/2 ) ) if i is odd 
 
 G I . L E . J , 
 AINV(I,J)=0 IF I IS EVEN & I.LE.J, 
 AINV( I , J )=AINV( J , I ) . 
 N=ODD; AINV( I , J )=(-l)**( IMT(I/2)+INT(J/2)+l) 
 
 IF I IS ODD £ I .LE.J, 
 AIMV(I,J)=0 IF I IS EVEN £ I.LE.J, 
 AINV( I,J)=AINV(J,I ). 
 EXAMPLE ( N = 5 ) 
 
 -1 1 1-1 -1 
 
 10 
 
 AINV= 10-111 
 
 -10 10 
 
 -1 1 0-1 
 
 EIGENVALUES : EI VAL ( I )=2*C0S( 2* I *P I / ( 2*N + 1 ) ) 
 
 FOR 1=1,2, ...,N. 
 EXAMPLE (IM = 5) 
 
 EIVAL(1)= 1.68250 70656 62362 33772 
 EIVAL(2)= 0.83083 00260 03772 85106 
 EI VAL (3 )=-0. 28462 96765 46570 28089 
 EIVAL (4 )=-l. 30972 14678 90570 12811 
 EIVAL (5)=-l. 91898 59472 28994 77978 
 
 EIGENVECTORS: EIVEC(I,J)= S IN ( 2*I*J*P I / ( 2*NI+1 ) ) 
 
 FOR I ,J = 1,2,...,N. 
 EXAMPLE (M=5) 
 
El VEC ( 1tD = 
 EIVEC(2t1 ) = 
 EIVEC(3»1)= 
 EIVEC(4,1 )= 
 EIVEC ( 5,1 >= 
 FIVEC(1,2)= 
 EIVFC(2,2 ) = 
 EIVEC (3,2)= 
 EI VEC (A, 2 ) = 
 EIVFC(5,2)= 
 FIVEC(1t3)= 
 EI VEC (2.3)= 
 EIVEC(3 t 3)= 
 E I V E C ( 4 , 3 ) = 
 FIVEC(5,3)= 
 EIVEC (1»4)= 
 EIVEC (2,4)= 
 EIVEC (3, 4)= 
 EIVEC (4, 4) = 
 EIVFC (5, 4) = 
 EIVEC (1,5)= 
 FIVEC(2,5) 
 EIVEC(3,5 ) 
 EIVEC (A, 5) 
 F1VEC ( 5,5 ) 
 
 determinant: he 
 
 0.54064 
 0.90963 
 0.98982 
 0.75574 
 0.28173 
 0.90963 
 0.75574 
 -0.28173 
 -0.98982 
 -0.54064 
 0.98982 
 -0.28173 
 -0.90963 
 0.54064 
 0.75574 
 0.75574 
 ,-0.98982 
 = 0.54064 
 = 0.28173 
 =-0.90963 
 = 0.28173 
 =-0.54064 
 = 0.75574 
 =-0.90963 
 = 0.98982 
 
 08174 
 
 19 9 5 3 
 
 14418 
 
 95743 
 
 2556H 
 
 19953 
 
 95743 
 
 25568 
 
 14418 
 
 08174 
 
 14418 
 
 25568 
 
 19953 
 
 08174 
 
 95743 
 
 95743 
 
 14418 
 
 08174 
 
 2 5 5 6 8 
 
 19953 
 
 2 5 5 6 8 
 
 08174 
 
 95 7 43 
 
 19953 
 
 14418 
 
 55597 
 
 54518 
 
 80932 
 
 54258 
 
 41429 
 
 54518 
 
 54258 
 
 41^29 
 
 80932 
 
 55597 
 
 80932 
 
 41429 
 
 54518 
 
 55597 
 
 54258 
 
 54258 
 
 80932 
 
 55597 
 
 41429 
 
 54518 
 
 41429 
 
 55597 
 
 54258 
 
 54518 
 
 80932 
 
 58211 
 
 37141 
 
 73238 
 
 28377 
 
 69771 
 
 37141 
 
 28377 
 
 69771 
 
 73238 
 
 58211 
 
 73238 
 
 69771 
 
 37141 
 
 58211 
 
 28377 
 
 28377 
 
 732^8 
 
 58211 
 
 69771 
 
 37141 
 
 h9771 
 
 5 8 211 
 
 28377 
 
 37141 
 
 73238 
 
 T(A) = (-1) — <1 + I,X,T( < N " 1)/2)) 
 
 CONDITION NUMBERS 
 EXAMPLES 
 
 N 
 5 
 20 
 4 
 60 
 HO 
 100 
 
 C ( 1 ) 
 
 1 .00 ( 1 ) 
 4.00(1 ) 
 8.00( 1 ) 
 1.20(2) 
 1.60(2) 
 
 2 .00(3 ) 
 
 C(2) 
 6.74 
 
 ,60( 1) 
 ,15(1) 
 ,70(1) 
 .02(2) 
 .28(2) 
 
 C(E) 
 
 1.16(1) 
 
 9 .05( 1 ) 
 
 2.55(2) 
 
 4.67(2) 
 
 7.18(2) 
 
 1.00( 3) 
 
 THE INVERSE OF THIS 
 (REF. 1 ) FOR TESTING 
 THF elEMFMTS OF THIS 
 
 MATRIX HAS BEEN USED *Y 
 EIGENVALUF ALGORITHMS. 
 MATRIX AND ITS INVERSE 
 
 REMARKS 
 FRANK 
 ALL OF 
 ARE 1 » 
 TIOM S 
 TION '/» ILL 
 I r F G E R S . 
 
 riF THE ELEMENTS 
 _, D AT FVFRY 
 
 ™ F FLF - MEM DFCOMPOSITION OF 
 
 TH p fLFMHM 5 nr inio . ir-, ..-- amcc ci CMTMA- 
 
 o ,m npmMpnSITinM AND GAUSS ELFWima 
 -1 OR 0. LU DECUMHuai i i SUBSTITII- 
 
 IN THE LU DECOMPOSITION ARE 1, 
 INT ERMEOIATF STAGE OF 
 REMAIN FOIJAL 
 
 rHls [S rRUl FOR 
 
 THF > nF AINVj, , MINAT I0N 
 
 LU DECOMPOSITION OR^GAUSS^L IMINAT ^ 
 
 1, -1 OR 
 
 AND FOR 
 
 Tv^TtnEDBY PERFORMING 
 I' «*"" THF MATR!X 
 
 r BE EXCHANGED 
 
 
 PAIRS 
 GAD 
 
 IS 
 
 THE (liD 
 THE 
 AINV 
 
 IN 
 
 I 3 , 4 ) » 
 
... AND IN GENERAL (2*1-1,2*1). 
 
 NO ROW EXCHANGES ARE REQUIRED IN THE COURSE OF GAUSS 
 
 ELIMINATION OF AINV WHEN PARTIAL PIVOTING FOR SIZE 
 
 IS USED. 
 
 ALL OF THE EIGENVALUES ARE LOCATED IN THE INTERVAL 
 
 (-2, 2). AS N INCREASES, THE MINIMUM EIGFMVALUE 
 
 APPROACHFS -2, AND THE MAXIMUM EIGENVALUE APPROACHES 
 
 2. 
 
 REFERENCES : 
 
 1. WERNER L. FRANK, COMPUTING EIGENVALUES OF 
 COMPLFX MATRICFS 8Y DETERMINANT EVALUATION 
 AND BY METHODS OF DANILFWSKI AND WIELANDT, 
 JSIAM, V6, PP378-3 92, YR1958. 
 
 2. C. W. GEAR, A SIMPLE SET OF TEST MATRICES 
 FUR EIGENVALUE PROGRAMS, MATH COMP t V23, 
 PP119-125, YR1969. 
 
 IDENTIFICATION: 001003 
 
 MATRIX: A( I ,1 )=A( 1 , I )=1 
 
 A(I,J)=A(I-1,J)+A(I,J-1) 
 
 OTHERWI SE. 
 
 THE ARRAY IS A SQUARE 
 
 WITH THE APEX 
 EXAMPLE (N=5) 
 
 IN THE 
 
 SEGMENT OF A PASCAL TRIANGLE 
 JPPER LEFT CORNER. 
 
 1 
 
 3 
 
 6 
 
 10 
 
 15 
 
 1 
 
 4 
 10 
 20 
 35 
 
 1 
 
 5 
 IS 
 35 
 70 
 
 CHARACTERISTICS: REAL, SYMMETRIC, NON-SINGULAR, 
 POSITIVE OF FINITE. 
 
 INVERSE: THE FOLLOWING FORTRAN STATEMENTS WILL 
 PRODUCE THE ELFMENTS AINV(I,J) OF THE INVERSE 
 DF THE MATRIX A. 
 
 DO 40 11=1, N 
 
 I=N-I 1+1 
 
 AIMV( I , I )=1 .0 
 
 IF ( I . FQ.M) GO TO 2 
 
 DO 10 J=2 , I I 
 10 AINV( I , I )=AIMV( I , I )-A( I , J)*AINV( I+J-l, I ) 
 
 IF ( I .FO.l ) GO TO 5 
 20 DO 40 K=2, I 
 
 AINV( I-K + l , I ) =0 
 [K=N-I +K 
 
 DO 30 J=2,NIK 
 30 AINV( I-K+l, I )=AINV( I-K+l, I ) 
 
 1 -A( I -1 ,J)*AINV( I-K+J, I ) 
 
 ^0 AIMV( I , I-K + l )=AIMV( I-K+l, I ) 
 
50 CONTINUE 
 EXAMPLF (N=5) 
 
 5-10 10 -5 
 
 -10 30 -35 19 
 
 AIMV= 10 -35 46 -27 
 
 -5 19 -27 17 
 
 1-4 6-4 
 
 EIGENVALUES: NOT KNOWN, 
 
 EIGENVECTORS: MOT KNOWN. 
 DETERMINANT: DET(A)= 1 
 
 1 
 
 -4 
 
 6 
 -4 
 
 1 
 
 CONDITION 
 
 NUMBERS : 
 
 
 
 EXAMPLES 
 
 
 
 
 N 
 
 C( 1 ) 
 
 C( 2) 
 
 C( E) 
 
 5 
 
 1.56(4) 
 
 8.52( 3) 
 
 8.55(3) 
 
 6 
 
 2.05(5) 
 
 1.11(5) 
 
 1.11(5) 
 
 7 
 
 2.87(6) 
 
 1.49(6) 
 
 1.50(6) 
 
 8 
 
 3.96(7) 
 
 2.06(7) 
 
 2.07(7 ) 
 
 9 
 
 5.72(8 ) 
 
 2.91(H) 
 
 2.91 (8) 
 
 10 
 
 8.13(9) 
 
 4.16(9) 
 
 4.16(9) 
 
 REMARKS: THF ELEMENTS OF A 
 
 HENCE ROTH CAM BE EXACTLY REPRESENTED. 
 THE CONDITION NUMBERS GIVEN ABOVE 
 
 MD AINV ARE IMTEGERSt 
 CDDCC ENTFD. 
 
 E INDICATE THAT THIS 
 
 2 . 
 
 JOHN CAFFRFY, ANOTHER TEST MATRIX FOR DETERMIN- 
 ANTS AMD INVERSES, CACM, V6,P310, YR1963. 
 FRANK J. STOCKMALr THE INVERSE Of 
 CACM, V6, P615, YR1963. 
 
 IF A TEST MATRIX, 
 
 • MTI FICAT ION: 001004 
 
 MATR IX : A ( I , J ) = OR ( I +J ,P ) 
 
 WHERI 'K(K,P) IS THE L EGENDRE- J ACOB I OOADRATIC SYMBOL 
 -JHICH IS DEFINED AS FOLLOWS: 
 FQR M=P-] WHERE P IS AM ODD PRIME 
 OR(K,P)= IF P DIVIDES K, 
 
 (K,P)= 1 IF K IS CONGRUENT TO A SOUARF 
 INTEGER MODULO P, 
 " (K,P )=-l OTHFRWISE. 
 
EXAMPLE (N = 6) 
 
 1-1 1-1-1 
 
 -1 1-1-1 1 
 
 A= 1-1-1 1 1 
 
 -1-1 1 1-1 
 
 -10 11-11 
 
 11-11-1 
 
 CHARACTERISTICS: REAL* SYMMETRIC, MOM-SINGULAR. 
 
 INVERSE: A1NV(I,J)= ( OR ( I + J , P ) -OR ( I , P ) -OR ( J , P ) ) / P 
 EXAMPLE (N=6) 
 
 -1 -3 1-3-1 
 -3 -1 -1 -3 1 
 AINV=(l/7)* 1-11033 
 -3 -3 -1 1 -1 
 -10 3 113 
 13-131 
 
 EIGENVALUES: RIVAL ( I )= 
 
 EACH OF THE LAST TWO 
 - EXAMPLE (N = 6 ) 
 EIVAL ( 1 )= I 
 EIVAL (2 )=-l 
 EIVAL (3)= SORT (7) 
 FIVAL(4)= S0RTI7) 
 EIVAL ( 5)=-SORT(7) 
 EIVAL (6 )=-SORT( 7) 
 
 1, -1. 
 
 HAVING 
 
 SORTIP ) , -SORT(P ) , 
 MULTIPLICITY (M-2)/2 
 
 EIGENVECTORS: MOT KNOWN. 
 
 DETERMINANT: DET(A)= P**((N-2)/2) EOR 
 DET(A)=-P**( (N-2) /2) FOR 
 
 (M-2)/2 onn 
 
 (M-2J/2 EVEN 
 
 CONOITION 
 
 NUMBERS : 
 
 
 
 FXAMPLES 
 
 
 
 
 N 
 
 C(l ) 
 
 C(2) 
 
 C( F) 
 
 6 
 
 6.43 
 
 2.65 
 
 8.78 
 
 16 
 
 2.21(1) 
 
 4.12 
 
 2.60( 1 ) 
 
 36 
 
 5.20(1 ) 
 
 6.08 
 
 6.06( 1 ) 
 
 58 
 
 8.41 ( 1 ) 
 
 7.68 
 
 9.87(1) 
 
 78 
 
 1.14(2 ) 
 
 8.89 
 
 1.33(2) 
 
 96 
 
 1.42 (2 ) 
 
 9.85 
 
 1 .65(2) 
 
 REMARKS: THE ELEMENTS OF THE INVERSE WILL ALWAYS HAVE 
 THE FORM 1/P, -1/P, 3/P, -3/P, OR 0. THUS* EXCEPT 
 FOR 0, OR P = 3, THEY CAN NOT RE REPRESENTED EXACTLY 
 IN A BINARY MACHINE. 
 
 THE CONOITION NUMBER C(2) IS CONSIDERABLY SMALLER 
 THAN C( 1 ) AND C(E) . 
 
 THOUGH WE HAVE NOT PROVED IT, IT SEEMS CERTAIN THAT 
 THE NUN-ZERO COMPONENTS OF THE EIGENVECTOR WITH 
 THE EIGENVALUE 1 OR -1 ARE EQUAL, AND THAT IF THF 
 I-TH COMPONENT OF THE EIGENVECTOR WITH EIGENVALUE 1 
 IS NOT ZERO, THEN THE I-TH COMPONENT OF THE EIGEN- 
 VECTOR WITH EIGENVALUE -1 IS AND CONVERSELY. 
 
THEREFORE WE CAN GET THE TWO EIGENVECTORS WITH 
 EIGENVALUES 1 AND -1 FROM THE COLUMN VFCTOR WHICH IS 
 THE PRODUCT OF A AND THE COLUMN VFCTOR WHOSE 
 COMPONENTS ARE ALL 1. FOR EXAMPLE IN THE CASE OF 
 N = 6 THE TRANSPOSE OF THE PRODUCT OF A AND THE 
 TRANSPOSE OF (1, 1, 1, 1, 1, 1) IS (-1 t-1 t 1,-1, 1, 
 1), AND THERFFORE THE EIGENVECTORS WITH EIGENVALUES 
 1 AMD -1 ARE (0, 0, 1, 0, 1, 1) AND (1, 1, 0, 1,0, 
 0) RESPECTIVELY. 
 
 REFERENCES: 
 
 1. MORkIS NEWMAN AND JOHN TODD, EVALUATION OF MATRIX 
 INVERSION PROGRAMS, JSIAM, V6 , PP466-476, YR1958. 
 
 IDENTIFICATION: 001005 
 
 MATRIX: A(I,J)= I/J FOR I.LE.J, 
 
 A ( I ,J ) = J/I FOR I .GT.J. 
 EXAMPLE ( N = 5 ) 
 
 1 1/2 1/3 1/4 1/5 
 
 1/2 1 2/3 2/4 2/5 
 
 A= 1/3 2/3 1 3/4 3/5 
 
 1/4 2/4 3/4 1 4/5 
 
 1/5 2/5 3/5 4/5 1 
 
 CHARACTERISTICS: RFAL, SYMMFTRIC, NON-SINGULAR, 
 POSITIVE DFFINITE, OSCILLATORY. 
 
 INVERSE: AIMV(I,I)= (4*1**3) / (4*1**2 - 1) FOR I.NE.N, 
 
 AIMV(M,M)= (M**2)/(2*N - 1), 
 
 AINV(I,J)= -I*(I+1)/(2*I + 1) FOR J.EO.I+1, 
 
 AINV( I ,J ) = FOR J.GT.I+1, 
 
 AIMV( I , J )= AINV( J, I ) . 
 F X A M P L F ( M = 5 ) 
 
 4/3 -2/3 
 -2/3 32/15 
 AINV= -6/5 
 
 
 -6/5 
 108/35 
 -12/7 
 
 
 
 
 
 
 
 -12/7 
 
 256/63 -20/9 
 
 -20/9 25/9 
 
 1 [GENVALUES : NOT KNOWN. 
 EIGENVECTORS: NOT KNOWN. 
 DPI | ■■••■ I 'ANT : NOT KNOWN. 
 
 MQIT I UN NUMBERS: 
 
 | KAMPI 
 
 ' 
 
 5 
 20 
 40 
 
 C ( 1 ) 
 
 2 . 6 8 ( ] ) 
 
 4 . / 3 ( 2 ) 
 
 C(2) 
 
 1.971 1 ) 
 3.72 (2 ) 
 
 C( F) 
 2.53( 1 ) 
 7.43(2) 
 
 L.92(3) 1.5/(3) 4.16(3) 
 

 
 10 
 
 
 60 
 
 4.33(3) 
 
 3.62(3) 
 
 1.14(4) 
 
 80 
 
 7.71(3) 
 
 6.51 (3) 
 
 2.35(4) 
 
 100 
 
 1.21 (4) 
 
 1.03(A) 
 
 4.09(4) 
 
 REMARKS: NO SIMPLE 
 EIGENVECTORS, OR 
 
 EXPRESSIONS FOR 
 THE DETERMINANT 
 
 EIGENVALUES AMD 
 ARE KNOWN TO US. 
 
 REFERENCES : 
 
 1. MORRIS NEWMAN AMD JOHN TODD, EVALUATION OF MATRIX 
 INVERSION PROGRAMS, JSIAM, V6 , PP466-476, YR195B. 
 
 IDENTIFICATION: 001006 
 
 MATRIX : A ( I , I )= 2 
 A(l,l)= 3 
 
 A ( N , M ) = 1 
 A( I ,J )=-l 
 A( I , J )= 
 EXAMPLE (N=5) 
 
 FOR I.NE.l AMD I.NE.N 
 
 FOR ABS(I-J) 
 OTHERWISE. 
 
 .EO.lt 
 
 A = 
 
 3 
 
 -1 
 
 
 
 -1 
 
 2 
 
 -1 
 
 
 
 
 -1 
 
 2 
 -1 
 
 
 
 
 
 -1 
 
 2 
 
 
 
 
 
 
 
 -1 
 
 0-1 1 
 
 CHARACTERISTICS: REAL, SYMMETRIC, NON-SINGULAR, 
 POSITIVE DF EI MITE, TR I DIAGONAL. 
 
 INVERSE : AIMV( I ,J ) = 
 A I M V ( J , I ) = 
 EXAMPLE ( N = 5 ) 
 
 1 
 1 
 1 
 1 
 1 
 
 ( 2 * I - 1 ) / 2 
 AINV( I, J ) 
 
 FOR I.LE.J 
 
 AIN\/=( 1/2 )* 
 
 EIGENVALUES: EIVAL(I)= 
 
 FXAMPLE (N=5) 
 
 EIVAL ( 1 )= 9.78869 
 EIVAL (2 )= 8.24429 
 EIVAL (3)= 2.00000 
 EIVAL(4)= 3.17557 
 EIVAL (5)= 3.90211 
 
 2*( 1-C0S( (2*1-1 )*PI/ (2 
 
 FOR 1=1 ,2, ... ,N. 
 
 : N) ) ) 
 
 67409 
 49541 
 00000 
 05045 
 30325 
 
 69285 
 50537 
 00000 
 84946 
 90307 
 
 57671 
 41663 
 00000 
 25834 
 14423 
 
 (-2) 
 (-1 ) 
 
 EIGENVECTORS: EI VEC ( I , J)= SI N ( ( 2*1 -1 ) * ( 2* J-l ) *PI / ( 4*N ) ) 
 
 FOR I,J=1,2 
 
 N. 
 
 EXAMPLF ( N = 5 ) 
 FIVEC (1,1) = 
 EI VEC (2,1 ) = 
 EIVFC(3, 1 ) = 
 
 0. 15643 
 0.45399 
 0.70710 
 
 44650 
 04997 
 67811 
 
 40230 
 39546 
 86547 
 
 86901 
 79156 
 52440 
 
11 
 
 EIVEC 
 BIVEC 
 EIVEC 
 EIVEC 
 EIVEC 
 FIVEC 
 EIVEC 
 EIVEC 
 EIVEC 
 EIVEC 
 EIVEC 
 EIVEC 
 FIVEC 
 EIVEC 
 EIVEC 
 EIVEC 
 EIVEC 
 EIVEC 
 EIVEC 
 EIVEC 
 FIVEC 
 EIVFC 
 
 (4,1 ) 
 (5,1 ) 
 (1,2) 
 (2,2 ) 
 (3,2 ) 
 (4,2 ) 
 (5,2 ) 
 (1,3) 
 (2,3) 
 (3,3) 
 (4,3) 
 (5,3) 
 (1,4) 
 (2,4) 
 (3,4) 
 (4,4) 
 ( 5,4) 
 (1,5) 
 (2,5 ) 
 (3,5) 
 (4,5 ) 
 (5,5) 
 
 0.89100 
 
 0.98768 
 
 0.45399 
 
 0.98768 
 
 0.70710 
 
 ■0. 15643 
 
 •0.89100 
 
 0.70710 
 
 0.70710 
 
 •0.70710 
 
 •0.7 0710 
 
 0.70710 
 
 0.89100 
 
 ■0. 15643 
 
 ■0.70710 
 
 0.98768 
 
 ■0.45399 
 
 0.98768 
 
 ■0.89100 
 
 0.70710 
 
 ■0.45399 
 
 0. 15643 
 
 65241 
 83405 
 04997 
 83405 
 67811 
 44650 
 65241 
 67811 
 67811 
 67811 
 67811 
 67811 
 65241 
 44650 
 67811 
 83405 
 04997 
 83405 
 65241 
 67811 
 04997 
 44650 
 
 38367 
 95137 
 39546 
 95137 
 86547 
 40230 
 88367 
 86547 
 86547 
 86547 
 86547 
 86547 
 88367 
 40230 
 86547 
 95137 
 39546 
 95137 
 88367 
 86547 
 39546 
 40230 
 
 86236 
 72619 
 79156 
 72619 
 52440 
 86901 
 86236 
 52440 
 52440 
 52440 
 52440 
 52440 
 86236 
 86901 
 52440 
 72619 
 79156 
 72619 
 86236 
 52440 
 79156 
 86901 
 
 DETERMINANT: DFT(A)= 2 
 
 CnMDITIOM NUMBERS: 
 FXAMPLES 
 M 
 
 5 
 20 
 40 
 60 
 80 
 100 
 
 C ( 1 ) 
 5.001 1 ) 
 8.00(2 ) 
 3.20(3 ) 
 7.20(3 ) 
 1 .28(4) 
 2.00(4) 
 
 C (2 ) 
 
 3.99( 1 ) 
 
 6.48(2) 
 
 2.59(3) 
 
 5.84(3) 
 
 1.04(4) 
 
 1.62(4) 
 
 C(E) 
 
 5.6 5 ( 1 ) 
 1.79(3) 
 1.01(4) 
 2.79(4) 
 5.72(4) 
 1.00(5) 
 
 REMARKS : 
 OF MAT 
 THF F|_ 
 R F P R F S 
 E L E M E N 
 THF RE 
 ALL OF 
 IN THF 
 EIGENV 
 A P PROA 
 W H F M L 
 
 ^FOR 
 SII 
 THFRFF 
 
 JATI 
 S I i 
 
 (TIN 
 
 E N L 
 
 'FOR 
 THE 
 EQUAL 
 TH^ 
 
 THIS M 
 R I C F S D 
 F M E M T S 
 FNTFD I 
 T (2N - 
 PRFSFMT 
 
 THF EI 
 
 I N T F R V 
 ALUF AP 
 CHFS FG 
 I J DFCOM 
 MED ON 
 THF COM 
 ORF, US 
 UN R 
 ILL MOT 
 E. 
 
 i) DECOM 
 M( D 
 
 pp [MC 1 
 TO 
 
 I ', 
 
 ATRIX IS 
 ESCRIBED 
 OF THIS I 
 M A BINAR 
 
 1 )/2 DDE 
 AT ION. 
 GENVALUES 
 AL (0, 4) 
 PROACHES 
 UR. 
 
 POSITION 
 THIS MATR 
 P U T F D P I V 
 F OF A AS 
 INF WHICH 
 
 TEST THF 
 
 A SPFCIA 
 BY GEAR 
 NVERSE C 
 Y MACHIN 
 S NOT EX 
 
 OF THIS 
 . AS N I 
 ZERO AND 
 
 OR GAUSS 
 
 IX MO RO 
 
 OTS ARE 
 
 A TEST 
 
 OSFS PA 
 
 ROW EXC 
 
 POSITION OR GAUSS 
 AINV THE ELFMFNTS 
 PAL DIAGONAL AND 
 
 FPT THF FIRST 
 ARF f OUAL TO 1 /?) 
 
 L CASF OF A WIDE CLASS 
 
 (REF. 2) . 
 
 AN BE EXACTLY 
 
 E PROVIDFD THE LARGEST 
 
 CEED THF CAPACITY OF 
 
 MATRIX ARE CONTAINED 
 NCRFASFS THE SMALLFST 
 THF LARGEST EIGENVALUE 
 
 ELIMINATION IS 
 W EXCHANGES ARE MADFt 
 ALWAYS MAXIMUM. 
 MATRIX IN A LINEAR 
 RTIAL PIVOTING FOR 
 HANGE PORTION OF THE 
 
 ELIMINATION IS 
 IN THE PIVOT COLUMN 
 BELOW WILL ALL BF 
 COLUMN WHERE ALL OF 
 
12 
 
 REFERENCES: 
 
 1. MORRIS NEWMAN AND JOHN TODD, EVALUATION OE MATRIX 
 INVERSION PROGRAMS* JSIAM, V6 , PP466-476, YR1958. 
 
 2. C. W. GEAR, A SIMPLE SET OF TEST MATRICES 
 FOR EIGENVALUE PROGRAMS, MATH COMP , V23, 
 PP119-125, YR1969. 
 
 IDENTIFICATION: 001007 
 
 MATRIX: A ( I ,J )= 
 EXAMPLE (M=5) 
 
 A = 
 
 l/( I+J-l ) 
 
 1 
 1/2 
 1/3 
 1/4 
 
 1/2 
 1/3 
 1/4 
 1/5 
 
 1/3 
 1/4 
 1/5 
 1/6 
 
 1/4 
 1/5 
 1/6 
 1/7 
 
 1/5 
 
 1/6 
 1/7 
 1/8 
 
 1/5 1/6 1/7 1/8 1/9 
 
 CHARACTERISTICS: REAL, SYMMETRIC* MOM-SINGULAR. 
 
 I MVERSE : A INV ( I , J ) = ( -1 ) ** ( I +J ) / ( I + J-1 ) *H ( I ,M ) *H ( J ,N ) 
 WHERE H(K,N)= FAC ( N+K-l ) / ( F AC ( N-K ) * ( F AC ( K-l ) **2 ) ) 
 FAC(P)=P*(P-1 )*...*! 
 
 F X A M P L E ( M = 5 ) 
 
 AINV = 
 
 25 
 -300 
 
 1050 
 
 -1400 
 
 630 
 
 -300 
 
 4800 
 
 -18900 
 
 26880 
 
 -12600 
 
 1050 
 -18900 
 
 79380 
 •1 17600 
 
 5 6700 
 
 -1400 
 
 26880 
 
 -117600 
 
 179200 
 
 -88200 
 
 630 
 -12600 
 
 56700 
 -88200 
 
 44100 
 
 EIGENVALUES: NO EXPRESSION 
 HAVE BEEN MADE (REF. 4). 
 FXAMPLE (N=5, FROM REF. 
 EIVAL ( 1 )= 1.56705 06910 
 EIVAL (2 )= 2.08534 21861 
 EIVAL(3)= 1.14074 91623 
 FIVAL(4)= 3.05898 04015 
 FIVAL(5)= 3.28792 87721 
 
 IN CLOSED FORM. TABULATIONS 
 
 4) 
 
 98231 
 
 10133 
 
 41981 
 
 11917 
 
 71863 
 
 (-1 ) 
 (-2) 
 (-4) 
 (-6) 
 
 EIGENVECTORS: NO EXPRESSION 
 TIOMS HAVE BEEN MADE (REF. 
 FXAMPLE (N = 5, FROM REF. 4) 
 
 FIVEC( 1,1) = 
 FI VEC (2,1 ) = 
 EIVEC(3,1 ) = 
 EI VEC (4,1 ) = 
 EI VEC (5,1 ) = 
 EIVEC( 1,2)= 
 FIVEC(2,2 ) = 
 EIVEC(3,2 ) = 
 EI VEC (4,2)= 
 EIVEC(5,2 ) = 
 
 00000 
 80566 
 18800 
 30061 
 
 2.73258 
 1.00000 
 -4.58425 
 -7.05925 
 -7.37537 
 -7.12798 
 
 IN CLOSED 
 4) . 
 
 FORM. TABULA- 
 
 00000 
 92249 
 95256 
 05409 
 24401 
 00000 
 80576 
 82907 
 92074 
 94314 
 
 00000 
 80478 
 90560 
 17674 
 62320 
 00000 
 61740 
 15063 
 31147 
 80946 
 
 (- 
 (- 
 (- 
 (- 
 
 (- 
 (- 
 (- 
 (- 
 
13 
 
 EIVEC 
 EIVFC 
 FIVEC 
 EIVEC 
 EIVEC 
 EIVEC 
 EIVEC 
 EIVEC 
 EIVEC 
 EIVEC 
 EIVEC 
 EIVEC 
 EIVEC 
 EIVEC 
 EIVEC 
 
 (1,3) 
 (2,3) 
 (3,3 ) 
 (4,3) 
 (5,3) 
 (1,4) 
 (2,4) 
 (3,4) 
 (4,4 ) 
 (5,4) 
 (1,5) 
 (2,5) 
 (3,5) 
 (4,5) 
 (5,5) 
 
 ■2.95833 
 1.00000 
 1 .66348 
 
 •4.27528 
 
 ■7.80543 
 7.06702 
 
 -6.48336 
 1 .00000 
 3.49178 
 
 ■8.35542 
 
 ■8.04735 
 1.52103 
 
 •6.59762 
 1 .00000 
 
 -4.90419 
 
 43954 
 00000 
 46563 
 04665 
 77407 
 26210 
 02593 
 00000 
 63233 
 93387 
 96573 
 86654 
 08136 
 00000 
 53143 
 
 91379 
 00000 
 67509 
 91248 
 62442 
 87525 
 66261 
 00000 
 06241 
 42830 
 69526 
 52718 
 21921 
 00000 
 50719 
 
 (-1 ) 
 
 ■1 ) 
 1 ) 
 1 ) 
 2) 
 1 ) 
 
 1 ) 
 1) 
 3) 
 1) 
 1 ) 
 
 (-1 ) 
 
 DETERMINANT 
 
 DET( A) = E/F 
 
 E = ( F AC ( 1 ) * E AC ( 2 ) * . . . * F A C ( N- 1 ) ) 
 
 "f v 2 
 
 F= N**N 
 
 t(N**2-l**2 )**(N-1 ) 
 MN**2-2**2)**(N-2) 
 J . ..*('N**2-(N-1 )**2 )**1 
 
 CONDI T I DM 
 
 NUMBERS : 
 
 
 
 EXAMPLES 
 
 
 
 
 N 
 
 cm 
 
 C(2) 
 
 C(E) 
 
 2 
 
 2.70(1) 
 
 1.93( 1 ) 
 
 1.93( 1 ) 
 
 3 
 
 7.48(2 ) 
 
 5.24(2) 
 
 5.26(2) 
 
 4 
 
 2.84(4) 
 
 1.55(4) 
 
 1.56(4) 
 
 5 
 
 9.44( 5 ) 
 
 4.77(5 ) 
 
 4.81(5) 
 
 6 
 
 2.91 (7) 
 
 1.50(7) 
 
 1.51(7) 
 
 8 
 
 3.39( 10) 
 
 1.53( 10) 
 
 1.55(10) 
 
 10 
 
 3.54( 13) 
 
 1.60( 13) 
 
 1.63( 13) 
 
 12 
 
 4.12( 16) 
 
 1.7K 16) 
 
 1 .75( 16) 
 
 REMARKS: THIS MATRIX IS A FINITE SEGMENT OF A HUBERT 
 MATRIX. IT IS PROBABLY THE MOST OFTEN CITED EXAMPLE 
 0!= AN ILL-CONDITIONED MATRIX. THIS MATRIX ARISES IN 
 LEAST-SQUARE APPROXIMATION (REF. 5). THE INVERSE IS 
 TABULATED FOR N=2(l)10 IN REF. 1. FOR GENERALIZATION 
 SEE TM010001, TM001010. 
 
 REFFRENCFS : 
 
 1 . 
 
 2. 
 
 RICHAR 
 INVEKS 
 MATW I X 
 JOHN I 
 OF THF 
 YR1954 
 THOMAS 
 FFI 
 
 HENRY 
 
 TUG 
 
 ANAI YS 
 
 n\\ FY 
 
 D SAVAGE 
 FS OF FI 
 , N B S A M S 
 ODD, THE 
 HI LBERT 
 
 • 
 
 C. DOYL 
 CIFNT MA 
 I [C FORM 
 F. FETTI 
 IVECTO 
 H If), MA 
 
 ! '.AACSO 
 IS OF NU 
 
IDENTIFICATION: 001008 
 
 Ik 
 
 MATRIX: A( I , I )= 1-K*I##2 
 A(I,N)= K*I 
 A ( N , I ) = A ( I , N ) 
 A(N,N)=-K 
 A( I ,J )=-K*I*J 
 K= 6/ (N*(N+1 ) 
 
 FOR I=1,2,...,N-1 
 
 WHERE 
 
 EXAMPLE (N=5) 
 
 A = ( 1 / 2 5 ) * 
 
 24 
 -2 
 -3 
 -4 
 
 1 
 
 FOR 1=1,2 
 
 , • • • , 
 
 (2 
 
 -2 
 21 
 -6 
 -8 
 2 
 
 OTHERWI SE» 
 ; N-5 ) ) . 
 
 -3 
 
 -6 
 
 16 
 
 -12 
 
 3 
 
 -4 
 
 -8 
 
 -12 
 
 9 
 
 A 
 
 1 
 2 
 3 
 4 
 -1 
 
 N-l 
 
 CHARACTERISTICS: REAL, SYMMETRIC, 
 MULTIPLE EIGENVALUES. 
 
 NON-SINGULAR, 
 
 INVERSE : AINV( I , I ) = 
 
 A I N V ( I , N ) = 
 
 A I M V ( N , I ) = 
 
 A I N V ( I , J ) = 
 
 EXAMPLE (N=5) 
 
 A I NV = 
 
 FOR 1 = 1 ,2, . . .,N-1 , 
 
 1 
 I 
 
 A I M V ( I , N ) 
 
 OTHERWISF. 
 
 
 
 
 1 
 
 
 
 3 
 
 EIGENVALUES 
 
 WHERE 
 
 X = 
 
 Y = 
 
 EXAMPLE (N=5) 
 EIVAL (1 )= 1 
 EIVAL (2 )= 1 
 EIVAL (3)= 1 
 EIVAL (4)= 1/(3 + 
 E I V A L ( 5 ) = 1/(3 - 
 
 EIVAL(I)= 1 FOR 1=1,2,... ,N-2, 
 EIVAL(N-1)= (X + S0RT(Y))**(-1) 
 EIVAL(N) = (X- SORT(Y) )**(-l) 
 
 ( N + 1 ) / 2 
 
 1/K + X*#2 
 
 SORT( 34) ) 
 S0RT(34) ) 
 
 F IGENVECTORS : NOT KNOWN. 
 
 DETERMINANT: DET(A)= -6/ ( N* ( N+l ) * ( 2*N~5 ) ) 
 
 CniMDITION NUM3ERS: 
 
 EXAMPLES 
 
 
 
 
 N 
 
 C ( 1 ) 
 
 C(2 ) 
 
 C(F) 
 
 5 
 
 2.40(1 ) 
 
 8.83 
 
 1.67(1) 
 
 20 
 
 4.59(2 ) 
 
 6.11(1) 
 
 3.11(2) 
 
 40 
 
 1.92(3) 
 
 1.65(2 ) 
 
 1.27(3) 
 
 60 
 
 4.37(3 ) 
 
 2.97(2 ) 
 
 2.89( 3) 
 
 80 
 
 7.83(3) 
 
 4.52( 2 ) 
 
 5.16(3) 
 
 100 
 
 1.23(4) 
 
 6.26(2) 
 
 8.08(3) 
 
 REMARKS: THIS MATRIX IS FAIRLY WELL CONDITIONED. THE 
 INVERSE IS COMPOSED OF INTEGERS AND THUS IS EXACTLY 
 
15 
 
 REPRESENTABLE IN A BINARY MACHINE. 
 
 THE EIGENVALUE 1 HAS MULTIPLICITY N-2 , SO IT CAM 
 
 EXPECTED THAT THIS MATRIX IS USEFUL IN TESTING 
 
 EIGENVALUE ALGORITHMS IN THE PRESENCE OF MULTIPLF 
 
 EIGENVALUES. 
 
 BE 
 
 REFERENCES : 
 
 1. M. J. AEGFRTER, 
 MATRICES, CACM, 
 
 CONSTRUCTION 
 V2, PP10-12 T 
 
 OF A SET OF TEST 
 YR1959, AUG. 
 
 IDENTIFICATION: 001009 
 
 MATRIX: A(I,J)= S0RT(2/ (N+l ) )*SIN( I*J*PI / (N+l ) ) 
 FXAMPLE (M = 5) LET T= SQRT(3) 
 
 1/2 T/2 1 T/2 1/2 
 
 T/2 T/2 -T/2 -T/2 
 
 A = ( 1 / T ) * 1 - 1 1 
 
 T/2 -T/2 T/2 -T/2 
 
 1/2 -T/2 1 -T/2 1/2 
 
 CHARACTERISTICS: REAL, SYMMFTRIC, NON-SINGULAR, 
 ORTHOGONAL . 
 
 INVERSE: AINV( I , J )=A{ I , J ) 
 
 EIGENVALUES: M=EVEN; 
 
 EIVAL(I)= 1 FOR 1 = 1 ,2, . . .,N/2, 
 EIVAL(I)=-1 FOR I=N/2+l, . .. ,N. 
 N=ODD; 
 
 FIVAL(I)= 1 FOR 1 = 1,2,..., (N+l )/2, 
 EIVAL(I)=-1 FOR I=(M+3)/2,... ,N. 
 EXAMPLE (N=5) 
 
 EIVAL ( 1 )= 1 
 
 EIVAL (2 )= 1 
 
 EIVAL (3)= 1 
 
 EIVAL (4)=-l 
 
 FIVAL ( 5)=-l 
 
 EIGENVECTORS: EXPRESSION IN TERMS OF ELEMENTARY 
 FUNCTIONS IS UNKNOWN. 
 
 DETERMINANT: +1 OR -1. 
 
 +1 IF N EVEN OR IF N=l (MOD 4), 
 -1 OTHERWISE. 
 
 CONDITION NUMBERS: 
 
 C( 1 )= ( (SIN( PI/ (N+l ) ) ) /( 1-C0S(PI/(M+1 ) ) ) )**2 
 C ( 2 ) = 1 
 C ( F ) = N 
 FXAMPLES 
 
 N C ( 1 ) C ( 2 ) C(F) 
 5 4.64 1.00 5.00 
 
16 
 
 20 
 
 1.70(1 
 
 1 1.00 
 
 2.00( 1 ) 
 
 40 
 
 3.32 ( 1 
 
 I 1.00 
 
 4.00( 1 ) 
 
 60 
 
 4.94( 1 
 
 1 1.00 
 
 6.00( 1 ) 
 
 80 
 
 6.56(1 
 
 1 1.00 
 
 8.00( 1) 
 
 100 
 
 8.19(1 
 
 1.00 
 
 1 .00(2) 
 
 REMARKS: THIS MATRIX WAS SUGGESTED BY NEWMAN AND TODD 
 (REF. 1). IT HAS THE SMALLEST POSSIBLE CONDITION 
 NUMBER C(2) HENCE MATRIX INVERSION ROUTINES CAN BE 
 EXPECTED TO PRODUCE THE INVERSE OF THIS MATRIX VERY 
 ACCURATELY. 
 
 REFERENCES : 
 
 1. MORRIS NEWMAN AMD JOHN TODD, THE EVALUATION OF 
 MATRIX INVERSION PROGRAMS, JSIAM, V6, PP466-476, 
 YR1958. 
 
 IDENTIFICATION: 001010 
 
 MATRIX: A(I,J)= 1/(P+I+J-1) FOR 
 
 WHERE P IS A NONNEGATIVE INTEGER. 
 EXAMPLE (P=2, N=5) 
 
 I,J= 1,2 
 
 . ,N 
 
 A = 
 
 1/3 
 
 1/4 
 1/5 
 1/6 
 1/7 
 
 1/4 
 1/5 
 1/6 
 1/7 
 1/8 
 
 1/5 
 1/6 
 1/7 
 1/8 
 1/9 
 
 1/6 
 1/7 
 1/8 
 1/9 
 1/10 
 
 1/7 
 
 1/8 
 
 1/9 
 
 1/10 
 
 1/11 
 
 CHARACTERISTICS: REAL, SYMMETRIC, NON-SINGULAR. 
 
 INVERSE: AINV(I,J)= (-1 )**( I+J)*F( I )*F( J )/ (P + I+J-l ) 
 F(K)= FAC( P+K+N-l )/ ( FAC(K-1 )*FAC(N-K )*FAC( P+K-l ) ) 
 EXAMPLE (P=2, N=5) 
 
 AINV^ 
 
 3675 
 
 -29400 
 
 79380 
 
 -88200 
 
 34650 
 
 29400 
 
 250880 
 
 -705600 
 
 806400 
 
 -323400 
 
 79380 
 
 -705600 
 
 2041200 
 
 -2381400 
 
 970200 
 
 88200 
 
 806400 
 
 -2381400 
 
 2822400 
 
 -1164240 
 
 34650 
 
 -323400 
 
 970200 
 
 -1164240 
 
 485100 
 
 EIGENVALUES: NOT KNOWN. 
 
 EIGENVECTORS: NUT KNOWN. 
 
 DETERMINANT: NOT KNOWN. 
 
 CONDITION NUMBERS: 
 FXAMPLES 
 
 N C ( 1 ) C ( 2 ) C ( E ) 
 
 2 8.17(1) 6.63( 1 ) 6.63( 1 ) 
 
 3 4.61(3) 3.09(3) 3.09(3) 
 
 4 2.11(5) 1 .24( 5 ) 1.24( 5) 
 
 5 7.97(6) 4.64(6) 4.64(6) 
 

 
 17 
 
 
 6 
 
 8 
 
 10 
 
 12 
 
 2.95(8 ) 
 4.01 (11 ) 
 4.99( 14) 
 5.85( 17) 
 
 1.67(8) 
 2.05( 11 ) 
 2.43( 14) 
 2.82( 17) 
 
 1.67(8) 
 2.06( 11 ) 
 2.44( 14) 
 2.83( 17) 
 
 REMARKS: THIS IS A GENERALIZED FORM OF 
 MEMT OF A HUBERT MATRIX (TM001007). 
 GENERALIZATION SEE TM010001. 
 LIKE THE HIL8ERT MATRIX, THIS MATRIX 
 CONDITIONED. 
 
 A FINITE SEG- 
 FOR AMOTHFR 
 
 IS VFRY ILL 
 
 REFERENCES: 
 
 1. A. R. COLLAR, ON THE RECIPROCATION OF CERTAIN 
 MATRICFS, PROC. ROY. SOC. EDINBURGH, V59 , PP195- 
 206, YR1939. 
 
 2. A. R. COLLAR, ON THE RECIPROCAL OF A SEGMENT OF 
 A GENERALIZED HILBERT MATRIX, PROC. CAMB. PHIL. 
 SOC, V47, PP11-17, YR1951. 
 
 IDENTIFICATION: 001011 
 
 MATRIX: A(I,J)= M-ABS(I-J) 
 EXAMPLE (M=5) 
 
 5 4 3 2 1 
 
 4 5 4 3 2 
 
 A= 3 4 5 4 3 
 
 2 3 4 5 4 
 
 12 3 4 5 
 
 CHARACTERISTICS: REAL, SYMMETRIC, NON-SINGULAR, 
 POSITIVE DEFINITE. 
 
 INVERSE: AINV(I,I)= ( N + 2 ) / ( 2-N + 2 ) FOR 1=1 OR N, 
 AINV( I , I )= 1 OTHERWISE, 
 
 AINV(I,J)= -1/3 FOR ABS(I-J)=1 AND N=2, 
 AINV(I,J)= -1/2 FOR ABS(I-J)=1 AND N.ME.2, 
 AIMV(I,J)= l/(2*N+2) FOR ABS( I-J )=N-1, 
 AIMV( I , J ) = OTHERWISE. 
 FXAMPLF ( M = 5 ) 
 
 7/12 -1/2 1/12 
 -1/2 1 -1/2 
 AINV= -1/2 1 -1/2 
 -1/2 1 -1/2 
 1/12 -1/2 7/12 
 
 • I iENVALUES : NOT KNOWN. 
 
 rORS: NOT KNOWN. 
 DFTER M I 'ANT: MOT KNOWN. 
 i!i I R S : 
 
18 
 
 EXAMPLES 
 
 N 
 
 5 
 
 20 
 
 40 
 
 60 
 
 80 
 
 100 
 
 C (1 ) 
 3.80( 1 ) 
 6.00(2 ) 
 2.40(3 ) 
 5.40(3 ) 
 9.60(3 ) 
 1.50(4) 
 
 C (2 ) 
 
 3.13(1) 
 
 5.38(2) 
 
 2.16(3) 
 
 4.86( 3) 
 
 8.64(3 ) 
 
 1.35(4) 
 
 C(E) 
 
 4.30( 1 ) 
 1.50(3) 
 8.62( 3) 
 2.39(4) 
 4.92(4) 
 8.60(4) 
 
 REMARK 
 REEE 
 IF T 
 08SE 
 MATR 
 IS A 
 FOR 
 
 BY T 
 
 WHIC 
 IE N 
 VECT 
 I J S I N 
 RE S 
 V( I ) 
 
 odd, 
 
 S: THIS 
 
 RENCE 1 
 
 HE EIGE 
 
 RVATION 
 
 IX, IT I 
 
 N EIGEN 
 
 THE SAM 
 
 V( I ) = 
 
 HE FOLL 
 
 V( 
 
 V( 
 
 H IMPL I 
 
 IS HDD 
 
 OR SUCH 
 
 G THE J 
 
 HOWN TH 
 
 = V ( M- 
 
 RESPEC 
 
 MATRIX WAS USED AS A TEST MATRIX IN 
 
 NVALUE 
 S HOLO 
 S CLEA 
 VECTOR 
 E EIGE 
 V(N-I+ 
 OWI NG 
 I ) = C 
 N-I+l ) 
 ES C** 
 AND V 
 THAT 
 AC06I 
 AT THE 
 1+1 ) I 
 TI VELY 
 
 S ARE DISTINCT THEN THE FOLLOWING 
 . BY THE GIVEN FORM OF THE N BY N 
 R THAT IF ( V( 1 ) t V(2), ...» V(N) ) 
 
 THEN SO IS (V(N),..., V(2), V(l)) 
 NVALUE. THEREFORE 
 
 1 ) OR -V( N-I + l ) , I = 1 , 2, .. . t N 
 ARGUMENTS: 
 *V< N-I+l ) 
 
 = C*V( I ) t 
 2=1. HENCE 
 
 AMD 
 I = 
 C = 
 
 N 
 
 It 2 » • • • » 
 
 1 OR -1. 
 
 (I) ARE THE COMPONENTS OF AN EIGEN- 
 V(I) =-V(N-I+l), THEM V( ( N+l )/2 )=0. 
 EXPANSION OF THE DETERMINANT IT CAN 
 
 NUMBER OF EIGENVECTORS SUCH THAT 
 S N/2 OR (N+l)/2 WHEN N IS EVEN OR 
 
 REFEREMCFS : 
 
 1. M. H. LIETZKE, R. W. STOUGHTON AND MARJORIF D. 
 LIETZKE, A COMPARISON OF SEVERAL METHODS FOR 
 INVERTING LARGE SYMMETRIC POSITIVE DFFINITE 
 MATRICES, MATH COMP, V18, PP449-456, YR1964. 
 
 IDENTIFICATION: 000001 
 
 MATRIX 
 
 A = 
 
 611 196 -192 407 
 
 -192 113 899 196 
 
 -8 -52 -49 
 
 196 899 113 -192 -71 
 
 61 
 
 -43 
 49 
 
 29 
 
 -8 -44 
 8 52 
 
 407 
 
 -192 
 
 196 
 
 611 
 
 8 
 
 44 
 
 59 
 
 -23 
 
 -8 
 
 -71 
 
 61 
 
 8 
 
 411 
 
 -599 
 
 208 
 
 208 
 
 -52 
 
 -43 
 
 49 
 
 44 
 
 -599 
 
 411 
 
 208 
 
 208 
 
 -49 
 
 -8 
 
 8 
 
 59 
 
 208 
 
 208 
 
 99 
 
 -911 
 
 29 
 
 -44 
 
 52 
 
 -23 
 
 208 
 
 208 
 
 -911 
 
 99 
 
 CHARACTERISTICS: REAL, SYMMFTRIC, SINGULAR, 
 SOME CLOSELY SPACED EIGENVALUES, A PAIR OF 
 EQUAL EIGENVALUES. 
 
 INVERSE: NONE. 
 
19 
 
 EIGENVALUES: 
 
 EIVAL(1)= 10*SQRT(10405) 
 
 = 1.02004 90184 29996 
 EIVAL(2)= 1020 
 EIVAL(3)= 510+100*SQRT(26) 
 
 = 1.01990 19513 59278 
 EIVAL(4)= 1000 
 FIVAL(5)= 1000 
 EIVAL(6)= 510-100*SQRT (26 ) 
 
 = 9.80486 40721 51699 
 EIVAL (7 )= 
 EI VAL (8 )=-10*SQRT( 10405 ) 
 
 =-1.02004 90184 29996 
 
 82385 (3) 
 
 48300 (3) 
 
 71776 (-2) 
 
 82385 (3) 
 
 EIGENVECTORS: 
 EIVEC( 1,1 ) = 
 EIVEC(2,1 )= 
 EIVEC(3,1 ) = 
 EIVEC(4,1 )= 
 EIVFC(5,1 ) = 
 
 FI VEC (6,1)= 
 
 EIVEC(7,1 )=■ 
 
 EI VEC (8,1)=- 
 
 E I V E C 
 
 F I V E C 
 EIVEC 
 EIVEC 
 EIVEC 
 FIVEC 
 EIVEC 
 FIVEC 
 FIVEC 
 FIVEC 
 FIVEC 
 FIVEC 
 EIVEC 
 
 (1,2) 
 (2,2 ) 
 (3,2 ) 
 (4,2 ) 
 (5,2 ) 
 (6,2 ) 
 (7,2 ) 
 (8,2 ) 
 (1,3) 
 (2,3) 
 (3,3) 
 (4,3) 
 (5,3) 
 
 FIVEC (6,3)=- 
 
 EIV6C (7,3)=- 
 
 EI VEC (8,3)= 
 
 FIVEC( 1,4)= 
 EI VEC (2, 4 )=- 
 EI VEC (3,4)=- 
 F I V F C ( 4 , 4 ) = 
 EIVEC(5,4)=- 
 E I VEC (6,4)= 
 FIVEC(7,4)=- 
 E I VE C ( 8 , 4 ) = 
 EIVEC(] ,5) = 
 
 2 
 
 1 
 
 1 
 
 2 
 
 102-S 
 
 4.901 
 
 102-S 
 
 4.901 
 
 204 + 2 
 
 9.803 
 
 204 + 2 
 
 9.803 
 
 1 
 
 2 
 
 2 
 
 1 
 
 2 
 
 2 
 
 1 
 
 1 
 
 2 
 
 1 
 
 1 
 
 2 
 
 SORT( 
 
 5.099 
 
 5-SOR 
 
 1 .009 
 10-2* 
 
 2 .019 
 10 + 2* 
 2.019 
 1 
 
 2 
 2 
 1 
 2 
 2 
 1 
 1 
 7 
 
 ORT( 10405) 
 84 29996 82384 
 ORT( 10405) 
 84 29996 82384 
 *SQRT( 10405 ) 
 68 59993 64769 
 *SQRT( 10405) 
 68 59993 64769 
 
 63137 (-3) 
 
 63137 (-3) 
 
 26275 (-3) 
 
 26275 (-3) 
 
 26) 
 
 01 95135 92784 83002 
 
 T(26) 
 
 90 19513 59278 48300 
 
 S0RK26) 
 
 80 39027 18556 96601 ( 1 ) 
 
 S0RK26) 
 
 80 39027 18556 96601 ( 1 ) 
 
20 
 
 EIVEC(2 ,5) 
 FIVEC(3,5 ) 
 EIVEC(4,5) 
 EIVEC(5,5) 
 FIVEC(6,5 ) 
 EIVEC (7, 5) 
 EIVEC(8,5) 
 EIVFC( 1 ,6) 
 EIVEC (2,6) 
 EIVEC(3,6 ) 
 EIVEC(4,6) 
 EIVEC (5,6) 
 
 >-S0KT ( 26 ) 
 .90195 13592 78483 00282 (-2) 
 + SQRK26) 
 
 .90195 13592 78483 00282 (-?) 
 -10+2*SQRT(26) 
 
 .98039 02718 55696 60056 (-1) 
 
 0-2*SQRT( 26) 
 
 .98039 02718 55696 60056 (-1) 
 
 EIVEC (6,6)= 
 
 EIVEC(8,6 )= 
 
 EIVEC( 
 EIVEC( 
 EIVEC( 
 EIVEC ( 
 EIVEC ( 
 EIVEC ( 
 EIVEC( 
 EIVEC( 
 EIVECI 
 EIVEC ( 
 FIVEC ( 
 EIVEC( 
 EIVEC( 
 
 1,7) 
 
 2,7) 
 3,7 ) 
 4,7 ) 
 5,7 ) 
 
 6.7 ) 
 7,7) 
 8,7) 
 1,8) 
 
 2.8 ) 
 3,8 ) 
 4,8 ) 
 5,8 ) 
 
 EIVEC(6,8 ) = 
 FIVEC(7,8 )=- 
 EIVEC (8,8 )=■ 
 
 02 + SORK 10405 ) 
 .04004 90184 29996 
 02 + SORK 10405 ) 
 .04004 90184 29996 
 04-2*SQRT( 10405 ) 
 .08009 80368 59993 
 04-2*SQRT( 10405 ) 
 .08009 80368 59993 
 
 82385 (2) 
 
 82385 (2) 
 
 64769 (2) 
 
 64769 (2) 
 
 riETERMIMAMT: DET(A)= 
 
 Cn\)()ITIGM NUMBERS: C(R)= 1.040(4) 
 
 REMARKS: THIS MATRIX WAS CONSTRUCTED BY ROSSER, 
 
 LANCZOS, HESTENES, AND KARUSH (REE. 1) FUR TESTING 
 CERTAIN EIGENVALUE EIGENVECTOR ALGORITHMS. IT HAS 
 SEVERAL CLOSE EIGENVALUES AND THEREFORE IT IS USEFUL 
 FOR TESTING THE ABILITY OF AM ALGORITHM TO SEPARATE 
 CLOSE EIGENVALUES. 
 
 REFERENCES: 
 
 1. ROSSER, J. B., LANCZOS, C, H F ST ENE S , M .R . , AND 
 
 KARUSH, W., SEPARATION OF CLOSE EIGENVALUES OF A 
 REAL SYMMETRIC MATRIX, J. RES. NBS, V47 , 
 PP291-297, YR1951. 
 
21 
 
 IDENTIFICATION: 002001 
 
 MATRIX: A ( I t J )=1 FOF 
 
 . I 
 
 .NE 
 
 Jt 
 
 
 A( I f I )=1 + X 
 
 
 
 
 
 EXAMPLE (N = 5, X=3) 
 
 
 
 
 
 4 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 4 
 
 1 
 
 1 
 
 1 
 
 A = 1 
 
 1 
 
 4 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 4 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 4 
 
 CHARACTERISTICS: REAL, SYMMETRIC, SINGULAR IF X=0 OR -M . 
 
 INVERSE: AINV(I,J)= -1/(X**2 + N * X) 
 AIMV( I , I )= 1/X - 1/(X*#2 + N 
 EXAMPLF (N = 5, X = 3 ) 
 
 FOR I. ME. J, 
 X ) 
 
 
 
 7 
 
 -1 
 
 -1 -] 
 
 L -1 
 
 
 
 -1 
 
 7 
 
 -1 -' 
 
 L -1 
 
 INV = 
 
 ■■ ( 1 / 2 4 ) * 
 
 -1 
 
 -1 
 
 7 -; 
 
 L -1 
 
 
 
 -1 
 
 -1 
 
 -l " 
 
 1 -1 
 
 
 
 -1 
 
 -1 
 
 -l -] 
 
 L 7 
 
 EIGENVALUES: FIVAL(I)= X FOR 1= 1 , 2» . . • ♦ N-l , 
 EIVAL(M)= X+N 
 EXAMPLE (N = 5, X=3) 
 EIVAL ( 1 )= 3 
 EIVAL (2 )= 3 
 EIVAL (3 )= 3 
 EIVAL (4)= 3 
 EIVAL (5)= 8 
 
 EIGENVECTORS: FOR THE ISOLATED EIGENVALUE (X + N) t 
 THE EIGENVECTOR IS 
 
 EIVEC(I,M)= 1 FOR I=l t 2r...tNt 
 AMD ANY VECTOR ORTHOGONAL TO THIS ONE IS AN 
 EIGENVECTOR WITH EIGENVALUE X. 
 
 DETERMINANT: DFT(A)= ( X+N ) #X#* ( N-l ) 
 
 CONDITION NUMBERS: ANY CONDITION! NUMBER GREATER THAN 
 ONE CAM BF OBTAINED BY APPROPRIATE CHOICE OF A 
 POSITIVE VALUE FOR X. 
 C(l)= 1 + (2*N-2)/X FOR X.GT.O, 
 1 + N/X FOR X.GT.O, 
 S(ORT(U*V) FOR X.GT.O 
 
 
 C ( 2 ) = 
 
 C ( F ) = 
 
 U = 
 
 v= 
 
 EXAMPLES 
 N 
 5 
 20 
 
 60 
 
 1 00 
 : <AMPLES 
 
 ( X**2+2#X+N ) 
 
 ( ( X+N)**2-2*X-fM) / (X- 
 
 (X = 3 ) 
 
 C ( 1 ) 
 
 3.67 
 1.37(1 
 2.70( 1 
 4.03( 1 
 5,37( 1 
 6,70(1 
 (X = l ) 
 
 C ( 1 ) 
 
 C(2) 
 
 2.67 
 
 7.67 
 
 1 .43( 1 ) 
 
 2.10(1) 
 
 2.77(1) 
 
 3,43(1) 
 
 C(2) 
 
 (X + N) )*#2. 
 
 C(F) 
 
 6.78 
 
 3.85( 1 ) 
 
 9.76( 1 ) 
 
 1.72(2) 
 
 2.58(2) 
 
 3.56(2) 
 
 C(F ) 
 
22 
 
 5 
 
 2 
 
 40 
 
 60 
 
 80 
 
 100 
 
 EXAMPLES 
 
 M 
 
 5 
 
 20 
 
 40 
 
 60 
 
 80 
 
 100 
 
 9.00 
 3.90( 1 ) 
 
 7.90( 1 ) 
 1.19(2 ) 
 1 .59(2) 
 1.99(2 ) 
 (X=0.01 ) 
 C ( 1 ) 
 8.01(2) 
 3.80(3) 
 7.80(3) 
 1.18(4) 
 1 .58(4) 
 1.98 (4) 
 
 6.00 
 2.101 1 ) 
 
 4.10(1) 
 6.10(1) 
 8.10( 1 ) 
 1.01(2) 
 
 C(2 ) 
 5.01 (2 ) 
 2.00( 3) 
 4.00(3 ) 
 6.00(3) 
 8.00(3) 
 1.00(4) 
 
 1.27(1) 
 
 9.35( 1 ) 
 2.59( 2) 
 4.72( 2) 
 7.24(2) 
 1.01(3) 
 
 C( F) 
 
 1.00( 3) 
 
 8.72(3) 
 
 2.50(4) 
 
 4.61 (4) 
 
 7.11(4) 
 
 9.95(4) 
 
 REMARKS: AMY CONDITION NUMBER MAY 
 MATRIX BY APPROPRIATE CHOICE OF 
 IS USEFUL FOR TESTING THE SENSIT 
 TO ILL-CONDITIONED SYSTEMS. THIS 
 PDINTED OUT BY NEWMAN AMD TODD ( 
 IS AN EXAMPLE OF A CIRCULANT OR 
 FOR THE DETERMINATION OF THE INV 
 AMD EIGENVECTORS ARE KNOWN FOR T 
 MATRICES (REF. 1,2). A BRIEF FLU 
 3,4,5,6) APPFARFD IN THE CACM BY 
 UNAWARE OF THE KNOWN FACTS ABOUT 
 MATRICES. ANOTHER GENERALIZATION 
 POINTED OUT BY CLINE (REF. 7). 
 
 BE ASSIGNFD TO THIS 
 X. HENCE THIS MATRIX 
 IVITY OF ALGORITHMS 
 
 ADVANTAGE WAS 
 REF. 1). THIS MATRIX 
 CYCLIC MATRIX. RULES 
 ERSF, EIGENVALUES, 
 HIS CLASS OF 
 RRY OF PAPERS (REF. 
 
 PEOPLF WHO SEEMED 
 
 THIS CLASS OF 
 
 OF THIS FORM WAS 
 
 REFERENCES 
 
 1. MORR 
 MATR 
 YR19 
 
 2. R. Z 
 LAG, 
 
 3 . M . L 
 
 URES 
 
 4 . W I L L 
 
 CACM 
 
 5. RUBE 
 CES 
 
 6. A. C 
 CACM 
 
 7 R A N D 
 IMVE 
 YR19 
 
 IS 
 
 IX 
 
 58 
 
 UR 
 
 Y 
 
 I A 
 
 » 
 
 PT 
 
 FO 
 
 . 
 
 » 
 
 AL 
 
 RS 
 
 64 
 
 NEWMAN AND JOHN TODD, THE EVAL 
 INVERSION PROGRAMS, JSIAM, \/6, 
 
 IATION OF 
 PP466-476, 
 
 MUHL » 
 R1964 
 PEI, 
 CACM, 
 M S. 
 V6, P 
 D. R 
 R INV 
 R. ME 
 V6, P 
 L E. 
 ION 
 
 MATRIZEN, 4TH ED., P231, SPRINGER-VER 
 
 A TFST MATRIX FOR INVERSION PROCED- 
 
 V5, P508, YR1962. 
 LASOR, TFST MATRIX FOR INVERSION, 
 102, YR1963. 
 
 ODMAN, A NOTE ON A SET OF TFST MATRI- 
 ERSION, CACM, V6 , P515, YR1963. 
 WBERRY, PEI MATRIX EIGENVECTORS, 
 515, YR1963. 
 
 CLINE, A CLASS OF MATRICES TO TFST 
 ROCEDURES, CACM, V7, PP724-725, 
 
23 
 
 IDENT 1 F I CATI ON: OllOOl 
 
 MATRIX: A( I , J )= 1 
 
 FDR 
 
 ( J-I )=1 
 
 A ( M , 1 ) = E 
 
 E. 
 
 ,NE 
 
 .0 
 
 A ( I , J ) = 
 
 OTHERWISE. 
 
 EXAMPLE (N=5) 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 A= 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 1 
 
 E 
 
 
 
 
 
 
 
 CHARACTERISTICS: REAL, NON-SYMMETRIC, NON-SINGULAR. 
 
 INVERSE: AINV(I,J)= 1 FOR (I-J)=l, 
 AINV(1,N)= 1/E 
 AINV( I , J )= OTHERWISE. 
 FXAMPLF (M=5) 
 
 1/E 
 10 
 AINV = 10 
 10 
 10 
 
 EIGENVALUES: 
 
 E IS POSITIVE; 
 
 EI VAL( I )=(E*EXP(2*( 1-1 )*PI*W) ) 
 FOR 1 = 1,2, .•• ,N AND W 
 
 E 
 
 FOR 1 = 1,2, ...,N 
 c IS NEGATIVE; 
 EI VAL ( I ) = ( A6S( EM 
 
 **( 1/ 
 
 = SQRT 
 
 N) 
 (-1 ) 
 
 E = 
 
 EXP ( (2*1-1 )*PI*W) )*#< 1/M) 
 FOR 1 = 1,2, ... ,N. 
 0.00001 ) 
 
 EXAMPLE (N=5, 
 
 FIVAH 1. )= 0.1 
 
 EI VAL (2 )= 0.1*(C0S(2*PI/5)+W*SIN(2*PI/5) ) 
 
 E I V A L ( 3 ) = 
 
 EI VAL (4)= 
 
 FIVAL (5)= 
 
 0.1*(C0S(2*PI/5)+W*SIN(2*PI/5) ) 
 0.1*(COS(4* PI/5) +W*SIN(4*P 1/5)) 
 . 1 * ( C OS ( 6*P I / 5 ) + W*S I N ( 6*P I / 5 ) ) 
 0.1*(CnS(8*PI/5)+W*SIN(8*PI/5)) 
 
 FIGFNVFCTORS: FIVEC(J,I)= ( E I VAL ( I ) ) ** ( J-l ) 
 
 FOR I ,J= 1,2,..., 
 00 001 ) 
 
 FXAMP 
 FIVFC 
 EIVEC 
 E I V E C 
 FIVFC 
 FIVFC 
 FIVEC 
 FIVFC 
 EIVEC 
 FIVFC 
 FIVFC 
 
 /EC 
 
 FIVEC 
 FIVEC 
 F IV F C 
 FIVEC 
 El ' 
 EIVEC 
 
 (M 
 tl ) : 
 
 LF 
 ( 1 
 (2,1 ) 
 (3,1 ) 
 (4,1 ) 
 (5,1 ) 
 (1,2) 
 (2,2 ) 
 (3,2 ) 
 (A, 2 ) 
 (5,2 ) 
 (1,3) 
 (2,3) 
 ( 3 , 3 ) 
 (4,3) 
 (5,3) 
 (1,4) 
 (2,4) 
 
 5, 
 1. 
 0. 
 0. 
 0. 
 0. 
 
 1. 
 
 0. 
 0. 
 0. 
 0. 
 
 1 . 
 
 0. 
 0. 
 0. 
 0. 
 
 1 . 
 
 0. 
 
 E = 0. 
 
 
 
 1 
 
 01 
 001 
 0001 
 
 1 * ( c 
 
 1 * ( 
 001* 
 0001 
 
 
 1 * ( c 
 
 1 * ( 
 
 00 1* 
 0001 
 
 
 1 -:= ( C 
 
 OS(2*PI/5 )+W*SIN(2*PI/5 ) ) 
 C0S(4#PI /5)+W#SIN(4*PI/5) ) 
 (C0S(6*PI /5)+W*SIN(6*PI/5) ) 
 * ( COS ( 8 *P I / 5 ) + W*S I N ( 8*P I / 5 ) ) 
 
 OS (4*PI /5)+W*SIN(4*PI/5) ) 
 CDS(8#PI/5)+W*SIN(8#PI/5) ) 
 (C0S(2*PI /5)+W*SIN(2*PI /5) ) 
 * ( COS ( 6#P I /5 ) +W*S I N ( 6*P I / 5 ) ) 
 
 0S(6*P 1/5 )+W*SIN(6*PI /5 ) ) 
 
2k 
 
 EIVEC(3,4)= 
 EI VEC(4,4)= 
 E I V E C ( 5 , 4 ) = 
 EIVECI 1,5)= 
 EIVEC(2,5)= 
 EIVFC(3,5 ) = 
 EIVEC(4,5)= 
 EIVEC(5,5)= 
 
 0.01*(COS(2*PI/5)+W*SIN(2#PI/5) ) 
 0.001*(COS(8*PI/5)+W*SIN(8*PI/5) ) 
 0.0001*(C0S(4#PI/5)+W*SIN(4#PI/5 ) ) 
 
 1.0 
 
 0.1*(CriS(8*PI /5)+W*SIN(8*PI/5) ) 
 
 0.01*(COS(6*PI/5)+W*SIN(6*PI/5 ) ) 
 0.001 s MCnS(4*PI/5)+W*SIN(4*PI/5)) 
 0.0001* (COS (2*P I/5)+W*SIN(2*PI/5) ) 
 
 DETERMINANT: OET(A)= (- 1 ) ** ( N-l ) *E 
 
 REMARKS: THIS IS A GENERALIZATION OE A CYCLIC OR 
 
 CIRCULANT MATRIX (REF. 2). ACCORDING TO REFERENCE 1 
 IT WAS SUGGESTED AS A TEST MATRIX BY FORSYTHF. THE 
 CONDITION NUMBER OF THIS MATRIX CAN BE MADE ARBITR- 
 ARILY LARGE BY CHOICE OF E. IT WAS USED AS THE BASIS 
 OE A TEST MATRIX BY PARLETT (REF. 3). 
 WHEN A PARTIAL PIVOTING STRATEGY IS USED IN GAUSS 
 FLEMINATIOM OR LU DECOMPOSITION ROW EXCHANGE WILL BE 
 REQUIRED IN THE SELECTION OF EVERY PIVOT. 
 
 CONDITION NUMBERS: 
 
 EXAMPLES (E = 0. 00001) 
 
 M 
 5 
 20 
 40 
 60 
 80 
 100 
 
 C ( 1 ) 
 
 1 .00( 5) 
 
 1.00(5) 
 
 1.00(5) 
 
 1.00(5) 
 
 1 .00( 5 ) 
 
 1.00(5) 
 
 C (2 ) 
 1.00(5 ) 
 1.00( 5) 
 1.00(5 ) 
 1.00(5 ) 
 1.00(5 ) 
 1.00(5) 
 
 C(E) 
 2.00(5) 
 4.36(5 ) 
 6.24( 5 ) 
 7.68(5) 
 8.89( 5 ) 
 9.95( 5 ) 
 
 REFERENCES: 
 
 OF A MATRIX, JSIAM, 
 4TH ED,, P231, 
 
 PAUL A. WHITF, THE COMPUTATION OF 
 
 AND EIGENVECTORS 
 
 437, YR1958. 
 
 R. ZURMUHL, MATRIZEM, 
 
 LAG, YR1964. 
 
 BFRESFORD PARLETT, LAGUERRF'S METHOD 
 
 THE MATRIX EIGENVALUE PROBLEM, MATH 
 
 PPA6^-485, YR1964. 
 
 EIGENVALUES 
 V6, PP393- 
 
 SPRINGER-VER 
 
 APPLIED TO 
 COMP, V18, 
 
 IDENTIFICATION: 010001 
 
 MATR IX : A ( I , J ) = 
 
 1 FOR 1=1: 
 
 ; J = ] 
 
 L , 2 , . . . , M , 
 
 A ( I , J ) = 
 
 l/( I+J-l ) 
 
 FOR 
 
 I = 2 , 3 , . . . , N ; 
 J=1,2,...,M. 
 
 EXAMPLF (M=5) 
 
 
 
 
 
 1 1 1 
 
 1 
 
 1 
 
 
 1/2 1/3 1/4 
 
 1/5 
 
 1/6 
 
 A = 
 
 1/3 1/4 1/5 
 
 1/6 
 
 1/7 
 
 
 1/4 1/5 1/6 
 
 1/7 
 
 1/8 
 
 
 1/5 1/6 1/7 
 
 1/8 
 
 1/9 
 
25 
 
 CHARACTERISTICS: REAL* NON-SYMMETR I C , EIGENVALUE OE 
 MAXIMUM MODULUS IS POSITIVE, EIGENVALUE OF MINIMUM 
 MODULUS IS NEGATIVE, ELEMENTS OF THE INVERSE ARE 
 INTEGERS. 
 
 INVERSE : AINV( I , 1 )= 
 
 AINV( I , J ) = 
 
 EXAMPLE (M=5) 
 
 A I M V : 
 
 (-1 )**(N-I )*C(N+I-1, 1-1 )*C(N, I ) 
 
 FOR 1 = 1,2, .. .,N, 
 
 (-1 )**( I-J-1)*I*C< I+J-1,J-1 ) • 
 
 *C( I+J-2,J-2)*C(N+I-1, I+J-l ) 
 
 *C(N+J-1, I + J-l ) 
 
 FOR 1=1,2 
 
 ,N; J=2,3, 
 
 N, 
 
 5 
 
 300 
 
 -2100 
 
 4200 
 
 -2520 
 
 -60 
 
 -2400 
 
 18900 
 
 -40320 
 
 25200 
 
 210 
 
 6300 
 
 -52920 
 
 117600 
 
 -75600 
 
 -280 
 
 -6720 
 
 58800 
 
 -134400 
 
 88200 
 
 126 
 
 2520 
 
 -22680 
 
 52920 
 
 -35280 
 
 EIGENVALUES: NOT KNOWN. 
 
 EIGENVECTORS: NOT KNOWN. 
 
 DETERMINANT: DFT(A)= ( -1 )**(N-1 )/DEL(N) , 
 
 WHERE DEL(M) IS GIVEN BY THE RECURRENCE RELATION 
 DEL(1)= 1 
 DELIN+1 )= C(2*N,N-1)*C( 2*N, N ) * ( 2*N+1 )*DEL(N) 
 
 AMD C(I,J) IS THF COMBINATORIAL SYMBOL. 
 EXAMPLE (N=5) 
 
 DEL (5)= 5 33433 60000 
 
 DET(A)= 1/DEL(5) 
 
 CONDITION 
 
 NUMBERS : 
 
 
 
 EXAMPLES 
 
 
 
 
 N 
 
 C ( 1 ) 
 
 C (2 ) 
 
 C(E) 
 
 2 
 
 1 .80(1 ) 
 
 1.41(1) 
 
 1.42(1) 
 
 3 
 
 8.64(2 ) 
 
 4.83(2) 
 
 4.85(2) 
 
 4 
 
 3.79(4) 
 
 1.70(4) 
 
 1.71(4) 
 
 5 
 
 1.44(6) 
 
 5.92(5) 
 
 5.96(5) 
 
 6 
 
 5.63(7) 
 
 2.04(7) 
 
 2.06(7) 
 
 8 
 
 8.16( 10) 
 
 2.38( 10) 
 
 2.41 ( 10) 
 
 10 
 
 1 .09( 14) 
 
 2.77( 13) 
 
 2.80( 13) 
 
 12 
 
 1.37(17) 
 
 3.20( 16) 
 
 3.2M 16) 
 
 REMARKS: THIS MATRIX IS CLOSELY RELATED TO A FINITE 
 SEGMENT OF THE HILBERT MATRIX (TM 001007). IT IS 
 ILL-CONDITIONED, LIKE THE HILBERT MATRIX, AND IT 
 IS NON-SYMMETRIC, UNLIKF THE HILBERT MATRIX. 
 
 RFFFREMCFS : 
 
 1. MARK L DTK IN, A SET OF TEST MATRICES, MTAC, V9, 
 PP153-161, YR1955. 
 
 2. P. J. FBERLFIN, A JACOBI-LIKE METHOD FOR THE 
 AUTOMATIC COMPUTATION OF EIGENVALUES AND 
 EIGENVECTORS OF AN ARBITRARY MATRIX, JSIAM, V10, 
 PP74-88, YR1962. 
 
26 
 
 IDENTIFICATION: 010002 
 
 MATRIX: A IS THE BLOCK MATRIX 
 A = B 2*8 
 V-:=B 3*B 
 
 AMD B IS THE MATRIX, BELOW E.NE.O 
 
 10 
 
 10 
 
 B= 1 
 
 1 
 
 F 
 
 CHARACTERISTICS: REAL, NON-SYMMETRIC, NON-SINGULAR. 
 
 INVERSE: AINV IS THE BLOCK MATRIX 
 AINV=(l/5)* -3-BINV 2*BINV 
 
 4-BINV -1-BINV 
 AND BINV IS THE MATRIX 
 1/E 
 10 
 BIN\/= 10 
 10 
 10 
 
 EIGENVALUES: E IS POSITIVF; 
 
 EI VAL (K ) =5*( E*EXP( 2*K*PI*W) )**0.2 
 
 FOR K=l,2,3,4,5, AND W=SQRT(-1), 
 EI VAL (K)= -( E*EXP(2*K*PI*W) )**0.2 
 
 FOR K=6,7,8,9 ,10. 
 E IS NEGATIVE; 
 
 EIVALtK )=5*(ABS(E)*EXP( (2K + 1 )*PI*W) )**0.2 
 FOR K=l,2,3,4,5, AND W=S0RT(-1), 
 FI VAL (K )= -( ABS(E )*EXP( ( 2K+1 )*PI*W) )**0.2 
 FOR K=6,7,8,9, 10. 
 EXAMPLF (E=0. 00001) 
 
 EIVAL(K)= 0.5*C0S(2*P I*K/5 ) +0. 5*W*S I N ( 2*P I *K/5 ) 
 
 FOR K=l ,2,3,4,5, 
 E I V A L ( K ) =-0 . 1 *Cf)S ( 2*- 1 *K /5 ) -0 . 1 * W*S I N ( 2 *- 1 *K / 5 ) 
 FOR K=6, 7,8,9,10. 
 
 F IGENVECTORS 
 
 EI VEC ( J,K )= (EIVAL(K) )**( J-l ) 
 
 FOR J=l,2,3,4,5; K=l,2,3,4,5, 
 EIVEC( J,K)=2*( EIVAL(K) )**( J-l ) 
 
 FOR J=6,7,8,9,10; K=l,2,3,4,5, 
 EIVEC( J,K) = ( EIVAL(K) )**( J-l ) 
 
 FOR J=l,2,3,4,5; K=6,7,8,9, 10, 
 EIVECt J,K)= -( EIVAL(K) ) ** ( J-l ) 
 
 FOR J=6,7,8,9,10;K=6,7,8,9, 10. 
 
 DETERMINANT: DFT(A )=-312 5*E**2 
 
 CONDITION NUMBERS: 
 
 EXAMPLE (E=0. 00001) 
 
 N C ( 1 ) C ( 2 ) C ( E ) 
 
 10 7.00(5) 5.83(5) 1.20(6) 
 
 REMARKS: A SPECIAL CASE OF THE MATRIX A WITH E=0. 00001 
 
27 
 
 WAS USED 
 (REF. 1,2 
 THE MATRI 
 WITH SOME 
 (SEE MATR 
 THE FORMU 
 ARE DERIV 
 LET A BE 
 M BY M MA 
 MATRIX. I 
 COEFFICIE 
 V=(V(1 ) , 
 V • ( iM ) ) A R 
 MATRICES, 
 (V (1 )*V, 
 EIGENVECT 
 
 BY PARL 
 
 ) . 
 
 X B IS 
 
 MATRIX 
 ICES 01 
 LAS FOR 
 EO BY T 
 AN M*N 
 TRIX BL 
 F D AMD 
 NT MATK 
 V(2) ,.. 
 E THE C 
 THEN D 
 V • ( 2 ) * 
 OR OF A 
 
 ETT AND EBERLEIN AS A TEST MATRIX 
 
 THE MATRIX 011001, AND, REPLACING B 
 , WE CAN GET ANOTHER TEST MATRIX 
 0004,010005 ) . 
 
 THE EIGENVALUES AND EIGENVECTORS 
 HE FOLLOWING PROPOSITION: 
 BY M*N BLOCK MATRIX WHOSE (I,J)-TH 
 OCK IS C ( I, J)*B, WHERE B IS AN MBY M 
 
 D' ARE EIGENVALUES OF B AND THE 
 IX C, RESPECTIVELY, AND 
 ., V(M)) AND V'=(VM1), VM2),..., 
 ORRESPONDING EIGENVECTORS OF THE TWO 
 •*D IS AN EIGENVALUE OF A AND 
 V,..., V'(N)-V) IS THE CORRESPONDING 
 
 REFERENCES : 
 
 1. BERESFORD PARLETT, LAGUERRE'S METHOD APPLIED TO 
 THE MATRIX EIGENVALUE PROBLEM, MATH COMP, V18, 
 PP464-485, YR1964. 
 
 2. P. J. EBERLEIN, A JACOBI-LIKE METHOD FOR THE 
 AUTOMATIC f 
 VECTORS OF AN 
 PP74-88, YR1962. 
 
 HE MATRIX EIGENVALUE PROBLEM, MATH COMP, V18, 
 P464-485, YR1964. 
 
 . J. EBERLEIN, A JACOBI-LIKE METHOD FOR THE 
 UTOMATIC COMPUTATION OF EIGENVALUES AND EIGEN- 
 FCTORS OF AN ARBITRARY MATRIX, JSIAM, V10, 
 P74-88, YR1962. 
 
 IDENTIFICATION: 010003 
 
 MATRIX: A(I,J)= 
 A( I , J ) = 
 
 A ( I , J ) = 
 EXAMPLE (N=5) 
 
 A = 
 
 5 
 4 
 3 
 2 
 1 
 
 FOR J-I .GT.l , 
 N-I FOR J-I. EQ.lt 
 N-I + l FOR I.GE.J. 
 
 CHARACTERISTICS: HESSENBFRG FORM. 
 
 INVFKSE : AINV( I ,J ) =-1 
 
 A I N V ( I , J ) = 
 A I N V ( 1 , 1 ) = 
 
 A IN V ( 1 , J ) = 
 AINV( I ,J ) = 
 
 M P L E 
 
 A I N V = 
 
 ( N = 5 ) 
 
 1 -4 
 
 -1 
 
 
 
 
 
 5 
 
 -1 
 
 
 
 12 
 -IS 
 
 4 
 -1 
 
 
 
 FOR I-J.EO.l, 
 FOR I-J.GT.l, 
 
 
 
 1 
 
 (-1 )**( J+l )*FAC(N-1 )/FAC (N-J ) 
 
 (-1 )**( I+J )*<N-I+2)#FAC(N-I ) 
 
 /FAC(N-J) 
 
 -24 
 
 3 
 
 -8 
 
 3 
 
 -1 
 
 24 
 
 -30 
 
 8 
 
 -3 
 
 ? 
 
28 
 
 p k;fnvalues : not known. 
 
 EIGENVECTORS: MOT KNOWN. 
 
 DETERMINANT: DET(A)= 1 
 
 ON 01 T I tin 
 
 NUMBERS : 
 
 
 
 M 
 
 C(l ) 
 
 0(2) 
 
 C(E) 
 
 2 
 
 9.00 
 
 6.85 
 
 7.00 
 
 3 
 
 3.50(1) 
 
 2.76( 1 ) 
 
 2.87( 1 ) 
 
 4 
 
 1.68 (2 ) 
 
 1.20(2) 
 
 1.27(2) 
 
 5 
 
 8.91(2) 
 
 6.47(2 ) 
 
 6.91(2) 
 
 6 
 
 5.87(3) 
 
 4.21 (3) 
 
 4.55 ( 3) 
 
 8 
 
 3.76(5) 
 
 2.76(5 ) 
 
 3.03(5) 
 
 10 
 
 3.84(7) 
 
 2.85(7) 
 
 3.18(7) 
 
 12 
 
 5.68(9) 
 
 4.27(9) 
 
 4.79(9) 
 
 REMARKS: THIS MATRIX WAS USFD BY FRANK (RFF. 1) AND 
 BY PARLETT (REF. 2) AS A TEST MATRIX. IT HAS BFFM 
 DISCUSSED BY WILKINSON. 
 
 REFERENCES: 
 
 1. 
 
 2. 
 
 3. 
 
 5 EIGENVALUES OF COMPLEX 
 VALUATION AMD BY METHODS 
 
 FRANK, COMPUTING 
 
 EVALUATION AN 
 
 V6, PP378-392, 
 
 WERNER L- 
 
 MATRICES BY DETERMINANT 
 
 OF DAMILEWSKI AMD WIELANDT, JSIAM, 
 
 YR1958. 
 
 LAGUERRES METHOD APPLIED TO 
 THF MATRIX EIGENVA! ' 
 PP464-485, YR1964. 
 
 J. H. WILKINSON, ERROR ANALYSIS DF FLOATING-POINT 
 COMPUTATION, 
 
 BERESFORD PARLETT, LAGUERRES METHOD APPLIED TO 
 THF MATRIX EIGENVALUE PROBLEM, MATH COMP, V18, 
 PP464-485, YR1964. 
 
 ON, ERROR ANALYSIS HF FLOATING-P 
 MUM MATH, V2, PP319-340, YR1960. 
 
 IDENTIFICATION: 010004 
 
 MATRIX: A IS THE BLOCK MATRIX 
 A = B 2 * B 
 4*B 3*B 
 AND B IS THF BLOCK MATRIX 
 B= 5*C -C 
 5*C C 
 AND C IS THF MATRIX 
 -2222 
 C= -3 3 2 2 
 -2042 
 -1 5 
 
 CHARACTERISTICS: REAL, NON-SYMMETRIC, NON-SINGULAR, 
 ALL EIGENVALUES ARE COMPLEX. 
 
 INVERSE: AINV IS THF BLOCK MATRIX 
 AINV= (1/5 )* -3*8 INV 2*BINV 
 
29 
 
 AND BIMV IS THF 
 BINV=(1/10)* 
 
 AMD CINV IS THE 
 
 CINV=(1/12 )* 
 
 4*BINV -BINV 
 BLOCK MATRIX 
 
 CINV 
 
 -5*C INV 
 
 MATRIX 
 
 30 -20 
 
 18 -8 
 
 12 -8 
 
 6 -4 
 
 CINV 
 5*CINV 
 
 -5 
 
 -5 
 
 1 
 
 -1 
 
 E IGENVALOES : 
 
 EIVAL ( 1 ) = 
 
 EIVAL (2 ) = 
 
 FIVAL(3) = 
 
 EIVAL (4) = 
 
 E I V A L ( 5 ) = 
 
 EIVAL (6 ) = 
 
 FIVAL(7) = 
 
 E I V A L ( 8 ) = 
 
 FIVAL (9 ) = 
 
 BELOW W = 
 60+W*20 
 60-W-20 
 45+W*15 
 45-W*15 
 30+W*10 
 3 0-W*10 
 15+W*5 
 15-W*5 
 -12+W*4 
 
 SORTI-1 ) 
 
 EIVAL ( 10 )=-12-W*4 
 EIVAL ( 11 ) = -9+W*3 
 E I V A L ( 1 2 ) = 
 FIVAL ( 13 ) = 
 EIVAL (14)= 
 EIVAL (15)= 
 E I V A L ( 1 6 ) = 
 THE ARHVF 
 
 -9-W*3 
 -6+W*2 
 -6-W*2 
 
 -3+W 
 
 -3-W 
 
 EIGENVALUES 
 
 -2 
 
 -2 
 
 -2 
 
 2 
 
 CAN BE OBTAINED BY 
 
 THE 
 
 PROPOSITION GIVEN IN THE REMARKS FOR THE MATRIX 
 010002 COMBINING THF FIGENVALOES GIVEN BELOW. 
 
 EIVAL(1)=-1 FOR THF COEFFICIENT MATRIX OF A 
 
 F I V A L ( 2 ) = 5 
 
 EI VAL ( 1 )= 3+W 
 E IVAL (2 )= 3-W 
 
 FOR THE COEFFICIENT MATRIX OF B 
 
 EIVAL(1)= 1 
 
 EIVAL (2)= 2 
 
 E IVAL (3)= 3 
 
 F I V A I. ( 4 ) = 4 
 
 FOR THE MATRIX C 
 
 EIGENVECTORS: THE CORRESPONDING EIGENVECTORS OF EACH 
 OF THE ABOVE MATRICFS ARE GIVEN BELOW, AND THE 
 EIGENVECTORS OF THE MATRIX A CAN BE OBTAIMFD BY THF 
 PROPOSITION RFFFERED TO ABOVF. 
 
 EI\/EC(1,1) = 1 FOR THE COEFFICIENT MATRIX OF A 
 
 EIVFC (2, 1. )=-l 
 
 EIVEC(1,2)= 1 
 
 1 f V F C ( 2 , 2 ) = 2 
 
 ! VEC (1,1)= 1 
 
 (2,1)= 2-W 
 
 F I V F C ( 1 , ? ) = 1 
 
 I [VEC(2 f 2 )= 2 + W 
 
 FOR THE COEFFICIFNT MATRIX OF B 
 
 E I V E C ( 1 , 1 ) = 4 
 B I V E C ( 2 f 1 ) a 3 
 
 FOR THF MATRIX C 
 
30 
 
 F I VEC (3,1 )= 2 
 
 FIVEC (A, 1 )= 1 
 EIVEC (1,2)= 3 
 EIVEC(2,2 )= 3 
 EIVEC (3,2 ) = 2 
 EIVEC (4,2)= 1 
 EI VEC (1,3 )= 2 
 EIVEC (2,3)= 2 
 EIVEC (3,3 )= 2 
 EIVEC (4,3)= 1 
 EIVEC (1,4)= 1 
 EIVEC (2,4)= 1 
 EIVEC (3,4)= 1 
 EIVEC(4,4)= 1 
 
 DETERMINANT: DET(A)=1296(16) 
 
 CONDITION NUMBERS: 
 
 N C ( 1 ) C ( 2 ) C ( E ) 
 16 1.99 (3 ) 8.99(2 ) 1.12(3) 
 
 REMARKS: THIS MATRIX WAS USED AS A TEST MATRIX BY 
 EBERLFIN (RFF. 1) AND BY PARLETT (REF. 2). 
 
 1 
 
 2. 
 
 IN, A JACOBI-LIKE METHOD FOR THF 
 MPUTATION OF EIGENVALUES AMD EIGEN- 
 N ARBITRARY MATRIX, JS1AM, V10, 
 962. 
 
 REFERENCES: 
 
 P. J. FBERLE 
 
 AUTOMATIC COi 1 
 
 VECTORS OF AM A.. 
 
 PP74-88, YR1962. 
 
 BERESFORD PARLETT, LAGUERRE'S 
 
 THE MATRIX EIGENVALUE PROBLEM, 
 
 PP464-485, YR1964. 
 
 METHOD APPLIFD TO 
 MATH COMP, V1R, 
 
 IOEN'TIFICATIOM: 010005 
 
 MATRIX: A IS THE BLOCK MATRIX 
 A = B 2 * B 
 4*B 3*8 
 AMD B IS THE BLOCK MATRIX 
 B= 3*C 3*C 
 5*C C 
 AND C IS THE BLOCK MATRIX 
 6*D -D D 
 C= 8*0 D 2*D 
 -2*0 D 2*D 
 5*0 -D -D D 
 AND IS THE MATRIX 
 -2222 
 D= -3322 
 -2042 
 -10 5 
 
31 
 
 CHARACTERISTICS: REAL, NON-SYMMETRIC, NGN-SINGULAR, 
 ALL EIGENVALUES ARE COMPLEX. 
 
 THE BLOCK MATRIX 
 -3*BINV 2*BINV 
 4*BINV -BINV 
 BLOCK MATRIX 
 -CINV 3*CINV 
 5*CINV -3*CINV 
 AND CINV IS THE BLOCK MATRIX 
 
 INVERSE: AINV IS 
 AlNV=(l/5)* 
 
 ANO BINV IS THE 
 BINV=( 1/12)* 
 
 
 
 
 DINV 
 
 -DINV 
 
 CINV=(1/10)* -6*0 INV 
 4*01 NV 
 
 6*DINV 
 
 
 -4*DINV -4*01 NV 
 2*DINV -4*01 NV 
 
 
 -2*DINV 
 
 DINV 
 
 3*DINV 2*DIMV 
 
 AMD 01 NV IS 
 
 THE MATRIX 
 
 
 
 
 30 -20 
 
 -5 -2 
 
 
 DINV = (1/12 )* 
 
 -5 -2 
 
 
 
 12 -8 
 
 1 -2 
 
 
 
 6 -4 
 
 -1 2 
 
 
 EIGENVALUES: I 
 
 }ELOW W=SQRT( 
 
 -1 ) 
 
 
 EI VAL ( 1 ) = 
 
 360+W*12 
 
 
 
 EIVAL(2 ) = 
 
 3 60- W* 120 
 
 
 
 E I V A L ( 3 ) = 
 
 2 70+W*90 
 
 
 
 EIVAL(4) = 
 
 270-W*90 
 
 
 
 E I V A L ( 9 ) = 
 
 ].80+W*60 
 
 
 
 EIVAL( 10)= 
 
 18 0-W*60 
 
 
 
 F I V A L ( 5 ) = 
 
 12 0+W*24 
 
 
 
 E I V A L ( 6 ) = 
 
 12 0-W*2 40 
 
 
 
 EI VAL (7) = 
 
 90+W*180 
 
 
 
 EIVAL (8 ) = 
 
 90-W*180 
 
 
 
 FIVAL (13)= 
 
 -1 20+W*40 
 
 
 
 E I V A L ( 1 4 ) = 
 
 -120-W*40 
 
 
 
 E I V A L ( 1 1 ) = 
 
 60+W*12 
 
 
 
 F I V A L ( 1 2 ) = 
 
 60-W*120 
 
 
 
 E I V A L ( 1 5 ) = 
 
 90+W*30 
 
 
 
 E I V A L ( 1 6 ) = 
 
 -90+W-30 
 
 
 
 E I V A L ( 1 7 ) = 
 
 -90-W*30 
 
 
 
 FIVAL ( 18 ) = 
 
 90-W*3 
 
 
 
 F I V A L ( 2 1 ) = 
 
 -72+W*24 
 
 
 
 E I V A L ( 2 2 ) = 
 
 -72-W*24 
 
 
 
 EIVAL ( 27 ) = 
 
 -60+W*20 
 
 
 
 FIVAL (28 ) = 
 
 -60-W*2 
 
 
 
 PI VAL (19)= 
 
 -40+W*80 
 
 
 
 FIVAL (20)= 
 
 -40-W*80 
 
 
 
 FIVAL (29)= 
 
 -54+W*18 
 
 
 
 FIVAL ( 30 ) = 
 
 -54-W*18 
 
 
 
 EI VAL (23 ) = 
 
 30+W*60 
 
 
 
 F I V A L ( 2 4 ) = 
 
 - 3 + W * 6 
 
 
 
 FIVAL (25)= 
 
 -30-W*60 
 
 
 
 //•.I. (26) = 
 
 -30+W*60 
 
 
 
 ML ( 37 ) = 
 
 -36+W*1 2 
 
 
 
 E I V A L ( 38 ) = 
 
 -36-W*12 
 
 
 
 | 7AL(31 ) = 
 
 -24+W*48 
 
 
 
 EI VAL (32) 
 
 _24-W*48 
 
 
 
 1 i in (39) = 
 
 -30+W*10 
 
 
 
 I ( 40 ) = 
 
 -30-W*10 
 
 
 
32 
 
 FIVAL(33 ) = 
 FIVAL(34)= 
 EIVAL(35)= 
 FI VAL ( 36 ) = 
 E I V A L ( 4 1 ) = 
 FIVAL(42 ) = 
 FIVAL(43)= 
 FIVAL (44)= 
 FI VAL (45 ) = 
 FIVAL (46 ) = 
 FIVAL (47)= 
 FIVAL (48 ) = 
 FIVAL (49 ) = 
 F I V A L ( 5 ) = 
 EIVAL(51 ) = 
 FIVAL ( 52 ) = 
 FIVAL(53)= 
 E I V A L ( 5 4 ) = 
 FIVAL (55 ) = 
 FIVAL (56)= 
 E I V A L ( 5 7 ) = 
 EIVAL (58 ) = 
 EIVAL ( 59 ) = 
 FIVAL (60)= 
 EIVAL(61 ) = 
 EIVAL(62 ) = 
 FIVAL (63 ) = 
 FIVAL (64)= 
 THE ABOVE 
 
 -2 0+W*40 
 -20-W*40 
 -18+W*36 
 -18-W*36 
 -12+W*24 
 -12-W*24 
 24+W*8 
 24-W*8 
 10+W*20 
 10-W*20 
 18+W*6 
 -18 + W-I-6 
 -18-W*6 
 18-W*6 
 8+W*16 
 8-W*16 
 6+W*12 
 -6+W*12 
 -6-W*12 
 6-W*12 
 12+W*4 
 12-W*4 
 4 + W*8 
 4-W*8 
 6 + W*2 
 6-W*2 
 2+W*4 
 2-W-4 
 EIGENVALUES 
 
 CAM BE OBTAINED BY 
 
 THE 
 
 PROPOSITION GIVEN IN THE REMARKS FOR THE MATRIX 
 010002 COMBINING THE EIGENVALUES GIVEN BELOW. 
 
 EIVAL(1)=-1 FOR THE COEFFICIENT MATRIX OF A 
 
 EIVAL (2 )= 5 
 
 EIVAL ( 1 )=-2 
 EIVAL (2 )= 6 
 
 FOR THE COEFFICIENT MATRIX OF B 
 
 FIVAL(1)= l + W*2 FOR THE COEFFICIENT MATRIX OF C 
 
 EIVAL (2)= 1-W*2 
 
 E I V A L ( 3 ) = 3 + W 
 
 EIVAL (4)= 3-W 
 
 EIVAL ( 1 )= 1 
 
 EIVAL(2 )= 2 
 
 FIVAL (3 )= 3 
 
 F I V A L ( 4 ) = 4 
 
 FOR THE MATRIX D 
 
 EIGENVECTORS: THE CORRESPONDING EIGENVECTORS OF EACH 
 OF THF ABOVE MATRICES ARE GIVEN BELOW, AND THE 
 EIGENVECTORS OF THE MATRIX A CAM BE OBTAINED BY THE 
 PROPOSITION RFFFFRFD TO ABOVE. 
 
 EIVEC(1,1)= 1 FOR THE COEFFICIENT MATRIX OF A 
 
 EIVEC (2,1 )=-l 
 
 EIVFC (1,2)= 1 
 
 EIVEC (2 ,2 )= 2 
 
 EIVEC( 1,1)= 3 
 
 FOR THF COEFFICIENT MATRIX OF B 
 
33 
 
 FIVEC(2,1 )=-5 
 EIVEC( 1 t 2 )= 1 
 EIVEC(2,2)= 1 
 
 EIVEC(1,1 )= 
 EIVEC(2f 1 )= W 
 EIVEC(3»1 )= W 
 EIVEC(4,1 )=-l 
 
 FOR THE COEFFICIENT MATRIX OF C 
 
 EIVEC(1,2)= 
 
 
 
 
 FIVEC (2,2 ) = 
 
 W 
 
 
 EIVFC(3,2)= 
 
 W 
 
 
 FIVFC(4,2 ) = 
 
 1 
 
 
 EIVEC(1,3)= 
 
 1 
 
 
 EIVFC (2,3 ) = 
 
 3-W 
 
 
 EIVFC (3,3)= 
 
 
 
 
 EIVEC(4,3)= 
 
 1 
 
 
 FIVECd ,4) = 
 
 1 
 
 
 FIVFC(2,4)= 
 
 3 + W 
 
 
 EIVFC (3,4)= 
 
 
 
 
 EIVEC(4,4)= 
 
 1 
 
 
 FIVFC ( 1 ,1 ) = 
 
 4 FOR THE MATRIX 
 
 FIVFC (2, 1 ) = 
 
 3 
 
 
 EIVEC (3,1 ) = 
 
 2 
 
 
 FIVEC (4,1 ) = 
 
 1 
 
 
 FIVEC ( 1 ,2 ) = 
 
 3 
 
 
 FIVFC (2,2 ) = 
 
 3 
 
 
 FIVFC (3,2 ) = 
 
 2 
 
 
 FIVEC (4,2)= 
 
 1 
 
 
 FIVEC (1,3)= 
 
 2 
 
 
 FIVFC (2,3 ) = 
 
 2 
 
 
 FIVFC ( 3,3 ) = 
 
 2 
 
 
 FIVEC (4,3)= 
 
 1 
 
 
 EI VEC ( 1 ,4)= 
 
 1 
 
 
 FIVEC (2,4)= 
 
 1 
 
 
 FIVFC (3,4)= 
 
 1 
 
 
 F I V F C ( 4 , 4 ) = 
 
 1 
 
 
 DETERMINANT: DET(A)= 12**32*3**16(64) 
 
 CONDITION 
 N 
 64 
 
 NUMBERS: 
 C( 1 ) 
 2.93(4) 
 
 C(2) 
 7.27(3) 
 
 C( F ) 
 1.20(4) 
 
 R E M A R K S : T H I S 
 
 TEST MATRIX 
 
 MATRIX HAS 
 (RFF. 1 ) . 
 
 BEEN USED BY PARLFTT AS A 
 
 REFERENCES: 
 
 1. BERESFORD PARLFTT, LAGUERRE'S METHOD APPLIFD TO 
 THE MATRIX FIGFNVALOE PROBLEM, MATH COMP, V1B, 
 PP464-485, YR1964. 
 
3h 
 
 IDENTIFICATION: 010006 
 
 MATRIX: A IS THE BLOCK MATRIX 
 A = 8*B 4*B 
 -5*B -B 
 AMD B IS THF BLOCK MATRIX 
 B= C 2*C 
 4*C 3*C 
 AND C IS THF BLOCK MATRIX 
 C= 4*0 3*D 
 -2*D -D 
 AND D IS THE MATRIX 
 
 -110 
 -10 10 
 0= -1 1 
 -10 1 
 -10 
 
 CHARACTERISTICS: REAL, NON-SYMMETRIC, NON-SINGULAR, 
 REAL AND COMPLFX EIGENVALUES, 
 
 INVFRSE: AINV IS 
 AINV= (1/12 )* 
 
 AMD BINV IS THF 
 B4NV= (1/5)* 
 
 AMD CINV IS THE 
 CIMV=( 1/2 )* 
 
 THF BLOCK MATRIX 
 
 -BINV -4*BINV 
 
 5 * B I N V 8 * B I N V 
 
 BLOCK MATRIX 
 
 -3 -CINV 2*CINV 
 
 4*CINV -CINV 
 
 BLOCK MATRIX 
 
 -DINV -3*DINV 
 
 AND DINV 
 
 DINV = 
 
 IS 
 
 
 
 1 
 
 
 
 
 
 THE 
 
 
 
 
 
 1 
 
 
 
 
 
 2*01 NV 
 
 MATR IX 
 
 
 
 
 
 1 
 
 4*DIMV 
 
 IGENVALUES: BFLOW W= SORT(-l) 
 EIVAL ( 1 ) = 3 
 EIVAL(2) = 4 
 F I V A L ( 3 ) = 6 
 EIVAL (4) = 1.5 
 FIVAL (5) = 1.5 
 EIVAL(6) =-1.5 
 FI\/AL(7) =-1.5 
 EIVAL (B) = 6 
 EIVAL (9) = 2+W 
 EIVAL ( 10 )= 2-W 
 FIVAL ( 11 )=-2-W 
 FIVAL ( 12 )=-2+W 
 EIVAL (13)= 3+W 
 EIVAL (14)= 3-W 
 EIVAL ( 15)=-3-W 
 EIVAL ( 16)=-3+W 
 EIVAL(17 )=-15 
 FIVAL (22 ) =-20 
 EIVAL(23)=-30 
 
 +W*1.5*SQRT( 3 ) 
 -W*l .5*S0RT ( 3) 
 -W*1.5*SQRT(3 ) 
 +W*1.5*S0RT( 3) 
 
 *2*S0RT(3) 
 *2*S0RT(3) 
 *2*S0RT(3) 
 *2*S0RT(3) 
 *3*S0RT(3) 
 *3*S0RT(3) 
 *3*S0RT( 3 ) 
 *3*S0RT(3) 
 
35 
 
 EIVAL( 
 EIVAL( 
 EIVAK 
 EIVAK 
 EIVAK 
 EIVAK 
 EIVAL ( 
 EIVAK 
 EIVAL ( 
 EIVAL ( 
 EIVAL ( 
 EIVAL( 
 EIVAL( 
 EIVAL( 
 FIVAL ( 
 EIVAL ( 
 EIVAL ( 
 EIVAL ( 
 E I V A L ( 
 EIVAL ( 
 EIVAL ( 
 THE AB 
 PROPOS 
 01000 2 
 EIV 
 EIV 
 
 24) = 
 
 25) = 
 26)=- 
 27)=- 
 
 28 )=- 
 
 29 ) = 
 
 30 ) = 
 
 31 )=- 
 
 32 )=- 
 
 33 ) = 
 34) = 
 35)=- 
 36)=- 
 
 37 ) = 
 
 38 ) = 
 
 39 )=- 
 
 40 )=- 
 
 18 ) = 
 
 19 ) = 
 20)=- 
 21 )=- 
 OVF E 
 I T I N 
 
 COMB 
 AL (1 ) 
 AL (2 ) 
 
 7.5 
 7.5 
 7.5 
 7.5 
 
 40 
 10 + 
 10- 
 10- 
 10 + 
 15 + 
 15- 
 15- 
 15 + 
 20 + 
 20- 
 20- 
 20 + 
 4 + W 
 4-W 
 4-W 
 4 + W 
 IGE 
 GI 
 I MI 
 = 3 
 = 4 
 
 +W*7.5*SQRT(3 ) 
 -W*7.5*SQRT(3) 
 -W*7.5*SQRT(3) 
 +W*7.5*S0RT(3) 
 
 W*2 
 *4* 
 
 *4* 
 
 ^r <+ 5 ! c 
 
 *4* 
 
 MVA 
 VEM 
 
 NG 
 
 0*S 
 
 o*s 
 
 0*S 
 
 o*s 
 
 5*S 
 5*S 
 5*S 
 5*S 
 0*S 
 
 o-*s 
 
 0*S 
 0*S 
 
 SOR 
 SOR 
 SOR 
 SOR 
 LUE 
 IN 
 THE 
 F 
 
 ORT( 
 ORT( 
 ORT( 
 ORT( 
 ORT( 
 ORT( 
 ORT( 
 ORT( 
 0RT( 
 ORT( 
 ORT( 
 QRT( 
 T(3) 
 T(3) 
 T(3) 
 T(3) 
 S CA 
 THE 
 EIG 
 OR T 
 
 3) 
 
 3) 
 3) 
 3) 
 3) 
 3) 
 3) 
 3) 
 3 ) 
 3) 
 3) 
 3) 
 
 N BE OBTAINED BY THE 
 
 REMARKS FOR THE MATRIX 
 ENVALDES GIVEN BELOW. 
 HE COEFFICIENT MATRIX OF 
 
 EIVAL ( 1 )=-l 
 EIVAL (2 )= 5 
 
 FOR THE COEFFICIENT MATRIX OF B 
 
 FIVAL ( 1 )= 1 
 EIVAL (2 )= 2 
 
 FOR THE COEFFICIENT MATRIX OF C 
 
 EIVAL(1)=-1 FOR THE MATRIX 
 EIVAL(2)= 0.5+W*0.5*SQRT (3) 
 EIVAL(3)= 0.5-W*0.5*S0RT(3) 
 ElVAL(4)=-0.5+W*0.5*SQRT(3) 
 EIVAK 5 )=-0.5-W*0.5*SQRT( 3 ) 
 
 EIGENVECTORS: THE CORRESPONDING EIGENVECTORS OF EACH 
 flF THE ABOVE MATRICES ARE GIVEN BELOW, AND THE 
 FIGENVECTORS OF THE MATRIX A CAN BE OBTAINED BY THE 
 PROPOSITION RFFFERED TO ABOVE. 
 
 FIVEC(ltl)= 4 FOR THF COEFFICIENT MATRIX OF A 
 
 EIVEC (2, 1 )=-5 
 
 ElVEC(l t 2)= 1 
 
 EIVEC(2,2 )=-l 
 
 I IVEC(1,1)= 1 
 FIVFC(2,1 )=-l 
 EIVEC(1,2 )= 1 
 E I V E C ( 2 r 2 ) = 2 
 
 FOR THF COEFFICIENT MATRIX OF B 
 
 EIVEC ( 1 ,1 )= 1 
 I [VEC(2»1 ) =-1 
 EIVEC (1,2 )= 3 
 EIVEC(2t2) =-2 
 
 FOR THF COEFFICIENT MATRIX OF C 
 
36 
 
 FOR THF. MATRIX D 
 
 EIVECdt 1 )= 1 
 
 EIVEC(2,1 )= 
 
 EIVEC(3,1 )= 1 
 
 EIVEC (4,1 )= 
 
 EIVEC(5,1 )= 1 
 
 ElVECd ,2 )=-l-W*SQRT(3) 
 
 EIVEC(2,2)= -W*2*SORT(3) 
 
 EIVEC(3,2)= 2-W*2*SQRT(3) 
 
 EIVEC(4,2)= 3-W*SQRT(3) 
 
 EIVEC(5,2 )= 2 
 
 EIVEC (1,3)=-1+W*SQRT( 3) 
 
 EIVEC (2,3)= W*2*SQRT(3) 
 
 ElVEC(3t3)= 2+W*2*SQRT(3) 
 
 3+W*SQRT(3) 
 
 2 
 
 1-W*SQRT(3 ) 
 
 2 
 
 EIVEC(4,3)= 
 EIVEC (5t 3)= 
 
 EIVEC( 1 ,4) = 
 EIVEC(2,4)= 
 EIVEC (3,4)= 
 EIVEC(4,4)= 
 EIVEC( 5,4)= 
 EIVEC(1,5)= 
 EIVEC(2,5 )= 
 EIVEC (3,5)= 
 EIVEC(4,5 )= 
 EIVEC(5,5)= 
 
 
 
 1-W*SQRT( 3) 
 
 2 
 
 1+W*SQRT( 3) 
 
 2 
 
 
 
 1+W*SQRT(3) 
 
 2 
 
 DFTERMINANT: DET(A)= 12**20(20) 
 
 CONDITION 
 N 
 40 
 
 NUMBERS: 
 C( 1 ) 
 7.64(3) 
 
 C(2) 
 
 4.42(3) 
 
 C(E) 
 7.15(3) 
 
 REMARKS: THIS MATRIX HAS BEEN USED AS A TEST MATRIX 
 BY EBERLEIN (REF. 1) AND BY PARLETT (REF. 2). IN 
 EBERLEIN'S PAPER THE EIGENVALUES LISTED SHOULD BE 
 MULTIPLIED BY 2. IN PARLETT'S PAPER FOUR EIGENVALUES 
 ARE OMITTED, THEY ARE IDENTIFIED HERE AS EIVAL(18), 
 EIVALU.9), EIVAL(20), AND EIVAL(21). 
 
 REFERENCES: 
 
 1. P. J. FBERLEIM, A JACOBI-LIKE METHOD FOR THE 
 AUTOMATIC COMPUTATION OF EIGENVALUES AND EIGEN- 
 VECTORS OF AM ARBITRARY MATRIX, JSIAM, V10, 
 PP74-88, YR1962. 
 
 2. BERESFORD PARLETT, LAGUERRE'S METHOD APPLIED TO 
 THE MATRIX EIGENVALUE PROBLEM, MATH COMP, V18, 
 PP464-485, YR1964. 
 
}rAEC _427 U.S. ATOMIC ENERGY COMMISSION 
 
 i6/68) UNIVERSITY-TYPE CONTRACTOR'S RECOMMENDATION FOR 
 
 a=m 3201 DISPOSITION OF SCIENTIFIC AND TECHNICAL DOCUMENT 
 
 ( See Instructions on Reverse Side ) 
 
 vEC REPORT NO. 
 
 COO-1U69-0166 
 
 2. TITLE 
 
 TEST MATRICES I 
 
 ». YPE OF DOCUMENT (Check one): 
 
 E3 a. Scientific and technical report 
 "2 b. Conference paper not to be published in a journal: 
 
 Title of conference 
 
 Date of conference 
 
 Exact location of conference 
 
 Sponsoring organization 
 
 □ c. Other (Specify) 
 
 RECOMMENDED ANNOUNCEMENT AND DISTRIBUTION (Check one): 
 
 O a AEC's normal announcement and distribution procedures may be followed. 
 2 b. Make available only within AEC and to AEC contractors and other U.S. Government agencies and their contractors. 
 T c. Make no announcement or distrubution. 
 
 5 REASON FOR RECOMMENDED RESTRICTIONS: 
 
 6 SUBMITTED BY: NAME AND POSITION (Please print or type) 
 
 C. W. Gear, Professor 
 
 and Principal Investigator 
 
 Organization 
 
 Department of Computer Science 
 university of Illinois 
 Urbana, Illinois 618OI 
 
 Signature 
 
 JZiktp- 
 
 Date June 19 TO 
 
 FOR AEC USE ONLY 
 
 AEC CONTRACT ADMINISTRATOR'S COMMENTS, IF ANY. ON ABOVE ANNOUNCEMENT AND DISTRIBUTION 
 RECOMMENDATION: 
 
 PATENT CLEARANCE: 
 
 [~| a. AEC patent clearance has been granted by responsible AEC patent group. 
 O b. Report has been sent to responsible AEC patent group for clearance. 
 M c. Patent clearance not required. 
 
MAY 17 1974