C CANO 10
C ..................................................................CANO 20
C CANO 30
C SUBROUTINE CANOR CANO 40
C CANO 50
C PURPOSE CANO 60
C COMPUTE THE CANONICAL CORRELATIONS BETWEEN TWO SETS OF CANO 70
C VARIABLES. CANOR IS NORMALLY PRECEDED BY A CALL TO SUBROU- CANO 80
C TINE CORRE. CANO 90
C CANO 100
C USAGE CANO 110
C CALL CANOR (N,MP,MQ,RR,ROOTS,WLAM,CANR,CHISQ,NDF,COEFR, CANO 120
C COEFL,R) CANO 130
C CANO 140
C DESCRIPTION OF PARAMETERS CANO 150
C N - NUMBER OF OBSERVATIONS CANO 160
C MP - NUMBER OF LEFT HAND VARIABLES CANO 170
C MQ - NUMBER OF RIGHT HAND VARIABLES CANO 180
C RR - INPUT MATRIX (ONLY UPPER TRIANGULAR PORTION OF THE CANO 190
C SYMMETRIC MATRIX OF M X M, WHERE M = MP + MQ) CANO 200
C CONTAINING CORRELATION COEFFICIENTS. (STORAGE MODE CANO 210
C OF 1) CANO 220
C ROOTS - OUTPUT VECTOR OF LENGTH MQ CONTAINING EIGENVALUES CANO 230
C COMPUTED IN THE NROOT SUBROUTINE. CANO 240
C WLAM - OUTPUT VECTOR OF LENGTH MQ CONTAINING LAMBDA. CANO 250
C CANR - OUTPUT VECTOR OF LENGTH MQ CONTAINING CANONICAL CANO 260
C CORRELATIONS. CANO 270
C CHISQ - OUTPUT VECTOR OF LENGTH MQ CONTAINING THE CANO 280
C VALUES OF CHI-SQUARES. CANO 290
C NDF - OUTPUT VECTOR OF LENGTH MQ CONTAINING THE DEGREES CANO 300
C OF FREEDOM ASSOCIATED WITH CHI-SQUARES. CANO 310
C COEFR - OUTPUT MATRIX (MQ X MQ) CONTAINING MQ SETS OF CANO 320
C RIGHT HAND COEFFICIENTS COLUMNWISE. CANO 330
C COEFL - OUTPUT MATRIX (MP X MQ) CONTAINING MQ SETS OF CANO 340
C LEFT HAND COEFFICIENTS COLUMNWISE. CANO 350
C R - WORK MATRIX (M X M) CANO 360
C CANO 370
C REMARKS CANO 380
C THE NUMBER OF LEFT HAND VARIABLES (MP) SHOULD BE GREATER CANO 390
C THAN OR EQUAL TO THE NUMBER OF RIGHT HAND VARIABLES (MQ). CANO 400
C THE VALUES OF CANONICAL CORRELATION, LAMBDA, CHI-SQUARE, CANO 410
C DEGREES OF FREEDOM, AND CANONICAL COEFFICIENTS ARE COMPUTED CANO 420
C ONLY FOR THOSE EIGENVALUES IN ROOTS WHICH ARE GREATER THAN CANO 430
C ZERO. CANO 440
C CANO 450
C SUBROUTINES AND FUNCTION SUBPROGRAMS REQUIRED CANO 460
C MINV CANO 470
C NROOT (WHICH, IN TURN, CALLS THE SUBROUTINE EIGEN.) CANO 480
C CANO 490
C METHOD CANO 500
C REFER TO W. W. COOLEY AND P. R. LOHNES, 'MULTIVARIATE PRO- CANO 510
C CEDURES FOR THE BEHAVIORAL SCIENCES', JOHN WILEY AND SONS, CANO 520
C 1962, CHAPTER 3. CANO 530
C CANO 540
C ..................................................................CANO 550
C CANO 560
SUBROUTINE CANOR (N,MP,MQ,RR,ROOTS,WLAM,CANR,CHISQ,NDF,COEFR, CANO 570
1 COEFL,R) CANO 580
DIMENSION RR(1),ROOTS(1),WLAM(1),CANR(1),CHISQ(1),NDF(1),COEFR(1),CANO 590
1 COEFL(1),R(1) CANO 600
C CANO 610
C ...............................................................CANO 620
C CANO 630
C IF A DOUBLE PRECISION VERSION OF THIS ROUTINE IS DESIRED, THE CANO 640
C C IN COLUMN 1 SHOULD BE REMOVED FROM THE DOUBLE PRECISION CANO 650
C STATEMENT WHICH FOLLOWS. CANO 660
C CANO 670
C DOUBLE PRECISION RR,ROOTS,WLAM,CANR,CHISQ,COEFR,COEFL,R,DET,SUM CANO 680
C CANO 690
C THE C MUST ALSO BE REMOVED FROM DOUBLE PRECISION STATEMENTS CANO 700
C APPEARING IN OTHER ROUTINES USED IN CONJUNCTION WITH THIS CANO 710
C ROUTINE. CANO 720
C CANO 730
C THE DOUBLE PRECISION VERSION OF THIS SUBROUTINE MUST ALSO CANO 740
C CONTAIN DOUBLE PRECISION FORTRAN FUNCTIONS. SQRT IN STATEMENT CANO 750
C 165 MUST BE CHANGED TO DSQRT. ALOG IN STATEMENT 175 MUST BE CANO 760
C CHANGED TO DLOG. CANO 770
C CANO 780
C ...............................................................CANO 790
C CANO 800
C PARTITION INTERCORRELATIONS AMONG LEFT HAND VARIABLES, BETWEEN CANO 810
C LEFT AND RIGHT HAND VARIABLES, AND AMONG RIGHT HAND VARIABLES. CANO 820
C CANO 830
M=MP+MQ CANO 840
N1=0 CANO 850
DO 105 I=1,M CANO 860
DO 105 J=1,M CANO 870
IF(I-J) 102, 103, 103 CANO 880
102 L=I+(J*J-J)/2 CANO 890
GO TO 104 CANO 900
103 L=J+(I*I-I)/2 CANO 910
104 N1=N1+1 CANO 920
105 R(N1)=RR(L) CANO 930
L=MP CANO 940
DO 108 J=2,MP CANO 950
N1=M*(J-1) CANO 960
DO 108 I=1,MP CANO 970
L=L+1 CANO 980
N1=N1+1 CANO 990
108 R(L)=R(N1) CANO1000
N2=MP+1 CANO1010
L=0 CANO1020
DO 110 J=N2,M CANO1030
N1=M*(J-1) CANO1040
DO 110 I=1,MP CANO1050
L=L+1 CANO1060
N1=N1+1 CANO1070
110 COEFL(L)=R(N1) CANO1080
L=0 CANO1090
DO 120 J=N2,M CANO1100
N1=M*(J-1)+MP CANO1110
DO 120 I=N2,M CANO1120
L=L+1 CANO1130
N1=N1+1 CANO1140
120 COEFR(L)=R(N1) CANO1150
C CANO1160
C SOLVE THE CANONICAL EQUATION CANO1170
C CANO1180
L=MP*MP+1 CANO1190
K=L+MP CANO1200
CALL MINV (R,MP,DET,R(L),R(K)) CANO1210
C CANO1220
C CALCULATE T = INVERSE OF R11 * R12 CANO1230
C CANO1240
DO 140 I=1,MP CANO1250
N2=0 CANO1260
DO 130 J=1,MQ CANO1270
N1=I-MP CANO1280
ROOTS(J)=0.0 CANO1290
DO 130 K=1,MP CANO1300
N1=N1+MP CANO1310
N2=N2+1 CANO1320
130 ROOTS(J)=ROOTS(J)+R(N1)*COEFL(N2) CANO1330
L=I-MP CANO1340
DO 140 J=1,MQ CANO1350
L=L+MP CANO1360
140 R(L)=ROOTS(J) CANO1370
C CANO1380
C CALCULATE A = R21 * T CANO1390
C CANO1400
L=MP*MQ CANO1410
N3=L+1 CANO1420
DO 160 J=1,MQ CANO1430
N1=0 CANO1440
DO 160 I=1,MQ CANO1450
N2=MP*(J-1) CANO1460
SUM=0.0 CANO1470
DO 150 K=1,MP CANO1480
N1=N1+1 CANO1490
N2=N2+1 CANO1500
150 SUM=SUM+COEFL(N1)*R(N2) CANO1510
L=L+1 CANO1520
160 R(L)=SUM CANO1530
C CANO1540
C CALCULATE EIGENVALUES WITH ASSOCIATED EIGENVECTORS OF THE CANO1550
C INVERSE OF R22 * A CANO1560
C CANO1570
L=L+1 CANO1580
CALL NROOT (MQ,R(N3),COEFR,ROOTS,R(L)) CANO1590
C CANO1600
C FOR EACH VALUE OF I = 1, 2, ..., MQ, CALCULATE THE FOLLOWING CANO1610
C STATISTICS CANO1620
C CANO1630
DO 210 I=1,MQ CANO1640
C CANO1650
C TEST WHETHER EIGENVALUE IS GREATER THAN ZERO CANO1660
C CANO1670
IF(ROOTS(I)) 220, 220, 165 CANO1680
C CANO1690
C CANONICAL CORRELATION CANO1700
C CANO1710
165 CANR(I)= SQRT(ROOTS(I)) CANO1720
C CANO1730
C CHI-SQUARE CANO1740
C CANO1750
WLAM(I)=1.0 CANO1760
DO 170 J=I,MQ CANO1770
170 WLAM(I)=WLAM(I)*(1.0-ROOTS(J)) CANO1780
FN=N CANO1790
FMP=MP CANO1800
FMQ=MQ CANO1810
175 CHISQ(I)=-(FN-0.5*(FMP+FMQ+1.0))*ALOG(WLAM(I)) CANO1820
C CANO1830
C DEGREES OF FREEDOM FOR CHI-SQUARE CANO1840
C CANO1850
N1=I-1 CANO1860
NDF(I)=(MP-N1)*(MQ-N1) CANO1870
C CANO1880
C I-TH SET OF RIGHT HAND COEFFICIENTS CANO1890
C CANO1900
N1=MQ*(I-1) CANO1910
N2=MQ*(I-1)+L-1 CANO1920
DO 180 J=1,MQ CANO1930
N1=N1+1 CANO1940
N2=N2+1 CANO1950
180 COEFR(N1)=R(N2) CANO1960
C CANO1970
C I-TH SET OF LEFT HAND COEFFICIENTS CANO1980
C CANO1990
DO 200 J=1,MP CANO2000
N1=J-MP CANO2010
N2=MQ*(I-1) CANO2020
K=MP*(I-1)+J CANO2030
COEFL(K)=0.0 CANO2040
DO 190 JJ=1,MQ CANO2050
N1=N1+MP CANO2060
N2=N2+1 CANO2070
190 COEFL(K)=COEFL(K)+R(N1)*COEFR(N2) CANO2080
200 COEFL(K)=COEFL(K)/CANR(I) CANO2090
210 CONTINUE CANO2100
220 RETURN CANO2110
END CANO2120